Investments, tenth edition

  Time-Weighted Returns versus Dollar-Weighted Returns 

 When we consider investments over a period during which cash was added to or with-

drawn from the portfolio, measuring the rate of return becomes more difficult. To continue 

our example, suppose that you were to purchase a second share of the same stock at the 

end of the first year, and hold both shares until the end of year 2, at which point you sell 

each share for $54. 

 Total cash outlays are  

  Time  

  Outlay  

 0  

$50 to purchase first share 

 1  

$53 to purchase second share a year later 

  

  Proceeds  

 1  

$2 dividend from initially purchased share 

 2  

$4 dividend from the 2 shares held in the second year, plus 

  

 $108 received from selling both shares at $54 each 

  

1

 This formula gives the geometric average as a quarterly rate of return, consistent with the quarterly rates used to 

compute it. When the observation period is of length  h  years (1/4 in this example), the  annualized   compounded 

rate is defined by 1  1   r  

 GA 

   5  (1  1   r  

 Gh 

 ) 

1/ h 

 . In general, the annualized geometric average of  T  observations, each of 

length  h,  is    1

1 r

GA

5 a q

T

t

51

(1

1 r

t

)

b

1/hT

  where ∏ is the product operator. In our example with  T   5  20  quarterly 

observations, each of length  h   5  ¼ year, 1/ hT   5  1/5, so to find the annualized geometric average, we would take 

the fifth root of the cumulative return over the 5-year investment period. 

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  C H A P T E R  

2 4

  Portfolio Performance Evaluation 

837

 Using the discounted cash flow (DCF) approach, we can solve for the average return 

over the 2 years by equating the present values of the cash inflows and outflows:

   50

1

53

1

1 r

5

2

1

1 r

1

112

(1

1 r)

2

  

 resulting in  r   5  7.117%. 

 This value is called the internal rate of return, or the    dollar-weighted rate of return    on 

the investment. It is “dollar weighted” because the stock’s performance in the second year, 

when two shares of stock are held, has a greater influence on the average overall return 

than the first-year return, when only one share is held. 

 The time-weighted (geometric average) return is 7.81%:

   

r

1

5

53

1 2 2 50

50

5 .10 5 10%    r

2

5

54

1 2 2 53

53

5 .0566 5 5.66%

r

G

5 (1.10 3 1.0566)

1/2

2 1 5 .0781 5 7.81%

  

 The dollar-weighted average is less than the time-weighted average in this example because 

the return in the second year, when more money was invested, is lower.   

 Shares of XYZ Corp. pay a $2 dividend at the end of every year on December 31. An investor buys two 

shares of the stock on January 1 at a price of $20 each, sells one of those shares for $22 a year later on 

the next January 1, and sells the second share an additional year later for $19. Find the dollar- and time-

weighted rates of return on the 2-year investment. 

 CONCEPT CHECK