ch03 by Salim Kasap

Finance and Derivatives: Theory and Practice Sébastien Bossu and Philippe Henrotte

Chapter 3 Bonds

1. Financial Markets 1.1. Securities and Portfolios 



A financial security is a contract in which two or more parties exchange cash flows A portfolio is a combination of n financial securities. A portfolio cash flow at time t is the sum of the securities’ cash flows at time t weighted by their quantities.

Chapter 3 Bonds

Exhibit p. 1: Portfolio Security

Quantity

Time (years)

t=1

t=2

10,000

Cash flows A (¥) 100

200

5,000

Cash flows B (¥) —

Portfolio P

Cash flows P (¥) 1,000,000

2,250,000

Chapter 3 Bonds

1.2. Value and price 



Value: positive or negative amount for the anticipated change in wealth. There are several valuation methods which will yield different values for the same security. When cash flows are certain, the standard method is present value. Price: amount of money agreed upon by two parties to trade a security. Note that the buyer and seller may not agree on ‘the’ value of the security; and even if they do nothing forces them to set the price at such value.

Chapter 3 Bonds

1.3. Financial markets 



Financial markets are marketplaces — physical or virtual — where one can buy and sell financial securities. Investors are usually authorized to short sell securities they do not own. This way, market participants can buy and sell securities at any time according to their views and the law of supply and demand is continuously verified. Note that the shortseller must pay all the security cash flows to the buyer.

Chapter 3 Bonds

1.4. Arbitrage 



An arbitrage opportunity is a strategy for buying and selling securities which has no cost and yields positive profits with absolute certainty (either today or at a future date.) Arbitrage opportunities are extremely uncommon and the absence of arbitrage opportunities is always assumed in financial theory.

Chapter 3 Bonds

Exhibit p.25: An arbitrage opportunity Cash flows Transaction

Today

6 months later

Sell A

€+99

€–100

Buy B

€–99

€+110

Total

0 (no cost)

+€10 (positive profit)

Chapter 3 Bonds

1.5. Price of a portfolio (‘law of one price’) 



Assume no arbitrage and infinite liquidity (i.e. one can buy and sell securities in any quantity without influencing their prices) Then the arbitrage price of a portfolio of securities is simply the sum of each security’s price multiplied by its respective quantity.

Chapter 3 Bonds

1.5. Price of a portfolio (2) 

 

Consider a portfolio P of two securities A and B with prices of €100 and €50 respectively : Security

Quantity

Unit price (€)

Time (years)

t=1

t=2

100

Cash flows A (€)

110

Cash flows B (€)

Portfolio P

250

Cash flows P (€)

250

The arbitrage price of P is €250: Suppose that P had a market price X > 250. Then we would have an arbitrage by shortselling P at X and buying 2 units of A and 1 unit of B for €250

Chapter 3 Bonds

Exhibit p.26: portfolio arbitrage Transaction

t=0

t=1

t=2

Sell P

-50

-250

Buy 2 A

-200

+20

+220

Buy B

-50

+30

Total

X – 250 >0

Chapter 3 Bonds

1.5. Price of a portfolio (3) 



Conversely, if we suppose that the market price of P is X < 250, investors could also make an arbitrage by buying P, shortselling 2 units of A and 1 unit of B Thus, the only price for P which does not lead to an arbitrage is X = 250

Chapter 3 Bonds

2. Bonds 2.1. Treasury bonds  



Bonds are debt securities. The debtor’s capacity for repayment is the first issue with bonds. A debtor who fails to repay is said to default on the bond. Treasury bonds, which are issued by the government, bear a low risk of default. We will only consider Treasury bonds and assume that they are defaultfree.

Chapter 3 Bonds

2.1. Treasury bonds (2)  



Bonds are defined by: Face value N (also called par amount, principal amount or sometimes notional amount): the amount borrowed. When N is not specified it is conventionally set at 100. Maturity date T: the date when the principal must be repaid Coupons Ct1, Ct2, …, CT: interest amounts paid by the borrower at dates t1, t2, …, T. Typically all equal and given as a percentage of the face value.

Chapter 3 Bonds

Exhibit p.27: Bond Cash Flows Time Cash flow

…

… Cash flows of a bond Ct1

Ct2

T N + CT

Date 25 Sep 06 25 Sep 07 25 Sep 08 25 Sep09 Cash flow (£) +8 +8 +8 +108 Example: UK Gilt Bond 8% 25 September 2009

Chapter 3 Bonds

2.2. Zerocoupons 



Zerocoupon bonds, also called zerocoupons, are bonds which pay no coupon: only the principal is repaid at maturity. Shortterms bonds (whose maturity is shorter than one year) are usually zerocoupon bonds when issued.

Chapter 3 Bonds

2.3. Bond markets 



Bonds are issued by a government’s treasury department through auctions on the primary market at a price usually close to the par amount N. Once issued they are traded on the secondary market at a fluctuating price. At each coupon date (typically every anniversary date of the issue date) coupons are ‘detached’ and bondholders receive the coupon amount from the issuer. At maturity, bondholders surrender their securities to the issuer who repays the par amount N.

