Fixed Income Securities Valuation, Risk, and Risk Management Test Bank

Fixed Income Securities Valuation, Risk, and Risk Management By Veronesi

Chapter 1 1. Is the following an arbitrage opportunity? A gift that makes me feel good just by having it.

Ans. This is an arbitrage opportunity because it doesn’t cost anything at initiation and it generates a positive profit by a certain date in the future (i.e. ’you feel good’).

2. Is the following an arbitrage opportunity? A bond that cost nothing but will payoff zero with certainty in the future.

Ans. This is not an arbitrage opportunity, since it doesn’t give a positive payoff in the future.

3. Is the following an arbitrage opportunity? A free car that if I repair well, I won’t have to spend money on gasoline or maintenance costs (i.e. repairs) ever.

Ans. This is not an arbitrage opportunity, since I have to pay money (to repair the car) in order to be free of future costs.

4. Is the following an arbitrage opportunity? Suppose you are in the desert and are given a bag of ice with a penny inside. Assume that the ice will melt instantly and the cost of disposing of the bag is zero.

Ans. This is an arbitrage opportunity because even though I can’t take advantage of the ice, I gain the penny for free.

5. Is the following an arbitrage opportunity? A security that cost zero and might pay a dollar in the future, but pays zero otherwise.

Ans. This is an arbitrage opportunity because I get for free the chance of getting a dollar in the future.

6. What steps would you follow in order to take advantage of the following arbitrage opportunity (if there is one)? Security A costs $3 and pays $5 in 2 years, while security B costs $3 and pays $4 in 2 years.

Ans. You borrow security B and sell it, which means you receive $3, with these proceeds you buy security A. In 2 years you receive $5 and have to pay $4. You make a $1 profit.

7. What steps would you follow in order to take advantage of the following arbitrage opportunity (if there is one)? Security A costs $100 and pays $120 in 3 years. Security B costs $100 and pays $110 in one year. Your friend tells you that he would like you to lend him $110 in a year and that he would give $130 the following year. Finally you know that in two years, with $130, you can invest in a security that will pay you either $140 or $121 (with equal probability) after a year.

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Ans. You borrow security A and sell it, with the proceeds you buy security B. After a year you lend the money to your friend. The next year when he pays back, you invest in the risky security. After the third year, this will give you either $140 or $121, while you have to pay $120. So you either have a profit of $20 or $1.

8. What steps would you follow in order to take advantage of the following arbitrage opportunity (if there is one)? Security A costs $100 and pays $110 in 2 years. Security B costs $100 and pays $109 in one year. You know that in a year with $109 you can invest in a security that pays $120 or $109 (with equal probability) the following year.

Ans. This is not an arbitrage opportunity.

9. Intuitively, is the Federal Funds rate generally higher, lower or the same as LIBOR? Why? Ans. The LIBOR rate should be higher than the Federal Funds rate, since it should include a higher probability of default by the banks trading LIBOR. 10. Intuitively, is LIBOR generally higher, lower or the same as the repo rate?

Why? Ans. LIBOR should be higher than the repo rate since it is not collateralized with another security, as it occurs with the repo rate.

11. What are the steps to take a long position on a given U.S. security via the repo market?

Ans. The trader must take the following steps (repo): At time t: i. Buy bond at Pt and deliver the bond to the repo dealer. ii. The repo dealer will pay Pt − haircut to the trader, which is made whole (minus the haircut) for the cost of the bond.

At time T : iii. The trader gets the bond from the repo dealer and sells it for PT . iv. With the proceeds PT the trader pays back (Pt −haircut)×(1+Repo) to the repo dealer.

12. What are the steps to take a short position on a given U.S. security via the repo market?

Ans. The trader must take the following steps (reverse repo): At time t:

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i. Borrows the bond from the repo dealer and sells it at Pt . ii. Receives Pt which he posts as collateral with the repo dealer.

At time T : iii. The trader buys the bond for PT and gives it back to the repo dealer. iv. The repo dealer pays Pt × (1+Repo). The dealer makes a profit if this is larger than PT .

13. What’s the return on capital for a trader who entered into a one-month repo where Pt = 98.5, PT = 99.01, Repo= 5% and haircut= 0.8?

Ans. The return on capital is 0.13.

14. What’s the profit for a trader who entered into a one-week reverse repo where Pt = 99.40, PT = 99.48 and Repo= 6%?

Ans. The profit is 0.0347.

15. You find a bond that has a repo rate substantially lower than the GCR. Is this, for certain, an arbitrage opportunity?

Ans. No, it might be that the bond is short on supply (hard to find), which means that the trader might be thinking that the bond is overpriced and is speculating that the price will fall. By entering in a reverse repo, the trader might make a substantial profit, even willing to forgo part of the rate to be in the transaction.

16. You are told that there is an ample supply for the bond mentioned in question 13. Does this affect your previous answer?

Ans. Yes, since it shows that the lower rate is not due to scarcity of a specific type of bond.

17. What are the gains from trade of entering into a swap for these two firms?

Fixed Rate Floating Rate

Firm A 13% LIBOR + 3%

Firm B 16% LIBOR + 6%

Ans. Gains from trade are zero.

18. What are the gains from trade of entering into a swap for these two firms?

Fixed Rate Floating Rate

Firm A 10% LIBOR + 2%

Ans. Gains from trade are 3.5%.

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Firm B 15.5% LIBOR + 4%

19. What are the gains from trade of entering into a swap for these two firms?

Fixed Rate Floating Rate

Firm A 9% LIBOR + 7%

Ans. Gains from trade are 1%.

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Firm B 5% LIBOR + 4%

Chapter 2 1. Do discount factors depend on compounding frequency? Why? Ans. No, discount factors do not depend on compounding frequency, since they are terms of exchange (prices) between having money at time t versus having at a later date T . If they changed with compounding frequency, there would be an arbitrage opportunity.

2. What effect does inflation have on discount factors?

Ans. Higher inflation makes less appealing money in the future, so discount factors will go down.

3. Can a bond be quoted in more than one interest rate?

Ans. Yes. It can be quoted in various compounding frequencies.

4. From the following data obtain the discount curve: a. A zero coupon bond Pz (0, 0.5) = 99.20. b. A coupon bond paying 3% quarterly P (0, 0.25) = 100.5485. c. A coupon bond paying 6% quarterly P (0, 0.75) = 100.1655. d. A coupon bond paying 5% semiannually P (0, 1) = 103.0325.

Ans. The discount factors are the following: t 0.25 0.50 0.75 1.00

Z(0, t) 0.9980 0.9920 0.9870 0.9810

5. Using the previous discount curve price the following: A zero coupon bond expiring at t = 0.75.

Ans. The price of the bond is 98.70.

6. Using the previous discount curve price the following: A 1-year coupon bond paying 4% quarterly.

Ans. The price of the bond is 102.0580.

7. Using the previous discount curve price the following: A 6-month coupon bond paying 7% semiannually.

Ans. The price of the bond is 102.6720.

8. Using the previous discount curve price the following: A 9-month coupon bond paying 5% semiannually.

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Ans. The price of the bond is 103.6625.

9. For the following scenario, check if there is a mispriced security: a. A zero coupon bond Pz (0, 0.5) = 99.00. b. A coupon bond paying 6% quarterly P (0, 0.25) = 101.1955. c. A coupon bond paying 4% quarterly P (0, 0.50) = 102.0830. d. A coupon bond paying 7% semiannually P (0, 0.75) = 105.8440.

Ans. The mispriced bond is [a.] the zero coupon bond.

10. For the following scenario, check if there is a mispriced security: a. A zero coupon bond Pz (0, 0.25) = 99.40. b. A zero coupon bond Pz (0, 0.50) = 98.00. c. A coupon bond paying 3% quarterly P (0, 0.50) = 100.4880.

Ans. The mispriced security is [b.].

11. For the following scenario, check if there is a mispriced security: a. A coupon bond paying 1% quarterly P (0, 0.25) = 100.6498. b. A coupon bond paying 4% semiannually P (0, 0.25) = 101.8980. c. A coupon bond paying 3% quarterly P (0, 0.50) = 101.2978. d. A coupon bond paying 5% quarterly P (0, 0.75) = 103.4425. e. A coupon bond paying 4% semiannually P (0, 1.00) = 103.5880.

Ans. The mispriced bond is [a.].

12. For the following scenario, check if there is a mispriced security: a. A zero coupon bond Pz (0, 0.5) = 99.50. b. A coupon bond paying 3% quarterly P (0, 0.50) = 100.9948. c. A coupon bond paying 5% quarterly P (0, 0.75) = 102.7288. d. A coupon bond paying 2% semiannually P (0, 1.25) = 102.8720. e. A zero coupon bond Pz (0, 1.25) = 98.4.

Ans. The mispriced bond is [d.].

13. For the following scenario, check if there is a mispriced security: a. A zero coupon bond Pz (0, 0.25) = 99.30. b. A zero coupon bond Pz (0, 0.50) = 98.70. c. A coupon bond paying 3% semiannually P (0, 0.50) = 100.1850. d. A coupon bond paying 2% semiannually P (0, 0.75) = 101.4880.

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Ans. The mispriced bond is [d.], since it requires the discount factor Z(0, 0.75) to be larger than the previous ones.

14. What is the price on a 4.5-year floating rate bond that pays a semiannual coupon (no spread)?

Ans. The price of the coupon is 100.

15. What is the price on a 5.75-year floating rate bond that pays a semiannual coupon (no spread)? We know the following: a. There is a coupon bond paying 3% quarterly P (0, 0.25) = 100.0448. b. Last quarter the semiannually compounded rate was 3%.

Ans. The price of the floating rate bond is 100.7895.

16. What is the price of a 0.5-year floating rate bond that pays a quarterly coupon equal to floating rate plus a 1% spread? We know the following: a. There is a zero coupon bond Pz (0, 0.25) = 99.80. b. There is a coupon bond paying 2% quarterly P (0, 0.5) = 100.3960.

Ans. The price of the floating rate bond is 100.4980.

17. What is the price of a 0.75-year floating rate bond that pays a semiannual coupon equal to floating rate plus 2% spread? We know the following: a. There is a zero coupon bond Pz (0, 0.25) = 99.70. b. There is a zero coupon bond Pz (0, 0.50) = 99.20. c. There is a coupon bond paying 3% quarterly P (0, 0.75) = 101.7380.

Ans. The price is 102.3145.

18. You have two coupon bonds with same maturity, one pays 5% semiannually and the other 5% quarterly. Which one has a higher yield?

Ans. The second bond has a higher yield because it compounds more frequently than the first one, increasing the expected return.

19. A Treasury dealer quotes the following 182-day bill at a 3.956% discount.

What is the price of the security?

Ans. The price of the Treasury is 98.00.

20. A Treasury dealer quotes the following 91-day bill at a 3.956% discount.

What is the price of the security?

Ans. The price of the Treasury is 99.00.

8

Chapter 3

Use the following discount factors when needed. t 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00

Z(0, t) 0.9840 0.9680 0.9520 0.9360 0.9190 0.9040 0.8880 0.8730 0.8587 0.8445 0.8308 0.8175 0.8047 0.7924 0.7806 0.7691

1. Calculate the duration of the following security: 5-year zero coupon bond.

Ans. The duration of the security is 5.00.

2. Calculate the duration of the following security: 2-year fixed coupon paying 5% quarterly.

Ans. The duration of the security is 1.9138.

3. Calculate the duration of the following security: 1.25-year floating coupon paying float + 50 bps semiannually. You know that last quarter the semiannual rate was 6.4%.

Ans. The duration of the security is 0.2534.

4. Calculate the duration of the following portfolio: i. 5 units of a 2-year fixed rate bond paying 6% quarterly. ii. 2 units of a 1.75-year floating rate bond paying float + 80 bps semiannually. You know that the reference rate was 6.5% three months ago. iii. 6 units of a 1-year zero coupon bond. iv. 5 units of a 1.5-year floating rate bond with no spread paid semiannually.

Ans. The duration of the portfolio is 0.8805.

5. Calculate the duration of the following portfolio: 9

i. 3 units of a 0.75-year fixed rate bond paying 6% quarterly. ii. 4 units of a 2-year fixed rate bond paying 3% semiannually. iii. 7 units of a 1.75-year zero coupon bond. iv. 1 unit of a 2-year floating rate bond with no spread paid semiannually.

Ans. The duration of the portfolio is 1.4617.

6. You have two bond coupon with the same maturity, one has a 9% coupon paid semiannually and the other a 8% coupon paid semiannually. Which one has a higher duration?

Ans. The one with the lower coupon has a higher duration. Higher coupons mean that a higher proportion of the total cash flows from the bond will be paid more quickly. These cash flows will also be less sensitive to interest rates as cash flows paid later are more exposed to interest rate variation.

7. Calculate the MacCaulay Duration for the following security: 1-year fixed rate coupon bond paying 6% semiannually. You know that the yield of the bond is 6.72%.

Ans. The MacCaulay Duration for the bond is 0.9854.

8. Calculate the Modified Duration for the same security.

Ans. The Modified Duration for the bond is 0.9533.

9. What is the duration of the following portfolio? i. Long a 1.5-year zero coupon bond. ii. Short a 2-year fixed coupon bond paying 1% quarterly.

Ans. Trying to compute duration from this long-short portfolio brings many problems since we can’t adequately weigh the securities within the portfolio.

10. What is the dollar duration of the following portfolio? i. Long a 1.5-year zero coupon bond. ii. Short a 2-year fixed coupon bond paying 1% quarterly.

Ans. Dollar duration for the portfolio is -41.0462.

11. What is the dollar duration of the following portfolio: i. Long a 1-year fixed coupon bond paying 4% quarterly. ii. Long a 1.75-year floating rate bond paying float plus 80 bps semiannually. You know that the reference rate was set at 6% six months ago. iii. Short a 2-year zero coupon bond.

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Ans. The dollar duration of the portfolio is -51.8169.

12. What is the dollar duration of the following portfolio: i. Long a 2-year fixed coupon bond paying 7% quarterly. ii. Short three 1.25-year floating rate bonds paying float plus 80 bps semiannually. You know that the reference rate was set at 7% six months ago. iii. Short two 0.5-year zero coupon bonds.

Ans. The dollar duration is 13.2098.

13. What is the PV01 of the following portfolio? i. Long a 1-year fixed coupon bond paying 4% quarterly. ii. Long a 1.75-year floating rate bond paying float plus 80 bps semiannually. You know that the reference rate was set at 6% six months ago. iii. Short a 2-year zero coupon bond.

Ans. The PV01 is -0.0052.

14. What is the PV01 of the following portfolio? i. Long a 2-year fixed coupon bond paying 7% quarterly. ii. Short three 1.25-year floating rate bonds paying float plus 80 bps semiannually. You know that the reference rate was set at 7% six months ago. iii. Short two 0.5-year zero coupon bonds.

Ans. The PV01 is 0.0013.

15. Compute the 95% VaR for the following portfolio: i. A 1.5-year fixed rate bond paying 2% quarterly. ii. A 0.75-year floating rate bond paying float plus 80 basis points semiannually. You know that the reference rate was set to 6% six months ago. iii. A 0.25 zero coupon bond.

Additionally you know that μdr = 0 and σdr = 0.4233.

Ans. The 95%V aR = 1.3116.

16. Suppose that you calculate VaR from Duration. In your many results you find that: i. using historical data (of whatever length) or a normal distribution does not affect the result; 11

ii. you find that kurtosis between historical data and the normal distribution is almost identical; iii. You find the expected change in the portfolio μP = 0, with very small standard errors.

Given the above, can you say that this Duration based VaR is an appropriate approach to measure risk?

Ans. No, it is not. It is still internally inconsistent: Duration measures small changes, but VaR uses extreme values (large changes).

17. Mr. Brown wants to invest $100,000 for the next five years. He purchases an annuity from a financial institution. Currently the term structure is flat at 10% (yearly compounded). i. If the payments are made yearly, what is the amount that the financial institution will agree to pay Mr. Brown? ii. Assume that there is a 5-year fixed coupon bond that pays 12% coupon every year. What is the price and duration of the bond? iii. How much must the financial institution invest in the long-term bond in order to hedge the position? What should it do with the remainder of the money?

Ans. The results for the immunization exercise are: i. The financial institution would pay a yearly amount of $26,379.75. ii. The price of the security is $107.58 with 4.074 duration. iii. The financial institution should invest 93.05% of the proceeds from the investment in the long-term bond and the rest in the money market account.

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Chapter 4

Use the following discount factors when needed.

T 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00

Z(0, T ) 0.9840 0.9680 0.9520 0.9360 0.9190 0.9040 0.8880 0.8730 0.8587 0.8445 0.8308 0.8175 0.8047 0.7924 0.7806 0.7691

1. Calculate the convexity of the following security: a 5-year zero coupon bond.

Ans. The convexity of the security is 25.

2. Calculate the convexity of the following security: a 3-year fixed rate bond paying 4% coupon on a semiannual basis.

Ans. The convexity of the security is 8.3780.

3. Calculate the convexity of the following security: a 3-year floating rate bond with no spread paid quarterly.

Ans. The convexity of the security is 0.

4. Calculate the convexity of the following portfolio: i. 1 unit of a 2-year fixed coupon bond paying 10% coupon quarterly. ii. 1 unit of a 2-year fixed coupon bond paying 1% coupon semiannually. iii. 1 unit of a 2-year zero coupon bond.

Ans. The convexity of the portfolio is 3.8230.

5. Calculate the convexity of the following portfolio: i. 2 units of a 1.5-year fixed rate bond paying 6% quarterly. ii. 4 units of a 1.75-year floating rate bond paying float + 80 bps semiannually. You know that the reference rate was 7% three months ago.

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iii. 6 units of a 2-year zero coupon bond. iv. 1 units of a 1.5-year floating rate bond with no spread paid semiannually.

Ans. The convexity of the portfolio is 2.0655.

6. Calculate the convexity of the following portfolio: i. 4 units of a 1.5-year fixed rate bond paying 4% quarterly. ii. 5 units of a 1.5-year fixed rate bond paying 5% semiannually. iii. 10 units of a 1.5-year zero coupon bond. iv. 3 units of a 1.5-year floating rate bond with no spread paid semiannually.

Ans. The convexity of the portfolio is 1.8916.

7. Calculate annualized expected returns (including convexity) for a 5-year zero coupon bond, when E[dr] = 0 and E[dr 2 ] = 6 × 10−07 (on a daily basis).

Ans. Annualized expected return on the bond is: 0.189%.

8. Calculate annualized expected returns (including convexity) for a 30-year zero coupon bond, when E[dr] = 0 and E[dr 2 ] = 7 × 10−07 (on a daily basis).

Ans. Annualized expected return on the bond is: 7.938%.

9. Calculate annualized expected returns (including convexity) for a 3-year fixed rate bond paying 2% coupon semiannually, when E[dr] = 0 and E[dr 2 ] = 7.5 × 10− 07 (on a daily basis).

Ans. Annualized expected return on the bond is 0.0815%.

10. Suppose you hold a bond and interest rates suddenly fall. Duration says that bond prices will raise a given amount. If Convexity is included in this estimate, will bond prices go above or below what Duration predicts?

Ans. Prices will go up even more than what Duration predicts.

11. Suppose you hold a bond and interest rates suddenly rise. Duration says that bond prices will fall a given amount. If Convexity is included in this estimate, will bond prices go above or below what Duration predicts?

Ans. Prices will go up, partially countering the decrease predicted by Duration.

12. Compute the Term Spread and the Butterfly Spread for the following data.

What shape does the yield curve have?

1-month yield 5.0%

5-year yield 8.0% 14

10-year yield 10.0%

Ans. The term spread is 5% and the Butterfly Spread is 1%. The shape of the yield cure is increasing.

13. Compute the Term Spread and the Butterfly Spread for the following data.

What shape does the yield curve have?

1-month yield 6.0%

5-year yield 4.5%

10-year yield 4.0%

Ans. The term spread is -2% and the Butterfly Spread is -1%. The shape of the yield cure is decreasing.

14. Compute the Term Spread and the Butterfly Spread for the following data.

What shape does the yield curve have?

1-month yield 3.0%

5-year yield 4.0%

10-year yield 3.0%

Ans. The term spread is 0% and the Butterfly Spread is 2%. The shape of the yield cure is a hump.

15. Compute the Term Spread and the Butterfly Spread for the following data.

What shape does the yield curve have?

1-month yield 4.0%

5-year yield 3.0%

10-year yield 5.0%

Ans. The term spread is 1% and the Butterfly Spread is -3%. The shape of the yield cure is an inverted hump.

16. You currently hold a 7-year fixed rate bond paying 5% annually. You would like to hedge against changes in the level and the slope of the yield curve and you plan to use a 1-year zero coupon bond and a 7-year zero coupon bond. Use the following table to compute the adequate positions in the hedging instruments. maturity 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00

β1 1.1150 0.9940 0.9640 0.9330 0.9300 0.9260 0.9270 0.9270

β2 -0.2540 -0.3010 -0.1470 0.0080 0.1620 0.3160 0.4230 0.5300

Z(t, T ) 0.9800 0.9600 0.9300 0.8900 0.8500 0.8100 0.7700 0.7300

Ans. The position in the short term bond should be -0.4651, while the position in the long term bond should be -1.1231.

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17. You currently hold a 7-year fixed rate bond paying 1% annually. You would like to hedge against changes in the level and the slope of the yield curve and you plan to use a 2-year zero coupon bond and a 6-year zero coupon bond as hedging instruments. Use the following table to compute the adequate positions in the hedging instruments. maturity 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00

β1 1.1150 0.9940 0.9640 0.9330 0.9300 0.9260 0.9270 0.9270

β2 -0.2540 -0.3010 -0.1470 0.0080 0.1620 0.3160 0.4230 0.5300

Z(t, T ) 0.9800 0.9600 0.9300 0.8900 0.8500 0.8100 0.7700 0.7300

Ans. The position in the short term bond should be 0.4233, while the position in the long term bond should be -1.4132.

18. You currently hold a 2-year fixed rate bond paying 1% annually. You would like to hedge against changes in the level and the slope of the yield curve and you plan to use a 1-year zero coupon bond and a 8-year zero coupon bond as hedging instruments. Use the following table to compute the adequate positions in the hedging instruments. maturity 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00

β1 1.1150 0.9940 0.9640 0.9330 0.9300 0.9260 0.9270 0.9270

β2 -0.2540 -0.3010 -0.1470 0.0080 0.1620 0.3160 0.4230 0.5300

Z(t, T ) 0.9800 0.9600 0.9300 0.8900 0.8500 0.8100 0.7700 0.7300

Ans. The position in the short term bond should be -1.9395, while the position in the long term bond should be 0.0317.

19. What is the advantage of a factor model?

Ans. A factor model has the advantage that it is adjusted to the data. The factors are not generated beforehand, but derived from the data itself. It so happens that embedded in the interest rate data there are three factors: slope, level and curvature (in addition to some more negligible factors).

Additionally, by obtaining factors this way, they are automatically independent. This makes the hedging easier.

20. How many securities do you need to hedge three factors? Why? 16

Ans. To hedge three factors you need three securities, because the three factors generate a system of equations with three unknowns. In order to solve it you must include a security for each of these unknowns.

21. If you need three securities to hedge three factors, can you do the following? Take two securities and make a third ”synthetic” security from these two (i.e. it is the average of both prices). Use it to solve the system of equations. Is this valid?

Ans. This is wrong, because it may lead to a singular matrix. This means that the system of equations cannot be solved because one the equations is just a transformation of another one. In order to be used to hedge a factor, the instrument should be independent from the other two.

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Chapter 5

Use the following table when needed: T 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 8.00

Z(t, T ) 0.9940 0.9880 0.9740 0.9620 0.9460 0.9330 0.9170 0.8950 0.8770 0.8580 0.8340 0.8130 0.7990 0.7760 0.7570 0.7360

1. What is a forward discount factor?

Ans. A forward discount factor at time t defines the time value of money between two future dates, T1 and T2 > T1 .

2. Compute F (0, 3, 5), f2 (0, 3, 5) and f(0, 3, 5).

Ans. F (0, 3, 5) = 0.9196, f2 (0, 3, 5) = 4.2343% and f(0, 3, 5) = 4.1901%.

3. Compute F (0, 0.5, 1), f2 (0, 0.5, 1) and f(0, 0.5, 1).

Ans. F (0, 0.5, 1) = 0.9940, f2 (0, 0.5, 1) = 1.2146% and f(0, 0.5, 1) = 1.2109%.

