Investments, tenth edition

 Figure 15.3 

Short rates versus spot rates  

bod61671_ch15_487-514.indd   493

bod61671_ch15_487-514.indd   493

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494 

P A R T   I V

 Fixed-Income 

Securities

 Equation 15.3 has a simple interpretation. The numerator on the right-hand side is the 

total growth factor of an investment in an  

n 

-year zero held until maturity. Similarly, 

the denominator is the growth factor of an investment in an ( n   2  1)-year zero. Because the 

former investment lasts for one more year than the latter, the difference in these growth 

factors must be the rate of return available in year  n  when the ( n   2  1)-year zero can be 

rolled over into a 1-year investment. 

 Of course, when future interest rates are uncertain, as they are in reality, there is no 

meaning to inferring “the” future short rate. No one knows today what the future interest 

rate will be. At best, we can speculate as to its expected value and associated uncertainty. 

Nevertheless, it still is common to use Equation 15.3 to investigate the implications of the 

yield curve for future interest rates. Recognizing that future interest rates are uncertain, 

we call the interest rate that we infer in this matter the    forward  interest  rate    rather than 

the  future short rate,  because it need not be the interest rate that actually will prevail at the 

future date. 

 If the forward rate for period  n  is denoted  f  

 n 

 , we then define  f  

 n 

  by the equation

 

   (1

1 f

n

)

5

(1

1 y

n

)

n

(1

1 y

n

21

)

n

21

 

 (15.4)   

 Equivalently, we may rewrite Equation 15.4 as

 

   (1

1 y

n

)

n

5 (1 1 y

n

21

)

n

21

(1

1 f

n

) 

 (15.5)   

 In this formulation, the forward rate is  defined  as the “break-even” interest rate that equates 

the return on an  n -period zero-coupon bond to that of an ( n   2  1)-period zero-coupon bond 

rolled over into a 1-year bond in year  n.  The actual total returns on the two  n -year  strate-

gies will be equal if the short interest rate in year  n  turns out to equal  f  

 n 

 . 

 Suppose a bond trader uses the data presented in  Table 15.1 . The forward rate for year 4 

would be computed as

   1

1 f

4

5

(1 1 y

4

)

4

(1 1 y

3

)

3

5

1.08

4

1.07

3

5 1.1106  

 Therefore, the forward rate is  f  

4

   5  .1106, or 11.06%.