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	(48%)
Below average	0.1	(11)
Average	0.3	16
Above average	0.3	21
Strong	0.2	49
	1.0

Assume the risk-free rate is 3%. 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	Return
Weak	0.1	-48%
Below Average	0.1	-11%
Average	0.3	16%
Above average	0.3	21%
Strong	0.2	49%

We have the following data:

p1 = 0.1, p2 = 0.1, p3 = 0.3, p4 = 0.3, p5 = 0.2

R1 = -48%, R2 = -11%, R3 = 16%, R4 = 21%, R5 = 49%

Risk-free rate = RF = 3%

Stock's expected return

Expected return is calculated using the formula:

Expected Return = E[R] = p1*R1 + p2*R2 + p3*R3 + p4*R4 + p5*R5 = 0.1*(-48%) + 0.1*(-11%) + 0.3*16% + 0.3*21% + 0.2*49% = -4.8% + (-1.1%) + 4.8% + 6.3% + 9.8% = 15%

Standard deviation

Variance of the returns 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 = 0.1*(-48%-15%)2 + 0.1*(-11%-15%)2 + 0.3*(16%-15%)2 + 0.3*(21%-15%)2 + 0.2*(49%-15%)2 = 0.03969 + 0.00676 + 0.00003 + 0.00108 + 0.02312 = 0.07068

Standard deviation is square root of variance

Standard deviation = σ = (​​​​​​​0.07068)1/2 = 26.5857104475318% ~ 26.59%

Coefficient of Variation

Coefficient of variation = CV = Standard deviation/Expected return = 26.5857104475318%/15% = 1.77238069650212 ~ 1.77 (Rounded to two decimals)

Sharpe Ratio

Sharpe ratio is calculated using the formula:

Sharpe ratio = (E[R] - RF)/σ = (15%-3%)/26.5857104475318% = 0.45137029622295 ~ 0.45 (Rounded to two decimals)

Answers

Stock's expected return (%) = 15

Standard deviation (%) = 26.59

Coefficient of variation = 1.77

Sharpe ratio = 0.45