4. Compute F (0, 4, 8), f2 (0, 4, 8) and f(0, 4, 8).

Ans. F (0, 4, 8) = 0.8223, f2 (0, 4, 8) = 4.9501% and f(0, 4, 8) = 4.8898%.

5. Compute F (0, 5, 6), f2 (0, 5, 6) and f(0, 5, 6).

Ans. F (0, 5, 6) = 0.9720, f2 (0, 5, 6) = 2.8573% and f(0, 5, 6) = 2.8371%.

6. Compute F (0, T − 0.5, T ), f2 (0, T − 0.5, T ) and f(0, T − 0.5, T ), up to year four.

Ans. The following table summarizes the answers:

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T 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

F (0, T − 0.5, T ) 0.9940 0.9940 0.9858 0.9877 0.9834 0.9863 0.9829 0.9760

f2 (0, T − 0.5, T ) 1.2072% 1.2146% 2.8747% 2.4948% 3.3827% 2.7867% 3.4896% 4.9162%

f(0, T − 0.5, T ) 1.2036% 1.2109% 2.8543% 2.4794% 3.3544% 2.7675% 3.4595% 4.8568%

7. The term structure of interest rates can be upward sloping or downward sloping. If this is so, can discount factors also be upward sloping or downward sloping? E.g. Z(t, T1 ) < Z(t, T2 ) as well as Z(t, T1 ) > Z(t, T2 ).

Ans. No, this can’t be so. It must always be the case that Z(t, T1 ) > Z(t, T2 ), otherwise the forward rate for an investment between T1 and T2 will be negative.

8. Today you notice that forward rates are well above the spot rate. What shape must the yield curve have?

Ans. It must be upward sloping, since spot rate is an average of forward rates.

9. You notice that forward rates are below the spot rate. What can you say about the yield curve?

Ans. It must be downward sloping, since spot rate is an average of forward rates.

10. What is the value of a Forward Rate Agreement with waiting period of on year for a six-month loan (we have f2 (0, 1, 1.5) = 2.00%), at time t (six-months after inception). For t we have the discount factors presented at the beginning of this section. Assume a notional of $100 million.

Ans. The value of the FRA is -427,139.

11. What is the Forward Price to purchase a 1.5-year fixed rate Treasury paying 5% semiannually, a year from now? At t = 0 we have the following discounts: T 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 19

Z(0, T ) 0.9680 0.9360 0.9040 0.8730 0.8445 0.8175 0.7924 0.7691

Ans. The Forward Price is 97.2262.

12. Suppose you have entered into the Forward Contract from the previous exercise, what is the value of the contract 6-months after initiation? Assume that the discount factors are now the ones presented at the beginning of this section.

Ans. The Value of the Forward Contract is now: 6.8671.

13. Determine the swap rate for the following maturities: 0.50, 1.00, 1.50, 2.00. Use the discount factors provided at the beginning of this section.

Ans. The values are c(0.50) = 1.21%, c(1.00) = 1.21%, c(1.50) = 1.76%, c(2.00) = 1.94%.

14. Determine the swap rate for the following maturities: 0.25, 0.50, 0.75, 1.00, 1.50, 2.00. Use the following discount factors: T 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

Z(0, T ) 0.9840 0.9680 0.9520 0.9360 0.9190 0.9040 0.8880 0.8730

Ans. The values are c(0.25) = 6.50%, c(0.50) = 6.56%, c(0.75) = 6.61%, c(1.00) = 6.67%, c(1.50) = 6.78%, c(2.00) = 6.84%.

15. What is the value of a swap at initiation?

Ans. The value of the swap at initiation, using the appropriate swap rate, is zero.

16. Value a 1.5 year swap, with swap rate 5.52%. Notional is 100 million. Use the following discount factors.

T 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

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Z(0, T ) 0.9848 0.9745 0.9618 0.9490 0.9353 0.9215 0.9084 0.8953

You are told that this is a swap at initiation. Is the value accurate? Be sure to take into account any payment frequency conventions on the swap.

Ans. The value of the swap is zero. You have to keep in mind that the swap’s fixed leg is paid semiannually, not quarterly.

17. Consider the same swap as in the previous question. What is the value of the swap three months after initiation, where the discount factors are now: T 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

Z(0, T ) 0.9840 0.9680 0.9520 0.9360 0.9190 0.9040 0.8880 0.8730

Ans. The value of the swap is now: 222,408.

18. If short term rates go up, what would happen to the swap rate?

Ans. If these rates go up, we would expect for obligations linked to these rates to go up as well. This means that there will be a greater demand for swappingg these obligations through a fixed-for-floating swaps. This would drive up swap rates as well, as there is a greater demand for these agreements.

19. Assume that the swap spread at the moment is large, compared to historical data. What would your expectations be? Will it increase, decrease or stay the same? To what extent?

Ans. It is expected for the swap spread, the difference between Treasury and LIBOR discounts to be small. If this difference is large, we would expect it to shrink. Additionally it would shrink only to the extent that Treasury discounts are still more expensive than LIBOR discounts, since these include the probability of default by swap dealers, which the Treasuries don’t have.

20. Use a swap to hedge the following Balance Sheet, so that parallel shifts in the term structure don’t have an impact on the equity value: Assets Liabilities

Amount (billion) 3.00 2.50

D 15.60 1.95

The swap used to hedge is 1.5 year swap, you know the following discount factors: 21

T 0.50 1.00 1.50

Z(0, T ) 0.9745 0.9490 0.9215

In order to get the answer, compute the following: i. What is the adequate swap rate? ii. What is the dollar duration of the swap? iii. What is the value of equity and its dollar duration, prior to any hedging? iv. What is the value of notional needed so that the swap position hedges any impact that parallel shifts in the yield curve may have on the value of equity?

Ans. See the following: i. The adequate swap rate is 5.52%. ii. Dollar duration for the swap rate (assuming 100 notional) is -1.46. iii. The value of equity is 0.5 billion, with dollar duration 13.65. iv. Notional in order to cover make dollar duration for equity equal to zero is N = 9.3492.

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Chapter 6 1. What are the main differences between a forward contract and a futures contract?

Ans. The main differences are: i. Futures contracts are traded in a regulated exchange, while forward contracts are traded in the over-the-counter (OTC) market. ii. Futures contracts are ”standardized”, while forward contracts are customized to clients’ requests. iii. Profits and losses in futures contracts are marked-to-market. Forward contracts are not.

2. What does mark-to-market mean?

Ans. Profits and losses from the futures contract trading activity accrue to traders with daily frequency.

3. What is a margin call?

Ans. When someone enters a futures contract, they must put up a specific amount with the exchange in order to meet the possible losses accrued from the contract. If this account declines below a specific amount, called maintenance margin, the exchange issues a margin call and the trader must replenish the account to the initial margin.

4. What will be the value of a futures contract at maturity?

Ans. The value of the futures contract will be equal to the value of the underlying security.

5. Under what conditions are futures and forwards the same? Is this realistic?

Ans. Futures and forward contracts are the same under the following conditions: i. The difference in timing of cash flows between forward and a futures contract is ignored. ii. The futures and forward contract payoffs occur at the same date.

Neither of these conditions is realistic.

6. What are the shortcomings of futures, when compared to forward contracts?

Ans. The most important shortcoming of futures when compared to forward contracts are:

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i. Basis Risk: The available maturity of the bond or of the particular instrument we want to hedge may not coincide exactly with the one offered by the futures contract. This leaves some residual risk in the instrument, called basis risk (because we are not matching the basis exactly). ii. Cash flows arising from the futures position accrue over time, which imply the need of the firm to take into account the time value of money between the time at which the cash flow is realized and the maturity of the hedge position.

7. What are the advantages of futures contracts, when compared to forward contracts?

Ans. The advantages of futures contracts over forward contracts are: i. Liquidity. It is easier to get in and out of a position because of standardization. Forward contracts are not easily unwound, since they were specifically made for a client. ii. Credit Risk. The existence of a clearing house, which in addition imposes certain margin requirement reduces credit risk substantially.

8. What is to tail the hedge?

Ans. To tail the hedge means to incorporate the time value of money into the equation used to determine the optimal hedge ratio of a futures contract.

It usually involves multiplying the appropriate discount rate to the hedge ratio.

9. What is the difference between a European option and an American option?

Ans. A European option can only be exercised at a given date, while the American option can be exercised at any moment in time.

10. What is a European Call option?

Ans. Given an underlying variable F (t), maturity T and strike price K; a European Call option is a contract between two counterparties, option buyer and option seller, according to which: i. At maturity T the option buyer has the right to ask the option seller for the payment of the following effective payoff: Payoff = (max(F (T ) − K, 0) ii. The option seller has the obligation to pay this amount to the option buyer at T . iii. In return for this right to obtain this payment (exercise) at T , the option buyer pays an option premium to the option seller at time 0.

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11. What is a European Put option?

Ans. Given an underlying variable F (t), maturity T and strike price K; a European Put option is a contract between two counterparties, option buyer and option seller, according to which: i. At maturity T the option buyer has the right to ask the option seller for the payment of the following effective payoff: Payoff = (max(K − F (T ), 0) ii. The option seller has the obligation to pay this amount to the option buyer at T . iii. In return for this right to obtain this payment (exercise) at T , the option buyer pays an option premium to the option seller at time 0.

12. What is an American Call option?

Ans. Given an underlying variable F (t), maturity T and strike price K; an American Call option is a contract between two counterparties, option buyer and option seller, according to which: i. At any time before maturity T the option buyer has the right to ask the option seller for the payment of the following effective payoff: Payoff = (max(F (t) − K, 0) ii. The option seller has the obligation to pay this amount to the option buyer at T . iii. In return for this right to obtain this payment (exercise) at T , the option buyer pays an option premium to the option seller at time 0.

13. What is an American Put option?

Ans. Given an underlying variable F (t), maturity T and strike price K; an American Put option is a contract between two counterparties, option buyer and option seller, according to which: i. At any time before maturity T the option buyer has the right to ask the option seller for the payment of the following effective payoff: Payoff = (max(K − F (t), 0) ii. The option seller has the obligation to pay this amount to the option buyer at T . iii. In return for this right to obtain this payment (exercise) at T , the option buyer pays an option premium to the option seller at time 0.

14. Today you notice that Pc(t, TB ) = 104. What is the payoff for a European Bond Option maturing today with the following payoff function: max(Pc (t, TB ) − K, 0), with K = 105?

Ans. The payoff of the option is zero.

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15. Today you notice that Pc (t, TB ) = 98.5. What is the payoff for a European Bond Option maturing today with the following payoff function: max(K − Pc (t, TB ), 0), with K = 100?

Ans. The payoff of the option is 1.5.

16. Today you notice that r4L (t − 1) = 2.8863%. What is the payoff for a European Interest Rate Option maturing today with the following payoff function: max(K − r4L (t − 1), 0), with K = 2.5 and N = 1 million?

Ans. The payoff of the option is 966.

17. Today you notice that r4L (t − 1) = 2.1129%. What is the payoff for a European Interest Rate Option maturing today with the following payoff function: max(r4L (t − 1) − K, 0), with K = 1.5 and N = 1 million?

Ans. The payoff of the option is 1,532.

18. You are given the following discount factors: T 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

Z(0, T ) 0.9940 0.9880 0.9740 0.9620 0.9460 0.9330 0.9170 0.8950

You are told that the price of a European Call option on a 6-month zero coupon bond, with T = 0.5 and K = 99.35 is 0.13. While the price of a European Put option with the exact same specification is: 0.11. Are the securities adequately priced?

Ans. No, according to Put-Call parity the price of the Call option should be 0.1561.

19. You are given the following discount factors: T 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 26

Z(0, T ) 0.9940 0.9880 0.9740 0.9620 0.9460 0.9330 0.9170 0.8950

You are told that the price of a European Call option on a 2-year fixed rate bond paying 5% semiannually, with T = 2 and K = 101 is 4.6155. While the price of a European Put option with the exact same specification is: 3.0500. Are the securities adequately priced?

Ans. Yes, according to the Put-Call parity the price of the Call option should be 4.6155.

27

Chapter 7 1. Which are the main tools that the Federal Reserve have to conduct Monetary Policy?

Ans. Open Market Operations, Reserve Requirements and the Federal Discount Rate.

2. How does the Fed reach its desired target rate?

Ans. Through Open Market Operations.

3. Through Open Market Operations, how do you lower rates?

Ans. The Federal Reserve will buy Treasury Securities from banks and credit their accounts at the Federal Reserve. Banks can use these additional funds to lend money to other banks. These excess liquidity (i.e. increased supply of funds) will bring down interest rates (i.e. the price at which the funds are exchanged).

4. Through Open Market Operations, how do you raise rates?

Ans. The Federal Reserve will sell Treasury Securities from banks and credit their accounts at the Federal Reserve. Banks that want to borrow funds will have to pay a higher price, since funds have been used to purchase securities. Lower liquidity (i.e. diminished supply of funds) will increase interest rates.

5. What is the Fed’s mandate?

Ans. Promote employment, stable inflation and low long term interest rates.

6. Interpret the following regression explaining the Fed rate: FF = α + β2 × XtP ay + β3 × X Inf rt+1

where: rtF F is the current Fed funds rate; XtP ay is Payroll Growth; and XtInf is Inflation.

You are given the following table: parameter α β2 β3

coefficient 1.5 0.5 0.8

std. error 0.70 0.25 0.20

You are also given the value of R2 = 0.5500.

Answer the following: i. What is the interpretation of α? Does this make any economic sense? ii. How well does Payroll growth explain Fed fund rates?

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iii. How well does inflation explain Fed fund rates?

Ans. For the model: i. α is the intercept of the equation and it basically says that on average rates will be at least 1.5%. This does not have any economic interpretation, it is mostly a term that is capturing variation on the equation due to the weakness of the other variables. ii. Payroll growth does a terrible job explaining future rates. It is not statistically significant. iii. Inflation is statistically significant and is in line with what we would expect about the impact of inflation on rates. When inflation goes up, interest rates will go up to take liquidity out of the market.

7. Interpret the following regression explaining the Fed rate: FF rt+1 = α + β1 × rtF F + β2 × XtP ay + β3 × X Inf

where: rtF F is the current Fed funds rate; XtP ay is Payroll Growth; and XtInf is Inflation.

You are given the following table: parameter α β1 β2 β3

coefficient 0.4 0.6 0.4 0.3

std. error 0.30 0.08 0.14 0.20

You are also given the value of R2 = 0.5500.

Answer the following: i. What is the interpretation of α? Does this make any economic sense? ii. How well do past rates explain present rates? iii. How well does Payroll growth explain Fed fund rates? iv. How well does inflation explain Fed fund rates? v. Is this model better than the one presented previously?

Ans. For the model: i. α is not statistically significant as expected. ii. Past rates are statistically significant, and in line with the economic intuition. We would expect for rates not to change much from their past value, given the instability that frequent changes on the rate would generate.

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iii. Changes in payroll are statistically significant and are in line with what would be expected. When payrolls fall, the government would be under pressure to lower interest rates to increase liquidity and expand the economy. Not a very neoclassical approach, but the evidence point to this happening. iv. Inflation is not statistically significant in this case. The other variables capture the variation assigned to inflation in the previous model. v. This model is better at explaining rates than the previous one, due to the higher R2 .

8. Assume that you believe that the Fed funds rate is best explained by the following: FF = α + β1 × rtF F rt+1 where: rtF F is the current Fed funds rate; and each time step is a week.

You are given the following table: parameter α β1

coefficient 0.0001 0.9955

std. error 0.0002 0.0020

You also find that standard error equals 0.33, and that r0 = 6%.

Answer the following: i. What would your payoff be in a week for a Fed Funds Futures contract that pays 100 minus the current Fed funds rate. ii. What is the confidence interval for the Fed funds futures payoff? The confidence interval is defined as: [x̄ + 1.96 × SEx̄ , x̄ − 1.96 × SEx̄ ].

Ans. The answers: i. The payoff is 94.0270. ii. The confidence interval for the contract’s payoff is [93.3802, 94.6738].

9. Is using Fed funds futures to forecast the Fed rate unbiased?

Ans. No, it is not unbiased. Futures prices include market participants’ expectations on the Fed rate, but also their attitude towards risk.

10. What is the expectations hypothesis?

Ans. The expectations hypothesis links market participants expectations about future rates and the current shape of the yield curve. It supposes that future yields (being averaged with current ones) build the current term structure. In other words, the expectations hypothesis states that long term yields depend only on market participants expectations of future yields.

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11. What is missing from the expectations hypothesis?

Ans. The expectations hypothesis assumes that future rates are known and, therefore, not random. This means that there is no uncertainty about these rates. This is false, future interest rates are uncertain. Yet the principal problem is that by being uncertain and affecting the value of market participants’ securities they are also a source of risk that is priced in the securities. Thus the expectations hypothesis isn’t an adequate way of explaining interest rates.

12. If market participants don’t expect yield to go up a year from now, can the yield curve be upwards sloping? Why? Ans. Yes. If market participants change their attitude towards risk (require a higher premium to hold long term bonds) it may so happen that the yield curve be upwards sloping without changing expectations on future rates.

13. How does inflation affect bonds?

Ans. Inflation affects purchasing power of coupons and principal payment from bonds.

14. In what sense are bonds insurance against inflation? When does this not hold?

Ans. To the extent that returns on a bond are above or the same as inflation, we can say that bonds are insurance against inflation. When inflation surpasses the rate of return on a bond, then this doesn’t holds since we lost value.

15. What is a TIPS? Ans. A TIPS or Treasury Inflation Protected Security is a Treasury security whose principal is indexed to inflation, specifically, the Consumer Price Index.

16. Can nominal interest rates or real interest rates be negative? Why? Why not?

Ans. Negative interest rates can only occur in real variables. This is because real rates are the rate of an investment net of inflation. If inflation is high, then real interest rates will be negative. Nominal interest rates cannot be negative.

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Chapter 8 1. What is a mortgage?

Ans. A mortgage is a loan, usually used to finance an investment in real estate (e.g. a house), that ammortizes simultanously its principal payments and its interest payments.

2. What does securitization mean?

Ans. Securitization is a way of diversifying risk, in which several institutions pool similar assets together in order to sell them to investors.

3. What is a Special Purpose Vehicle (SPV)?

Ans. An SPV is a stand-alone firm through which financial institutions pool the assets together. It is created by these financial institutions in order to formally buy the assets by raising the necessary capital from investors.

4. Prior to 2008, under what assumption were mortgages backed by Fannie Mae and Freddie Mac considered to be ’safe’ investments? Was this assumptions well founded?

Ans. The assumption was that these government-sponsored agencies would be ’bailed out’ if they were at risk of becoming insolvent. This assumption proved correct when the government placed both firm under ’conservatorship’.

5. Fill data on the following tables representing the cash flow of a 2-year mortgage that is paid semiannually with principal 1,000 and interest 5%: t i Ai Coupon Principal (before) Interest paid Principal paid Principal (after)

0.0 0

0.5 1 0.9756

1.0 2 0.9518

1.5 3 0.9268

2.0 4 0.9060

0.0 0

0.5 1 0.9756 265.82 1,000 25.00 240.82 759.18

1.0 2 0.9518 265.82 759.18 18.98 246.84 512.34

1.5 3 0.9268 265.82 512.34 12.81 253.01 259.33

2.0 4 0.9060 265.82 259.33 6.48 259.33 0.00

Ans. The results are: t i Ai Coupon Principal (before) Interest paid Principal paid Principal (after)

1,000

1,000

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6. Price the previous mortgage, assuming that the term structure of interest rates is flat at 5%.

Ans. The mortgage is worth $1,052.29.

7. What is a prepayment option?

Ans. A prepayment option refers to the posibility that the mortgatge owner has of paying, at any time, the outstanding principal on the mortgage in order to terminate the contract.

8. When is it assumed that the prepayment option will be exercised, if mortgage owners act rationally?

Ans. It is assumed that a mortgage owner will prepay (refinance) the mortgage if the market value of the mortgage becomes higher than the remaining principal.

9. What does CPR stand for? What does it measure?

Ans. CPR is the Conditional Prepayment Rate and it is the annualized rate of monthly prepayment of outstanding mortgages in a pool.

10. What does 100% PSA mean? How does it compare against 50% PSA and 200% PSA? Ans. PSA refers to the Public Securities Association. Through its experience, this association set an industry benchmark on prepayment speed (based on CPR). At 100% CPR we have that: • i. In the first month CPR is 0.2%.

• ii. CPR increases by 0.2% in each of the following 30 months.

• iii. After this period, CPR stays at 6% until maturity Increasing or decreasing the PSA simply means that prepayment is ocurring at a slower or faster rate than this scenario, respectively. For example, 50% PSA means that prepayment occurs at half of the speed presented in the previous scenario, while 200% PSA means that prepayment occurs twice as fast than presented in the 100% PSA scenario.

11. What is the main difference between a pool of mortgages and a passthrough security from the same pool?

Ans. The main difference will be in the rate of interest charged. The pool of mortgages will receive a given rate of interest, but only a portion of this is passed along since it is used to pay off the institution in charge of the securitization. So pass through rates are smaller than the actual mortgage rate. The original pool and the pass through will have the same maturity.

12. What is negative convexity?

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Ans. Negative convexity refers to the effect that changes in interest rates have on some fixed income securities. For bonds it is usually thought that when interest rates decrease, prices go up. Yet for some securities the opposite occurs.

13. Do MBS have negative convexity? Why? Ans. MBS do have negative convexity because as interest rates fall, more mortgage owners will exercise the prepayment option. This means that these securities don’t gain value as regular bonds when interest rates fall, but instead converge to the value of the principal.

14. What is the TBA market?

Ans. It is the ”To-Be-Announced” market. Which means that traders don’t know the exact composition of these securiites when they start to bid for the security. It is similar to the forward market.

15. What is a Collateralized Mortgage Obligation (CMO)?

Ans. Collateralized Mortgage Obligations are securities derived from MBS that are structured with different risk-return characteristics in order to appeal to different investors’ clienteles.

16. What is a tranche?

Ans. A tranche is a division on the total pool of assets to be paid in a given sequence, sually named by A, B, C, and so on. A tranche will receive the payments from the mortgage pool until a given amount of principal is retired. The first tranches are less susceptible to risk (prepayment, and other types) than the following ones.

17. What is a Planned Amortization Class (PAC)?

Ans. PAC are also a type of tranche, these are modeleded to have a deterministic coupon given a range of PSA. The final tranche, called Companion Tranche aborbs all of the prepayment risk.

18. What is an Interest Only (IO) strip?

Ans. Interest Only strips are securities that receive all interest payments from the underlying pool of assets and none of the principal.

19. What is a Principal Only (PO) strip?

Ans. Principal Only strips are securities that receive all principal payments, scheduled and unscheduled, from the underlying pool of assets and none of the interest payments.

20. What happens to the value of an IO if interest rates fall?

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Ans. If interest rates fall, more mortgage owners will prepay their debt. In turn this means that future interest payments will not occur, so IO strips loose value.

21. What happens to the value of a PO if interest rates fall?

Ans. If interest rates fall, more mortgage owners will prepay their debt. In turn this means that principal payments will occur quicker. Given the time value of money, the PO strips increase in value.

35

Chapter 9

You are given the following interest rate tree. Use it when required in the exercises. i t

0 0.0 2%

1 0.5 4% 1%

1. What is the favored approach in the development of interest rate models?

CAPM? Ans. The favored approach is to use no arbitrage, with this methodology we mainly use simple traded securities to price more complicated derivatives.

2. What is a replicating portfolio?

Ans. A replicating portfolio of a security with payoffs V1,u and V1,d in the two nodes u and d at time i = 1 is a portfolio of bonds that exactly replicates the values of the security at time i = 1. That is, if Πi,j denotes the value of the portfolio at time i in node j, we have Π1,u = V1,u and Π1,d = V1,d .

The value of the option at i = 0 equals the value of the portfolio Π0 = V0 3. Given the tree at the begining of this chapter, what is the value of a zero coupon bond maturing in six months?

Ans. The value is 99.0050.

4. What values can a one year zero coupon bond take at t = 0.5?

Ans. It can be either 98.0199 or 99.5012.

5. What is the replicating portfolio for a European option on interest rates with maturity at t = 0.5, rK = 2.5% and payoff: 100 × max(rt − rK , 0).

You use the two year zero coupon to replicate, which is currently trading at 97.4790. Give values of N1 and N2 , as well as the price of the option under this procedure.

