Question

Statistical measures of stand-alone risk

Remember, the expected value of a probability distribution is a statistical measure of the average (mean) value expected to occur during all possible circumstances. To compute an asset’s expected return under a range of possible circumstances (or states of nature), multiply the anticipated return expected to result during each state of nature by its probability of occurrence.

Consider the following case:

James owns a two-stock portfolio that invests in Celestial Crane Cosmetics Company (CCC) and Lumbering Ox Truckmakers (LOT). Three-quarters of James’s portfolio value consists of CCC’s shares, and the balance consists of LOT’s shares.

Each stock’s expected return for the next year will depend on forecasted market conditions. The expected returns from the stocks in different market conditions are detailed in the following table:

Market Condition	Probability of Occurrence	Celestial Crane Cosmetics	Lumbering Ox Truckmakers
Strong	0.50	30%	42%
Normal	0.25	18%	24%
Weak	0.25	-24%	-30%

Calculate expected returns for the individual stocks in James’s portfolio as well as the expected rate of return of the entire portfolio over the three possible market conditions next year.

•	The expected rate of return on Celestial Crane Cosmetics’s stock over the next year is     .
•	The expected rate of return on Lumbering Ox Truckmakers’s stock over the next year is     .
•	The expected rate of return on James’s portfolio over the next year is     .

The expected returns for James’s portfolio were calculated based on three possible conditions in the market. Such conditions will vary from time to time, and for each condition there will be a specific outcome. These probabilities and outcomes can be represented in the form of a continuous probability distribution graph.

For example, the continuous probability distributions of rates of return on stocks for two different companies are shown on the following graph:

  

Based on the graph’s information, which statement is false?

Company H has lower risk.

Company G has lower risk.

Answer 1

(1)-The expected rate of return on Celestial Crane Cosmetic's stock over the year

Expected Return = Sum of [Returns x Probability]

= [30.00% x 0.50] + [18.00% x 0.25] + [-24.00% x 0.25]

= 15.00% + 4.50% - 6.00%

= 13.50%

(2)-The expected rate of return on Lumbering Ox Truckmaster's stock over the next year

Expected Return = Sum of [Returns x Probability]

= [42.00% x 0.50] + [24.00% x 0.25] + [-30.00% x 0.25]

= 21.00% + 6.00% - 7.50%

= 19.50%

(3)-The expected rate of return on James’s portfolio over the next year

Expected Return = [Expected Rate of return on Celestial Crane Cosmetics stock x Proportion of the Investment in Celestial Crane Cosmetics] + [Expected Rate of return on Lumbering Ox Truckmakers stock x Proportion of the Investment in Lumbering Ox Truckmakers]

= [13.50% x ¾] + [19.50% x ¼]

= 10.12% + 4.88%

= 15.00%