CREDIT RISK AND BOND MARKETS
Module 2
Bond Prices and Yields
Outline • • • • • • • • • • •
Bond Pricing Time-Value of Money Present Value Formula Interest Rates Frequency Continuous Compounding Coupon Rate Current Yield Yield-to-Maturity Bank Discount Rate Forward Rates
Bond Pricing • Bond pricing is a 2 steps process – Step 1: find the cash-flows the bondholder is entitled to – Step 2: find the bond price as the discounted value of the cash-flows
• Step 1 - Example – Government of Canada bond issued in the domestic market pays one-half of its coupon rate times its principal value every six months up to and including the maturity date – Thus, a bond with an 8% coupon and $5,000 face value maturing on December 1, 2005 will make future coupon payments of 4% of principal value every 6 months – That is $200 on each June 1 and December 1 between the purchase date and the maturity date
Bond Pricing • Step 2 is discounting T
Ft P0 = ∑ t ( 1 + r ) t =1 • Does it make sense to discount all cash-flows with same discount rate? • Notion of the term structure of interest rates – see next chapter • Rationale behind discounting: time value of money
Time-Value of Money • Would you prefer to receive $1 now or $1 in a year from now? • Chances are that you would go for money now • First, you might have a consumption need sooner rather than later – That shouldn’t matter: that’s what fixed-income markets are for – You may as well borrow today against this future income, and consume now
• In the presence of money market, the only reason why one would prefer receiving $1 as opposed to $1 in a year from now is because of time-value of money
Present Value Formula • If you receive $1 today – Invest it in the money market (say buy a one-year T-Bill) – Obtain some interest r on it – Better off as long as r strictly positive: 1+r>1 iif r>0
• How much is worth a piece of paper (contract, bond) promising $1 in 1 year? – Since you are not willing to exchange $1 now for $1 in a year from now, it must be that the present value of $1 in a year from now is less than $1 – Now, how much exactly is worth this $1 received in a year from now? – Would you be willing to pay 90, 80, 20, 10 cents to acquire this dollar paid in a year from now?
• Answer is 1/(1+r) : the exact amount of money that allows you to get $1 in 1 year • Chicken is the rate, egg is the value
Interest Rates • Specifying the rate is not enough • One should also specify – Maturity – Frequency of interest payments – Date of interest rates payment (beginning or end of periods)
• Basic formula
– After 1 period, capital is C1= C0 (1+ r ) – After n period, capital is Cn = C0(1+ r )n – Interests : I = Cn - C0
• Example – Invest $10,000 for 3 years at 6% with annual compounding – Obtain $11,910 = 10,000 x (1+ .06)3 at the end of the 3 years – Interests: $1,910
Frequency • Watch out for – Time-basis (rates are usually expressed on an annual basis) – Compounding frequency
• Examples – Invest $100 at a 6% two-year annual rate with semi-annual compounding • 100 x (1+ 3%) after 6 months • 100 x (1+ 3%)2 after 1 year • 100 x (1+ 3%)3 after 1.5 year • 100 x (1+ 3%)4 after 2 years – Invest $100 at a 6% one-year annual rate with monthly compounding • 100 x (1+ 6/12%) after 1 month • 100 x (1+ 6/12%)2 after 2 months • …. 100 x (1+ 6/12%)12 = $106.1678 after 1 year • Equivalent to 6.1678% annual rate with annual compounding
Frequency • More generally – – – – –
Amount x invested at the interest rate r Expressed in an annual basis Compounded n times per year For T years Grows to the amount r nT
x(1 + ) n
• The effective equivalent annual (i.e., compounded once a year) rate ra is defined as the solution to r x(1 + ) nT = x(1 + r a )T n or n r r a = 1 + − 1 n
Continuous Compounding • What happens if we get continuous compounding • The amount of money obtained per dollar invested after T years is
r nT lim x(1 + ) = xe rT n →∞ n