Ans. N1 = 1.0075 and N2 = -1.0126, when we input into the portfolio formula: Π0 = N1 × P0 (1) + N2 × P0 (2) = 1.0452 6. What is the replicating portfolio for a European option on interest rates with maturity at t = 0.5, rK = 3.0% and payoff: 100 × max(rK − rt , 0).

You use the two year zero coupon to replicate, which is currently trading at 97.4790. Give values of N1 and N2 , as well as the price of the option under this procedure.

Ans. N1 = -1.3234 and N2 = 1.3501, when we input into the portfolio formula: Π0 = N1 × P0 (1) + N2 × P0 (2) = 0.5865 36

7. What is the replicating portfolio for an interest rate swap with maturity at t = 0.5, rK = 2.0% and payoff: 100 × (rK − rt )/2. You use the two year zero coupon to replicate, which is currently trading at 97.4790. Give values of N1 and N2 , as well as the price of the option under this procedure.

Ans. N1 = -0.9975 and N2 = 1.0126, when we input into the portfolio formula: Π0 = N1 × P0 (1) + N2 × P0 (2) = −0.0551 8. What is the market price of risk underlying the tree presented?

Ans. We have λ0 = −0.2018.

9. Using the risk adjusted discounted present value of future cash flows price a European option on interest rates with maturity at t = 0.5, rK = 2.0% and payoff: 100 × max(rt − rK , 0).

Ans. The price is 1.3936.

10. Using the risk adjusted discounted present value of future cash flows price a European option on interest rates with maturity at t = 0.5, rK = 2.5% and payoff: 100 × max(rK − rt , 0).

Ans. The price is 0.4399.

11. Using the risk adjusted discounted present value of future cash flows price an interest rate swap with maturity at t = 0.5, rK = 1.5% and payoff: 100 × (rK − rt )/2.

Ans. The price is 0.1924.

12. What is the market price of risk?

Ans. The market price of risk is defined by the ratio between risk premium and risk that is common to all interest rate securities. In other words it is the amount of compensation that market participants expect to receive per unit of risk that they hold.

13. For pricing purposes how important is it to know the true probabilities?

Ans. It is not important to know the true probabilities, as long as they are consistent. This is shown by the fact the we use risk neutral probabilities to price, knowing that they are not the true (risk natural) probabilities.

14. What is risk neutral pricing?

Ans. Risk neutral pricing means to deliberately modify the probabilities on a tree or model, in order to set the market price of risk to zero. This simplifies the calcualations made when pricing securities.

15. What is a risk neutral probability?

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Ans. Risk neutral probability is the a special type of probability on a model that makes the market price of risk equal to zero.

16. Assuming that there is a risk premium in the market (people worry about risk and expect to be compensated for it), is risk neutral probability for an up state (high interest rates) higher, lower or the same as the risk natural probability?

Ans. Risk neutral probabilities tend to be higher than risk natural probabilities since they should compensate market participants enough (by giving them a higher probability of higher interest rates) in order to make them risk neutral.

17. True or False: from a given risk neutral tree can you compute the market participants’ expectation on the level of interest rates in the future?

Explain.

Ans. False. You cannot use a tree to compute market expectations since the tree is computed in the risk neutral world, where probabilities are not the the true probabilities. In order to predict the true value we need to use the true probabilities.

18. Are forward interest rates equal to the market’s expectation of future interest rates?

Ans. No. Current high forward rates might mean two things: either market participants expect higher interest rates; or they are strongly averse to risk, and thus the price of long term bonds is low today.

19. Why are forward interst rates and the risk neutral expected future interest rates not the same?

Ans. The key to understand this is that prices are the relevant values, not interest rates. But it should be noted that the relationship between interest rates and prices is not linear but convex. When comparing the prices derived from these rates we see that forward rates have to higher than risk neutral expected future interst rates.

20. What is the risk neutral probability p∗ of the tree presented?

Ans. We have that p∗ = 0.7038.

21. Using risk neutral pricing obtain the value for a European option on interest rates with maturity at t = 0.5, rK = 1.5% and payoff: 100 × max(rt − rK , 0).

Ans. The price is 1.7419.

22. Using risk neutral pricing obtain the value for a European option on interest rates with maturity at t = 0.5, rK = 2.0% and payoff: 100 ×max(rK − rt , 0).

38

Ans. The price is 0.2933.

23. Using risk neutral pricing obtain the value for an interest rate swap with maturity at t = 0.5, rK = 1.0% and payoff: 100 × (rK − rt )/2.

Ans. The price is 0.4399.

39

Chapter 10

You are given the following interest rate tree. Use it when required in the exercises. i t

0 0.0 2.00%

1 0.5 4.00% 1.00%

2 1.0 6.00% 2.60% 0.12%

1. Using risk neutral pricing obtain the value for a 1.5 year zero coupon bond.

Assume that p∗ = 0.7038 is constant over time.

Ans. The price is 95.5267.

2. Using risk neutral pricing obtain the value for a call option on a 1.5 year zero coupon bond with K = 99.00, maturity at t = 1. Assume that p∗ = 0.7038 is constant over time.

Ans. The price is 0.0813.

3. Using risk neutral pricing obtain the value for a put option on a 1.5 year zero coupon bond with K = 97.40, maturity at t = 1. Assume that p∗ = 0.7038 is constant over time.

Ans. The price is 0.1709.

4. When talking about options, what is a straddle?

Ans. A straddle is a combination of long a call option and long a put option with the same strike price.

5. Which of the following prices should be higher: a call option, a put option or a straddle. All of them have the same maturity, underlying security and strike price. Explain.

Ans. The most expensive one is the straddle, since it is actually the sum of the other two options.

6. Using risk neutral pricing obtain the value for a straddle on a 1.5 year zero coupon bond with K = 98.00, maturity at t = 1. Assume that p∗ = 0.7038 is constant over time.

Ans. The price is 0.9158.

7. Compute the values NtL and NtS for the dynamic replication strategy for a call option on a 1.5 year zero coupon bond with K = 99.00, maturity at t = 1. Assume that p∗ = 0.7038 is constant over time.

Ans. The following give a summary of the answer:

40

NtS -0.0891

0.0000 -0.7534

NtL 0.0932

0.0000 0.7632

8. Compute the values NtL and NtS for the dynamic replication strategy for a put option on a 1.5 year zero coupon bond with K = 97.40, maturity at t = 1. Assume that p∗ = 0.7038 is constant over time.

Ans. The following give a summary of the answer: NtS 0.0813

0.2109 0.0000

NtL -0.0825

-0.2136 0.0000

9. Compute the values NtL and NtS for the dynamic replication strategy for a straddle on a 1.5 year zero coupon bond with K = 98.00, maturity at t = 1. Assume that p∗ = 0.7038 is constant over time.

Ans. The following give a summary of the answer: NtS -0.0567

0.1536 -0.9800

NtL 0.0683

-0.1485 1.0000

10. Why do we say that the dynamic replication strategy is self-financing?

Ans. Because with the capital gain on the replicating portfolio from a previous period we have enough to purchase the new (rebalanced) portfolio for the next period.

11. What is the difference between risk neutral probability and risk natural probability?

Ans. Risk natural probability explains the state of the world, where market participants are willing to take on risk for a given price. Risk neutral probability is a probability that is deliberately altered in order to make market participants risk neutral, this is done specifically for pricing purposes.

12. In order to compute the spot rate duration do you use risk neutral probabilities or risk natural probabilities?

Ans. You use risk neutral probabilities, since it is derived from pricing bonds in different scenarios (see spot rate duration formula).

13. What is one major drawback from using empirical estimates to fit the ”true” interest rate tree?

Ans. One major drawback is that it may generate negative nominal interest rates.

41

14. How realistic is it to speak about negative interest rate in real terms?

Ans. Real negative interst rate can occur and have occured, as inflation may be higher than nominal interest rates.

15. How realistic is it to speak about negative interest rate in nominal terms?

Ans. This is economically unreasonable since it means that investors are willing to pay the goverment in order to loose money in the future.

16. Compute the spot rate duration for a call option on a 1.5 year zero coupon bond with K = 99.00, maturity at t = 1. Assume that p∗ = 0.7038 is constant over time.

Ans. Spot rate duration is -9.2354.

17. Compute the spot rate duration for a put option on a 1.5 year zero coupon bond with K = 97.40, maturity at t = 1. Assume that p∗ = 0.7038 is constant over time.

Ans. Spot rate duration is 8.1734.

18. Compute the spot rate duration for a straddle on a 1.5 year zero coupon bond with K = 98.00, maturity at t = 1. Assume that p∗ = 0.7038 is constant over time.

Ans. Spot rate duration is -6.7695.

42

Chapter 11

You are given the following interest rate tree. Use it when required in the exercises. t

0 3.00%

1 6.00% 2.00%

2 9.00% 4.00% 1.00%

1. What is the benefit of using an interest rate model when compared to empirical estimates?

Ans. Interest rate models such as Ho-Lee and Black-Derman-Toy eliminate the posibilities for negative interest probabilities that may occur in simple empirical estimates.

2. What advantage does the Black-Derman-Toy model have over the Ho-Lee model, when comparing the plausibility of the modeled interest rates?

Ans. Ho-Lee allows for the posilibity of negative interest rates.

3. What are the main differences between the Ho-Lee model and the BlackDerman-Toy model?

Ans. The following are the differences: i. The Ho-Lee model gives non-zero probability to negative interest rates, ans small probability to high interest rates. ii. The Black-Derman-Toy model gives essentially zero probability to interest rates below 1%, but it assigns a much higer probability to high interest rates.

4. You find that the Black-Derman-Toy model predicts a rise in the short rate for both the next up period and the next down period. Given this information you decide to short Treasuries, since a future rise in interest rates will bring bond prices down. Is this right?

Ans. This is wrong, since the Black-Derman-Toy model is a risk neutral model.

The probabilities given are only good for pricing purposes, they are not an indication of the level of future interest rates.

5. Assume that after you estimate the risk neutral model for the continously compounded rate you arrive at the tree presented at the beginning of this chapter. There is equal probability of moving up or down on the tree.

Compute the current zero coupon spot curve for all possible maturities.

Ans. We have: Z(0, 1) = 0.9704; Z(0, 2) = 0.9326; and Z(0, 3) = 0.8923.

43

6. Assume that after you estimate the risk neutral model for the continously compounded rate you arrive at the tree presented at the beginning of this chapter. There is equal probability of moving up or down on the tree.

Price a floor that pays at time t + 1 the following cash flow: CF (t + 1) = N × max(rK − r1 (i), 0) where N = 100 and rK = 4%.

Ans. The price is 2.5731.

7. Assume that after you estimate the risk neutral model for the continously compounded rate you arrive at the tree presented at the beginning of this chapter. There is equal probability of moving up or down on the tree.

Price a floor that pays at time t + 1 the following cash flow: CF (t + 1) = N × max(rK − r1 (i), 0) where N = 100 and rK = 3%.

Ans. The price is 0.9357.

8. Assume that after you estimate the risk neutral model for the continously compounded rate you arrive at the tree presented at the beginning of this chapter. There is equal probability of moving up or down on the tree.

Price a 2-year cap that pays at time t + 1 the following cash flow: CF (t + 1) = N × max(rK − r1 (i), 0) where N = 100 and rK = 4%.

Ans. The price is 2.1645.

9. Assume that after you estimate the risk neutral model for the continously compounded rate you arrive at the tree presented at the beginning of this chapter. There is equal probability of moving up or down on the tree.

Price a 2-year cap that pays at time t + 1 the following cash flow: CF (t + 1) = N × max(rK − r1 (i), 0) where N = 100 and rK = 3%.

Ans. The price is 3.3233.

10. How do you compute the swap rate at initiation?

Ans. The swap rate at initiation is given by:

1 − Z(0, TM ) c = n × M j=1 Z(0, Tj ) 44

11. Assume that after you estimate the risk neutral model for the continously compounded rate you arrive at the tree presented at the beginning of this chapter. There is equal probability of moving up or down on the tree.

Price a 2-year swap with N = 100 and c = 3%.

Ans. The price is 0.9899.

12. Suppose you want to hedge the cap with the swap, what is the hedge ratio?

Ans. The hedge ratio is: 0.7431.

13. What is the difference between empirical volatility and implied volatility?

Ans. Empirical volatility refers to the standard deviation of past realized changes in the short term interest rate. Implied volatility is the volatility needed so that the price derived from the modeled matches exactly the value of a security in the market.

14. Does empirical σ (based on past realizations) price caps, floors and swaptions acurately? On average does it overprice or underprice these securities?

Ans. Empirical volatility tends to underprice caps, floors and swaptions.

15. You find the implied volatility for a 5-year cap and you use it as an input for your model (Ho-Lee). Does this solve the problem with volatility when you want to price 1-year securities?

Ans. No it doesn’t. The 5-year cap volatility will work only for securities with that maturity, but it will not price correctly securities with different maturities. This is mainly because volatility is not constant over time.

16. What is the difference between flat volatility and forward volatility?

Ans. Forward volatility is the level of volatility at each step of the BlackDerman-Toy model that matches the cap price for that step. Forward volatility assumes that volatility changes at each step. Flat volatility can be thought as an average of forward volatilities, and it is the single value of volatility needed in order to match the model price with the observed price for a given security. Note that flat volatility doesn’t assume that volatility changes at each step, but instead maintains constant volatility in order to price a single security.

17. In the context of the futures market, what does ’cheapest-to-deliver’ mean?

Ans. In a T-bond and T-note futures, the short side has the option to deliver any security within a given class of deliverable treasury. Alsthough an adjustment is made to standarize securities, still there is a bond that is the least expensive to deliver at maturity. This bond is the chepest-todeliver.

45

18. If you use caps and bonds to fit the Black-Derman-Toy model, can you use the model to price the same caps and bonds?

Ans. No, you cannot. You need one set of securities to calibrate the model, and then you can use the model to price other securities.

46

Chapter 12

You are given the following interest rate tree. Use it when required in the exercises. t

0 3.00%

1 6.00% 2.00%

2 9.00% 4.00% 1.00%

1. What is an American Call option?

Ans. An American Call option is a contract between two counterparties in which one party (option buyer) has the right, but not the obligation, to buy a given security at a predetermined price on or before a given maturity, and the other party (option seller) has the obligation to sell such security.

2. Does a callable bond have an American option embedded in it?

Ans. Yes it does, because the issuer has the right to buyback the option at any given time before the bond reaches maturity.

3. Does a student loan have an American option embedded in it?

Ans. Yes it does, becuase the person who takes out the loan can payback the remaining principal at any time before maturity.

4. Does a mortgage have an American option embedded in it?

Ans. Yes it does, becuase the person who takes out the loan can payback the remaining principal at any time before maturity.

5. What effect does an embedded option have on a security’s price?

Ans. An option is a security since it is a promise on contingent cash flows.

This means that, under no arbitrage, its price should be incorporated into the price of the original security, thus, increasing the value of the original security for the holder of the option.

6. Using the information provided at the beginning of this chapter, price a 3-year callable bond with coupon rate 5% and strike price at par.

Ans. The price of the callable bond is 100.

7. Using the information provided at the beginning of this chapter, price a 3-year callable bond with coupon rate 5% and strike price at par. There is also a lockout period through which the option cannot be exercised until t = 1.

Ans. The price of the callable bond is 100.4335.

47

8. Using the information provided at the beginning of this chapter, price a 3-year callable bond with coupon rate 5% and strike price at par. There is also a lockout period through which the option cannot be exercised until t = 2.

Ans. The price of the callable bond is 101.8507.

9. You notice that when no lockout period is present and for coupons above 5%, the callable bond with strike at par is priced at par. What is happening?

Ans. Higher coupons increase the present value of the bond, this means that it is more profitable to exercise the option early. So the amount received for the bond is the strike price, which is par.

10. Why is it that as the lockout period becomes longer the price of the bond increases?

Ans. This happens because an increase in the lockout increases restrictions on the option, which will fall in value. As the option reduces the value of the underlying bond, a fall in its price increases the value of the bond.

11. Using the information provided at the beginning of this chapter, price a 3-year callable bond with coupon rate 5% and strike price at par. There is no lockout period.

Ans. The price of the callable bond is 100.0000.

12. What is negative convexity?

Ans. Negative convexity refers to the effect that changes in interest rates have on some fixed income securities. For bonds it is usually thought that when interest rates decrease, prices go up. Yet for some securities the opposite occurs.

13. How is it that negative convexity is present on a callable bond?

Ans. Callable bonds allow the issuer to call back the security by paying face value to the owner of the bond. When interest rates fall the value of the bond normally increase, yet in this case there is a difference. When interest rates fall and make the value of the bond higher than face value the issuer will prefer to call it back and pay face value. This means that as interest rates fall, the value of the bond converges to face value.

14. What happens when the issuer of an American option does not exercise optimally?

Ans. Not exercising optimally increases the value of the bond for the holder of the bond. This occurs because the holder now holds a security that has appreciated in value above face value, which is what he would receive under optimal exercise.

48

15. Which is more valuable and American call option or a European call option?

Ans. The posiblity of exercising in more dates gives additional value to the holder. So an American call option is worth more than an European call option.

16. Why is a mortgage a type of callable bond?

Ans. Because at any time the mortgage owner may pay off the remaining principal in order to get out of the contract, with no obligation to cover the lost interest payments.

17. Do pass through securities have negative convexity?

Ans. Yes, the have negative convexity. As with the underlying pool of mortgages, pass through securities are exposed to prepayment risk which makes the value of the securities to converge to the face value when interest rates go down.

18. What is the relationship between changes in interest rates and changes in value of an Interest Only (IO) strip?

Ans. IO strips move in the same direction as the interest rate (the opposite than normal bonds). As interest rates fall, there is a higher probability of prepayment. Higher prepayment means that interest rates originally expected to be received will be forgone. This in turn reduces the value of the IO strips.

19. What is the relationship between changes in interest rates and changes in value of a Principal Only (PO) strip?

Ans. As interest rates fall Principal Only strips increase in value, as a higher prepayment means that the bulk of the cash to be received is paid more quickly. Given the time value of money, this earlier cash is bares additional value.

49

Chapter 13 1. What is a Monte Carlo Simulation?

Ans. A Monte Carlo Simulation is a methodology of predicting the behavior of a variable by simulating a large number of paths under which the random component of the variable can take any value. The result is a large sample of possible values for the variable from which we can infer its expected value and other moments.

2. What is an Asian Interest Rate Option?

Ans. An Asian Interest Rate Option is an option whose payoff at maturity is given by a. For an Asian Call max(average rate from 0 to T − rK , 0) b. For an Asian Put max(rK − average rate from 0 to T, 0) 3. Is the traditional tree methodology well-suited to price the following: an option where the owner has the right to buy a bond at its lowest price over some specified period.

Ans. No, this security, called a ”lookback” option is not well suited for the traditional tree methodology because it is path dependant.

4. Is the traditional tree methodology well-suited to price the following: a fixed-for-floating swap where LIBOR is the underlying rate.

Ans. Yes, tress can price well these types of securities.

5. When pricing zero coupon bonds, are results from the Monte Carlo simulations on a tree the same as risk neutral pricing on a tree? Why? Ans. They might not be the same, but as the number of simulations increase the results will converge.

6. What is a standard error?

Ans. Standard error is the standard deviation of an estimate (e.g. the √ ones obtained through Monte Carlo Simuations). It is defined as SE = σ/ N .

7. What is a confidence interval?

Ans. A confidence interval presents an interval under which there is 95% probability (or any other probability) that the estimate will be within such interval. For the 95% interval we have: [p̂ − 1.96 × SE, p̂ + 1.96 × SE] 50

8. How many simulations are enough?

Ans. As many as we need to get standard errors which we feel comfortable with.

9. Given that simulations do not offer a closed form solution, can we still calculate a price’s sensitivity to interest rate movements?

Ans. Yes, we can still do so through spot rate duration that prices the security again with an increment in interest rates (e.g. 1 basis point).

10. How is spot rate duration defined in Monte Carlo simulations?

Ans. Spot rate duration is approximated by: −

1 P̂ (r0 )

P̂ (r0 + dr) − P̂ (r0 ) dr

11. When pricing through Monte Carlo simulations on trees we are implicitly using risk neutral probabilities, this is also so when computing the spot rate duration. Is this correct? Shouldn’t measures of sensitivity be computed with risk natural probabilities?

Ans. No, when we review the spot rate duration formula we see that we are actually pricing. Here we are not interested so much in estimating what might happen, but in describing the security’s price sensitivity.

12. Why is it useful to price Mortgage Backed Securities (MBS) through Monte Carlo Simulations?

Ans. Because we can incorporate additional factors affecting the prepayment decision into the pricing model.

13. What additional factors may affect the prepayment decision?

Ans. Some additional factors affecting the prepayment decision are: i. Random Event. For example the sale of the house. ii. Seasonality. Homeowners tend to move much more over the summer than in the winter. iii. Forgetfulness. Some homeowners don’t pay attention to the fact that interest rates are sufficently low to refinance their mortgage.

14. In the context of the prepayment of mortgages, what is seasonality?

Ans. Seasonaility is a factor that affects the prepayment decision, in the sense that people don’t move as much in the winter as they do in other seasons.

15. How does seasonality affect the prepayment option? Is the link direct?

51

Ans. This factor does not affect the decision directly, instead it says that given the fact that people decide to move from their house (because of some random event as changing jobs) they will prefer to do it outside the winter season.

16. How effective is pricing of Collateralized Mortgage Obligations (CMO) on a risk neutral tree? Why? Ans. The tree methodology has many problems for mortgages because it maintains constant the level of PSA, which is directly linked to the interest rate through the prepayment option.

17. What advantages do Monte Carlo simulatons on a tree provide when pricing MBS tranches?

Ans. Through Monte Carlo simulations we can adjust the level of PSA faced under different interest rate scenarios by including this adjustment in the paths to be taken by the simulation, incorporating the sensitivity of prepayment to changes in the interest rates.

18. What is a prepayment model?

Ans. A prepayment model is a model that predicts the amount of prepayment to occur under different market conditions. It may include the factors mentioned before (random events, seasonality and forgetfulness) as well as others.

52

Chapter 14 1. Why do we need continous time models?

Ans. Because it allows us to consider the case in which underlying variables, such as interest rates, move at a high frequency (e.g. daily or intradaily).

This brings realism and simplicity into our models, and, in some cases, allows us to obtain analytical formulas to price and hedge securities.

2. What is a Brownian motion?

Ans. First, we define Δ = t/n where t is a lenght of time, and n is the number of intervals into which we divide that length. Second, we √ define Zi as a √ random variable with equal probability of being Δ or − Δ. We then consider the following quantity: Xt =

n

Zi

i=1

So a Brownian motion is given by the stochastic variable Xt as n increases to infinity, and thus Δ converges to zero.

3. What are the properties of a Brownian motion?

Ans. The main property is that if we take any two times t1 and t2 , then the difference (Xt2 − Xt1 ) is normally distributed (as Δ goes to zero).

4. Show that: V ar[dXt ] = dt.

Ans. We know that: Xt2 − Xt1 =

m

Zi

i=1

which leads to: V ar[(Xt2 − Xt1 )] = V ar

m

Zi

i=1

From statistics we know that if some random variables are identically and independently distributed, then the variance of the sum is equal to the sum of the variances, so: m

V ar[Zi ] =

i=1

m

Δ = m×Δ

i=1

In this case, given that the interval is [t1 , t2 ] we have:

t2 − t 1 Δ= m So we get:

V ar[(Xt2 − Xt1 )] = m ×

53

t2 − t1 m

= t2 − t1

5. What is the Martingale property?

Ans. The Martingale Property states that the best forecast of the value of a Brownian motion in the future is the value today. That is, if we know that at t = 0 the value of the Brownian motion X0 , then for every t > 0: E[Xt |X0 ] = X0 6. You put money into the stock market because you expect to make a profit (although you might loose money). Does your capital follow a Martingale?

Ans. No, it doesn’t. Not unless you are unrational. You put your money in because on average the stock market gives positive returns, not zero.

7. What is a differential equation?

Ans. A differential equation establishes a relation between the rate of change of a function B(t) and its value at time t, B(t).

8. What is a solution to an Ordinary Differential Equation?

Ans. Let f(t) be a function of a variable t, and let G(·) a function that relates f(t) to the rate of change of f(t) (this is an Ordinary Differential Equation): df = G(f(t)) dt Given and initial condition f(0) = k, f(t) is the solution, or satisfies, the Ordinary Differential Equation if for every t its derivative df/dt equal G(f(t)) and if indeed f(0) = k.

