Use the rate-of-return data for the stock and bond funds presented in Spreadsheet 6.1, but now assume the probability of each scenario is as follows: severe recession: 0.10; mild recession: 0.20; normal growth:0.35; boom: 0.35.
(a) Would you expect the mean return and variance of the stock fund to be more than, less than, or equal to the values computed in Spreadsheet 6.2? Why?
The variance is expected to increase because the probabilities of the extreme outcomes are now higher.
The mean return would be probably similar to the original one.
(b) Calculate the new values of mean return and variance for the stock fund using a format similar to Spreadsheet 6.2. Confirm ...view middle of the document...
8 | Correlation Coefficient = Covariance / (StdDev(Stocks)*StdDev(Bonds)) = | -0.23356 |
Covariance has increased because the stock returns are more extreme in the recession and boom periods. This makes the tendency for stock returns to be poor when bond returns are good (and vice versa) even more dramatic.
Stocks offer an expected rate of 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 with respect to both mean return and volatility, would anyone hold gold? If so, demonstrate graphically why one would do so.
Although stocks seem to dominate gold in both expected return and standard deviation, it still might be an attractive asset for investor to hold as a part of a portfolio for the purpose of diversification. If the correlation between gold and stocks is sufficiently low, it will be held as an element in a portfolio --the optimal tangency portfolio, i.e. point P.
(b) How would you answer (a) if the correlation coefficient between gold and stocks were 1? Draw a graph illustrating why one would or would not hold gold. Could these expected returns, standard deviations, and correlation represent an equilibrium for the security market?
If gold had a correlation coefficient of +1 with stocks, it would not be held. The optimal CAL would be comprised of stocks and risk-free asset only. Since the set of risk / return combinations of stocks and gold would plot as a straight line with a negative slope (see the following graph), it would be dominated by the stocks portfolio. But this situation could not persist. If no one desired gold, its price would fall and its expected rate of return would Increase until equilibrium is reached.
Suppose that many stocks are traded in the market and that it is possible to borrow at the risk-free rate, rf. The characteristics of two of the stocks are as follows:
Stock | Expected Return | Standard Deviation |
A | 10% | 25% |
B | 18% | 75% |
Correlation = -1 | | |
Could the equilibrium rf be greater than 12%? (Hint: Can a particular stock portfolio be substituted for the risk-free asset?)
Since Stock A and Stock B are perfectly negatively correlated, a risk-free portfolio can be created and the rate of return for this portfolio in equilibrium will always be the risk-free rate. To find the proportions of this portfolio [with wA invested in Stock A and wB = (1 –wA ) invested in Stock B], set the standard deviation equal to zero. With perfect negative correlation, the portfolio standard deviation reduces to:
P = lwAA – wBBl 0 = 0.25wA + 0.75(1 – wA) wA = 0.75, wB = 0.25
The expected rate of return on this risk-free portfolio is:
E(r) = (0.75 0.1) + (0.25 0.18) = 12.0%
Therefore, the risk-free rate must also be 12.0%, if it is greater than 12%, then an arbitrage opportunity exists.
The following figure shows plots of...