Investments, tenth edition

 Portfolio  Performance 

Evaluation 

 CHAPTER TWENTY-FOUR 

     24.1 

The Conventional Theory of Performance 

Evaluation 

   Average Rates of Return 

 We defined the holding-period return (HPR) in Section 5.1 of Chapter 5 and explained 

the differences between arithmetic and geometric averages. Suppose we evaluate the per-

formance of a portfolio over a period of 5 years from 20 quarterly rates of return. The 

arithmetic average of this sample of returns would be the best estimate of the expected 

bod61671_ch24_835-881.indd   835

bod61671_ch24_835-881.indd   835

7/25/13   3:13 AM

7/25/13   3:13 AM

Final PDF to printer

---

836 

P A R T   V I I

  Applied Portfolio Management

rate of return of the portfolio for the next quarter. In contrast, the geometric average is the 

constant quarterly return over the 20 quarters that would yield the same total cumulative 

return. Therefore, the geometric average,  r  

 G 

 , is defined by

   (1

1 r

G

)

20

5 (1 1 r

1

)(1

1 r

2

)

 

c

 

(1

1 r

20

)  

 The right-hand side of this equation is the compounded final value of a $1 investment earn-

ing the 20 quarterly rates of return over the 5-year observation period. The left-hand side is 

the compounded value of a $1 investment earning  r  

 G 

   each  quarter. We solve for 1  1   r  

 G 

   as  

1

   

   1

1 r

G

5 3(1 1 r

1

)(1

1 r

2

)

 

c

 

(1

1 r

20

)

4

1/20

  

 Each return has an equal weight in the geometric average. For this reason, the geometric 

average is referred to as a    time-weighted  average    .  

 To set the stage for discussing the more subtle issues that follow, let us start with a 

trivial example. Consider a stock paying a dividend of $2 annually that currently sells for 

$50. You purchase the stock today, collect the $2 dividend, and then sell the stock for $53 

at year-end. Your rate of return is

   

Total proceeds

Initial investment

5

Income

1 Capital gain

50

5

2

1 3

50

5 .10, or 10%  

 Another way to derive the rate of return that is useful in the more difficult multiperiod 

case is to set up the investment as a discounted cash flow problem. Call  r  the rate of return 

that equates the present value of all cash flows from the investment with the initial outlay. 

In our example the stock is purchased for $50 and generates cash flows at year-end of $2 

(dividend) plus $53 (sale of stock). Therefore, we solve 50   5   (2   1   53)/(1   1     r ) to find 

again that  r   5  10%.