Investments, tenth edition

  Horizon,   T  

  Price,   P  (  T   )  

  [100/  P  (  T   )]   2   1  

  Risk-Free Return 

for Given Horizon  

 Half-year  

$97.36 

 100/97.36  2  1  5   .0271 

 

 r  

 f 

  (.5)    5  2.71% 

 1 year 

 $95.52 

 100/95.52  2  1  5   .0469 

 

 r  

 f 

  (1)    5  4.69% 

 25 years 

 $23.30 

 100/23.30  2  1  5  3.2918  

 r  

 f 

  (25)  5  329.18% 

 Example  5.2 

Annualized Rates of Return 

 Not surprisingly, longer horizons in Example 5.2 provide greater total returns. How 

should we compare returns on investments with differing horizons? This requires that we 

express each  total  return as a  rate  of return for a common period. We typically express all 

investment returns as an    effective annual rate (EAR),    defined as the percentage increase 

in funds invested over a 1-year horizon. 

 For a 1-year investment, the EAR equals the total return,  r  

 f 

   (1), and the gross return, 

(1   1 

  EAR), is the terminal value of a $1 investment. For investments that last less 

than 1 year, we compound the per-period return for a full year. For the 6-month bill in 

Example 5.2, we compound 2.71% half-year returns over two semiannual periods to obtain 

a terminal value of 1  1  EAR  5  (1.0271) 

2

   5  1.0549, implying that EAR  5  5.49%. 

 For investments longer than a year, the convention is to express the EAR as the annual 

rate that would compound to the same value as the actual investment. For example, the 

  

4

 Yields on Treasury bills and bonds of various maturities are widely available on the Web, for example at Yahoo! 

Finance, MSN Money, or directly from the Federal Reserve. 

  

5

 The U.S. Treasury issues T-bills, which are pure discount (or zero-coupon) securities with maturities of up to 

1 year. However, financial institutions create zero-coupon Treasury bonds called Treasury strips with maturities 

up to 30 years by buying coupon-paying T-bonds, “stripping” off the coupon payments, and selling claims to the 

coupon payments and final payment of face value separately. See Chapter 14 for further details. 

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  C H A P T E R  

5

  Risk, Return, and the Historical Record 

123

investment in the 25-year bond in Example 5.2 grows by its maturity by a factor of 4.2918 

(i.e., 1  1  3.2918), so its EAR is found by solving   

 

(1 1 EAR)

25

5 4.2918

1 1 EAR 5 4.2918

1/25

5 1.0600

   

 In general, we can relate EAR to the total return,  r  

 f 

   ( T ), over a holding period of length 

 T  by using the following equation:   

 

1 1 EAR 5

31 1 r

f

(T )

4

1/T

 

 (5.7)   

 We illustrate with an example.  

 For the 6-month Treasury in Example 5.2,  T   5  ½, and 1/ T   5  2. Therefore,

   

1 1 EAR 5 (1.0271)

2

 5 1.0549 and EAR 5 5.49%  

 For the 25-year Treasury in Example 5.2,  T   5  25. Therefore,

   

1 1 EAR 5 4.2918

1/25

 5 1.060 and EAR 5 6.0%

  

 Example  5.3 

Effective Annual Rate versus Total Return 

In  Table 5.1  we use Equation 5.8 to find the APR corresponding to an EAR of 5.8% 

with various compounding periods. Conversely, we find values of EAR implied by an 

APR of 5.8%.