Topic+4+The+Characterstic+of+the+Opportunity+Set+under+Risk+-+Questions

Topic 4. The Characteristics of the Opportunity Set under Risk Questions

Question 4.1. Assume that the average variance of return for an individual security is 50 and that the average covariance is 10. What is the expected variance of an equally weighted portfolio of 5, 10, 20, 50 and 100 securities?

Question 4.2. In Question 4.1 how many securities need to be held before the risk of a portfolio is only 10% more than the minimum?

Question 4.3. Assume that you are considering selecting assets from among the following four candidates.

Asset 1 Market Condition Good Average Poor

Return 16 12 8

Probability ¼ ½ ¼

Asset 2 Market Condition Good Average Poor

Return 4 6 8

Probability 1/4 1/2 1/4

Asset 3 Market Condition Good Average Poor

Return 20 14 8

Probability 1/4 1/2 1/4

Asset 4 Rainfall Good Average Poor

Probability 1/3 1/3 1/3

Return 16 12 8

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Assume that there is no relationship between the amount of rainfall and the condition of the stock market. i. ii. iii.

Solve for the expected return and standard deviation for each separate investment. Solve for the correlation coefficient and the covariance between each pair of investments. Solve for the expected return and variance of each of the portfolios shown below.

Asset 1 1/2 1/2 1/2

A B C D E F G H I

Asset 2 1/2

1/3 1/4 iv.

Asset 4

1/2 1/2 1/2

1/3

Asset 3

1/3 1/3 1/4

½ ½ 1/3 1/3 1/3 ¼

1/2 1/3 1/3 1/4

Plot the original assets and each of the portfolios of part iii in expected return standard deviation space.

Question 4.5. You manage an equity fund with an expected risk premium of 10% and an expected standard deviation of 14%. The rate on the risk-free asset is 6%. Your client chooses to invest R600, 000 of her portfolio in your equity fund and R400, 000 in the risk-free asset. i. ii.

What is the expected return and standard deviation of return on your client’s portfolio? What is the reward to volatility ratio in part i?

Question 4.6. Suppose that you manage a risky portfolio with expected return of 18% and standard deviation of 28%. The return on a riskless bond is 8%. i. ii.

Your client chooses to invest 70% in your fund and 30% in the risk-free asset. What is the expected return and the standard deviation of his portfolio? Suppose that your risky asset includes the following investments in the given proportions. Stock A 25% Stock B 32% Stock C 43%

What are the investment proportions of your clients overall portfolio including the position in the risk-free asset?

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iii.

iv.

Suppose that your client decides to invest in your portfolio a proportion y of the total investment budget so that the overall portfolio has an expected rate of return of 16%. a. What is the proportion y? b. What are the client’s investment in the three stocks and the risk-free asset? c. What is the standard deviation of the return on your client’s portfolio? Suppose that your client prefers to invest in your fund a proportion y that maximises the expected return subject to the constraint that the complete portfolio’s standard deviation does not exceed 18% a. What is the investment proportion y? b. What is the expected rate of return on the complete portfolio?

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