Determining the Term Structure of Interest Rates throughBootstrapping

Assume that coupon interest is paid annually and all bonds have aface value of $100. You’re given the following yields to maturity

A. A one-year 13% coupon bond has a YTM of y = 10%

B. A two-year 11.5% coupon bond has a YTM of y = 9.5%

C. A three-year 9% coupon bond has a YTM of y = 9%

Compute 2f3, i.e. the forward rate for a of a 1-year investmentstarting in 2 years’ time.

In order to calculate the arbitrage-free value of 2f3 we need to figure out theterm structure of interest rates (or at least some relevant part thereof). Wecan infer the term structure from the observed bond prices by “bootstrapping”,i.e. through process of sequentially inferring spot rates from the prices of bondswith increasing maturity. Recall that the yield to maturity of a bond is definedas the constant discount rate that ensures that the sum of a bond’s discountedfuture cash flows equals its price:

P =T∑

t=1

CFt

(1 + y)t

Also recall that the arbitrage-free price of a (risk-free) bond is the sum of thebond’s future cash flows discounted at the appropriate spot-rates, i.e. the spot-rates that match the maturity of each cash flow:

P =T∑

t=1

CFt

(1 + yt)t

Combining these equations, we get

T∑t=1

CFt

(1 + y)t=

T∑t=1

CFt

(1 + yt)t

Solving these equations for bonds of different maturities will get us the termstructure of interest rates, which we can use to infer the arbitrage-free value of2f3. Let’s start with the one-year bond:

1

CF1

(1 + y) = CF1

(1 + y1)

Since there is only one CF involved here, we can immediately see that y = y1 =10%. Let’s move on to the two-year bond:

CF1

(1 + y) + CF2

(1 + y)2 = CF1

(1 + y1) + CF2

(1 + y2)2

11.51.095 + 111.5

1.0952 = 11.51.1 + 111.5

(1 + y2)2

y2 =

√111.5

/( 11.51.095 + 111.5

1.0952 − 11.51.1

)− 1 ≈ 9.47%

We follow the same procedure to figure out the arbitrage-free value of y3:

CF1

(1 + y) + CF2

(1 + y)2 + CF3

(1 + y)3 = CF1

(1 + y1) + CF2

(1 + y2)2 + CF3

(1 + y3)3

91.09 + 9

1.092 + 1091.093 ≈ 9

1.1 + 9(1.0947)2 + 109

(1 + y3)3

y3 ≈[109/( 9

1.09 + 91.092 + 109

1.093 − 91.1 − 9

1.09472

)]1/3− 1 ≈ 8.94%

Having inferred the relevant term-structure, we apply our usual replication ar-gument to determine the arbitrage-free forward rate. That is, we note that aone dollar investment between t = 2 and t = 3 at the relevant forward rate 2f3

would achieve a negative cash flow of one dollar at t = 2 (CF2 = −1), and apositive cash flow of that dollar plus interest rate at t = 3 [CF3 = 1 · (1 +2 f3)].We replicate CF2 by borrowing 1

(1+y2)2 for two years at the relevant spot-rate,y2. We reinvest this money for three years at the appropriate spot-rate, y3,achieving a cash flow at t = 3 of CF3 = (1+y3)3

(1+y2)2 . Since (in the absence of ar-bitrage) this certain cash flow must be equal to the certain cash flow resultingfrom the investment in the forward rate, we get the following equation:

1 +2 f3 = (1 + y3)3

(1 + y2)2

2

2f3 ≈ (1.0894)3

(1.0947)2 − 1 ≈ 7.88%

3