Topic 2+3 - Portfolio Theory

Question 1

In forming a portfolio of two risky assets, what must be true of the correlation coefficient between their returns if there are to be gains from diversification? Explain.

Answer 1

So long as the correlation coefficient is less than 1.0, the portfolio will contain diversification benefits. These combinations will result in a portfolio with a lower standard deviation than the sum of the asset standard deviations.

Question 2

When adding a risky asset to a portfolio of many risky assets, which property of the asset is more important, its standard deviation or its covariance with the other assets? Explain.

Answer 2

The covariance with the other assets is more important. Diversification is accomplished via correlation with other assets. Covariance helps determine that number.

Question 3

Shares offer an expected return of 10% with a standard deviation of 20% and gold offers an expected return of 5% with a standard deviation of 25%.

(a) In light of the apparent inferiority of gold to shares with respect to both mean return and volatility, would anyone hold gold? If so, demonstrate graphically why one would do so.
(b) How would you answer (a) if the correlation coefficient between gold and shares were 1.0? Draw a graph illustrating why one would or would not hold gold. Could these expected returns, standard deviations and correlations represent an equilibrium for the security market?

Answer 3

a. Although it appears that gold is dominated by shares, gold can still be an attractive diversification asset. If the correlation between gold and shares is sufficiently low, gold will be held as a component in the optimal portfolio.
b. If gold had a perfectly positive correlation with shares, gold would not be a part of efficient portfolios. The set of risk/return combinations of shares and gold would plot as a straight line with a negative slope. (See the following graph.) The graph shows that the share-only portfolio dominates any portfolio containing gold. This cannot be an equilibrium; the price of gold must fall and its expected return must rise.

Question 4

The problem with covariance:

Answer 4

• Covariance does not tell us the intensity of the comovement of the share returns, only the direction.
• We can standardise the covariance however and calculate the correlation coefficient which will tell us not only the direction but provides a scale to estimate the degree to which the shares move together.

Question 5

Systematic risk

Answer 5

Systematic risk arises from events that affect the entire economy, such as a change in interest rates or GDP or a financial crisis such as occurred in 2007 and 2008

Question 6

Unsystematic or firm specific risk:

Answer 6

risk not related to the macro factor or market index

Question 7

Sharpe ratios

Answer 7

• When ranking portfolios and security performance we must consider both return and risk • also called as “reward-to-variability ratio”
• ‘Well performing’ diversified portfolios provide high Sharpe ratios: Sharpe = (rp – rf) / σp
• You can also use the Sharpe ratio to evaluate an individual stock if the investor does not diversify.