Investments, tenth edition

     Forward Rates 

 The following equation generalizes our approach to inferring a future short rate from the 

yield curve of zero-coupon bonds. It equates the total return on two  n -year  investment 

strategies: buying and holding an  n -year zero-coupon bond versus buying an ( n   2  1)-year 

zero and rolling over the proceeds into a 1-year bond.

 

   (1

1 y

n

)

n

5 (1 1 y

n

21

)

n

21

3 (1 1 r

n

) 

 (15.2)   

 where   n  denotes the period in question, and  y  

 n 

  is the yield to maturity of a zero-coupon 

bond with an  n -period maturity. Given the observed yield curve, we can solve Equation 15.2 

for the short rate in the last period:

 

   (1

1 r

n

)

5

(1

1 y

n

)

n

(1

1 y

n

21

)

n

21

 

 (15.3)   

The 2-year bond can be bought for $1,000/1.06 

2

   5  $890.00. Next year, the bond will 

have a remaining maturity of 1 year and the 1-year interest rate will be 7.01%. Therefore, 

its price next year will be $1,000/1.0701  5  $934.49, and its 1-year holding-period rate 

of return will be ($934.49  2  $890.00)/$890.00  5  .05, for an identical 5% rate of return. 

 Show that the rate of return on the 3-year zero in  Table 15.1  also will be 5%. Hint: Next year, the bond will 

have a maturity of 2 years. Use the short rates derived in  Figure 15.3  to compute the 2-year spot rate that 

will prevail a year from now. 

 CONCEPT CHECK 

15.3 

Year

Short Rate in Each Year

Current Spot Rates 

(Yields to Maturity) 

for Various Maturities

1-Year Investment

2-Year Investment

3-Year Investment

4-Year Investment

1

2

3

4

r

2

 

= 7.01%

r

1

 

= 5%

r

3

 

= 9.025%

r

4

 

= 11.06%

y

1

 

= 5%

y

2

 

= 6%

y

3

 

= 7%

y

4

 

= 8%