Investments, tenth edition

 Example  5.9 

Sharpe Ratio 

 Using the annual returns for years 2003–2005 in  Spreadsheet 5.2 ,

     a.   Compute the arithmetic average return.  

    b.   Compute the geometric average return.  

    c.   Compute the standard deviation of returns.  

    d.   Compute the Sharpe ratio assuming the risk-free rate was 6% per year.    

 CONCEPT CHECK 

5.5 

bod61671_ch05_117-167.indd   135

bod61671_ch05_117-167.indd   135

6/18/13   8:03 PM

6/18/13   8:03 PM

Final PDF to printer

---

136

P A R T   I I

  Portfolio Theory and Practice

one bad day,” which can happen in two ways (first a good day, or first a bad day). The 

probability of this outcome is .5. Less likely are the two extreme outcomes (both good 

days or both bad days) with probability .25 each. 

 What is the distribution of profits at the end of 200 days? There are 201 possible out-

comes and, again, the midrange outcomes are the most likely because there are more 

sequences that lead to them. While only one sequence can result in 200 consecutive bad 

days, an enormous number of sequences result in 100 good days and 100 bad days. The 

probability distribution will eventually take on the familiar bell shape.  

11

  

  

  Figure 5.4  shows a graph of the normal curve with mean of 10% and standard deviation 

of 20%. A smaller SD means that possible outcomes cluster more tightly around the mean, 

while a higher SD implies more diffuse distributions. The likelihood of realizing any 

particular outcome when sampling from a normal distribution is fully determined by the 

number of standard deviations that separate that outcome from the mean. Put differently, 

the normal distribution is completely characterized by two parameters, the mean and SD.  

  

 

 Investment management is far more tractable when rates of return can be well approxi-

mated by the normal distribution. First, the normal distribution is symmetric, that is, the 

probability of any positive deviation above the mean is equal to that of a negative devia-

tion of the same magnitude. Absent symmetry, measuring risk as the standard deviation 

of returns is inadequate. Second, the normal distribution belongs to a special family of 

distributions characterized as “stable,” because of the following property: When assets 

with normally distributed returns are mixed to construct a portfolio, the portfolio return 

also is normally distributed. Third, scenario analysis is greatly simplified when only two 

parameters (mean and SD) need to be estimated to obtain the probabilities of future sce-

narios.   Fourth, when constructing portfolios of securities, we must account for the statis-

tical dependence of returns across securities. Generally, such dependence is a complex, 

23s

22s

21s

0

11s

12s

13s

250

230

210

30

50

70

68.26%

95.44%

99.74%

10