Question

2. 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 Happy Dog Soap Company (HDS) and Black Sheep Broadcasting (BSB). Three-quarters of James’s portfolio value consists of HDS’s shares, and the balance consists of BSB’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	Happy Dog Soap	Black Sheep Broadcasting
Strong	0.25	50%	70%
Normal	0.45	30%	40%
Weak	0.30	-40%	-50%

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 Happy Dog Soap’s stock over the next year is ____________.
•	The expected rate of return on Black Sheep Broadcasting’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

Expected Rate of return = w1 * r1 + w2 * r2 + w3 * r3

For Happy Dogs, Expected rate = 0.25 * 50% + 0.45 * 30% + 0.30 * -40% = 14.00%

For Black Sheep Broadcasting, Expected rate = 0.25 * 70% + 0.45 * 40% + 0.30 * -50% = 20.50%

For James portfolio = (3/4) * 14% + (1/4) * 20.5% = 15.6250%

Question 2:

This is incomplete and the graph is missing.

If this is how the graph looks like, in this graph, Company A has lower risk since it is spread in a lower area of X-axis, which indicates lower variability and hence lower standard deviation.