8.) During the past five years, you owned two stocks that had the following annual rates of return:

Year Stock T Stock B

1 0.16 0.08

2 0.08 0.03

3 -0.12 -0.09

4 -0.03 0.02

5 0.15 0.04

a) compute the arithmetic mean annual rate of return for each stock, which stock is most desirable by this measure?

b) Compute the standard deviation of the annual rate of return for each stock. Which stock is most desirable by this measure?

C) Compute the coefficient of variation for each stock. Which stock is most desirable by this measure?

D) Compute the geometric mean rate of return for each stock. Dicuss the difference between the arithmetic mean and the geometric mean return for each stock.

Discuss the differences in the mean returns relative to the standard deviation of the return for each stock?

## EXPERT ANSWER

**a) Arithmetic mean** (nothing but average rate of return) annual rate of return for each stock are as below,

FOR Stock T, AM = (0.16 + +0.08+(-0.12)+(-0.03)+0.15)/ 5 =0.24/5 = 0.048 = 4.8%

FOR Stock B, AM = ( 0.08+0.03+(-0.09)+0.02+0.04)/5 = 0.08/5 = 0.016= 1.6%

Stock T is the most desirable as it gives higher return on average i.e 4.6%

**b) Standard Deviation (SD)** of the annual rate of return for each stock,

SQRT is nothing but Square root,

FOR Stock T, SD = SQRT [[ (0.16-0.048)^2+(0.03–0.048)^2+(-0.12-0.048)^2+(-0.03-0.048)^2+(0.15-0.048)]/(5-1)]]

= SQRT[ (0.095144)/4)] = SQRT [0.023786] = 0.154227105 = 15.42%

FOR Stock B, SD = SQRT[[ (0.08 -0.016)^2 + (0.03-0.016)^2+(-0.09 -0.016)^2+(0.02-0.016)^2+(0.04-0.016)^2]/(5-1)]

= SQRT[[(0.020216/4)] = SQRT [0.005054] = 0.07109149 = 7.11%

Standard deviation is the measure of the volatility of return, higher the SD, higher the Risk and volatility in the stock. So, Stock B is the most desirable to less risk taking investor since it offers a lower standard deviation

**C) Coefficient of variation** is nothing but total risk per unit of return of an investment. It is calculated by dividing the standard deviation of an investment by Arithmetic mean or expected rate of return

Coefficient of variation,

FOR STOCK T: = 0.154227105/0.048 = 3.21

FOR STOCK B = 0.07109149 /0.016 = 4.44

Stock T is the most desirable to a risk-averse investor according to this measure as it offers the lowest coefficient of variation.

**D) Geometric mean,**

FOR STOCK T: = [(1.16*1.08*0.88*0.97*1.15)^(1/5) ] -1 = 1.042 -1 = 0.042 = 4.2%

FOR STOCK B = [(1.08*1.03*0.91*1.02*1.04)^(1/5) ] -1 =1.0143-1= 0.0143=1.43%

The geometric mean for stock T and B is lower than the arithmetic mean. This is because Arithmetic mean captures some of the variation in return.

**The entire problem solution can be find in the below table:**

Stock T | Stock B | ||||||

Rt | Rb | (Rt-AM)^2 | (Rb-AM)^2 | ||||

0.16 | 0.08 | 0.0125 | 0.0041 | ||||

0.08 | 0.03 | 0.0010 | 0.0002 | ||||

-0.12 | -0.09 | 0.0282 | 0.0112 | ||||

-0.03 | 0.02 | 0.0061 | 0.0000 | ||||

0.15 | 0.04 | 0.0104 | 0.0006 | ||||

Total | 0.24 | 0.08 | 0.0369 | 0.0041 | |||

Mean (AM) | 0.048 | 0.016 | Total | 0.0951 | 0.0202 | ||

in % | 4.8% | 1.6% | Total/(n-1) | 0.0238 | 0.0051 | ||

SD (Standard dev) | 0.1542 | 0.0711 | SQRT of [Total/(n-1)] | ||||

in % | 15.42% | 7.11% | |||||

Coefficient of variation | 3.21 | 4.44 | |||||

Stock T | Stock B | Geometric mean | |||||

Rt | Rb | (1+Rt) | (1+Rb) | ||||

0.16 | 0.08 | 1.16 | 1.08 | ||||

0.08 | 0.03 | 1.08 | 1.03 | ||||

-0.12 | -0.09 | 0.88 | 0.91 | ||||

-0.03 | 0.02 | 0.97 | 1.02 | ||||

0.15 | 0.04 | 1.15 | 1.04 | ||||

Multiplication | 1.22980 | 1.07383 | |||||

0.2 | 0.2 | ||||||

Geometric mean | 1.042 | 1.014 | |||||

in % | 4.22% | 1.43% |