9. What two components do we include in a Continous Time Stochastic Process?

Ans. We include a Brownian motion and a Differential Equation.

10. What is the main difference between the Ho-Lee and the Vasicek model?

Ans. Vasicek model includes a mean reverting process, while Ho-Lee doesn’t.

11. When dealing with a stochastic process what do we mean by a ’drift’ ?

Ans. The drift term of the stochastic process represents the predictable component of the stochastic process.

12. When dealing with a stochastic process what do we mean by a ’diffusion’ ?

Ans. The diffusion term is the unpredictable component of the stochastic process, due to the lack of predictability of the Brownian motion dXt .

13. What problem do both Vasicek and Ho-Lee models share?

Ans. They assign postive probability to the occurrence of negative interest rates (nominal interest rates).

54

14. For the Vasicek model, show that the as t → ∞ we have: μ(r0 , t) = r̄ σ 2 (t) =

σ2 2γ

Ans. The distribution of interest rates under the Vasicek model is: rt N (μ(r0 , t), σ 2(t)) where:

μ(r0 , t) = r̄ + (r0 − r̄)e−γt

σ2 (1 − e−2γt ) 2γ As t → ∞ all terms elevated to −t will go to zero. So we get the values above. σ 2 (t) =

15. What is Ito’s Lemma?

Ans. Ito’s Lemma provides the ”rules” of calculus to link the variation of an underlying stochastic variable, such as the interest rate rt , to the price of securities that depend on it.

16. According to Ito’s Lemma, what are the three components of the drift term in an asset?

Ans. The three components are: i. Capital gain (loss) due to the passage of time. ii. Capital gain (loss) due to variation of interest rates. iii. Convexity effect adjusted for variance (higher variance, higher impact of convexity effect).

17. Show that: E Ans. We know that:

dPt = rt dt Pt

dPt = rt dt − (T − t)σdXt Pt

So: E

dPt = E[rt dt−(T −t)σdXt ] = E[rt dt]−E[(T −t)σdXt ] = rt dt−(T −t)σE[dXt ] Pt

Recall that E[dXt ] = 0, so: E

dPt = rt dt Pt 55

18. Show that: E Ans. We have:

E

2

dPt Pt

dPt Pt

2

√ = (T − t)σ dt

= E[(rt dt − (T − t)σdXt )2 ]

= E[rt2 dt2 − 2rt (T − t)σdXt dt + (T − t)2 σ 2 dXt2 ] = rt2 dt2 − 2rt (T − t)σE[dXt ]dt + (T − t)2 σ 2 E[dXt2 ] Given that dt2 = 0, E[dXt ] = 0, and E[dXt2 ] = dt so:

2 dPt E = (T − t)2 σ 2 dt Pt So taking square root of this we get: E

19. Show that: E

dPt Pt

2

√ = (T − t)σ dt

dPt × drt = −(T − t)σ 2 dt Pt

Ans. Recall that drt = σdXt , so: E

dPt × drt = E[(rt dt − (T − t)σdXt )(σdXt )] Pt

= E[rt σdtdXt − (T − t)σ 2 dXt2 ] = rt σdtE[dXt ] − (T − t)σ 2 E[dXt2 ] Given that E[dXt ] = 0, and E[dXt2 ] = dt so: E

dPt × drt = −(T − t)σ 2 dt Pt

56

Chapter 15 1. For a deterministic interest rate model, what would be the rate of return on securities? Explain.

Ans. Given no arbitrage:

dZt = rt dt Zt

2. Why is it that in a deterministic interest rate world we have that: dZt = rt dt Zt Ans. Because a deterministic interest rate would not bear any risk on account of varying interest rates, this means that under no arbitrage it should give the same return as a the risk-free rate.

3. What is a Partial Differential Equation (PDE)?

Ans. A PDE is a differential equation on many variables, in this case r and t.

4. What is a solution to a PDE? Ans. Given the following PDE: ∂Z ∂Z + γ(r̄ − r) = rZ ∂t ∂r A solution to a PDE is a function of r and t such that: (i) if we take the derivatives on the left hand side we get the right hand side; (ii) it satisfies the boundary condition (e.g. Z(r, T ) = 1).

5. Does the following equality hold in the real world? dZ = rdt Z Why or why not?

Ans. It does not hold in the real world because the short term rate of return on a bond is risky and should give a premium above the risk-free rate.

6. What do we need to build in order to be able to price any security through no arbitrage?

Ans. We need to build a portfolio that is hedged against interest rate changes.

7. What relation should hold across all securities, given the no arbitrage condition?

57

Ans. The following relationship should hold: 2

2

1 ∂ Z1 2 1 ( ∂Z ( ∂Z2 + 12 ∂∂rZ22 σ 2 − rt Z2 ) ∂t + 2 ∂r2 σ − rt Z1 ) = ∂t ∂Z1 /∂r ∂Z2 /∂r

8. What’s the intuition behind the following relationship: 2

2

1 ∂ Z1 2 1 ( ∂Z ( ∂Z2 + 12 ∂∂rZ22 σ 2 − rt Z2 ) ∂t + 2 ∂r2 σ − rt Z1 ) = ∂t ∂Z1 /∂r ∂Z2 /∂r

Ans. The key terms are: i. (∂Zi /∂t) = Annualized dollar capital gain (or loss) due to the passage of time; ii. 12 (∂ 2 Zi /∂r 2 )σ 2 = Annualized dollar capital gain (or loss) due to convexity and the stochastic nature of interest rates; iii. rt Zi = Annualized dollar interest payments in order to borrow the value Zi to purchase the bond; iv. (∂Zi /∂r) = Annualized sensitivity of the bond price to change in the interest rate.

The numerator of each of the expressions gives the annualized capital gain return due to the passage of time or convexity of a leverage position in the security. The denominator provides the ”risk” of the long position in the security, expressed in terms of its sensitivity to changes in interest rates.

9. For the Vasicek model can we say that always m∗ (r, t) = m(r, t)? Explain.

Ans. No, m(r, t) is the drift parameter for the Interest Rate process, which tries to describe the way interest rates move in reality. On the other hand, m∗ (r, t) is constructed for pricing purposes and is defined by:

∂Z 1 ∂2Z 2 ∗ ∂t + 2 ∂r2 σ − rt Z m (r, t) = − ∂Z/∂r 10. In the Vasicek model de we have level, slope and curvature in the term structure?

Ans. Yes, we have level, slope and curvature.

11. How strong is the correlation among rates in the term structure obtained from the Vasicek model? Is this realistic?

Ans. Rates on the term structure are perfectly correlated, this is a problem since this is not true for the term structure of interest rates.

12. How can we obtain r̄ and γ in order to calibrate the Vasicek model?

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Ans. We can obtain r̄ and γ from the time series of short term interest rates. r̄ can be computed as the average short term interest rate over the sample period. γ can be approximated by regressing the changes in interest rate (rt+δ − rt ) on rt × δ, where δ is the time between observations.

13. How can we obtain r̄ ∗ and γ ∗ in order to calibrate the Vasicek model?

Ans. First we note that for any choice of these two parameters, we can compute the prices of zero coupons according to the Vasicek formula for every maturity. We then compare the Vasicek zero coupons to the Treasury STRIPS data we observe. We then search for parameters for which the model prices are close, to some degree, to the data. In other words, find values of r̄ ∗ and γ ∗ such that the following quantity is minimized (NonLinear Least Squares): J(r̄ ∗ , γ ∗ ) =

n

(Z M odel (0, r0; Ti ) − Z Data (0, Ti ))2

i=1

14. How can we obtain σ in order to calibrate the Vasicek model?

Ans. σ can be estimated directly from the time series of interest rates rt .

15. Does the Vasicek model match the term structure?

Ans. The Vasicek model is overly simple and doesn’t match the term structure.

16. You are given the following parameters: r̄ = 5%; γ = 0.32; r̄ ∗ = 6.5%; γ ∗ = 0.46; and σ = 2.00%. Currently r0 = 3%. Using the Vasicek model price a 2-year coupon bond paying 8% semiannually.

Ans. The price of the security is 107.1828.

17. What are the three steps for derivative pricing and hedging?

Ans. The three steps are the following: i. Select an interest rate model (e.g. Vasicek). ii. Estimate the parameters of the model, using available data, such as zero coupon bonds. iii. Price the derivative security using the model. iv. Hedge the option exposure through a position in the underlying security.

18. What problem does the Cox-Ingersol-Ross (CIR) model solve when compared to the Vasicek and Ho-Lee models?

Ans. It does not allow negative interest rates.

19. How strong is the correlation among rates with different maturities in the term structure obtained from the CIR model?

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Ans. As with the Vasicek model, the interest rates on the term structure are perfectly correlated.

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Chapter 16 1. What is a Replicating Portfolio?

Ans. A replicating portfolio (Pt ) is a portfolio, consisting of positions in an interest rate security (Z1,t ) and cash (Ct ), that moves in the same fashion as another interest rate security (Z2,t ). In other words we have, that if 1 /∂r Δ = ∂Z ∂Z2/∂r then: Pt = ΔZ2,t + Ct replicates the return on a security Z1,t between t and t + dt, namely: dPt = dZ1,t 2. What use does the Replicating Portfolio have?

Ans. It is a useful tool to hedge securities and to take a view on whether a security may be mispriced and, thus, an arbitrage opportunity exists.

3. What means to rebalance the Replicating Portfolio?

Ans. The replicating portfolio follows the movements of the asset from one time period to another, yet as the time interval increases it looses accuracy. In order to reduce this we need to recompute the values of Δ periodically and adjust the cash position.

4. How close should the value of the Replicating Portfolio be to the asset if we increase the rebalancing frequency?

Ans. The higher the rebalancing frequency the tighter the relationship between the replicating portfolio and the asset.

5. Can you think of any reason why rebalancing doesn’t always occur at extermely short frequencies (instantenous) in the real world?

Ans. More frequent rebalancing leads to higher transaction costs.

6. What is a relative value trade?

Ans. A relative value trade seeks to take advantage of differences found between the prices of zero coupon bonds observed and those computed from a model, such as Vasicek.

7. What steps do we follow for a relative value trade?

Ans. You do the following: i. Estimate the parameters for the interest rate model (e.g. Vasicek) in order to best match prices. ii. Confront the prices given by the model with the data and find out whether they are in line with each other.

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iii. If the model does not agree with the data, according to the model there is an arbitrage opportunity. You need to set up a strategy in which you buy the cheap security and sell the expensive one. iv. In the long run, both securities should converge so that cash coming from one, covers cash going to the other one.

8. What instrument is used to hedge derivative exposure?

Ans. For hedging derivatives on an underlying asset, we use the underlying asset itself.

9. What general principal do you follow, once an arbitrage opportunity is found?

Ans. Buy cheap, sell dear.

10. What is theta?

Ans. Theta is the capital gain (loss) on a bond due to the passage of time:

1 ∂Π Π ∂t 11. What is gamma?

Ans. Gamma is another way of calling convexity:

1 ∂2Π Π ∂r 2 12. What is the theta-gamma relation?

Ans. The theta-gamma relation states that large convexity is always counterbalanced by a large, opposite, theta.

13. Given the Fundamental PDE, explain the theta-gamma relation.

Ans. Suppose we sold a call option on a zero coupon bond and we put on a position Δ in the unerlying zero coupon bond to hedge the interest rate risk. The portfolio Π is riskless and earns the risk free rate, that is, dΠ = rΠdt The portfolio must satisfy the Fundamental PDE, that is: ∂Π ∂Π 1 ∂2Π 2 ν = rΠ + (η̃ − γr) + ∂t ∂r 2 ∂r 2 However since the portfolio is Delta Hedged (∂Π/∂r) = 0, the relation is: ∂Π 1 ∂ 2 Π 2 ν = rΠ + ∂t 2 ∂r 2 62

If we divied everything by Π we get:

1 ∂Π 1 1 ∂2Π + Π ∂t 2 Π ∂r 2

ν2 = r

The intuition for the Theta-Gamma relation is that a positive-value portfolio with a high Theta is expected to make money because of the passage of time. If it was to make more money than the risk free rate, it would be pure arbitrage, because a trader could borrow at the risk free rate, set up the portfolio, and wait. The negative convexity rebalances the pure arbitrage: the volatility in interest rates tends in average to depress the portfolio value.

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Chapter 17 1. Can the following equation rV =

1 ∂2V ∂V ∗ ∂V s(r, t)2 + m (r, t) + ∂t ∂r 2 ∂r 2

subject to the boundary condition V (rT , T ) = g(rT , T ), be always solved analytically?

Ans. No, for many models we can’t find a closed form solution and we need to apply numerical methods to solve.

2. What is the Feynman-Kac Theorem?

Ans. Let V (r, t) be the price of a security, with final payoff V (rT , T ) = g(rT , T ), satisfying the partial differential equation R(r)V =

∂V 1 ∂2V ∂V ∗ s(r, t)2 + m (r, t) + ∂t ∂r 2 ∂r 2

where R(r) is some function of r. Then V (r, t) is given by −

V (rt , t) = E ∗ e

T t

R(ru )du

g(rT , T )|rt

where the expectation E ∗ [·] is taken with respect to the probability distribution induced by the process drt = m∗ (rt , t)dt + s(rt , t)dXt 3. How is the Risk Neutral process obtained?

Ans. The Risk Neutral or Risk Adjusted interest rate process is obtained from the original interest rate process by substituting its drift rate m(r, t) with the coefficient that multiplies the term ”∂V /∂r” in the Fundamental Pricing Equation, which is m∗ (rt , t) (Risk Neutral Drift).

4. What is a Monte Carlo Simulation?

Ans. A Monte Carlo Simulation is a methodology of predicting the behavior of a variable by simulating a large number of paths under which the random component of the variable can take any value. The result is a large sample of possible values for the variable from which we can infer its expected value and other moments.

5. How can an interest rate process be simulated?

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Ans. You start from an interest rate process such as: drt = m∗ (rt , t)dt + s(rt , t)dXt then we discretize the time interval [0, T ] in N = T /δ intervals of size δ.

Let the initial condition be the current √ interest rate r0 . We approximate: drt ≈ rt+δ − rt ; dt ≈ δ; and dXt ≈ δ × t+δ where t+δ N (0, 1). With this all we have to follow recursively ”Euler’s discretization scheme”: √ rt+δ = rt + m∗ (rt , t)δ + s(rt , t) δ t+δ 6. Using Monte Carlo Simulations, what steps do you need to follow in order to price a coupon bond?

Ans. Once you have simulated the interest rate path, discount the payoff and take expectations.

7. Is the price obtained from Monte Carlos Simulations exaclty the same as the one obtained through an analytical formula?

Ans. No, when there is an analytical solution, the results are different but converge as the number of simulations increase.

8. How can you increase the accuracy of the price given by the model when using Monte Carlo Simulations?

Ans. By increasing the number of simulations.

9. What is a standard error?

J Ans. The standard error of the simulated value V (r0 , 0) = J1 j=1 V j (r0 , 0) is given by Standard Deviation √ Standard Error = J where the standard deviation is computed as Standard Deviation =

1 j (V (r0 , 0) − V (r0 , 0))2 J j=1 J

10. What is a confidence interval?

Ans. A confidence interval is an interval under which there is a given probability that the real value of the parameter is.

11. How is a 95% conficende interval defined?

Ans. Confidence interval = [V (r0 , 0) − 2 × std.err., V (r0 , 0) + r × std.err.]

12. What is a range floater?

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Ans. A range floater: i. pays higher coupons than standard floating rate bonds, ii. only pays when reference rate is within a range.

13. Why were range floaters so popular during the mid-nineties?

Ans. Because they appeared in a time where there were strong expectations for low and stable interest rates. This allowed investors to enhance their yields in a low interest environment as well as speculate on the movement of interest rates.

14. What risks are involved in a range floater?

Ans. If the reference rate increases beyond the range the bond would earn less than a fixed coupon bond.

15. From MCS, how can you compute ∂V /∂r ?

Ans. Using the Central Approximation we have: ∂V V (r0 + δ) − V (r0 − δ) ≈ ∂r 2δ 16. There is a ”Forward Approximation” and a ”Central Approximation”, is there a ”Backward Apporximation”? How can you compute it?

Ans. Yes, the Backward Approximation is: ∂V V (r0 − δ) − V (r0 ) ≈ ∂r δ 17. When computing ∂V /∂r, what is the difference between ”Central Approximation” and ”Forward Approximation”? Which one is closer to the true value?

Ans. The Central Approximation is better because it averages the values of the Forward Approximation and the Backward Approximation.

18. How can you compute gamma?

Ans. The formula to compute Gamma is: V (r0 + δ) + V (r0 − δ) − 2 × V (r0 ) ∂2V ≈ 2 ∂r δ2 19. How can you compute theta?

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Ans. Recall the Fundamental Pricing Equation: rV =

∂V 1 ∂2V 2 ∂V ∗ s (r, t) + m (rt , t) + ∂t ∂r 2 ∂r 2

We can compute the values for all the terms except the ∂V /∂t term, so all we have to do is: ∂V 1 ∂2V 2 ∂V ∗ s (r, t) = rV − m (rt , t) − ∂t ∂r 2 ∂r 2

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Chapter 18 1. How do we define the market price of risk?

Ans. The market price of risk is given by the quantitiy: λ(r, t) =

1 (γ(r̄ − rt ) − γ ∗ (r̄ ∗ − rt ) σ

Defining two constants λ0 = σ1 (γr̄ − γ ∗ r̄ ∗ ) and λ1 = σ1 (γ ∗ − γ), we get: λ(r, t) = λ0 + λ1 rt 2. You are planning to use Monte Carlo Simulations in order to simulate an interest rate one quarter from now. Which probability do you use?

Ans. You use risk natural probabilities.

3. You are planning to use Monte Carlo Simulations in order to compute the value of the range floater for different scenarios. Which type of probability do you use?

Ans. You use risk neutral probabilities.

4. How do we go from Monte Carlo Simulations to security prices?

Ans. We use the Feynman-Kac theorem that states that the solution to the fundamental pricing equation is given by an expectation. Then, thanks to the Central Limit Theorem, we know that the expectation can be approximated by an average of simulated payoffs.

5. What is the ”delta” approximation?

Ans. It is a common way to approximate risk of a security by making the firs order linear approximation, so the value of an interest rate security at an interest rate level rt+δ , is often approximated through the linear Taylor expansion: δV V (rt+δ ) ≈ V (rt ) + (rt+δ − rt ) δr 6. What problem does the ”delta” approximation have?

Ans. If the convexity of the security is particularly strong this approximation may yield quite different results.

7. What factors explain long term yields?

Ans. The following explain long term yields: i. Higher expected long term inflation. ii. Higher risk aversion of market participants.

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iii. Higher amount of risk.

8. Explain the intuition behind the link between high long term yields and higher expected long term inflation.

Ans. Higher expected inflation means that cash balances from long term bonds will be worth less. In order to counter this, market participants expect higher yields.

9. Explain the intuition behind the link between high long term yields and higher risk aversion of market participants.

Ans. As risk aversion increases, market participants will require higher yields in order to hold these securities.

10. Explain the intuition behind the link between high long term yields and higher amount of risk.

Ans. A higher amount of risk pushes the market price of risk up, as market participants expect to be compensated more.

11. Why is σy σi ρyi the amount of risk?

Ans. This is the ”economic risk” of holding nominal bonds. In particular, as already mentioned, a security is risky not when it has high volatility per se, but when it delivers little money when agents need it the most. During recessions or periods of low GDP, a security that pays little in these states is risky. This is captured with ρyi < 0, since if the correlation between expcected inflation and GDP growth is negative, it implies that the nominal bond will suffer a capital loss exactly during recessions, when investros need it the most. Given this negative correlation, the amount of risk increases with the volatility of expected inflation σi and with the severity of economic downturns, captured by the volatility of GDP growth σy .

12. On what does λ depend?

Ans. The market price of risk depends on the coefficent of risk aversion, the amount of risk and the convexity term.

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Chapter 19 1. If a model dosen’t fit the term structure what can be happening?

Ans. Two things might be happening: there is an arbitrage opportunity or the model is wrong. To know in which case we are in, we need to understand the advantages and limitations of the models we use.

2. What advantage does the Ho-Lee model hold over the Vasicek model?

Ans. Ho-Lee model chooses its parameter in order to match the current term structure of interest rates.

3. Can you engage on Relative Value Trades on zero coupons obtained from the Ho-Lee model? Explain.

Ans. No, you cannot since you used the zero coupons to calibrate the model.

4. Why are values obtained from the Ho-Lee model generally higher than does obtained from the Vasicek model?

Ans. In the Vasicek model, the interest rate is mean reverting which makes its variance converge to a steady value. In Ho-Lee variance increase with the lenght of maturity.

5. What are the drawbacks of the Ho-Lee model?

Ans. There are two major drawback for the Ho-Lee model: i. The model is non-stationary. The process follows essentially a random walk, and thus for very large T interest rates could grow to plus or minus infinity. ii. The term structure of volatility is flat.

6. Under the Ho-Lee model how does the volatility of long term bonds compare to short term bonds?

Ans. It is the same, since the term structure of volatility is flat.

7. How realistic is the the term structure of volatility in the Ho-Lee model?

Ans. Empirically we know that the term structure of volatility is not flat, which makes the Ho-Lee unrealistic in this sense.

8. What is the main difference of the Hull-White model with the Vasicek model?

Ans. The main difference is that the central tendency (from mean reversion) is time dependant (i.e. it changes over time).

9. In what aspect is the Hull-White model similar to the Ho-Lee model?

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Ans. As with the Ho-Lee model, the term structure of interest rates is matched exactly.

10. What happens to the term structure of volatility in the Hull-White model?

Ans. Under the Hull-White model we can also match the term structure of spot-rate volatility.

11. Explain how each paramet in the Hull-White model is determined.

Ans. The Hull-White model for interest rate dynamics is: drt = (θt − γ ∗ rt )dt + σdXt where θt , γ ∗ and σ need to be determined. θt is used to fit the model to the term structure of interest rates, while γ ∗ and σ are used to fit the spot-rate volatilities.

12. When pricing options under the Hull-White, what are the important differences with the Vasicek model?

Ans. The main differences are: i. There is no differences between model prices and market prices. ii. The parameters γ ∗ and σ have been estimated to best fit the term structure of volatility, and thus the option price will likely be more accurate. In contrast, in the Vasicek model, the parameter γ ∗ was estimated to fit the term structure of interest rates.

13. Why are Hull-White prices for call options different from the prices obtained in the Ho-Lee model?

Ans. The reason is twofold: First, the volatility estimate of yields at the one year horizon in the case of the Hull-White model is substantially smaller than in the Ho-Lee model. Second, the mean reversion of interest rates implies that the sensitivity to interest rates is smaller in the Hull-White model than in the Ho-Lee model.

14. Why are Hull-White prices for call options different from the prices obtained in the Vasicek model?

Ans. Because instead of adjusting γ ∗ to fit the term structure of interest rates we know use it to fit the term structure of volatility of spot rates.

15. What are the characteristics of Normal models?

Ans. The three important properties of Normal models are: i. The zero-coupon bond price has the form Z(r, 0; T ) = eA(0,T )−B(0,T )r0 . ii. The distribution of future interest rates is normally distributed.

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iii. The option pricing formula is essentially identical, with the only difference stemming from the volatility σZ (T0 ; TB ).

16. What are the main drawback of Normal models?

Ans. The main drawback is that they give positive probabilities for negative (nominal) interest rates.

17. What is a log-normal model? When are they used?

Ans. A log-normal model uses log(rt ) instead of rt for the model. This makes it so that only positive rates are allowed.

18. What is the main drawback for log-normal models such as Black-DermanToy and Black-Karasinsky?

Ans. The main drawback of these models is that they do not allow analytical solutions as some of theri Normal counterparts.

19. How are Black-Derman-Toy and Black-Karasinsky related?

Ans. Black-Derman-Toy is a special case of Black-Karasinsky.

20. What is the main characteristic of an afine model?

Ans. Afine models allow for a closed form formula for bond prices.

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Chapter 20 1. What is a caplet?

Ans. A caplet is an option on interest rates with the following payoff: N × Δ × max(rn (Ti ) − rK , 0) 2. What is a cap?

Ans. A cap is a set of caplets with same stike price and with different maturities occuring within a set period (e.g. semiannually).

3. How is the pricing of a cap related to a caplet?

Ans. The price of a cap equals the sum of the caplets forming it.

