Investments, tenth edition

  The Geometric (Time-Weighted) Average Return 

 The arithmetic average provides an unbiased estimate of the  expected  future return. But 

what does the time series tell us about the  actual  performance of a portfolio over the  past

sample period? Let’s continue Example 5.6 using its very short sample period just to illus-

trate. We will present results for meaningful periods later in the chapter. 

 Column F in  Spreadsheet 5.2  shows the wealth index from investing $1 in an S&P 500 

index fund at the beginning of 2001. The value of the wealth index at the end of 2005, 

$1.0275, is the terminal value of the $1 investment, which implies a  5 - year   holding-period 

return (HPR) of 2.75%. 

 An intuitive measure of performance over the sample period is the (fixed) annual HPR 

that would compound over the period to the same terminal value as obtained from the 

sequence of actual returns in the time series. Denote this rate by  g,   so  that   

Terminal value 5 (1 1 r

1

) 3 (1 1 r

2

) 3 ? ? ? 3 (1 1 r

5

) 5 1.0275

(1 1 g)

n

 5 Terminal value 5 1.0275  (cell F9 in Spreadsheet 5.2) 

 (5.14)   

g 5 Terminal value

1/n

 2 1 5 1.075

1/5

 2 1 5 .0054 5 .54%  (cell E14)

 where  1   1     g  is the geometric average of the gross returns (1   1     r ) from the time series 

(which can be computed with Excel’s GEOMEAN function) and  g  is the annual HPR that 

would replicate the final value of our investment.  

 Practitioners call  g  the  time-weighted  (as opposed to dollar-weighted) average return, 

to emphasize that each past return receives an equal weight in the process of averaging. 

This distinction is important because investment managers often experience significant 

changes in funds under management as investors purchase or redeem shares. Rates of 

return obtained during periods when the fund is large produce larger dollar profits than 

rates obtained when the fund is small. We discuss this distinction in the chapter on perfor-

mance evaluation.