Question

4. Statistical measures of standalone 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 Blue Llama Mining Company (BLM) and Hungry Whale Electronics (HWE). Three-quarters of James’s portfolio value consists of BLM’s shares, and the balance consists of HWE’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	Blue Llama Mining	Hungry Whale Electronics
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 Blue Llama Mining’s stock over the next year is     .
•	The expected rate of return on Hungry Whale Electronics’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 company’s returns exhibit the greater risk?

Company H

Company G

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Returns earned over a given time period are called realized returns. Historical data on realized returns is often used to estimate future results. Analysts across companies use realized stock returns to estimate the risk of a stock.

Consider the case of Blue Llama Mining Inc. (BLM):

Five years of realized returns for BLM are given in the following table. Remember:

1.	While BLM was started 40 years ago, its common stock has been publicly traded for the past 25 years.
2.	The returns on its equity are calculated as arithmetic returns.
3.	The historical returns for BLM for 2012 to 2015 are:

	2012	2013	2014	2015	2016
Stock return	20.00%	13.60%	24.00%	33.60%	10.40%

Given the preceding data, the average realized return on BLM’s stock is   .

The preceding data series represents   of BLM’s historical returns. Based on this conclusion, the standard deviation of BLM’s historical returns is   .

If investors expect the average realized return from 2012 to 2016 on BLM’s stock to continue into the future, its coefficient of variation (CV) will be   .

Answer 1

Expected Return = Sum of Probability*Return

Blue Llama Mining= 50%*0.25 + 30%*0.45 +-40%*0.30

= 14%

Hungry Whale Electronics = 70%*0.25 + 40%*0.45 +-50%*0.30

= 20.5%

Expected return on portfolio is equal to weighted average return

= 14%*3/4 + 20.5%*1/4

= 15.625%

Company with higher variation in returns has higher risk

Expected Return = Sum of Probability*Return

Blue Llama Mining= 50%*0.25 + 30%*0.45 +-40%*0.30

= 14%

Hungry Whale Electronics = 70%*0.25 + 40%*0.45 +-50%*0.30

= 20.5%

Expected return on portfolio is equal to weighted average return

= 14%*3/4 + 20.5%*1/4

= 15.625%

Company with higher variation in returns has higher risk