• Very convenient: present value of X is Xe-rT • One may of course easily obtain the effective equivalent annual ra 2 3 r r xe rT = x(1 + r a )T ⇔ r a = e r − 1 = r + + + ... 2 3! • The equivalent annual rate of a 6% continuously compounded interest rate is e6% –1 = 6.1837%
Bond Prices • Bond price
T
Ft P =∑ t ( 1 + r ) t =1
• Shortcut when cash-flows are all identical T F F 1 P=∑ = 1 − T t r (1 + r ) t =1 (1 + r ) • Coupon bond T cN N cN 1 N 1 − + P=∑ + = T T T t ( 1 + r ) r ( ) ( ) ( ) 1 + r 1 + r 1 + r t =1 – Note that when r=c, P=N (see next example)
Bond Prices - Example •
Example – – – –
•
Consider a bond with 5% coupon rate 10 year maturity $1,000 face value All discount rates equal to 6%
Present value
50 1,000 50 1 1,050 P=∑ + = 1− + = $926.3991 10 10 10 i (1 + 6% ) 6% (1 + 6% ) (1 + 6% ) i =1 (1 + 6%) 10
•
We could have guessed that price was below par –
•
You do not want to pay the full price for a bond paying 5% when interest rates are at 6%
What happens if rates decrease to 5%? –
Price = $1,000
Perpetuity • When the bond has infinite maturity (consol bond) cN 1 P= 1− r (1 + r ) T
N cN + T → T →∞ r (1 + r )
• Example – How much money should you be willing to pay to buy a contract offering $100 per year for perpetuity? – Assume the discount rate is 5% – The answer is
100 P= = $2,000 .05
– Perpetuities are issued by the British government (consol bonds)
Coupon Rate and Current Yield • Coupon rate is the stated interest rate on a security – It is referred to as an annual percentage of face value – It is usually paid twice a year – It is called the coupon rate because bearer bonds carry coupons for interest payments – It is only used to obtain the cash-flows
• Current yield gives you a first idea of the return on a bond cN yc = P • Example – A $1,000 bond has a coupon rate of 7 percent – If you buy the bond for $900, your actual current yield is
70 yc = = 7.78% 900
Yield to Maturity (YTM) • It is the interest rate that makes the present value of the bond’s payments equal to its price • It is the solution to (T is # of semester) T
Ft P=∑ t ( 1 + YTM ) t =1 • YTM is the IRR of cash-flows delivered by bonds – – – – –
YTM may easily be computed by trial-and-error YTM is a semi-annual rate because coupons usually paid semi-annually Each cash-flow is discounted using the same rate Implicitly assume that the yield curve is flat at point in time It is a complex average of pure discount rates (see below)
BEY versus EAY • Bond equivalent yield (BEY): obtained using simple interest to annualize the semi-annual YTM (street convention): y = 2 × YTM • One can always turn a bond yield into an effective annual yield (EAY), i.e., an interest rate expressed on a yearly basis with annual compounding • Example – What is the effective annual yield of a bond with a 5.5% annual YTM – Answer is 2
5.5% r a = 1 + − 1 = 5.5756% 2
One Last Complication • What happens if we don’t have integer # of periods? • Example – Consider the US T-Bond with coupon 4.625% and maturity date 05/15/2006, quoted price is 101.739641 on 01/07/2002 – What is the YTM and EAY?
• Solution (street convention) – There are 128 calendar days between 01/07/2002 and the next coupon date (05/15/2002) 8
101.739641 = ∑ t =0
4.625%
2
128 +t 181
(1 + YTM ) 2
+
102.3125 ⇒ YTM = 4.353% 128 +8 (1 + YTM ) 181 2
– Fed convention: =1+YTM/2*128/181 2 • EAY is 4 . 353 % 1 + − 1 = 4.40% 2
Quoted Bond Prices - Screen
Quoted Bond Prices • Bonds are – Sold in denominations of $1,000 par value – Quoted as a percentage of par value
• Prices – Integer number + n/32ths (Treasury bonds) or + n/8ths (corporate bonds) – Example: 112:06 = 112 6/32 = 112.1875% – Change -5: closing bid price went down by 5/32%
• Ask yield – YTM based on ask price (APR basis:1/2 year x 2) – Not compounded (Bond Equivalent Yield as opposed to Effective Annual Yield)
Examples • Example – Consider a $1,000 face value 2-year bond with 8% coupon – Current price is 103:23 – What is the yield to maturity of this bond?
