Investments, tenth edition

  Expected Returns and the Arithmetic Average 

 When we use historical data, we treat each observation as an equally likely “scenario.” 

So if there are  n  observations, we substitute equal probabilities of 1/ n  for each  p ( s )  in 

Equation 5.11. The expected return,  E ( r ), is then estimated by the arithmetic average of the 

sample rates of return:   

E(r) 5 a

n

s51

p(s)r (s) 5

1

n a

n

s51

r (s) 

(5.13)

  5 Arithmetic average of rates of return 

    

 

    5.5 

Time Series Analysis of Past Rates of Return 

 To illustrate,  Spreadsheet 5.2  presents a (very short) time series of annual holding-period 

returns for the S&P 500 index over the period 2001–2005. We treat each HPR of the 

 n   5  5 observations in the time series as an equally likely annual outcome during the sam-

ple years and assign it an equal probability of 1/5, or .2. Column B in  Spreadsheet 5.2  

therefore uses .2 as probabilities, and Column C shows the annual HPRs. Applying 

Equation 5.13 (using Excel’s SUMPRODUCT function) to the time series in  

Spread-

sheet 5.2  demonstrates that adding up the products of probability times HPR amounts 

to taking the arithmetic average of the HPRs (compare cells C10 and C11). 

 Example  5.6 

Arithmetic Average and Expected Return 

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  C H A P T E R  

5

  Risk, Return, and the Historical Record 

131

 Example 5.6 illustrates the logic for the wide use of the arithmetic average in invest-

ments. If the time series of historical returns fairly represents the true underlying prob-

ability distribution, then the arithmetic average return from a historical period provides a 

forecast of the investment’s expected future HPR.