Question

Consider the following probability distribution of returns estimated for a proposed project that involves a new ultrasound machine:

State of the Economy	Probability of Occurrence	Rate of Return
Very poor	0.10	-10.0%
Poor	0.20	-2.0%
Average	0.40	14.0%
Good	0.20	22.0%
Very good	0.10	34.0%

a. What is the expected rate of return on the project?

b. What is the project’s standard deviation of returns?

c. What is the project’s coefficient of variation (CV) of returns?

d. What type of risk do the standard deviation and CV measure?

e. In what situation is this risk relevant?

Answer 1

(a) Expected Rate of Return = Σ R_iP_i, where R_i is the return for Probability P_i (i is the state of economy)

State of the	Probability of	Rate of	Expected
Economy	Occurrence	Return	Return
Very Poor	0.1	-10%	-1.00%
Poor	0.2	-2%	-0.40%
Average	0.4	14%	5.60%
Good	0.2	22%	4.40%
Very Good	0.1	34%	3.40%
		ER	12.00%

Hence, Expected Returns = 12%

(b) Variance = Σ Squared Deviation from Expected Value * Probability of Occurance

State of the	Probability of	Rate of	Expected	Deviation from
Economy	Occurrence	Return	Return	Expected Returns	Squared	Variance
Very Poor	0.1	-0.10	-0.010	-0.220	0.048	0.00484
Poor	0.2	-0.02	-0.004	-0.140	0.020	0.00392
Average	0.4	0.14	0.056	0.020	0.000	0.00016
Good	0.2	0.22	0.044	0.100	0.010	0.00200
Very Good	0.1	0.34	0.034	0.220	0.048	0.00484
		ER	0.120		Variance	0.01576

Hence, Variance = 0.0156

(c) Standard Deviation = sqrt (variance) = 12.49%

Hence, CV = Standard Deviation/Expected Returns = 12.49/12 = 1.041

(d) Standard deviation measures the volatility of the investment

CV measures the amount of volatility in comparison to the expected return rate of an investment

(e) the risk is more relevant when compared to the returns i.e. CV

when comparing multiple investments, an investor would invest in the stock with lower CV