• To answer that question – First note that 103:23 means 103 + (23/32)%=103.72% – And obtain the following equation
40 40 40 1,040 1,037.2 = + + + y y y (1 + ) (1 + ) 2 (1 + ) 3 (1 + y ) 4 2 2 2 2 – With solution y/2 = 3% or y = 6%
Accrued Interest • The quoted price (or market price) of a bond is usually its clean price, that is its gross price (or dirty or full price) minus the accrued interest • Example – An investor buys on 12/10/01 a given amount of the US Treasury bond with coupon 3.5% and maturity 11/15/2006 – The current market price is 96.15625 – The accrued interest period is equal to 26 days; this is the number of calendar days between the settlement date (12/11/2001) and the last coupon payment date (11/15/2001) – Hence the accrued interest is equal to the last coupon payment (1.75) times 26 divided by the number of calendar days between the next coupon payment date (05/15/2002) and the last coupon payment date (11/15/2001) – In this case, the accrued interest is equal to $1.75x(26/181) = $0.25138 – The investor will pay 96.40763 = 96.15625 + 0.25138 for this bond
Bank Discount Rate (T-Bills) • Bank discount rate is the quoted rate on T-Bills
rBD
10,000 − P 360 = × 10,000 n
– where P is price of T-Bill – n is # of days until maturity
• Example: 90 days T-Bill, P = $9,800
rBD •
10,000 − 9,800 360 = × = 8% 10,000 90
Can’t compare T-bill directly to bond – 360 vs 365 days – Return is figured on par vs. price paid
Bond Equivalent Yield • Adjust the bank discounted rate to make it comparable 10,000 − P 365 rBEY = × P n • Example: same as before 10,000 − 9,800 365 rBEY = × = 8.28% 9,800 90 • BDR versus BEY
P 360 rBEY × × = rBD 10,000 365
Spot Zero-Coupon (or Discount) Rate • Spot Zero-Coupon (or Discount) Rate is the annualized rate on a pure discount bond 1 = B ( 0, t ) t (1 + R0 ,t ) – where B(0,t) is the market price at date 0 of a bond paying off $1 at date t – See Chapter 4 for how to extract implicit spot rates from bond prices
• General pricing formula T
T Ft P0 = ∑ = ∑Ft B ( 0, t ) t t =1 (1 + R0 ,t ) t =1
Bond Par Yield • Recall that a par bond is a bond with a coupon identical to its yield to maturity • The bond's price is therefore equal to its principal • Then we define the par yield c(n) so that a n-year maturity fixed bond paying annually a coupon rate of c(n) with a $100 face value quotes at par • Typically, the par yield curve is used to determine the coupon level of a bond issued at par
1 1− n n ( 1 + R ) 100c (n) 100 0,n 100 = ∑ + ⇒ c ( n ) = n i n 1 ( 1 + R ) ( 1 + R ) i =1 0 ,i 0,n ∑ i ( 1 + R ) i =1 0 ,i
Forward Rates • One may represent the term structure of interest rates as set of implicit forward rates • Consider two choices for a 2-year horizon: – Choice A: Buy 2-year zero – Choice B: Buy 1-year zero and rollover for 1 year
• What yield from year 1 to year 2 will make you indifferent between the two choices?
1 + F1,1 =
(1 + R0, 2 ) 2 (1 + R0,1 )
Forward Rates (continued)
• They are ‘implicit’ in the term structure • Rates that explain the relationship between spot rates of different maturity • Example: – Suppose the one year spot rate is 4% and the eighteen month spot rate is 4.5%
(1 + R0 ,18 m ) 3 / 2 = (1 + R0 ,1 )(1 + F12 m , 6 m )1/ 2 (1.045) 3 / 2 = (1.04)(1 + F12 m , 6 m )1/ 2 ; F12 m , 6 m = 5.51%
Recap: Taxonomy of Rates • • • • • •
Coupon Rate Current Yield Yield to Maturity Zero-Coupon Rate Bond Par Yield Forward Rate