4. What is a floorlet?

Ans. A floorlet is an option on interest rates with the follwing payoff: N × Δ × max(rK − rn (Ti ), 0) 5. What is a floor?

Ans. A floor is a set of floorlets with same stike price and with different maturities occuring within a set period (e.g. semiannually).

6. How is the pricing of a floorlet related to a floor?

Ans. The price of a floor equals the sum of the floorlets forming it.

7. What is an at-the-money cap?

Ans. An at-the-money cap is a cap for which strike rate rK equals the corresponding swap rate.

8. What is flat volatility?

Ans. The Flat Volatility of a cap with maturity T is the quoted volatility σf (T ) that must be entered in Black’s formula for each and every caplet that make up the cap, in order to obtain a dollar price for the cap.

9. What is forward volatility?

Ans. The Forward Volatility of a caplet with maturity T and strike rate rK is the volatility of σfF wd (T ) that characterizes that particularly caplet, independently of which cap the caplet belongs to.

10. You are given information on a 12-month cap, among which is the quoted volatility, you then want to obtain the price fo the 6 month caplet, can you simply take the quoted volatility and use as the input needed in Black’s formula?

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Ans. No, this is simply a quoted volatility it can’t be used directly as a measure of a caplet’s volatility.

11. What is the source of flat volatility?

Ans. The source of flat volatility is the need of a quoting reference that is somewhat homogenous within different maturities.

12. What is the advantage of quoting securities in flat volatilities instead of prices?

Ans. When coupons are added to a security or when we have different maturities prices are dificult to compare, quoting in volatilities can help solve this issue.

13. How do we extract forward volatilities from flat volatilities?

Ans. Follow these steps: i. Use the quoted flat volatilities to obtain cap prices for all maturities. ii. The shortes cap (maturing in six months) has consists of only one caplet, so in this case flat and forward volatilities are the same. iii. For the following maturities use the previously extracted forward volatilities in such a way that only one caplet’s volatility (obviously forward volatility) isn’t known. For this one, through some iterative method, find the volatility needed in order to match the model price with the observed price.

14. What determines the variation of the forward volatility and the shape of the forward volatility curve?

Ans. Recall that the forward volatility embedded in caps reflects an insurence premium, namely, the amoung of money that an investor is willing to pay to be covered against a run up in interest rates. Such an insurence is more valueble the higher is the uncertainty about future interest rates. This leads to different shapes in the forward volatility curve.

15. What effect does mean reversion have on the shape of the implied forward volatility curve?

Ans. The result is a hump shaped curve for the term structure of implied forward volatility, given that the mean reverting process reduces volatility at the long end.

16. What is the relation between forward volatilities from Black’s model and forward volatilities from the Black-Dermand-Toy model?

Ans. In theory they are the same thing, and they converge as dt becomes smaller.

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17. What is a swaption?

Ans. A swaption is an option to enter into a swap.

18. What is the forward swap rate?

Ans. The forward swap rate is the level of c that makes the value of the forward swap equal to zero.

19. What is the swaptions implied volatility?

Ans. As with caps, Swaptions dealers trade swaptions in terms of implied volatility, that is, the volatility to insert in Black’s formula to obtain the dollar value of a swaption.

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Chapter 21 1. According to the Fundamental Pricing Equation, what conditions must V (r, t; T ) satisfy?

Ans. The interest rate security must satisfy: rV =

∂V 1 ∂2V ∂V ∗ s(r, t)2 + m (r, t) + ∂t ∂r 2 ∂r 2

subject to the boundary condition V (r, T ) = gT .

2. What does the Feynman-Kac formula say on pricing securities?

Ans. The Feynman-Kac formula says that the price of a security is given by: T r du E ∗ e t u gT 3. What complications arise when computing T r du E ∗ e t u gT for interest rate derivatives such as options?

Ans. The problem is that it is hard to evaluate this expectation, as the interest rate rt enters twice in the formula: T r du i. It enters in the discount term e t u ii. It enters in the final payoff gT .

4. Given that gT0 = f(rt , ...), what new terms must we deal with when computing V (r, t; T )?

Ans. If we have the payoff a function of rt then when getting the expectation we have to take into account the covariance between the stochastic process in the discount and the stochastic process in the payoff.

5. What does the change in numeraire technique accomplish?

T r du Ans. The change in numaeraire technique allows us to separate E ∗ e t u and E ∗ [gT ]. It does so by setting the value of R(r) in the Fundamental Pricing Equation to zero, R(r) = 0.

6. From where does the change in numeraire technique get its name?

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Ans. It is called this because we normalize V (r, t; T ) by dividing it by another security. For example: Ṽ (r, t; T ) =

V (r, t; T ) Z(r, t; T )

We then use Ṽ (r, t; T ) instead of V (r, t; T ). We use one numeraire instead of the other.

7. What are the two important differences in the Fundamental Pricing Formula, when applying the change of numeraire technique?

Ans. The two differences are: i. The term ”rV ” disappears from the left hand side of the partial differential equation. ii. The coefficent of ∂∂rṼ now has one more term σZ (r, t)s(r, t).

The Fundamental Pricing Formula, thus, stands: 0=

1 ∂ 2 Ṽ ∂ Ṽ ∂ Ṽ s(r, t)2 + (m∗ (r, t) + σZ (r, t)s(r, t)) + ∂t ∂r 2 ∂r 2

8. What is a forward risk neutral process?

Ans. A forward risk neutral process is a risk neutral process that has been normalized by dividing it by another interest rate security.

9. Show that: V (r, t; T ) = Z(r, t; T )Ef∗ [gT ] Ans. We know that: Ṽ (r, t; T ) =

V (r, t; T ) Z(r, t; T )

If we rearrange this we get: V (r, t; T ) = Ṽ (r, t; T )Z(r, t; T ) We know that the solution to the Fundamental Pricing Equation after the numeraire change, subject to final condition: Ṽ (r, T ) = gT is: Ṽ (r, t; T ) = Ef∗ [gT ] so substituting this term we get: V (r, t; T ) = Z(r, t; T )Ef∗ [gT ] 10. In order to obtain forward risk neutral dynamics, must we always use zero coupons?

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Ans. No, we can use any traded security. It doesn’t matter if it has different maturity as the derivative we are evaluating.

11. Given forward risk neutral dynamics, what can be said of a forward price?

Ans. Under the T - forward risk neutral dynamics, the forward price for delivery at T is a martingale: F (t; T ) = Ef∗ [F (T ; T )] = Ef∗ [gT ] 12. What requirement must a numeraire fulfill?

Ans. It should be a traded security.

13. What underlying assumption is there in any form of the Fundamental Pricing Equation?

Ans. It assumes that there is a sufficent number of traded securities in order to create the riskless portfolio.

14. How strong is the consistency among prices for different securities, when using different numeraires?

Ans. Prices can become inconsistent, and lead to arbitrage opportunities.

15. In the most literal sense, are Heath-Jarrow-Morton type of models shortterm models?

Ans. No, Heath-Jarrow-Morton models concentrate on forward rate dynamics instead of the short term interst rate. In other words maturity is fixed as we move forward.

16. What is the only restriction that the Heath-Jarrow-Morton framework impose?

Ans. Heath-Jarrow-Morton models require that as we reach maturity volatility goes to zero, since bond prices become riskless.

17. What is the only input needed for pricing securities under the HeathJarrow-Morton framework?

Ans. We need a function for volatility v(t, T ), which determines the risk neutral process for the forward rate. From this process we can then price securities.

78

Chapter 22 1. What are multifactor models?

Ans. Multifactor models are models that allow for more than one factor (that is in addition to the interest rate) to be defined by a stochastic process.

2. When are multifactor models used?

Ans. When we want to allow independent variation in the level, slope and curvature components of the yield curve.

3. Under the Vasicek model, what degree of correlation do different interest rates have?

Ans. Rates on the term structure are perfectly correlated in the Vasicek model.

4. How is the multivariate Ito’s lemma defined?

Ans. This equation is defined as:

∂F ∂F 1 ∂2F ∂F m1,t + m2,t + + dPt = ∂t ∂φ1 ∂φ2 2 ∂φ21

∂F ∂F + s1,t dX1,t + s1,t dX1,t ∂φ1 ∂φ1

s21,t +

1 2

2 ∂ F ∂φ22

s22,t dt

5. Given that we now have two stochastic factors, when writing Ito’s lemma as the following

∂F ∂F 1 ∂2F 1 ∂2F ∂F 2 2 dPt = m1,t + m2,t + s1,t + s2,t dt + ∂t ∂φ1 ∂φ2 2 ∂φ21 2 ∂φ22

∂F ∂F + s1,t dX1,t + s1,t dX1,t ∂φ1 ∂φ1 what are we implicitly assuming about them?

Ans. We are assuming that there is no correlation between both factors.

6. What conditions must V (φ1 , φ2 , t) satisfy?

Ans. V (φ1 , φ2 , t) must satisfy the Fundamental Pricing Equation, now defined as: R(φ1 , φ2)V =

∂V ∂V ∗ 1 ∂2V 2 1 ∂2V 2 ∂V ∗ m1,t + m2,t + s + s + 1,t ∂t ∂φ1 ∂φ2 2 ∂φ21 2 ∂φ22 2,t

and the boundary condition: V (φ1 , φ2; T ) = gT .

79

7. Show that, given: dφ1,t = γ1∗ (φ̄∗1 − φ1,t )dt + σ1 dX1,t dφ2,t = γ1∗ (φ̄∗2 − φ2,t )dt + σ2 dX2,t through drt = dφ1,t + dφ2,t and rt = φ1,t + φ2,t, we get:

drt = γ1∗ (φ̄∗1 − rt ) + γ2∗ φ¯∗2 + (γ1∗ − γ2∗ )φ2,t dt + σ1 dX1,t + σ2 dX2,t Ans. First you add both, so: drt = γ1∗ (φ̄∗1 − φ1,t)dt + σ1 dX1,t + γ1∗ (φ̄∗2 − φ2,t )dt + σ2 dX2,t Adding +γ1∗ φ2,t −γ1∗ φ2,t and regrouping the parenthesis gives the solution.

8. In the Vasicek one factor model, the short rate determines the model. In the Vasicek two factor model, what rates drive the model?

Ans. In the two factor model it is the short rate and the long rate that determine the rest of the rate dynamics in the term structure.

9. Why is it considered that implied volatility of interest rate options has a ”hump” shape?

Ans. Because of mean reversion long-term interest rate don’t become infinitely volatile, this depresses this tail making the curve look like a hump.

10. What advantages does the 2-factor Vasicek have when fitting volatility?

Ans. Now the Vasicek model can better fit the term structure of volatility because it uses two factors to do so.

11. Is the 2-factor Vasicek model an afine model?

Ans. Yes, it is an afine model. It allows a closed form solution.

12. What modifications should be included to Ito’s lemma when we allow correlation between the two factors?

Ans. The following term should be included in the drift term: 2 ∂ F s1,t s2,t ρ ∂φ1 φ2 13. What conditions must V (φ1 , φ2 , t) satisfy, given correlation among the factors?

Ans. V (φ1 , φ2 , t) must satisfy the Fundamental Pricing Equation, now defined as: R(φ1 , φ2 )V =

∂V ∂V ∗ ∂V ∗ 1 ∂ 2 V 2 1 ∂2V 2 ∂2V m1,t + m2,t + s1,t + s2,t + s1,t s2,t ρ + 2 2 ∂t ∂φ1 ∂φ2 2 ∂φ1 2 ∂φ2 ∂φ1 ∂φ2

and the boundary condition: V (φ1 , φ2; T ) = gT .

80

14. Does the 2-factor Vasicek model fit the yield curve?

Ans. No, one of its main drawbacks is that it doesn’t fit the yield curve exactly.

15. What is the difference between using a model for finding arbitrage opportunities and using a model for pricing derivatives and other securities?

Ans. In the first case we assume that our model is correct and we check the data to find a way of making a riskless (costless) profit. In the second case, we hold the view that all prices are priced correctly and we use the model as a way of linking the prices of some observed securities to others that are more complex.

16. In the 2-factor Hull-White model, what is the benefit of introducing θt (time dependent central tendency)?

Ans. It provides two benefits: First it allows the term structure to be fitted perfectly. Second, it allows for the rest of the parameters to better fit the volatility structure.

17. Can a solution always be found in order to price a security as proposed by the Feynman-Kac formula?

Ans. It is not always posible to find a closed form solution, but when this fails numeric methods can step in and give a solution.

18. What peculiarity of a yield curve steepner makes the use of 2-factor models attractive?

Ans. A yield curve steepner depends on the relative value of different points on the term structure of interest rates. If we didn’t use this models we would have a deterministic term structure defined by the short rate. This wouldn’t be optimal to analyze a security that depends on different points along the yield curve. We would like for these two points to move, to a certain degree, freely. This is why 2-factor models are more attractive in this case.

81

n PT Pt repo haircut

12 0.083333 99.01 98.5 5% 0.8

n PT Pt repo haircut

52 0.019231 99.48 99.40 6%

Return

0.13

Profit

0.0347 0.001154 99.51469

zero Z(0,.25) Z(0,.5) Z(0,.75) Z(0,1)

0.9980 0.9920 0.9870 0.9810

3% 100.55

5%

99.20

2.48

99.20

100.55 103.0325

100.5485

6% 1.50 1.49 100.18 103.1655

4% 1.00 0.99 0.99 99.08 102.0580

7%

5% 2.50

102.67 101.17 102.6720

103.6625

zero Z(0,.25) Z(0,.5) Z(0,.75) Z(0,1) Z(0,1.25)

0.9970 0.9950 0.9910 0.9890 0.9860

zero

6% 101.20 99.5

3.48

98.9 98.90

7%

4% 1.00 1.00 100.09

102.36 99.50 99.00

101.1955

105.8440

102.0830

5%

zero Z(0,.25) Z(0,.5) Z(0,.75) Z(0,1) Z(0,1.25)

0.9980 0.9950 0.9900 0.9890 0.9840

99.50

3% 0.75 100.25

2% 1.00 0.99

99.50

100.9948

99.38 101.3720 102.8720

5% 1.25 1.24 100.24

102.7288

6%

98.40 98.4000

zero Z(0,.25) Z(0,.5) Z(0,.75) Z(0,1) Z(0,1.25)

0.9940 0.9900 0.9900 0.9890 0.9840

zero 99.40 99.00

99.40

99.00 98.00

3% 0.75 99.74

100.4880

Z(0,.25) Z(0,.5) Z(0,.75) Z(0,1) Z(0,1.25)

0.9990 0.9980 0.9970 0.9960 0.9950

1% 100.15

4% 101.90

3% 0.75 100.55

5% 1.25 1.25 100.95

4%

2% 1.00

2.00 1.00 101.59

100.1498 101.8980 100.6498

101.2978

103.4425 103.5880

100.50 102.4910

zero Z(0,.25) Z(0,.5) Z(0,.75) Z(0,1) Z(0,1.25)

0.9930 0.9870 0.9950 0.9890 0.9840

zero 99.30 98.70

3% 0.74 99.44

2% 0.99 100.50

99.30

98.70

100.1850 101.4880

4% 101.286

Z(0,.25) Z(0,.5) Z(0,.75) Z(0,1) Z(0,1.25)

0.9930 0.9870 0.9950 0.9890 0.9840

3% 3% 100.04 100.7895

100.0448 100.7895

zero Z(0,.25) Z(0,.5) Z(0,.75) Z(0,1) Z(0,1.25)

0.9980 0.9940 0.9950 0.9890 0.9840

99.80

2% 0.50 99.90

1% 0.25 0.25

2% 1.00 100.50

99.80

100.3960

0.4980 101.4930 100.4980

4% 101.796

zero Z(0,.25) Z(0,.5) Z(0,.75) Z(0,1) Z(0,1.25)

0.9970 0.9920 0.9950 0.9890 0.9840

4% 99.70 99.20

99.70

99.2000

3% 0.75 0.74 100.25

2% 0.37 0.37 0.37

3% 101.1955

101.7380

1.1190 102.3145 spread total

98.00 3.9560%

99.00 3.9560%

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00

0.9840 0.9680 0.9520 0.9360 0.9190 0.9040 0.8880 0.8730 0.8587 0.8445 0.8308 0.8175 0.8047 0.7924 0.7806 0.7691

0.9675 0.9360 0.9200 0.9048 0.8899 0.8752

CF 1.25 1.25 1.25 1.25 1.25 1.25 1.25 101.25

2yr @ 5% qrtr DCF w*(t,T) 1.23 0.0032 1.21 0.0063 1.19 0.0092 1.17 0.0121 1.15 0.0149 1.13 0.0176 1.11 0.0201 88.39 1.8304 96.58 1.9138

1.25 fl + 50bps semi CF DCF w*(t,T) 0.25 0.25 0.0862 0.00 0.0000 0.25 0.24 0.2501 0.00 0.0000 0.25 0.23 0.4024 0.00 0.0000 0.00 0.0000 0.00 0.0000 0.71 0.7386

yield 6.72% 6.72% 6.72% 6.72% 6.72% 6.72%

0.75yr @ 6% qrtr CF DCF w*(t,T) 3 2.90 0.0146 103 96.41 0.9708 0.0000 0.0000 0.0000 0.0000 99.31 0.9854

0.50 1.00 1.25 1.50 1.75 2.00

0.9680 0.9360 0.9190 0.9040 0.8880 0.8730

0.9533

2.904 96.408 0 0 0 0 99.312

0.00

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

0.9840 0.9680 0.9520 0.9360 0.9190 0.9040 0.8880 0.8730

CF 1.5 1.5 1.5 1.5 1.5 1.5 1.5 101.5

2yr @ 6% qrtr DCF w*(t,T) 1.48 0.0037 1.45 0.0074 1.43 0.0109 1.40 0.0143 1.38 0.0175 1.36 0.0207 1.33 0.0237 88.61 1.8003 98.44 1.8985

5 TOTAL

1,759.97 0.8805

492.18 0.279652

0.53091

1.75 fl + 80bps semi CF DCF w*(t,T) 0.4 0.39 0.0010 0.00 0.0000 0.4 0.38 0.0028 0.00 0.0000 0.4 0.37 0.0045 0.00 0.0000 0.4 0.36 0.0060 0.00 0.0000 1.50 0.0142 6.50% 101.60 0.2464 103.10 0.2606 2 206.1904 0.117156 0.030528

1 year zero DCFs 0.00 0.00 0.00 93.60 0.00 0.00 0.00 0.00 93.60

CFs

100

6

561.6 0.319096

1 year zero w*(t,T) 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 1.0000

1.5 fl semi DCFs 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 100

CFs

5 0.319096

500 0.284096

w*(t,T) 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

0.9840 0.9680 0.9520 0.9360 0.9190 0.9040 0.8880 0.8730

0.75yr @ 6% qrtr CF DCF w*(t,T) 1.5 1.48 0.0037 1.5 1.45 0.0073 101.5 96.63 0.7279 0.00 0.0000 0.00 0.0000 0.00 0.0000 0.00 0.0000 0.00 0.0000 99.56 0.7389

3 TOTAL

1,391.55 1.4617

298.668 0.214629 0.158598

2yr @ 3% semi DCF w*(t,T) 0.00 0.0000 1.5 1.45 0.0078 0.00 0.0000 1.5 1.40 0.0151 0.00 0.0000 1.5 1.36 0.0219 0.00 0.0000 101.5 88.61 1.9092 92.82 1.9541 CF

4

371.286 0.266814 0.521383

1.75 year zero DCFs 0.00 0.00 0.00 0.00 0.00 0.00 100 88.80 0.00 88.80 CFs

7

621.6 0.446695

1.75 year zero w*(t,T) 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.7500 0.0000 1.7500

2.0 fl semi DCFs 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 100

CFs

1 0.781716

100 0.071862

w*(t,T) 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

0.9840 0.9680 0.9520 0.9360 0.9190 0.9040 0.8880 0.8730

1.2440

CF 0.25 0.25 0.25 0.25 0.25 0.25 0.25 100.25

2yr @ 1% qrtr DCF w*(t,T) 0.25 0.0007 0.24 0.0014 0.24 0.0020 0.23 0.0026 0.23 0.0032 0.23 0.0038 0.22 0.0044 87.52 1.9633 89.16 1.9813 176.6462

CFs

100

1.5 year zero DCFs w*(t,T) 0.00 0.0000 0.00 0.0000 0.00 0.0000 0.00 0.0000 0.00 0.0000 90.40 1.5000 0.00 0.0000 0.00 0.0000 90.40 1.5000 135.6 -41.0462

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

0.9840 0.9680 0.9520 0.9360 0.9190 0.9040 0.8880 0.8730

CF 1 1 1 101

1yr @ 4% qrtr DCF w*(t,T) 0.98 0.0025 0.97 0.0050 0.95 0.0073 94.54 0.9702 0.00 0.0000 0.00 0.0000 0.00 0.0000 0.00 0.0000 97.44 0.9850

1.75 fl + 80bps semi CF DCF w*(t,T) 0.4 0.39 0.0010 0.00 0.0000 0.4 0.38 0.0028 0.00 0.0000 0.4 0.37 0.0045 0.00 0.0000 0.4 0.36 0.0060 0.00 0.0000 1.50 0.0142 6.00% 101.35 0.2464 102.85 0.2606

95.98

26.8031

PV -51.8169 -0.00518

CFs

100

2 year zero DCFs 0.00 0.00 0.00 0.00 0.00 0.00 0.00 87.30 87.30

174.6

2 year zero w*(t,T) 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 2.0000 2.0000

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

0.9840 0.9680 0.9520 0.9360 0.9190 0.9040 0.8880 0.8730

CF 1.75 1.75 1.75 1.75 1.75 1.75 1.75 101.75

2yr @ 7% qrtr DCF w*(t,T) 1.72 0.0043 1.69 0.0084 1.67 0.0125 1.64 0.0163 1.61 0.0200 1.58 0.0237 1.55 0.0271 88.83 1.7714 100.29 1.8837

1 188.9233 13.20981

0.0013

1.25 fl + 80bps semi CF DCF w*(t,T) 0.4 0.39 0.0010 0.00 0.0000 0.4 0.38 0.0028 0.00 0.0000 0.4 0.37 0.0045 0.00 0.0000 0.00 0.0000 0.00 0.0000 1.14 0.0082 7.00% 101.84 0.2472 102.99 0.2554 3 78.9135

CFs 100

.5 year zero DCFs 0.00 96.80 0.00 0.00 0.00 0.00 0.00 0.00 96.80

2 96.8

.5 year zero w*(t,T) 0.0000 0.5000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.5000

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

0.9840 0.9680 0.9520 0.9360 0.9190 0.9040 0.8880 0.8730

CF 0.5 0.5 0.5 0.5 0.5 100.5

1.5yr @ 2% qrtr DCF w*(t,T) 0.49 0.0013 0.48 0.0026 0.48 0.0038 0.47 0.0050 0.46 0.0062 90.85 1.4617 0.00 0.0000 0.00 0.0000 93.23 1.4806

0.75 fl + 80bps semi CF DCF w*(t,T) 0.4 0.39 0.0010 0.00 0.0000 0.4 0.38 0.0028 0.00 0.0000 0.00 0.0000 0.00 0.0000 0.00 0.0000 0.00 0.0000 0.77 0.0038 6.00% 101.35 0.2481 102.13 0.2519

293.76 0.317375 0.469919 0.641223 sig 0.42% sig_P 0.797346 -1.31163 95VaR 1.3116

0.347655 0.087562

CFs 100

0.25 year zero DCFs 98.40 0.00 0.00 0.00 0.00 0.00 0.00 0.00 98.40

0.25 year zero w*(t,T) 0.2500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.2500

0.33497 0.083742

Mr. Brown term (flat) Capital Annuity t

10.0% 100,000.00 26,379.75 CF 3.7908 1 0.9091 2 0.8264 3 0.7513 4 0.6830 5 0.6209

12%

10% pcnt:

93.05% CF2

12 12 12 12 112

Hedge

107.58 10.91 9.92 9.02 8.20 69.54

4.0740 0.1014 0.1844 0.2514 0.3047 3.2321

110

864.90 69.5266 93,047.34 6,952.66 100,000.00

100 100

26,379.75 26,379.75 26,379.75 26,379.75 26,379.75

23,981.59 21,801.44 19,819.50 18,017.72 16,379.75 100,000.00

r(t) 1/2/1980 1/9/1980 1/16/1980 1/23/1980 1/30/1980 2/6/1980 2/13/1980 2/20/1980 2/27/1980 3/5/1980 3/12/1980 3/19/1980 3/26/1980 4/2/1980 4/9/1980 4/16/1980 4/23/1980 4/30/1980 5/7/1980 5/14/1980 5/21/1980 5/28/1980 6/4/1980 6/11/1980 6/18/1980 6/25/1980 7/2/1980 7/9/1980 7/16/1980 7/23/1980 7/30/1980 8/6/1980 8/13/1980 8/20/1980 8/27/1980 9/3/1980 9/10/1980 9/17/1980 9/24/1980 10/1/1980 10/8/1980 10/15/1980 10/22/1980 10/29/1980 11/5/1980

r(t-1) 14.04 13.94 13.91 13.77 13.54 12.8 13.64 14.87 14.62 16.17 16.45 16.24 17.78 19.39 19.04 18.35 17.56 15.12 12.96 10.85 10.71 9.46 10.74 9.68 8.99 9.08 9.41 9.26 8.98 8.68 8.98 9.6 8.85 9.35 10.03 10.47 10.22 10.64 10.85 12.38 12.59 12.64 12.55 13.17 13.99

14.04 13.94 13.91 13.77 13.54 12.8 13.64 14.87 14.62 16.17 16.45 16.24 17.78 19.39 19.04 18.35 17.56 15.12 12.96 10.85 10.71 9.46 10.74 9.68 8.99 9.08 9.41 9.26 8.98 8.68 8.98 9.6 8.85 9.35 10.03 10.47 10.22 10.64 10.85 12.38 12.59 12.64 12.55 13.17

SUMMARY OUTPUT 100 Regression Statistics Multiple R 0.996236 R Square 0.992486 Adjusted R Square 0.992482 Standard Error 0.327871 Observations 1558

0.017735 0.995596

ANOVA df Regression Residual Total

SS MS F 1 22095.02 22095.02 205536.3 1556 167.2691 0.107499 1557 22262.29

Coefficients Standard Error t Stat P-value Intercept 0.017735 0.015679 1.131125 0.258177 X Variable 10.995596 0.002196 453.3611 0 Simplified: alpha beta r(0) std.err.

