Investments, tenth edition

  Sharpe’s Ratio Is the Criterion for Overall Portfolios 

 Suppose that Jane Close constructs a portfolio and holds it for a considerable period of 

time. She makes no changes in portfolio composition during the period. In addition, sup-

pose that the daily rates of return on all securities have constant means, variances, and 

covariances. These assumptions are unrealistic, and the need for them highlights the short-

coming of conventional applications of performance measurement. 

 Now we want to evaluate the performance of Jane’s portfolio. Has she made a good 

choice of securities? This is really a three-pronged question. First, “good choice” com-

pared with what alternatives? Second, in choosing between two dis-

tinct alternative portfolios, what are the appropriate criteria to evaluate 

performance? Finally, the performance criteria having been identified, 

is there a rule that will separate basic ability from the random luck of 

the draw? 

 Earlier chapters of this text help to determine portfolio choice cri-

teria. If investor preferences can be summarized by a mean-variance 

utility function such as that introduced in Chapter 6, we can arrive at 

a relatively simple criterion. The particular utility function that we 

used is

U 5 E(r

P

) 2    

1

⁄

2

As

P

2

  

 where   A  is the coefficient of risk aversion. With mean-variance pref-

erences, Jane wants to maximize the Sharpe ratio [ 

E ( r  

 P 

 )   2     r  

 f 

 ]/ s  

 P 

 . 

Recall that this criterion led to the selection of the tangency portfolio in 

 Chapter 7. Jane’s problem reduces to the search for the portfolio with 

the highest possible Sharpe ratio.  

 Using the data of Concept Check 2,  P  has a standard deviation of 42% versus a market 

standard deviation of 30%. Therefore, the adjusted portfolio  P * would be formed by 

mixing bills and portfolio  P  with weights 30/42  5  .714 in  P  and 1  2  .714  5  .286 in bills. 

The return on this portfolio would be (.286  3  6%)  1  (.714  3  35%)  5  26.7%, which is 

1.3% less than the market return. Thus portfolio  P  has an    M

2

P

  measure of  2 1.3%. 

 A graphical representation of  M  

2

  appears in  Figure 24.2 . We move down the capital 

allocation line corresponding to portfolio  P  (by mixing  P  with T-bills) until we reduce the 

standard deviation of the adjusted portfolio to match that of the market index.    M

2

P

  is then 

the vertical distance (the difference in expected returns) between portfolios  P * and  M.  

You can see from  Figure 24.2  that  P  will have a negative  M  

2

  when its capital allocation 

line is less steep than the capital market line, that is, when its Sharpe ratio is less than 

that of the market index.  

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