Question

An investor is faced with two risky asset portfolios (each of which is highly diversified within its asset class) – an equity fund and a bond fund.

The investor is aware that asset returns are not always normally distributed, but is nonetheless prepared to use the normal distribution as a tool for the estimation of approximate portfolio risks and expected returns.

The equity fund has a forecast expected return of +11% pa over the time horizon of 12 months, and the investor is more than 99.7% sure that the range of outcomes will lie between a worst case scenario of about -34%pa and a best case scenario of approximately+56%pa.

The bond fund has a forecast expected return over the time horizon of +4% pa, and an interval between worst and best case scenarios of -8%pa to +16%pa.

The covariance between the equity fund and the bond fund is believed to be approximately +6.

There is a perfectly competitive banking system, where interest rate margins for customers such as this investor have been driven down to zero. The current interest rate on 12 month deposits is 1%pa.

1. Calculate the forecast expected return and the estimated risk of an equally weighted portfolio, consisting entirely of the equity fund and the bond fund. Demonstrate and explain any risk reducing benefit from diversification between the two asset classes. (30%).
2. Calculate the expected return and the risk of a combined portfolio, which is equally weighted between bank deposits and the risky asset portfolio identified in part 1. (10%).
3. Suppose the investor wishes to maintain the balanced risky asset portfolio of shares and bonds, but has a target rate of return of 10.75% pa on her combined portfolio. How could the investor raise the expected return to 10.75%pa? What weights would she have for the equity fund, the bond fund and the risk-free asset in her combined portfolio? What level of risk would be involved? (15%).
4. What would be the risk and expected return of a portfolio consisting entirely of the equity and bond fund, but with weights chosen to minimise the portfolio variance? (15%).
5. Calculate the risk and expected return of portfolios with (i) weights of 25% for the equity fund and 75% for the bond fund and (ii) 75% for the equity fund and 25% for the bond fund. (10%).
6. Graph a minimum variance frontier, based on your answers to parts 1, 4 and 5 above. Provide a clear and concise explanation of what the frontier shows. (10%)
7. Add an approximate capital allocation line (CAL) to your diagram. Provide a clear and concise explanation of what the CAL shows. (10%).

Answer 1

Expected Return of Equity fund= 11%

Expected Return of Bond = 4%Weight of both equity and bond fund = 0.5Expected Return of Portfolio = (Weight of security 1 * Expected Return of Security 1) + (Weight
of Security 2 * Expected Return of Security 2)

= (0.5*11)+(0.5*4)

= 7.5%

For 99.7 % surety , in Z table the value is 0.4985

= = 2.97

= (56-11)/=2.97

There fore standard deviation = 15.15%

If we assume surety in bonds also to be 99.7%,

Standard deviation of bond would be 4.04%

Covariance = 6

Covariance = (Standard deviation of security 1) * (Standard deviation of security 2) * Correlation

6 = 15.15 * 4.04 * correlation

Correlation = 0.098

Variance of portfolio =

= (0.5*15.15)² + (0.5*4.04)²+2*0.5*15.15*0.5*4.04*0.098

= 64.46

standard deviation of portfolio = 8.03%

As we can see that with the increase in expected return the risk assumed is also higher as in equity portfolio . So if an investor needs to reduce risk then investment should be diverted into bond funds from equity funds.

Part 2:

Expected Return of Portfolio with equal weight in bank deposits and risky asset portfolio

= (0.5*7.5) + (0.5*1)

=4.25%

Risk of combined protfolio would be same as risk of risky protfolio i.e 8.03% because the standard deviation of bank deposits is 0.

Part 3:

For 10.75 % of expected return the weight of risky asset portfolio should be 1.5% and loan should be taken from risk free assets and hence the weight would be -0.5%.

The weight of equity and bond fund to be equal in risky asset portfolio.

Part 4:

If variance of portfolio is to be minimised then minimum weight to be allocated to equity funds and maximum weights to be allocated to bond funds.

Part 5:

If equity fund = 25% and bond fund = 75 % then Expected Return = 5.75%

and standard deviation (Risk) = 5.08%

If equity fund = 75% and bond fund = 25 % then expected Return = 9.25%

and standard deviation (Risk ) = 11.5%