0 0.9955 6 0.33

r(1)

11/12/1980 11/19/1980 11/26/1980 12/3/1980 12/10/1980 12/17/1980 12/24/1980 12/31/1980 1/7/1981 1/14/1981 1/21/1981 1/28/1981 2/4/1981 2/11/1981 2/18/1981 2/25/1981 3/4/1981 3/11/1981 3/18/1981 3/25/1981 4/1/1981 4/8/1981 4/15/1981 4/22/1981 4/29/1981 5/6/1981 5/13/1981 5/20/1981 5/27/1981 6/3/1981 6/10/1981 6/17/1981 6/24/1981 7/1/1981 7/8/1981 7/15/1981 7/22/1981 7/29/1981 8/5/1981 8/12/1981 8/19/1981 8/26/1981 9/2/1981 9/9/1981 9/16/1981 9/23/1981 9/30/1981

14.65 15.22 17.43 17.72 18.82 19.83 19.44 18.45 20.06 19.64 19.35 18.12 17.19 16.51 15.81 14.96 15.73 15.53 14.13 13.48 14.93 15.43 15.33 15.55 16.28 18.91 18.21 18.89 18.71 18.4 19.33 19.1 19.2 18.84 19.93 18.76 19.05 18.54 18.25 18.29 18.19 17.41 16.89 16.5 16.09 15.33 15

13.99 14.65 15.22 17.43 17.72 18.82 19.83 19.44 18.45 20.06 19.64 19.35 18.12 17.19 16.51 15.81 14.96 15.73 15.53 14.13 13.48 14.93 15.43 15.33 15.55 16.28 18.91 18.21 18.89 18.71 18.4 19.33 19.1 19.2 18.84 19.93 18.76 19.05 18.54 18.25 18.29 18.19 17.41 16.89 16.5 16.09 15.33

10/7/1981 10/14/1981 10/21/1981 10/28/1981 11/4/1981 11/11/1981 11/18/1981 11/25/1981 12/2/1981 12/9/1981 12/16/1981 12/23/1981 12/30/1981 1/6/1982 1/13/1982 1/20/1982 1/27/1982 2/3/1982 2/10/1982 2/17/1982 2/24/1982 3/3/1982 3/10/1982 3/17/1982 3/24/1982 3/31/1982 4/7/1982 4/14/1982 4/21/1982 4/28/1982 5/5/1982 5/12/1982 5/19/1982 5/26/1982 6/2/1982 6/9/1982 6/16/1982 6/23/1982 6/30/1982 7/7/1982 7/14/1982 7/21/1982 7/28/1982 8/4/1982 8/11/1982 8/18/1982 8/25/1982

15.46 14.93 15.32 14.87 14.79 14.01 13.17 12.42 12.48 12.04 12.26 12.43 12.54 12.98 12.42 12.96 13.98 14.77 15.19 15.61 13.86 14.07 14.35 14.89 14.48 14.99 15.15 14.68 15.01 14.72 15.53 14.97 14.67 13.7 13.43 13.6 14.24 14.17 14.81 14.47 13.18 12.14 11.02 11.15 10.9 10.11 9.04

15 15.46 14.93 15.32 14.87 14.79 14.01 13.17 12.42 12.48 12.04 12.26 12.43 12.54 12.98 12.42 12.96 13.98 14.77 15.19 15.61 13.86 14.07 14.35 14.89 14.48 14.99 15.15 14.68 15.01 14.72 15.53 14.97 14.67 13.7 13.43 13.6 14.24 14.17 14.81 14.47 13.18 12.14 11.02 11.15 10.9 10.11

9/1/1982 9/8/1982 9/15/1982 9/22/1982 9/29/1982 10/6/1982 10/13/1982 10/20/1982 10/27/1982 11/3/1982 11/10/1982 11/17/1982 11/24/1982 12/1/1982 12/8/1982 12/15/1982 12/22/1982 12/29/1982 1/5/1983 1/12/1983 1/19/1983 1/26/1983 2/2/1983 2/9/1983 2/16/1983 2/23/1983 3/2/1983 3/9/1983 3/16/1983 3/23/1983 3/30/1983 4/6/1983 4/13/1983 4/20/1983 4/27/1983 5/4/1983 5/11/1983 5/18/1983 5/25/1983 6/1/1983 6/8/1983 6/15/1983 6/22/1983 6/29/1983 7/6/1983 7/13/1983 7/20/1983

10.15 10.14 10.27 10.31 10.12 10.77 9.6 9.53 9.44 9.43 9.45 9.61 8.91 8.69 8.84 8.86 8.69 8.79 10.21 8.42 8.49 8.44 8.53 8.5 8.62 8.47 8.44 8.59 8.57 8.75 8.88 9.43 8.76 8.7 8.58 8.8 8.48 8.59 8.72 8.77 8.84 8.84 9.14 8.9 9.39 9.21 9.43

9.04 10.15 10.14 10.27 10.31 10.12 10.77 9.6 9.53 9.44 9.43 9.45 9.61 8.91 8.69 8.84 8.86 8.69 8.79 10.21 8.42 8.49 8.44 8.53 8.5 8.62 8.47 8.44 8.59 8.57 8.75 8.88 9.43 8.76 8.7 8.58 8.8 8.48 8.59 8.72 8.77 8.84 8.84 9.14 8.9 9.39 9.21

7/27/1983 8/3/1983 8/10/1983 8/17/1983 8/24/1983 8/31/1983 9/7/1983 9/14/1983 9/21/1983 9/28/1983 10/5/1983 10/12/1983 10/19/1983 10/26/1983 11/2/1983 11/9/1983 11/16/1983 11/23/1983 11/30/1983 12/7/1983 12/14/1983 12/21/1983 12/28/1983 1/4/1984 1/11/1984 1/18/1984 1/25/1984 2/1/1984 2/8/1984 2/15/1984 2/22/1984 2/29/1984 3/7/1984 3/14/1984 3/21/1984 3/28/1984 4/4/1984 4/11/1984 4/18/1984 4/25/1984 5/2/1984 5/9/1984 5/16/1984 5/23/1984 5/30/1984 6/6/1984 6/13/1984

9.46 9.59 9.66 9.67 9.41 9.44 9.53 9.54 9.48 9.04 10 9.46 9.36 9.36 9.4 9.36 9.42 9.26 9.27 9.49 9.52 9.62 8.96 10.06 9.53 9.54 9.53 9.41 9.58 9.53 9.6 9.62 9.74 9.79 10.04 9.97 10.41 10.13 10.37 9.98 10.7 10.46 10.52 9.75 10.3 10.72 10.85

9.43 9.46 9.59 9.66 9.67 9.41 9.44 9.53 9.54 9.48 9.04 10 9.46 9.36 9.36 9.4 9.36 9.42 9.26 9.27 9.49 9.52 9.62 8.96 10.06 9.53 9.54 9.53 9.41 9.58 9.53 9.6 9.62 9.74 9.79 10.04 9.97 10.41 10.13 10.37 9.98 10.7 10.46 10.52 9.75 10.3 10.72

6/20/1984 6/27/1984 7/4/1984 7/11/1984 7/18/1984 7/25/1984 8/1/1984 8/8/1984 8/15/1984 8/22/1984 8/29/1984 9/5/1984 9/12/1984 9/19/1984 9/26/1984 10/3/1984 10/10/1984 10/17/1984 10/24/1984 10/31/1984 11/7/1984 11/14/1984 11/21/1984 11/28/1984 12/5/1984 12/12/1984 12/19/1984 12/26/1984 1/2/1985 1/9/1985 1/16/1985 1/23/1985 1/30/1985 2/6/1985 2/13/1985 2/20/1985 2/27/1985 3/6/1985 3/13/1985 3/20/1985 3/27/1985 4/3/1985 4/10/1985 4/17/1985 4/24/1985 5/1/1985 5/8/1985

11.49 11.27 10.91 11.25 11.21 11.19 11.53 11.59 11.63 11.77 11.5 11.68 11.52 11.46 10.73 11.2 10.01 10.22 9.45 9.73 9.87 9.55 9.47 9 8.83 8.7 7.99 7.95 8.75 8.27 8.23 8.19 8.45 8.59 8.44 8.57 8.4 8.63 8.52 8.75 8.38 8.68 8.45 8.46 7.69 8.35 8.19

10.85 11.49 11.27 10.91 11.25 11.21 11.19 11.53 11.59 11.63 11.77 11.5 11.68 11.52 11.46 10.73 11.2 10.01 10.22 9.45 9.73 9.87 9.55 9.47 9 8.83 8.7 7.99 7.95 8.75 8.27 8.23 8.19 8.45 8.59 8.44 8.57 8.4 8.63 8.52 8.75 8.38 8.68 8.45 8.46 7.69 8.35

5/15/1985 5/22/1985 5/29/1985 6/5/1985 6/12/1985 6/19/1985 6/26/1985 7/3/1985 7/10/1985 7/17/1985 7/24/1985 7/31/1985 8/7/1985 8/14/1985 8/21/1985 8/28/1985 9/4/1985 9/11/1985 9/18/1985 9/25/1985 10/2/1985 10/9/1985 10/16/1985 10/23/1985 10/30/1985 11/6/1985 11/13/1985 11/20/1985 11/27/1985 12/4/1985 12/11/1985 12/18/1985 12/25/1985 1/1/1986 1/8/1986 1/15/1986 1/22/1986 1/29/1986 2/5/1986 2/12/1986 2/19/1986 2/26/1986 3/5/1986 3/12/1986 3/19/1986 3/26/1986 4/2/1986

8.14 7.91 7.6 7.75 7.62 7.13 7.46 8.06 8.07 7.77 7.88 7.64 7.92 7.88 8.06 7.78 7.88 7.8 7.85 7.96 8.12 7.84 8.03 8.14 7.89 8.3 7.95 8.13 7.71 8.49 8.03 8.05 8.02 9.55 8.2 7.94 7.87 7.83 7.97 7.85 7.84 7.82 7.89 7.52 7.47 7.25 7.39

8.19 8.14 7.91 7.6 7.75 7.62 7.13 7.46 8.06 8.07 7.77 7.88 7.64 7.92 7.88 8.06 7.78 7.88 7.8 7.85 7.96 8.12 7.84 8.03 8.14 7.89 8.3 7.95 8.13 7.71 8.49 8.03 8.05 8.02 9.55 8.2 7.94 7.87 7.83 7.97 7.85 7.84 7.82 7.89 7.52 7.47 7.25

4/9/1986 4/16/1986 4/23/1986 4/30/1986 5/7/1986 5/14/1986 5/21/1986 5/28/1986 6/4/1986 6/11/1986 6/18/1986 6/25/1986 7/2/1986 7/9/1986 7/16/1986 7/23/1986 7/30/1986 8/6/1986 8/13/1986 8/20/1986 8/27/1986 9/3/1986 9/10/1986 9/17/1986 9/24/1986 10/1/1986 10/8/1986 10/15/1986 10/22/1986 10/29/1986 11/5/1986 11/12/1986 11/19/1986 11/26/1986 12/3/1986 12/10/1986 12/17/1986 12/24/1986 12/31/1986 1/7/1987 1/14/1987 1/21/1987 1/28/1987 2/4/1987 2/11/1987 2/18/1987 2/25/1987

7.05 6.97 6.92 6.88 6.87 6.82 6.87 6.85 6.95 6.89 6.87 6.86 7.02 6.87 6.51 6.42 6.32 6.36 6.31 6.38 5.87 5.83 5.82 5.88 5.81 6.08 5.75 5.83 5.91 5.86 6.02 5.98 6.13 6 6.25 5.97 6.3 6.31 9.2 7.62 6.01 6.01 6.13 6.22 6.14 6.21 5.95

7.39 7.05 6.97 6.92 6.88 6.87 6.82 6.87 6.85 6.95 6.89 6.87 6.86 7.02 6.87 6.51 6.42 6.32 6.36 6.31 6.38 5.87 5.83 5.82 5.88 5.81 6.08 5.75 5.83 5.91 5.86 6.02 5.98 6.13 6 6.25 5.97 6.3 6.31 9.2 7.62 6.01 6.01 6.13 6.22 6.14 6.21

3/4/1987 3/11/1987 3/18/1987 3/25/1987 4/1/1987 4/8/1987 4/15/1987 4/22/1987 4/29/1987 5/6/1987 5/13/1987 5/20/1987 5/27/1987 6/3/1987 6/10/1987 6/17/1987 6/24/1987 7/1/1987 7/8/1987 7/15/1987 7/22/1987 7/29/1987 8/5/1987 8/12/1987 8/19/1987 8/26/1987 9/2/1987 9/9/1987 9/16/1987 9/23/1987 9/30/1987 10/7/1987 10/14/1987 10/21/1987 10/28/1987 11/4/1987 11/11/1987 11/18/1987 11/25/1987 12/2/1987 12/9/1987 12/16/1987 12/23/1987 12/30/1987 1/6/1988 1/13/1988 1/20/1988

6.06 6.12 6.08 6.14 6.21 6.13 6.41 6.26 6.5 7.3 6.75 6.77 6.8 6.65 6.7 6.75 6.79 6.61 6.64 6.52 6.57 6.63 6.75 6.58 6.74 6.76 6.85 6.95 7.21 7.26 7.56 7.43 7.59 7.37 7.03 6.43 6.68 6.77 6.78 6.89 6.84 6.58 6.75 6.81 7.02 6.81 6.89

5.95 6.06 6.12 6.08 6.14 6.21 6.13 6.41 6.26 6.5 7.3 6.75 6.77 6.8 6.65 6.7 6.75 6.79 6.61 6.64 6.52 6.57 6.63 6.75 6.58 6.74 6.76 6.85 6.95 7.21 7.26 7.56 7.43 7.59 7.37 7.03 6.43 6.68 6.77 6.78 6.89 6.84 6.58 6.75 6.81 7.02 6.81

1/27/1988 2/3/1988 2/10/1988 2/17/1988 2/24/1988 3/2/1988 3/9/1988 3/16/1988 3/23/1988 3/30/1988 4/6/1988 4/13/1988 4/20/1988 4/27/1988 5/4/1988 5/11/1988 5/18/1988 5/25/1988 6/1/1988 6/8/1988 6/15/1988 6/22/1988 6/29/1988 7/6/1988 7/13/1988 7/20/1988 7/27/1988 8/3/1988 8/10/1988 8/17/1988 8/24/1988 8/31/1988 9/7/1988 9/14/1988 9/21/1988 9/28/1988 10/5/1988 10/12/1988 10/19/1988 10/26/1988 11/2/1988 11/9/1988 11/16/1988 11/23/1988 11/30/1988 12/7/1988 12/14/1988

6.66 6.77 6.38 6.65 6.64 6.6 6.51 6.61 6.51 6.62 6.82 6.81 6.93 6.85 6.82 7.02 7.04 7.14 7.41 7.37 7.43 7.54 7.63 7.81 7.59 7.83 7.8 7.84 7.75 8.19 8.02 8.15 8.15 8.13 8.17 8.24 8.38 8.27 8.27 8.29 8.36 8.31 8.26 8.33 8.44 8.59 8.51

6.89 6.66 6.77 6.38 6.65 6.64 6.6 6.51 6.61 6.51 6.62 6.82 6.81 6.93 6.85 6.82 7.02 7.04 7.14 7.41 7.37 7.43 7.54 7.63 7.81 7.59 7.83 7.8 7.84 7.75 8.19 8.02 8.15 8.15 8.13 8.17 8.24 8.38 8.27 8.27 8.29 8.36 8.31 8.26 8.33 8.44 8.59

12/21/1988 12/28/1988 1/4/1989 1/11/1989 1/18/1989 1/25/1989 2/1/1989 2/8/1989 2/15/1989 2/22/1989 3/1/1989 3/8/1989 3/15/1989 3/22/1989 3/29/1989 4/5/1989 4/12/1989 4/19/1989 4/26/1989 5/3/1989 5/10/1989 5/17/1989 5/24/1989 5/31/1989 6/7/1989 6/14/1989 6/21/1989 6/28/1989 7/5/1989 7/12/1989 7/19/1989 7/26/1989 8/2/1989 8/9/1989 8/16/1989 8/23/1989 8/30/1989 9/6/1989 9/13/1989 9/20/1989 9/27/1989 10/4/1989 10/11/1989 10/18/1989 10/25/1989 11/1/1989 11/8/1989

8.87 8.86 9.22 9.08 9.13 9.06 9.16 9.1 9.27 9.39 9.8 9.83 9.83 9.86 9.88 9.71 9.82 9.95 9.86 9.88 9.86 9.75 9.74 9.84 9.68 9.35 9.48 9.58 9.58 9.31 9.24 9.14 8.95 8.98 9.04 9.01 8.96 8.96 8.96 9.05 9.02 9.18 8.93 8.76 8.72 8.8 8.69

8.51 8.87 8.86 9.22 9.08 9.13 9.06 9.16 9.1 9.27 9.39 9.8 9.83 9.83 9.86 9.88 9.71 9.82 9.95 9.86 9.88 9.86 9.75 9.74 9.84 9.68 9.35 9.48 9.58 9.58 9.31 9.24 9.14 8.95 8.98 9.04 9.01 8.96 8.96 8.96 9.05 9.02 9.18 8.93 8.76 8.72 8.8

11/15/1989 11/22/1989 11/29/1989 12/6/1989 12/13/1989 12/20/1989 12/27/1989 1/3/1990 1/10/1990 1/17/1990 1/24/1990 1/31/1990 2/7/1990 2/14/1990 2/21/1990 2/28/1990 3/7/1990 3/14/1990 3/21/1990 3/28/1990 4/4/1990 4/11/1990 4/18/1990 4/25/1990 5/2/1990 5/9/1990 5/16/1990 5/23/1990 5/30/1990 6/6/1990 6/13/1990 6/20/1990 6/27/1990 7/4/1990 7/11/1990 7/18/1990 7/25/1990 8/1/1990 8/8/1990 8/15/1990 8/22/1990 8/29/1990 9/5/1990 9/12/1990 9/19/1990 9/26/1990 10/3/1990

8.46 8.46 8.51 8.52 8.47 8.52 8.38 8.32 8.22 8.2 8.23 8.24 8.22 8.21 8.25 8.27 8.28 8.27 8.27 8.26 8.33 8.25 8.27 8.24 8.12 8.2 8.16 8.22 8.19 8.26 8.3 8.28 8.28 8.33 8.28 8.14 8.05 8.03 8.07 8.13 8.3 8.08 8.25 8.12 8.18 8.26 8.23

8.69 8.46 8.46 8.51 8.52 8.47 8.52 8.38 8.32 8.22 8.2 8.23 8.24 8.22 8.21 8.25 8.27 8.28 8.27 8.27 8.26 8.33 8.25 8.27 8.24 8.12 8.2 8.16 8.22 8.19 8.26 8.3 8.28 8.28 8.33 8.28 8.14 8.05 8.03 8.07 8.13 8.3 8.08 8.25 8.12 8.18 8.26

10/10/1990 10/17/1990 10/24/1990 10/31/1990 11/7/1990 11/14/1990 11/21/1990 11/28/1990 12/5/1990 12/12/1990 12/19/1990 12/26/1990 1/2/1991 1/9/1991 1/16/1991 1/23/1991 1/30/1991 2/6/1991 2/13/1991 2/20/1991 2/27/1991 3/6/1991 3/13/1991 3/20/1991 3/27/1991 4/3/1991 4/10/1991 4/17/1991 4/24/1991 5/1/1991 5/8/1991 5/15/1991 5/22/1991 5/29/1991 6/5/1991 6/12/1991 6/19/1991 6/26/1991 7/3/1991 7/10/1991 7/17/1991 7/24/1991 7/31/1991 8/7/1991 8/14/1991 8/21/1991 8/28/1991

8.2 7.96 7.99 8.17 7.97 7.94 7.8 7.56 7.6 7.25 7.29 7.16 7.17 6.4 6.77 6.88 7.46 6.32 6.29 6.26 6.31 6.47 6.17 6.1 6.1 6 5.9 5.69 5.92 5.92 5.79 5.78 5.79 5.72 5.91 5.75 5.78 5.79 6.34 5.79 5.85 5.75 5.79 5.83 5.62 5.68 5.58

8.23 8.2 7.96 7.99 8.17 7.97 7.94 7.8 7.56 7.6 7.25 7.29 7.16 7.17 6.4 6.77 6.88 7.46 6.32 6.29 6.26 6.31 6.47 6.17 6.1 6.1 6 5.9 5.69 5.92 5.92 5.79 5.78 5.79 5.72 5.91 5.75 5.78 5.79 6.34 5.79 5.85 5.75 5.79 5.83 5.62 5.68

9/4/1991 9/11/1991 9/18/1991 9/25/1991 10/2/1991 10/9/1991 10/16/1991 10/23/1991 10/30/1991 11/6/1991 11/13/1991 11/20/1991 11/27/1991 12/4/1991 12/11/1991 12/18/1991 12/25/1991 1/1/1992 1/8/1992 1/15/1992 1/22/1992 1/29/1992 2/5/1992 2/12/1992 2/19/1992 2/26/1992 3/4/1992 3/11/1992 3/18/1992 3/25/1992 4/1/1992 4/8/1992 4/15/1992 4/22/1992 4/29/1992 5/6/1992 5/13/1992 5/20/1992 5/27/1992 6/3/1992 6/10/1992 6/17/1992 6/24/1992 7/1/1992 7/8/1992 7/15/1992 7/22/1992

5.6 5.56 5.44 5.29 5.33 5.19 5.28 5.24 5.1 5.05 4.74 4.89 4.68 4.79 4.54 4.49 4.22 4.19 4.19 4.01 3.87 4.01 4.17 3.93 4.2 3.96 4.08 3.95 4.04 3.94 4.09 3.98 3.65 3.47 3.65 3.77 3.84 3.89 3.8 3.85 3.69 3.73 3.72 3.87 3.24 3.28 3.22

5.58 5.6 5.56 5.44 5.29 5.33 5.19 5.28 5.24 5.1 5.05 4.74 4.89 4.68 4.79 4.54 4.49 4.22 4.19 4.19 4.01 3.87 4.01 4.17 3.93 4.2 3.96 4.08 3.95 4.04 3.94 4.09 3.98 3.65 3.47 3.65 3.77 3.84 3.89 3.8 3.85 3.69 3.73 3.72 3.87 3.24 3.28

