Question

A stock's returns have the following distribution:

Demand for the Company's Products	Probability of This Demand Occurring	Rate of Return If This Demand Occurs
Weak	0.1	(28%)
Below average	0.1	(11)
Average	0.4	10
Above average	0.3	35
Strong	0.1	61
	1.0

Assume the risk-free rate is 4%. Calculate the stock's expected return, standard deviation, coefficient of variation, and Sharpe ratio. Do not round intermediate calculations. Round your answers to two decimal places.

Stock's expected return:   %

Standard deviation:   %

Coefficient of variation:

Sharpe ratio:

Answer 1

Demand	Probability	Rate of return
Weak	0.1	-28%
Below Average	0.1	-11%
Average	0.4	10%
Above average	0.3	35%
Strong	0.1	61%

We have the following data:

p1 = 0.1, p2 = 0.1, p3 = 0.4, p4 = 0.3, p5 = 0.1

R1 = -28%, R2 = -11%, R3 = 10%, R4 = 35%, R5 = 61%

Part 1

Stock's expected return is calculated using the formula:

Expected return = E[R] = p1*R1 + p2*R2 + p3*R3 + p4*R4 + p5*R5 = 0.1*(-28%) + 0.1*(-11%) + 0.4*10% + 0.3*35% + 0.1*31% = 16.7%

Part 2

Vraiance of the stock return is calculated using the formula:

Variance = σ2 = p1*(R1 - E[R])2 + p2*(R2 - E[R])2 + p3*(R3 - E[R])2​​​​​​​ + p4*(R4 - E[R])2​​​​​​​ + p5*(R5 - E[R])2​​​​​​​

σ2 = 0.1*(-28% - 16.7%)2 + 0.1*(-11% - 16.7%)2​​​​​​​ + 0.4*(10% - 16.7%)2​​​​​​​ + 0.3*(35% - 16.7%)2​​​​​​​ + 0.1*(61% - 16.7%)2​​​​​​​ = 0.0199809+0.0076729+0.0017956+0.0100467+0.0196249 = 0.059121

Standard deviation is the square-root of variance

Standard deviation of stock's return = σ = (0.059121)1/2 = 24.3148103015426% ~ 24.31% (Rounded to two decimals)

Part 3

Coefficient of Variation = Standard deviation/Mean = 24.3148103015426%/16.7% = 1.45597666476303 ~ 1.46 (Rounded to two decimals)

Part 4

Risk-free rate = RF​​​​​​​ = 4%

Sharpe ratio = (E[R] - RF)/σ = (16.7%-4%)/24.3148103015426% = 0.52231540540517 ~ 0.52 (Rounded to two decimals)

Answers

Stock's expected return (%) = 16.7

Standard deviation (%) = 24.31

Coefficient of variation = 1.46

Sharpe ratio = 0.52