Chapter 3 Bonds

3. Yield  

 

How do we: 1. Compare the relative value of two bonds given their prices? 2. Find the fair price of a bond? The classical approach relies on the concept of yield to maturity.

Chapter 3 Bonds

3.1. Yield to maturity 

Given the price of a bond we can calculate its internal rate of return (see Chapter 2) which is called yield to maturity or simply yield:

Ct1

Ct2

N + CT P= + +L+ t1 t2 T (1 + y ) (1 + y ) (1 + y ) 

Note that when the yield y increases the price P must decrease: ‘when rates go up, prices go down.’

Chapter 3 Bonds

3.2. Yield curve 

Investors tend to prefer short maturity bonds: 



Lower interest rate risk: The central bank often changes its shortterm interest rate to adjust to economic conditions, causing the price and yield of bonds to fluctuate. Shortterm bonds limit the risk of having the principal locked up at a comparatively lower interest rate for too long. Lower default risk: Even though governments have a very low default risk, the probability of default on a specific bond is still higher for long maturities than for short maturities.

Chapter 3 Bonds

3.2. Yield curve (2) 



But Issuers tend to prefer long maturities in order to postpone their debt repayments. The divergence between demand (investors) and supply (issuers) normally results in longmaturity yields being higher than shortterm ones. In other words the yield curve (also called the interest rate term structure, that is the structure of rates as a function of maturity or term), usually slopes upward.

Chapter 3 Bonds

Exhibit p.29: U.S. Yield curve 5 4.5 4 3.5 3 2.5

31-03-06 31-03-07 15 -0 2 15 -0 8 1505-0 - 9 1504-1 -0 0 15 2-1 1 -0 15 2-1 2 -0 15 2-1 3 -0 15 2-1 4 -02 -1 5

5.5

15-02-20

15-02-25

13-10-05 14-07-05

15-02-31

Chapter 3 Bonds

3.2. Yield curve (3) 

In practice there are three typical shapes of the yield curve: 



Flat: y ≡ r (where r is constant). This is a theoretical case in which all rates are assumed to be the same regardless of maturity. Upward sloping: This is the most common case: the longer the maturity the higher the interest rate and default risks for which the market demands a higher yield. Downward sloping: This is the case when the market expects rates to decrease in the future.

Chapter 3 Bonds

3.3. Approximate valuation 

Using the yield curve, we can compute the approximate value of a bond with maturity T whose price is unknown: 



(1) Linearly interpolate the yields of the two bonds with closest maturities:

y2 − y1 y% = y1 + (T − T1 ) T2 − T1

(2) Use the interpolated yield to compute the present value of the bond: Ct1 Ct2 N + CT % V= + + L + (1 + y% )t1 (1 + y% )t2 (1 + y% )T

Chapter 3 Bonds

Exhibit p.30: Yield interpolation 6 5.03

5.23

5.4

5 years

7 years

4.83

5 4 3

2.75 4 years

2 1 year

3 years

Chapter 3 Bonds

4. Zerocoupon yield curve 4.1. Yield to maturity: limits 

The concept of yield to maturity has flaws: 



It assumes that the bondholder will be able to reinvest the coupons at the same yield It fails to identify arbitrages

Arbitragefree bond analysis relies on the concept of zerocoupon yield

Chapter 3 Bonds

4.2. Zerorate curve 

The zerorate at maturity T is the yield of a zero coupon bond which matures at time T: 1 T

 N  z(T ) =   −1  p(T )  where N is the face value and p(T) the price of the zerocoupon.

Chapter 3 Bonds

4.5 4 3.5 3

15 /05 / 20 25

15/05/2008 15/05/2009 15 /05 15 / 201 0 /0 155/ 20 /0 1 1 155/ 20 /0 1 2 15 5/ 20 /0 1 3 15 5/ 20 /05 1 4 / 20 15

5.5

15 /05 / 20 20

Exhibit p.32: U.S. zerorate curve 15/05/2030

15/05/2007 15/05/2006

2.5 2

Chapter 3 Bonds

4.3. Arbitrage price 

Given a zerorate curve z(t) we can determine the arbitrage price of any bond:

Ct1

Ct2

N + CT P= + +L+ t1 t2 (1 + z (t1 )) (1 + z (t2 )) (1 + z (T ))T 

This generalizes the present value formula for a series of n cash flows (Ft) at future dates t1, t2, …, tn n

P=∑ i =1

Fti

( 1 + z (ti ) )

Chapter 3 Bonds

4.4. Zerorate calculation by inference (‘Bootstrapping’) 



We can infer the zero rates from standard bond prices if we assume that there is no arbitrage opportunity on the bond market. This is done by solving a linear system, with the provision that all bond cash flow dates must coincide.