7/29/1992 8/5/1992 8/12/1992 8/19/1992 8/26/1992 9/2/1992 9/9/1992 9/16/1992 9/23/1992 9/30/1992 10/7/1992 10/14/1992 10/21/1992 10/28/1992 11/4/1992 11/11/1992 11/18/1992 11/25/1992 12/2/1992 12/9/1992 12/16/1992 12/23/1992 12/30/1992 1/6/1993 1/13/1993 1/20/1993 1/27/1993 2/3/1993 2/10/1993 2/17/1993 2/24/1993 3/3/1993 3/10/1993 3/17/1993 3/24/1993 3/31/1993 4/7/1993 4/14/1993 4/21/1993 4/28/1993 5/5/1993 5/12/1993 5/19/1993 5/26/1993 6/2/1993 6/9/1993 6/16/1993

3.18 3.33 3.24 3.33 3.27 3.33 3.09 3.28 3.07 3.41 3.2 3.2 3.05 2.96 3.07 2.91 2.97 3.1 3.37 2.94 2.93 2.94 2.86 3.03 2.98 3.1 2.94 3.15 2.92 3.06 2.91 3.24 3.02 3.04 2.93 3.18 3.11 2.93 2.91 2.87 2.98 2.9 3.01 3.07 3.09 2.96 3.01

3.22 3.18 3.33 3.24 3.33 3.27 3.33 3.09 3.28 3.07 3.41 3.2 3.2 3.05 2.96 3.07 2.91 2.97 3.1 3.37 2.94 2.93 2.94 2.86 3.03 2.98 3.1 2.94 3.15 2.92 3.06 2.91 3.24 3.02 3.04 2.93 3.18 3.11 2.93 2.91 2.87 2.98 2.9 3.01 3.07 3.09 2.96

6/23/1993 6/30/1993 7/7/1993 7/14/1993 7/21/1993 7/28/1993 8/4/1993 8/11/1993 8/18/1993 8/25/1993 9/1/1993 9/8/1993 9/15/1993 9/22/1993 9/29/1993 10/6/1993 10/13/1993 10/20/1993 10/27/1993 11/3/1993 11/10/1993 11/17/1993 11/24/1993 12/1/1993 12/8/1993 12/15/1993 12/22/1993 12/29/1993 1/5/1994 1/12/1994 1/19/1994 1/26/1994 2/2/1994 2/9/1994 2/16/1994 2/23/1994 3/2/1994 3/9/1994 3/16/1994 3/23/1994 3/30/1994 4/6/1994 4/13/1994 4/20/1994 4/27/1994 5/4/1994 5/11/1994

3 3.13 3.1 3.01 3.09 3.03 3.1 2.98 3.06 2.98 3.08 2.99 3.03 3.12 3.05 3.24 2.91 2.97 2.97 3.04 2.96 3.03 2.98 3.09 2.92 2.94 2.99 2.99 3 2.98 3.13 2.97 3.17 3.2 3.25 3.25 3.28 3.25 3.19 3.31 3.49 3.69 3.37 3.59 3.59 3.76 3.7

3.01 3 3.13 3.1 3.01 3.09 3.03 3.1 2.98 3.06 2.98 3.08 2.99 3.03 3.12 3.05 3.24 2.91 2.97 2.97 3.04 2.96 3.03 2.98 3.09 2.92 2.94 2.99 2.99 3 2.98 3.13 2.97 3.17 3.2 3.25 3.25 3.28 3.25 3.19 3.31 3.49 3.69 3.37 3.59 3.59 3.76

5/18/1994 5/25/1994 6/1/1994 6/8/1994 6/15/1994 6/22/1994 6/29/1994 7/6/1994 7/13/1994 7/20/1994 7/27/1994 8/3/1994 8/10/1994 8/17/1994 8/24/1994 8/31/1994 9/7/1994 9/14/1994 9/21/1994 9/28/1994 10/5/1994 10/12/1994 10/19/1994 10/26/1994 11/2/1994 11/9/1994 11/16/1994 11/23/1994 11/30/1994 12/7/1994 12/14/1994 12/21/1994 12/28/1994 1/4/1995 1/11/1995 1/18/1995 1/25/1995 2/1/1995 2/8/1995 2/15/1995 2/22/1995 3/1/1995 3/8/1995 3/15/1995 3/22/1995 3/29/1995 4/5/1995

4.02 4.22 4.27 4.13 4.21 4.19 4.19 4.38 4.3 4.3 4.28 4.28 4.26 4.35 4.66 4.72 4.74 4.7 4.73 4.66 5.07 4.62 4.72 4.72 4.77 4.74 5.22 5.53 5.85 5.47 5.48 5.56 5.45 5.4 5.53 5.45 5.42 5.63 5.95 5.93 5.94 5.88 5.93 5.94 5.97 6.06 6.2

3.7 4.02 4.22 4.27 4.13 4.21 4.19 4.19 4.38 4.3 4.3 4.28 4.28 4.26 4.35 4.66 4.72 4.74 4.7 4.73 4.66 5.07 4.62 4.72 4.72 4.77 4.74 5.22 5.53 5.85 5.47 5.48 5.56 5.45 5.4 5.53 5.45 5.42 5.63 5.95 5.93 5.94 5.88 5.93 5.94 5.97 6.06

4/12/1995 4/19/1995 4/26/1995 5/3/1995 5/10/1995 5/17/1995 5/24/1995 5/31/1995 6/7/1995 6/14/1995 6/21/1995 6/28/1995 7/5/1995 7/12/1995 7/19/1995 7/26/1995 8/2/1995 8/9/1995 8/16/1995 8/23/1995 8/30/1995 9/6/1995 9/13/1995 9/20/1995 9/27/1995 10/4/1995 10/11/1995 10/18/1995 10/25/1995 11/1/1995 11/8/1995 11/15/1995 11/22/1995 11/29/1995 12/6/1995 12/13/1995 12/20/1995 12/27/1995 1/3/1996 1/10/1996 1/17/1996 1/24/1996 1/31/1996 2/7/1996 2/14/1996 2/21/1996 2/28/1996

5.98 6.07 5.99 6.05 6 6.02 5.99 6.02 6.03 6.02 6 5.95 6.21 5.81 5.72 5.75 5.83 5.73 5.74 5.7 5.71 5.77 5.73 5.78 5.8 6 5.72 5.71 5.76 5.76 5.71 5.74 5.81 5.91 5.75 5.73 5.9 5.48 5.35 5.53 5.61 5.44 5.53 5.21 5.09 5.17 5.31

6.2 5.98 6.07 5.99 6.05 6 6.02 5.99 6.02 6.03 6.02 6 5.95 6.21 5.81 5.72 5.75 5.83 5.73 5.74 5.7 5.71 5.77 5.73 5.78 5.8 6 5.72 5.71 5.76 5.76 5.71 5.74 5.81 5.91 5.75 5.73 5.9 5.48 5.35 5.53 5.61 5.44 5.53 5.21 5.09 5.17

3/6/1996 3/13/1996 3/20/1996 3/27/1996 4/3/1996 4/10/1996 4/17/1996 4/24/1996 5/1/1996 5/8/1996 5/15/1996 5/22/1996 5/29/1996 6/5/1996 6/12/1996 6/19/1996 6/26/1996 7/3/1996 7/10/1996 7/17/1996 7/24/1996 7/31/1996 8/7/1996 8/14/1996 8/21/1996 8/28/1996 9/4/1996 9/11/1996 9/18/1996 9/25/1996 10/2/1996 10/9/1996 10/16/1996 10/23/1996 10/30/1996 11/6/1996 11/13/1996 11/20/1996 11/27/1996 12/4/1996 12/11/1996 12/18/1996 12/25/1996 1/1/1997 1/8/1997 1/15/1997 1/22/1997

5.57 5.24 5.36 5.22 5.3 5.08 5.24 5.24 5.3 5.22 5.26 5.22 5.19 5.33 5.24 5.45 5.21 5.53 5.26 5.23 5.25 5.53 5.38 5.1 5.18 5.21 5.39 5.16 5.22 5.34 5.4 5.14 5.22 5.22 5.27 5.32 5.21 5.41 5.3 5.52 5.22 5.38 5.18 5.37 5.28 5.19 5.19

5.31 5.57 5.24 5.36 5.22 5.3 5.08 5.24 5.24 5.3 5.22 5.26 5.22 5.19 5.33 5.24 5.45 5.21 5.53 5.26 5.23 5.25 5.53 5.38 5.1 5.18 5.21 5.39 5.16 5.22 5.34 5.4 5.14 5.22 5.22 5.27 5.32 5.21 5.41 5.3 5.52 5.22 5.38 5.18 5.37 5.28 5.19

1/29/1997 2/5/1997 2/12/1997 2/19/1997 2/26/1997 3/5/1997 3/12/1997 3/19/1997 3/26/1997 4/2/1997 4/9/1997 4/16/1997 4/23/1997 4/30/1997 5/7/1997 5/14/1997 5/21/1997 5/28/1997 6/4/1997 6/11/1997 6/18/1997 6/25/1997 7/2/1997 7/9/1997 7/16/1997 7/23/1997 7/30/1997 8/6/1997 8/13/1997 8/20/1997 8/27/1997 9/3/1997 9/10/1997 9/17/1997 9/24/1997 10/1/1997 10/8/1997 10/15/1997 10/22/1997 10/29/1997 11/5/1997 11/12/1997 11/19/1997 11/26/1997 12/3/1997 12/10/1997 12/17/1997

5.18 5.3 5.05 5.22 5.16 5.36 5.19 5.26 5.4 5.86 5.37 5.48 5.48 5.61 5.55 5.49 5.52 5.43 5.54 5.48 5.62 5.42 5.82 5.48 5.44 5.43 5.57 5.62 5.45 5.59 5.56 5.64 5.48 5.58 5.45 5.58 5.46 5.45 5.54 5.5 5.6 5.5 5.51 5.49 5.58 5.4 5.66

5.19 5.18 5.3 5.05 5.22 5.16 5.36 5.19 5.26 5.4 5.86 5.37 5.48 5.48 5.61 5.55 5.49 5.52 5.43 5.54 5.48 5.62 5.42 5.82 5.48 5.44 5.43 5.57 5.62 5.45 5.59 5.56 5.64 5.48 5.58 5.45 5.58 5.46 5.45 5.54 5.5 5.6 5.5 5.51 5.49 5.58 5.4

12/24/1997 12/31/1997 1/7/1998 1/14/1998 1/21/1998 1/28/1998 2/4/1998 2/11/1998 2/18/1998 2/25/1998 3/4/1998 3/11/1998 3/18/1998 3/25/1998 4/1/1998 4/8/1998 4/15/1998 4/22/1998 4/29/1998 5/6/1998 5/13/1998 5/20/1998 5/27/1998 6/3/1998 6/10/1998 6/17/1998 6/24/1998 7/1/1998 7/8/1998 7/15/1998 7/22/1998 7/29/1998 8/5/1998 8/12/1998 8/19/1998 8/26/1998 9/2/1998 9/9/1998 9/16/1998 9/23/1998 9/30/1998 10/7/1998 10/14/1998 10/21/1998 10/28/1998 11/4/1998 11/11/1998

5.44 5.45 5.74 5.45 5.53 5.53 5.52 5.43 5.54 5.51 5.6 5.45 5.47 5.43 5.6 5.48 5.47 5.37 5.4 5.35 5.49 5.6 5.45 5.63 5.43 5.58 5.42 5.88 5.47 5.49 5.5 5.54 5.61 5.5 5.59 5.48 5.61 5.47 5.54 5.42 5.58 5.22 5.14 4.87 4.95 5.22 4.8

5.66 5.44 5.45 5.74 5.45 5.53 5.53 5.52 5.43 5.54 5.51 5.6 5.45 5.47 5.43 5.6 5.48 5.47 5.37 5.4 5.35 5.49 5.6 5.45 5.63 5.43 5.58 5.42 5.88 5.47 5.49 5.5 5.54 5.61 5.5 5.59 5.48 5.61 5.47 5.54 5.42 5.58 5.22 5.14 4.87 4.95 5.22

11/18/1998 11/25/1998 12/2/1998 12/9/1998 12/16/1998 12/23/1998 12/30/1998 1/6/1999 1/13/1999 1/20/1999 1/27/1999 2/3/1999 2/10/1999 2/17/1999 2/24/1999 3/3/1999 3/10/1999 3/17/1999 3/24/1999 3/31/1999 4/7/1999 4/14/1999 4/21/1999 4/28/1999 5/5/1999 5/12/1999 5/19/1999 5/26/1999 6/2/1999 6/9/1999 6/16/1999 6/23/1999 6/30/1999 7/7/1999 7/14/1999 7/21/1999 7/28/1999 8/4/1999 8/11/1999 8/18/1999 8/25/1999 9/1/1999 9/8/1999 9/15/1999 9/22/1999 9/29/1999 10/6/1999

4.89 4.54 4.86 4.68 4.97 4.69 4.48 4.3 4.75 4.64 4.66 4.75 4.77 4.75 4.75 4.85 4.8 4.79 4.79 4.84 4.8 4.68 4.61 4.79 4.9 4.7 4.76 4.73 4.65 4.71 4.73 4.71 4.95 5 4.97 4.96 5.01 5.06 4.96 5.03 5.02 5.34 5.16 5.24 5.16 5.27 5.27

4.8 4.89 4.54 4.86 4.68 4.97 4.69 4.48 4.3 4.75 4.64 4.66 4.75 4.77 4.75 4.75 4.85 4.8 4.79 4.79 4.84 4.8 4.68 4.61 4.79 4.9 4.7 4.76 4.73 4.65 4.71 4.73 4.71 4.95 5 4.97 4.96 5.01 5.06 4.96 5.03 5.02 5.34 5.16 5.24 5.16 5.27

10/13/1999 10/20/1999 10/27/1999 11/3/1999 11/10/1999 11/17/1999 11/24/1999 12/1/1999 12/8/1999 12/15/1999 12/22/1999 12/29/1999 1/5/2000 1/12/2000 1/19/2000 1/26/2000 2/2/2000 2/9/2000 2/16/2000 2/23/2000 3/1/2000 3/8/2000 3/15/2000 3/22/2000 3/29/2000 4/5/2000 4/12/2000 4/19/2000 4/26/2000 5/3/2000 5/10/2000 5/17/2000 5/24/2000 5/31/2000 6/7/2000 6/14/2000 6/21/2000 6/28/2000 7/5/2000 7/12/2000 7/19/2000 7/26/2000 8/2/2000 8/9/2000 8/16/2000 8/23/2000 8/30/2000

5.17 5.18 5.18 5.27 5.2 5.44 5.52 5.63 5.45 5.44 5.46 5.01 4.72 5.62 5.59 5.43 5.66 5.71 5.75 5.72 5.77 5.73 5.79 5.81 6.01 6.12 5.98 6.04 5.97 6.06 5.96 6.16 6.5 6.53 6.49 6.5 6.51 6.53 6.85 6.44 6.5 6.5 6.49 6.45 6.53 6.46 6.54

5.27 5.17 5.18 5.18 5.27 5.2 5.44 5.52 5.63 5.45 5.44 5.46 5.01 4.72 5.62 5.59 5.43 5.66 5.71 5.75 5.72 5.77 5.73 5.79 5.81 6.01 6.12 5.98 6.04 5.97 6.06 5.96 6.16 6.5 6.53 6.49 6.5 6.51 6.53 6.85 6.44 6.5 6.5 6.49 6.45 6.53 6.46

9/6/2000 9/13/2000 9/20/2000 9/27/2000 10/4/2000 10/11/2000 10/18/2000 10/25/2000 11/1/2000 11/8/2000 11/15/2000 11/22/2000 11/29/2000 12/6/2000 12/13/2000 12/20/2000 12/27/2000 1/3/2001 1/10/2001 1/17/2001 1/24/2001 1/31/2001 2/7/2001 2/14/2001 2/21/2001 2/28/2001 3/7/2001 3/14/2001 3/21/2001 3/28/2001 4/4/2001 4/11/2001 4/18/2001 4/25/2001 5/2/2001 5/9/2001 5/16/2001 5/23/2001 5/30/2001 6/6/2001 6/13/2001 6/20/2001 6/27/2001 7/4/2001 7/11/2001 7/18/2001 7/25/2001

6.56 6.5 6.5 6.5 6.58 6.47 6.49 6.51 6.55 6.49 6.52 6.51 6.5 6.57 6.47 6.53 6.48 5.88 5.91 6.02 5.96 5.94 5.51 5.47 5.5 5.5 5.49 5.46 5.33 5 5.21 4.96 4.98 4.42 4.53 4.43 4.37 3.98 3.98 4.08 4 3.95 3.91 3.89 3.67 3.76 3.81

6.54 6.56 6.5 6.5 6.5 6.58 6.47 6.49 6.51 6.55 6.49 6.52 6.51 6.5 6.57 6.47 6.53 6.48 5.88 5.91 6.02 5.96 5.94 5.51 5.47 5.5 5.5 5.49 5.46 5.33 5 5.21 4.96 4.98 4.42 4.53 4.43 4.37 3.98 3.98 4.08 4 3.95 3.91 3.89 3.67 3.76

8/1/2001 8/8/2001 8/15/2001 8/22/2001 8/29/2001 9/5/2001 9/12/2001 9/19/2001 9/26/2001 10/3/2001 10/10/2001 10/17/2001 10/24/2001 10/31/2001 11/7/2001 11/14/2001 11/21/2001 11/28/2001 12/5/2001 12/12/2001 12/19/2001 12/26/2001 1/2/2002 1/9/2002 1/16/2002 1/23/2002 1/30/2002 2/6/2002 2/13/2002 2/20/2002 2/27/2002 3/6/2002 3/13/2002 3/20/2002 3/27/2002 4/3/2002 4/10/2002 4/17/2002 4/24/2002 5/1/2002 5/8/2002 5/15/2002 5/22/2002 5/29/2002 6/5/2002 6/12/2002 6/19/2002

3.79 3.7 3.75 3.63 3.52 3.63 3.49 2.47 2.99 2.71 2.44 2.44 2.49 2.55 2.36 2.03 2.01 1.95 2.02 1.88 1.84 1.77 1.63 1.64 1.74 1.74 1.78 1.74 1.72 1.75 1.75 1.74 1.71 1.76 1.7 1.77 1.71 1.78 1.7 1.81 1.74 1.75 1.71 1.78 1.78 1.74 1.75

3.81 3.79 3.7 3.75 3.63 3.52 3.63 3.49 2.47 2.99 2.71 2.44 2.44 2.49 2.55 2.36 2.03 2.01 1.95 2.02 1.88 1.84 1.77 1.63 1.64 1.74 1.74 1.78 1.74 1.72 1.75 1.75 1.74 1.71 1.76 1.7 1.77 1.71 1.78 1.7 1.81 1.74 1.75 1.71 1.78 1.78 1.74

6/26/2002 7/3/2002 7/10/2002 7/17/2002 7/24/2002 7/31/2002 8/7/2002 8/14/2002 8/21/2002 8/28/2002 9/4/2002 9/11/2002 9/18/2002 9/25/2002 10/2/2002 10/9/2002 10/16/2002 10/23/2002 10/30/2002 11/6/2002 11/13/2002 11/20/2002 11/27/2002 12/4/2002 12/11/2002 12/18/2002 12/25/2002 1/1/2003 1/8/2003 1/15/2003 1/22/2003 1/29/2003 2/5/2003 2/12/2003 2/19/2003 2/26/2003 3/5/2003 3/12/2003 3/19/2003 3/26/2003 4/2/2003 4/9/2003 4/16/2003 4/23/2003 4/30/2003 5/7/2003 5/14/2003

1.75 1.75 1.73 1.74 1.72 1.72 1.74 1.72 1.73 1.76 1.81 1.73 1.73 1.72 1.8 1.73 1.75 1.72 1.79 1.7 1.21 1.28 1.27 1.24 1.23 1.27 1.23 1.2 1.2 1.26 1.23 1.24 1.29 1.22 1.3 1.24 1.29 1.21 1.27 1.22 1.28 1.23 1.27 1.26 1.28 1.26 1.25

1.75 1.75 1.75 1.73 1.74 1.72 1.72 1.74 1.72 1.73 1.76 1.81 1.73 1.73 1.72 1.8 1.73 1.75 1.72 1.79 1.7 1.21 1.28 1.27 1.24 1.23 1.27 1.23 1.2 1.2 1.26 1.23 1.24 1.29 1.22 1.3 1.24 1.29 1.21 1.27 1.22 1.28 1.23 1.27 1.26 1.28 1.26

5/21/2003 5/28/2003 6/4/2003 6/11/2003 6/18/2003 6/25/2003 7/2/2003 7/9/2003 7/16/2003 7/23/2003 7/30/2003 8/6/2003 8/13/2003 8/20/2003 8/27/2003 9/3/2003 9/10/2003 9/17/2003 9/24/2003 10/1/2003 10/8/2003 10/15/2003 10/22/2003 10/29/2003 11/5/2003 11/12/2003 11/19/2003 11/26/2003 12/3/2003 12/10/2003 12/17/2003 12/24/2003 12/31/2003 1/7/2004 1/14/2004 1/21/2004 1/28/2004 2/4/2004 2/11/2004 2/18/2004 2/25/2004 3/3/2004 3/10/2004 3/17/2004 3/24/2004 3/31/2004 4/7/2004

1.27 1.24 1.26 1.24 1.25 1.21 1.13 0.96 1.02 1.01 1.04 0.97 0.98 1.18 1 1.01 0.96 1.02 1 1.07 0.99 1.03 1 1 1.01 0.99 0.99 0.99 1 0.98 1 0.99 0.96 0.97 0.99 1 1.02 1.01 1 1.01 1 1.03 1 1 0.99 1.01 1.01

1.25 1.27 1.24 1.26 1.24 1.25 1.21 1.13 0.96 1.02 1.01 1.04 0.97 0.98 1.18 1 1.01 0.96 1.02 1 1.07 0.99 1.03 1 1 1.01 0.99 0.99 0.99 1 0.98 1 0.99 0.96 0.97 0.99 1 1.02 1.01 1 1.01 1 1.03 1 1 0.99 1.01

4/14/2004 4/21/2004 4/28/2004 5/5/2004 5/12/2004 5/19/2004 5/26/2004 6/2/2004 6/9/2004 6/16/2004 6/23/2004 6/30/2004 7/7/2004 7/14/2004 7/21/2004 7/28/2004 8/4/2004 8/11/2004 8/18/2004 8/25/2004 9/1/2004 9/8/2004 9/15/2004 9/22/2004 9/29/2004 10/6/2004 10/13/2004 10/20/2004 10/27/2004 11/3/2004 11/10/2004 11/17/2004 11/24/2004 12/1/2004 12/8/2004 12/15/2004 12/22/2004 12/29/2004 1/5/2005 1/12/2005 1/19/2005 1/26/2005 2/2/2005 2/9/2005 2/16/2005 2/23/2005 3/2/2005

1.01 1 1 1.02 0.99 1.02 0.99 1.01 0.99 1.01 1 1.11 1.28 1.25 1.25 1.26 1.27 1.36 1.42 1.51 1.53 1.5 1.49 1.64 1.76 1.82 1.73 1.76 1.74 1.78 1.79 2 2 2.02 2 2.15 2.24 2.27 2.14 2.25 2.28 2.27 2.43 2.5 2.5 2.52 2.51

1.01 1.01 1 1 1.02 0.99 1.02 0.99 1.01 0.99 1.01 1 1.11 1.28 1.25 1.25 1.26 1.27 1.36 1.42 1.51 1.53 1.5 1.49 1.64 1.76 1.82 1.73 1.76 1.74 1.78 1.79 2 2 2.02 2 2.15 2.24 2.27 2.14 2.25 2.28 2.27 2.43 2.5 2.5 2.52

3/9/2005 3/16/2005 3/23/2005 3/30/2005 4/6/2005 4/13/2005 4/20/2005 4/27/2005 5/4/2005 5/11/2005 5/18/2005 5/25/2005 6/1/2005 6/8/2005 6/15/2005 6/22/2005 6/29/2005 7/6/2005 7/13/2005 7/20/2005 7/27/2005 8/3/2005 8/10/2005 8/17/2005 8/24/2005 8/31/2005 9/7/2005 9/14/2005 9/21/2005 9/28/2005 10/5/2005 10/12/2005 10/19/2005 10/26/2005 11/2/2005 11/9/2005 11/16/2005 11/23/2005 11/30/2005 12/7/2005 12/14/2005 12/21/2005 12/28/2005 1/4/2006 1/11/2006 1/18/2006 1/25/2006

