Investments, tenth edition

      Expected Returns 

We use “expected value” and “mean” interchangeably. For an analysis with  n   scenarios, 

where the rate of return in scenario  i  is  r ( i ) with probability  p ( i ), the expected return is   

 

E(r)

5 a

n

i

51

p(i)r (i) 

 (7B.1)  

If you were to increase the rate of return assumed for each scenario by some amount D, 

then the mean return will increase by D. If you multiply the rate in each scenario by a 

 factor   w,  the new mean will be multiplied by that factor:  

 

  a

n

i

51

p(i)

3 3r

 

(i)

1 D4 5 a

n

i

51

p(i)

3 r

 

(i)

1 D a

n

i

51

p(i)

5 E(r) 1 D 

(7B.2)

  a

n

i

51

p(i)

3 3wr(i)4 5 w a

n

i

51

p(i)

3 r(i) 5 wE(r)

the optimal risky portfolio. Therefore, we add at the bottom of  Spreadsheet 7A.3  a row with 

entries obtained by multiplying the SD of each column’s portfolio by the Sharpe ratio of the 

optimal risky portfolio from cell H46. This results in the risk premium for each portfolio along 

the CAL efficient frontier. We now add a series to the graph with the standard deviations in 

B45–I45 as the  x -axis and cells B54–I54 as the  y -axis. You can see this CAL in  Figure 7A.2 .