Devoir d'anglais

Portfolio Theory

REVIEW QUESTIONS

Lectures 1-8

A: Measuring Risk and Return

Q1

Explain how to calculate the geometric mean annual rate of return, given a set of annual returns. Use a simple example to show how the geometric mean is superior to the arithmetic mean as a measure of the return.

Q2

Give reasons supporting the view that the standard deviation of the return is considered a good measure of risk.

Q3

Explain how the future standard deviation of a security’s return can be estimated with a subjective probability distribution.

Q4

Explain why the standard deviation of the return may not be a good measure of risk for all investors.

Q5

What are the dangers associated with using historical return variability to make estimates of future risk?

Q6

Interpret a correlation coefficient of +1 and -1. What is the shape of a 2-asset efficient frontier if we assume correlation coefficient of +1 and -1? Comment on the plausibility of these correlation coefficients in the real world equity markets.

B:Properties of Portfolios - Diversification

Q1

Explain how it is possible for the risk of a portfolio, as measured by its standard deviation, to be less than the standard deviation of any of the assets in the portfolio.

Q2

Which statement about portfolio diversification is correct?

a. Proper diversification can reduce or eliminate systematic risk.

b. Diversification reduces the portfolio’s expected return because it reduces a portfolio’s total risk.

c. As more securities are added to a portfolio, total risk typically would be expected to fall at a decreasing rate.

d. The risk –reducing benefits of diversification do not occur meaningfully until at least 30 individual securities are included in the portfolio.

Q3

Consider N assets with with the same expected return and same standard deviation .

a. Show that if the asset returns are uncorelated then the mean and variance of an equally weighted portfolio ( ) are

and

b. For the case compare the variance of portfolios of one, five and ten assets.

c. Now assume that all the assets are positively correlated with covariance . Show that the variance of an equally weighted portfolio ( ) is and it cannot become lower than .

Solution

Q4

The expected return and standard deviation of return on two stocks are given below, together with the correlation coefficient.

Expected Return (%) Standard Deviation (%)

Stock 1 15 36

Stock 2 12 15

Correlation = -1

Suppose that it is possible to borrow at the risk free rate . What must be the value of the risk-free rate?

Q5

Listed here are estimates of the standard deviations and correlation coefficients for three stocks

Stock Standard

Deviation Correlation with stock:

A B C

A 12% 1 -1 0.20

B 15% -1 1 0.6

C 10% 0.2 0.60 1

a. If a portfolio is composed of 30% of stock A and 70% of stock C, what is the portfolio's standard deviation?

b. If the portfolio is composed of 30% of stock A, 30% of stock B, and 40% of stock C, what is the portfolio's standard deviation?

c. If you were asked to design a portfolio using stocks A and B, what percentage investment in each stock would produce a zero standard deviation?

Q6

Both the covariance and the correlation coefficient measure the extent to which the returns on securities move together. What is the relationship between the two statistical measures? Why is the correlation coefficient a more convenient measure?

Q7

When is the standard deviation of a portfolio equal to the weighted average of the standard deviation of the component securities? Show this mathematically for a two-security portfolio.

Q8

When, if ever, would a stock with a large standard deviation be included in a portfolio?

Q9

Why would you expect most company shares to exhibit positive covariances? Give an example of two stocks that you would expect to have a very high positive covariance. Give an example of two stocks that you would expect to have a very low positive (or even negative) covariance.

Q10

The risk of an equity portfolio can be

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