2.5 2.55 2.71 2.77 2.81 2.76 2.78 2.74 2.96 2.99 3.01 3.01 3.02 2.98 3.02 2.98 3.11 3.29 3.22 3.26 3.26 3.29 3.48 3.54 3.52 3.55 3.51 3.49 3.65 3.77 3.85 3.68 3.76 3.76 3.94 4 3.98 4 4.02 4 4.19 4.24 4.22 4.16 4.23 4.29 4.26

2.51 2.5 2.55 2.71 2.77 2.81 2.76 2.78 2.74 2.96 2.99 3.01 3.01 3.02 2.98 3.02 2.98 3.11 3.29 3.22 3.26 3.26 3.29 3.48 3.54 3.52 3.55 3.51 3.49 3.65 3.77 3.85 3.68 3.76 3.76 3.94 4 3.98 4 4.02 4 4.19 4.24 4.22 4.16 4.23 4.29

2/1/2006 2/8/2006 2/15/2006 2/22/2006 3/1/2006 3/8/2006 3/15/2006 3/22/2006 3/29/2006 4/5/2006 4/12/2006 4/19/2006 4/26/2006 5/3/2006 5/10/2006 5/17/2006 5/24/2006 5/31/2006 6/7/2006 6/14/2006 6/21/2006 6/28/2006 7/5/2006 7/12/2006 7/19/2006 7/26/2006 8/2/2006 8/9/2006 8/16/2006 8/23/2006 8/30/2006 9/6/2006 9/13/2006 9/20/2006 9/27/2006 10/4/2006 10/11/2006 10/18/2006 10/25/2006 11/1/2006 11/8/2006 11/15/2006 11/22/2006 11/29/2006 12/6/2006 12/13/2006 12/20/2006

4.44 4.5 4.49 4.49 4.5 4.51 4.51 4.57 4.7 4.88 4.76 4.77 4.74 4.83 4.84 5 4.98 5.01 4.99 5 4.95 5 5.14 5.24 5.25 5.24 5.27 5.25 5.23 5.24 5.25 5.25 5.23 5.24 5.27 5.3 5.23 5.23 5.24 5.25 5.24 5.25 5.24 5.26 5.25 5.24 5.25

4.26 4.44 4.5 4.49 4.49 4.5 4.51 4.51 4.57 4.7 4.88 4.76 4.77 4.74 4.83 4.84 5 4.98 5.01 4.99 5 4.95 5 5.14 5.24 5.25 5.24 5.27 5.25 5.23 5.24 5.25 5.25 5.23 5.24 5.27 5.3 5.23 5.23 5.24 5.25 5.24 5.25 5.24 5.26 5.25 5.24

12/27/2006 1/3/2007 1/10/2007 1/17/2007 1/24/2007 1/31/2007 2/7/2007 2/14/2007 2/21/2007 2/28/2007 3/7/2007 3/14/2007 3/21/2007 3/28/2007 4/4/2007 4/11/2007 4/18/2007 4/25/2007 5/2/2007 5/9/2007 5/16/2007 5/23/2007 5/30/2007 6/6/2007 6/13/2007 6/20/2007 6/27/2007 7/4/2007 7/11/2007 7/18/2007 7/25/2007 8/1/2007 8/8/2007 8/15/2007 8/22/2007 8/29/2007 9/5/2007 9/12/2007 9/19/2007 9/26/2007 10/3/2007 10/10/2007 10/17/2007 10/24/2007 10/31/2007 11/7/2007 11/14/2007

5.24 5.22 5.23 5.24 5.25 5.27 5.25 5.26 5.25 5.28 5.25 5.25 5.26 5.26 5.26 5.28 5.24 5.23 5.25 5.23 5.27 5.24 5.28 5.24 5.26 5.25 5.25 5.28 5.23 5.27 5.26 5.27 5.25 4.79 4.91 5.11 5.03 4.98 5.12 4.78 4.72 4.75 4.74 4.73 4.78 4.33 4.54

5.25 5.24 5.22 5.23 5.24 5.25 5.27 5.25 5.26 5.25 5.28 5.25 5.25 5.26 5.26 5.26 5.28 5.24 5.23 5.25 5.23 5.27 5.24 5.28 5.24 5.26 5.25 5.25 5.28 5.23 5.27 5.26 5.27 5.25 4.79 4.91 5.11 5.03 4.98 5.12 4.78 4.72 4.75 4.74 4.73 4.78 4.33

11/21/2007 11/28/2007 12/5/2007 12/12/2007 12/19/2007 12/26/2007 1/2/2008 1/9/2008 1/16/2008 1/23/2008 1/30/2008 2/6/2008 2/13/2008 2/20/2008 2/27/2008 3/5/2008 3/12/2008 3/19/2008 3/26/2008 4/2/2008 4/9/2008 4/16/2008 4/23/2008 4/30/2008 5/7/2008 5/14/2008 5/21/2008 5/28/2008 6/4/2008 6/11/2008 6/18/2008 6/25/2008 7/2/2008 7/9/2008 7/16/2008 7/23/2008 7/30/2008 8/6/2008 8/13/2008 8/20/2008 8/27/2008 9/3/2008 9/10/2008 9/17/2008 9/24/2008 10/1/2008 10/8/2008

4.51 4.53 4.55 4.39 4.21 4.21 3.77 4.23 4.24 4 3.5 3.01 3 2.98 2.96 3 2.97 2.7 2.18 2.23 2.23 2.34 2.25 2.28 1.94 1.96 1.96 2.05 1.99 1.99 1.98 1.97 2.08 1.95 2.01 1.99 2.08 2.02 1.99 2.02 1.99 1.96 1.99 2.25 1.54 1.32 1.59

4.54 4.51 4.53 4.55 4.39 4.21 4.21 3.77 4.23 4.24 4 3.5 3.01 3 2.98 2.96 3 2.97 2.7 2.18 2.23 2.23 2.34 2.25 2.28 1.94 1.96 1.96 2.05 1.99 1.99 1.98 1.97 2.08 1.95 2.01 1.99 2.08 2.02 1.99 2.02 1.99 1.96 1.99 2.25 1.54 1.32

10/15/2008 10/22/2008 10/29/2008 11/5/2008 11/12/2008 11/19/2008 11/26/2008 12/3/2008 12/10/2008 12/17/2008 12/24/2008 12/31/2008 1/7/2009 1/14/2009 1/21/2009 1/28/2009 2/4/2009 2/11/2009 2/18/2009 2/25/2009 3/4/2009 3/11/2009 3/18/2009 3/25/2009 4/1/2009 4/8/2009 4/15/2009 4/22/2009 4/29/2009 5/6/2009 5/13/2009 5/20/2009 5/27/2009 6/3/2009 6/10/2009 6/17/2009 6/24/2009 7/1/2009 7/8/2009 7/15/2009 7/22/2009 7/29/2009 8/5/2009 8/12/2009 8/19/2009 8/26/2009 9/2/2009

0.96 0.69 0.82 0.24 0.28 0.36 0.56 0.49 0.13 0.15 0.11 0.1 0.1 0.1 0.2 0.19 0.23 0.23 0.23 0.2 0.22 0.2 0.17 0.17 0.16 0.14 0.15 0.14 0.17 0.21 0.17 0.16 0.17 0.19 0.2 0.19 0.24 0.19 0.17 0.14 0.15 0.15 0.18 0.17 0.16 0.16 0.15

1.59 0.96 0.69 0.82 0.24 0.28 0.36 0.56 0.49 0.13 0.15 0.11 0.1 0.1 0.1 0.2 0.19 0.23 0.23 0.23 0.2 0.22 0.2 0.17 0.17 0.16 0.14 0.15 0.14 0.17 0.21 0.17 0.16 0.17 0.19 0.2 0.19 0.24 0.19 0.17 0.14 0.15 0.15 0.18 0.17 0.16 0.16

9/9/2009 9/16/2009 9/23/2009 9/30/2009 10/7/2009 10/14/2009 10/21/2009 10/28/2009 11/4/2009 11/11/2009

0.15 0.16 0.16 0.12 0.13 0.12 0.12 0.11 0.12 0.12

0.15 0.15 0.16 0.16 0.12 0.13 0.12 0.12 0.11 0.12

Significance F 0 0.9999 0.991292

Lower 95% Upper 95% Lower 95.0%Upper 95.0% -0.0130195 0.0484898 -0.0130195 0.04849 0.9912883 0.9999033 0.9912883 0.999903 up 5.9730 94.0270

dn 6.6198 93.3802

5.3262 94.6738

3yr @ 4% qrtr DCF w*(t,T)^2 0.00 0.0000 2 1.94 0.0052 0.00 0.0000 2 1.87 0.0203 0.00 0.0000 2 1.81 0.0440 0.00 0.0000 2 1.75 0.0756 0.00 0.0000 2 1.69 0.1142 0.00 0.0000 102 83.39 8.1188 92.44 8.3780 CF

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00

0.9840 0.9680 0.9520 0.9360 0.9190 0.9040 0.8880 0.8730 0.8587 0.8445 0.8308 0.8175 0.8047 0.7924 0.7806 0.7691

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

0.9840 0.9680 0.9520 0.9360 0.9190 0.9040 0.8880 0.8730

2yr @ 10% qrtr CF DCF w*(t,T)^2 2.5 2.46 0.0015 2.5 2.42 0.0058 2.5 2.38 0.0129 2.5 2.34 0.0225 2.5 2.30 0.0346 2.5 2.26 0.0489 2.5 2.22 0.0654 100.25 87.52 3.3695 103.90 3.5611

2yr @ 1% qrtr DCF w*(t,T)^2 0.00 0.0000 0.5 0.48 0.0014 0.00 0.0000 0.5 0.47 0.0053 0.00 0.0000 0.5 0.45 0.0114 0.00 0.0000 100.5 87.74 3.9370 89.14 3.9550

0.370611 1.319787

0.317977 1.257604

CF

CFs

100

2 year zero DCFs 0.00 0.00 0.00 0.00 0.00 0.00 0.00 87.30 87.30

280.34

3.8230

0.311412

2 year zero w*(t,T)^2 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 4.0000 4.0000

1.245647

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

0.9840 0.9680 0.9520 0.9360 0.9190 0.9040 0.8880 0.8730

CF 1.5 1.5 1.5 1.5 1.5 101.5

1.5yr @ 6% qrtr DCF w*(t,T)^2 1.48 0.0009 1.45 0.0037 1.43 0.0081 1.40 0.0142 1.38 0.0218 91.76 2.0876 0.00 0.0000 0.00 0.0000 98.89 2.1363

2 TOTAL

1,234.95 2.0655

197.789 0.160159 0.342146

1.75 fl + 80bps semi CF DCF w*(t,T)^2 0.4 0.39 0.0002 0.00 0.0000 0.4 0.38 0.0021 0.00 0.0000 0.4 0.37 0.0056 0.00 0.0000 0.4 0.36 0.0105 0.00 0.0000 1.50 0.0184 7.00% 101.84 0.0616 103.34 0.0800 4 413.3648 0.334721 0.026774

2 year zero DCFs 0.00 0.00 0.00 0.00 0.00 0.00 0.00 87.30 87.30

CFs

100

6

523.8 0.424145

2 year zero w*(t,T)^2 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 4.0000 4.0000

1.5 fl semi DCFs w*(t,T)^2 0.00 0.0000 0.00 0.0000 0.00 0.0000 0.00 0.0000 0.00 0.0000 0.00 0.0000 0.00 0.0000 0.00 0.0000 100

CFs

1 1.696582

100 0.080975

0

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

0.9840 0.9680 0.9520 0.9360 0.9190 0.9040 0.8880 0.8730

CF 1 1 1 1 1 101

1.5yr @ 4% qrtr DCF w*(t,T)^2 0.98 0.0006 0.97 0.0025 0.95 0.0056 0.94 0.0097 0.92 0.0149 91.30 2.1385 0.00 0.0000 0.00 0.0000 96.06 2.1720

4 TOTAL

2,075.35 1.8916

384.252 0.18515 0.402139

1.5yr @ 5% semi DCF w*(t,T)^2 0.00 0.0000 2.5 2.42 0.0062 0.00 0.0000 2.5 2.34 0.0240 0.00 0.0000 102.5 92.66 2.1401 0.00 0.0000 0.00 0.0000 97.42 2.1703 CF

5

487.1 0.234707 0.509383

CFs

100

1.5 year zero DCFs 0.00 0.00 0.00 0.00 0.00 90.40 0.00 0.00 90.40

10

904 0.435589

1.5 year zero w*(t,T)^2 0.0000 0.0000 0.0000 0.0000 0.0000 2.2500 0.0000 0.0000 2.2500

1.5 fl semi DCFs w*(t,T)^2 0.00 0.0000 0.00 0.0000 0.00 0.0000 0.00 0.0000 0.00 0.0000 0.00 0.0000 0.00 0.0000 0.00 0.0000 100

CFs

3 0.980075

300 0.144554

0

E[dr] E[dr^2] D C E[dP/P] E[dP/P] n

0 6.00E-07 5 25 7.50E-06 daily 0.1890% annualized 252

E[dr] E[dr^2] D C E[dP/P] E[dP/P] n

0 7.00E-07 30 900 3.15E-04 daily 7.9380% annualized 252

3yr @ 2% semi DCF w*(t,T)^2 0.00 0.0000 1 0.97 0.0028 0.00 0.0000 1 0.94 0.0106 0.00 0.0000 1 0.90 0.0231 0.00 0.0000 2 1.75 0.0794 0.00 0.0000 1 0.84 0.0600 0.00 0.0000 101 82.57 8.4477 87.97 8.6236 CF

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00

0.9840 0.9680 0.9520 0.9360 0.9190 0.9040 0.8880 0.8730 0.8587 0.8445 0.8308 0.8175 0.8047 0.7924 0.7806 0.7691

0.00 0.01 0.00 0.01 0.00 0.02 0.00 0.04 0.00 0.02 0.00 2.82 2.91

E[dr] E[dr^2] D C E[dP/P] E[dP/P] n

0 7.50E-07 2.91 8.6236 3.23E-06 daily 0.0815% annualized 252

1-month 5-year 10-year 5.0% 8.0% 10.0% 6.0% 4.5% 4.0% 3.0% 4.0% 3.0% 4.0% 3.0% 5.0%

Slope Curvature 5.00% 1.00% -2.00% -1.00% 0.00% 2.00% 1.00% -3.00%

maturity beta1 beta2 Z(t,T) 1.00 1.1150 -0.2540 0.9800 2.00 0.9940 -0.3010 0.9600 3.00 0.9640 -0.1470 0.9300 4.00 0.9330 0.0080 0.8900 5.00 0.9300 0.1620 0.8500 6.00 0.9260 0.3160 0.8100 7.00 0.9270 0.4230 0.7700 8.00 0.9270 0.5300 0.7300

5% annual w factor1 factor2 4.9 0.0454 0.050611 -0.01153 4.8 0.0445 0.088396 -0.02677 4.65 0.0431 0.124574 -0.019 4.45 0.0412 0.153843 0.001319 4.25 0.0394 0.183071 0.03189 4.05 0.0375 0.208447 0.071133 80.85 0.7490 4.859987 2.217664 107.95

ks kL

5.6689

-0.4651 -1.1231

2.2647

1y factor1 1y factor2 8y factor1 8y factor2 1.1150 -0.2540

6.4890 98.00

77.00

2.9610

0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 8.00

beta1 beta2 Z(t,T) 1.1690 -0.1510 0.9900 1.1150 -0.2540 0.9800 1.0550 -0.2780 0.9700 0.9940 -0.3010 0.9600 0.9790 -0.2240 0.9400 0.9640 -0.1470 0.9300 0.9480 -0.0690 0.9100 0.9330 0.0080 0.8900 0.9320 0.0850 0.8700 0.9300 0.1620 0.8500 0.9280 0.2390 0.8300 0.9260 0.3160 0.8100 0.9260 0.3690 0.7900 0.9270 0.4230 0.7700 0.9270 0.4760 0.7500 0.9270 0.5300 0.7300

maturity beta1 beta2 Z(t,T) 1.00 1.1150 -0.2540 0.9800 2.00 0.9940 -0.3010 0.9600 3.00 0.9640 -0.1470 0.9300 4.00 0.9330 0.0080 0.8900 5.00 0.9300 0.1620 0.8500 6.00 0.9260 0.3160 0.8100 7.00 0.9270 0.4230 0.7700 8.00 0.9270 0.5300 0.7300

1% annual w factor1 factor2 0.98 0.0100 0.011157 -0.00254 96.96 0.9900 1.968108 -0.59598 0.0000 0 0 0.0000 0 0 0.0000 0 0 0.0000 0 0 0.0000 0 0

97.94

ks kL

1.9793

-1.9395 0.0317

-0.5985

1y factor1 1y factor2 8y factor1 8y factor2 1.1150 -0.2540

1.1150 98.00

-0.2540

7.4160 7.4160 77.00

4.2400 4.2400

maturity Z(t,T) F(0,3,5) f2(0,3,5) f(0,3,5) 0.50 0.9940 1.00 0.9880 1.50 0.9740 2.00 0.9620 2.50 0.9460 3.00 0.9330 0.9196 3.50 0.9170 4.00 0.8950 4.50 0.8770 5.00 0.8580 5.50 0.8340 6.00 0.8130 6.50 0.7990 7.00 0.7760 7.50 0.7570 8.00 0.7360 2.00 0.9196 4.2343% 4.1901%

maturity Z(t,T) F(0,.5,1) f2(0,.5,1) 0.50 0.9940 0.9940 1.00 0.9880 1.50 0.9740 2.00 0.9620 2.50 0.9460 3.00 0.9330 3.50 0.9170 4.00 0.8950 4.50 0.8770 5.00 0.8580 5.50 0.8340 6.00 0.8130 6.50 0.7990 7.00 0.7760 7.50 0.7570 8.00 0.7360 0.50 0.9940 1.2146%

f(0,.5,1)

1.2109%

maturity Z(t,T) F(0,4,8) f2(0,4,8) f(0,4,8) 0.50 0.9940 1.00 0.9880 1.50 0.9740 2.00 0.9620 2.50 0.9460 3.00 0.9330 3.50 0.9170 4.00 0.8950 0.8223 4.50 0.8770 5.00 0.8580 5.50 0.8340 6.00 0.8130 6.50 0.7990 7.00 0.7760 7.50 0.7570 8.00 0.7360 4.00 0.8223 4.9501% 4.8898%

maturity Z(t,T) 0.50 0.9940 1.00 0.9880 1.50 0.9740 2.00 0.9620 2.50 0.9460 3.00 0.9330 3.50 0.9170 4.00 0.8950 4.50 0.8770 5.00 0.8580 5.50 0.8340 6.00 0.8130 6.50 0.7990 7.00 0.7760 7.50 0.7570 8.00 0.7360 1.00

F(0,5,6)

f2(0,5,6)

f(0,5,6)

0.9720

0.9720 2.8573% 2.8371%

maturity Z(t,T) F(0,t-0.5,t) f2(0,t-0.5,t)f(0,t-0.5,t) 0.50 0.9940 0.9940 1.2072% 1.2036% 1.00 0.9880 0.9940 1.2146% 1.2109% 1.50 0.9740 0.9858 2.8747% 2.8543% 2.00 0.9620 0.9877 2.4948% 2.4794% 2.50 0.9460 0.9834 3.3827% 3.3544% 3.00 0.9330 0.9863 2.7867% 2.7675% 3.50 0.9170 0.9829 3.4896% 3.4595% 4.00 0.8950 0.9760 4.9162% 4.8568% 4.50 0.8770 5.00 0.8580 5.50 0.8340 6.00 0.8130 6.50 0.7990 7.00 0.7760 7.50 0.7570 8.00 0.7360 0.50

maturity F(0,t-0.5,t) f2(0,t-0.5,t)f(0,t-0.5,t) 0.50 0.9940 1.2072% 1.2036% 1.00 0.9940 1.2146% 1.2109% 1.50 0.9858 2.8747% 2.8543% 2.00 0.9877 2.4948% 2.4794% 2.50 0.9834 3.3827% 3.3544% 3.00 0.9863 2.7867% 2.7675% 3.50 0.9829 3.4896% 3.4595% 4.00 0.9760 4.9162% 4.8568%

maturity Z(t,T) F(0,t-0.5,t) f2(0,t-0.5,t)f(0,t-0.5,t) 0.50 0.9940 1.00 0.9880 1.50 0.9740 0.9858 2.8747% 2.8543% 2.00 0.9620 2.50 0.9460 3.00 0.9330 3.50 0.9170 4.00 0.8950 0.50

Notional f2(0,1,1.5)

100 2.00% -0.427139

T 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

T

Z(0,T) F(0,1,4) 5% semi 0.9680 0.9360 0.9040 0.9658 2.5 0.8730 0.9327 2.5 0.8445 0.9022 102.5 0.8175 0.8734 0.7924 0.8466 0.7691 0.8217

Z(t,T) F(0,0.5,3.5) 5% semi 0.50 0.9940 1.00 0.9880 0.9940 2.5 1.50 0.9740 0.9799 2.5 2.00 0.9620 0.9678 102.5 2.50 0.9460 0.9517 3.00 0.9330 0.9386 3.50 0.9170 0.9225 4.00 0.8950 0.9004

0.0000 0.0000 2.4145 2.3317 92.4800 0.0000 0.0000 0.0000 97.2262

V(fwd) 0.0000 2.4849 2.4497 99.2002 0.0000 0.0000 0.0000 0.0000 104.1348

6.8671

$T$ 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00

$Z(0,T)$ 0.9840 0.9680 0.9520 0.9360 0.9190 0.9040 0.8880 0.8730 0.8587 0.8445 0.8308 0.8175 0.8047 0.7924 0.7806 0.7691

6.50% 6.56% 6.61% 6.67% 6.81% 6.78% 6.84% 6.84% 6.82% 6.81% 6.80% 6.77% 6.75% 6.71% 6.67% 6.63%

0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

$Z(0,T)$ 0.9940 0.9880 0.9740 0.9620 0.9460 0.9330 0.9170 0.8950

1.21% 1.21% 1.76% 1.94% 2.22% 2.31% 2.47% 2.76%

$T$

$T$ 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00

$Z(0,T)$ 0.9848 0.9745 0.9618 0.9490 0.9353 0.9215 0.9084 0.8953 0.8826 0.8699 0.8582 0.8464 0.8350 0.8236 0.8121 0.8006

c 0.9745

5.23%

0.9490

5.30%

0.9215

5.52%

Value

5.52% 0.0000 2.6889 0.0000 2.6185 0.0000 94.6926 100.0000 0.0000

$T$ 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00

$Z(0,T)$ 0.9840 0.9680 0.9520 0.9360 0.9190 0.9040 0.8880 0.8730 0.8587 0.8445 0.8308 0.8175 0.8047 0.7924 0.7806 0.7691

0.9840 0.9520 0.9190

2.7151 0.0000 2.6268 0.0000 94.4357 99.7776 Value

0.222408

$T$ 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00

$Z(0,T)$ 0.9848 0.9745 0.9618 0.9490 0.9353 0.9215 0.9084 0.8953 0.8826 0.8699 0.8582 0.8464 0.8350 0.8236 0.8121 0.8006

0.50 1.00 1.50

$Z(0,T)$ 0.9745 0.9490 0.9215

$T$

c 0.9745

5.23%

0.9490

5.30%

0.9215

5.52%

Value

5.52% 0.0000 2.6889 0.0000 2.6185 0.0000 94.6926 100.0000

0.0000 0.0134 0.0000 0.0262 0.0000 1.4204 1.4600

0.0000 Duration

-1.4600

Assets

Liabilities

Equity

Swap

Amount Duration 3.00 15.6

2.5

1.95

0.5

0.00

9.3492

-13.65

13.65

$T$ 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

$Z(0,T)$ 0.9940 0.9880 0.9740 0.9620 0.9460 0.9330 0.9170 0.8950

N K r(t-1)

5% semi Z(0,0.5) Pfwd(0,0.5,3.5) K

Put Call Call(parity)

1 2.5000% 2.8863% 0.000966

0.9940 99.3964 99.35 0.0461 0.11 0.13 0.1561

1 1.5000% 2.1129% 0.001532

0.9834 0.9699 0.9532 0.9304

100

2.5 2.5 2.5 102.5

2.46 2.42 2.38 95.36 102.63

Z(0,0.5) 0.9620 Pfwd(0,0.5,3.5) 102.6273 K 101 1.5655 Put Call Call(parity)

3.05 4.6155 4.6155