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	(36%)
Below average	0.1	(15)
Average	0.3	16
Above average	0.3	21
Strong	0.2	56
	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

a) Expected Return- It refers to the sum of the average return of each stock multiplied by its weight or probability.

E(R) = P₁ * R₁ + P₂ * R₂ +.......P_{n *} R_n

where, P₁ ,P_{2 ...}P_n = Probabilty of return

R₁ ,R_{2 ...}R_n  = Return expectation

Here, E(R) = (0.1* -36%) + (0.1 * -15%) + (0.3 * 16%) + (0.3 * 21%) + (0.2 * 56%)

= -0.036 - 0.015 + 0.048 + 0.063 + 0.112 = 0.172 or 17.2%

b) Standard Deviation- It is a measure of volatility or risk. It is the squareroot of the variance.

_p = [P₁ * (R₁- E(R))² + P₂* (R₂- E(R))² + ......... Pn * Rn - E(R))²]

_p =  (0.1 * (-0.36-0.172)²) + (0.1 * (-0.15-0.172)²) + (0.3 * (0.16-0.172)²) + (0.3 * (0.21-0.172)² ) + (0.2 * (0.56-0.172)²)

_p=   (0.0283 + 0.0103684 + 0.0000432 + 0.0004332 + 0.0301)

_p= (0.0692448)

_p= 0.2631 or 26.31%

c) Coefficient of variation - It is the ratio of standard deviation to the mean (expected return)

COV = SD/ expected return

COV = 26.3%/ 17.2% = 1.52

d) Sharpe ratio - The ratio is the average return earned in excess of the risk-free rate per unit of volatility or total risk

Sharpe ratio = R_p - R_f / _p

where, R_p - Return on portfolio

R_f - Risk free rate

_{p -} Standard deviation of portfolio

Sharpe ratio = R_p - R_f / _p

= (0.172 - 0.03) / 0.263

= 0.53