Problem Set

1. Stocks offer an expected rate of return of 18%, with a standard deviation of 22%. Gold offers an expected return of 10% with a standard deviation of 30%.

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. Explain.

Answer:

Even though it seems that gold is dominated by stocks, gold might still be an attractive asset to hold as a part of a portfolio. If the correlation between gold and stocks is sufficiently low, gold will be held as a component in a portfolio, specifically, the optimal tangency portfolio.

Efficient frontier
Efficient frontier

b. Given the data above, re-answer part (a) with the additional assumption that the correlation coefficient between gold and stocks equals 1.0. Draw a graph illustrating why one would or would not hold gold in one’s portfolio. Could this set of assumptions for expected returns, standard deviations, and correlation coefficients represent an equilibrium for the financial markets? Explain.

Answer:

If the correlation between gold and stocks equals +1, then no one would hold gold. The optimal portfolio would be comprised of bills and stocks 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), these combinations would be dominated by the stock portfolio. Of course, this situation could not persist. If no one desired gold, its price would fall and its expected rate of return would increase until it became sufficiently attractive to include in a portfolio.

2.

Assume that the following two investment classes are available in the market:

Asset | E(ri) | σi | Debt | 0.18 | 0.20 | Equity | 0.24 | 0.35 |
Determine the portfolio standard deviation and expected return if you invest the following weights in the two investment classes and when ρD,E = +1.00, or ρD,E = +0.50, or ρD,E = 0.00, or ρD,E = −0.50, or ρD,E = −1.00:

Case | wD | wE | A | 0.00 | 1.00 | B | 0.10 | 0.90 | C | 0.20 | 0.80 | D | 0.30 | 0.70 | E | 0.40 | 0.60 | F | 0.50 | 0.50 | G | 0.60 | 0.40 | H | 0.70 | 0.30 | I | 0.80 | 0.20 | J | 0.90 | 0.10 | K | 1.00 | 0.00 |

Tabulate your results and graph the (expected return, standard deviation) pairs with expected returns on the vertical axis and standard deviations on the horizontal axis. In the graph, also show the location of the individual assets. Comment on the graph.

Note: You may use MS Excel for producing the graphs in this question.

Answer: | E(ri) | σ | | | Debt | 0.18 | 0.20 | | | Equity | 0.24 | 0.35 | | | | | | | | Corr(D.E) = +1.00 | Portfolio | wD | wE | E(rP) | σP | A | 0.00 | 1.00 | 0.240 | 0.35 | B | 0.10 | 0.90 | 0.234 | 0.34 | C | 0.20 | 0.80 | 0.228 | 0.32 | D | 0.30 | 0.70 | 0.222 | 0.31 | E | 0.40 | 0.60 | 0.216 | 0.29 | F | 0.50 | 0.50 | 0.210 | 0.28 | G | 0.60 | 0.40 | 0.204 | 0.26 | H | 0.70 | 0.30 | 0.198 | 0.25 | I | 0.80 | 0.20 | 0.192 | 0.23 | J | 0.90 | 0.10 | 0.186 | 0.22 | K | 1.00 | 0.00 | 0.180 | 0.20 | | | | | | Corr(D.E) = +0.50 | Portfolio | wD | wE | E(rP) | σP | A | 0.00 | 1.00 | 0.240 | 0.35 | B | 0.10 | 0.90 | 0.234 | 0.33 | C | 0.20 | 0.80 | 0.228 | 0.30 | D | 0.30 | 0.70 | 0.222 | 0.28 | E | 0.40 | 0.60 | 0.216 | 0.26 | F | 0.50 | 0.50 | 0.210 | 0.24 | G | 0.60 | 0.40 | 0.204 | 0.23 | H | 0.70 | 0.30 | 0.198 | 0.21 | I | 0.80 | 0.20 | 0.192 | 0.20 | J | 0.90 | 0.10 | 0.186 | 0.20 | K | 1.00 | 0.00 | 0.180 | 0.20 | | | | | | Corr(D.E) = 0.00 | Portfolio | wD | wE | E(rP) | σP | A | 0.00 | 1.00 | 0.240 | 0.35 | B | 0.10 | 0.90 | 0.234 | 0.32 | C | 0.20 | 0.80 | 0.228 | 0.28 | D | 0.30 | 0.70 | 0.222 | 0.25 | E | 0.40 | 0.60 | 0.216 | 0.22 | F | 0.50 | 0.50 | 0.210 | 0.20 | G | 0.60 | 0.40 | 0.204 | 0.18 | H | 0.70 | 0.30 | 0.198 | 0.18 | I | 0.80 | 0.20 | 0.192 | 0.17 | J | 0.90 | 0.10 | 0.186 | 0.18 | K | 1.00 | 0.00 | 0.180 | 0.20 | | | | | | | | | | | | | | | | | | | | | | | | | | Corr(D.E) = -0.50 | Portfolio | wD | wE | E(rP) | σP | A | 0.00 | 1.00 | 0.240 | 0.35 | B | 0.10 | 0.90 | 0.234 | 0.31 | C | 0.20 | 0.80 | 0.228 | 0.26 | D | 0.30 | 0.70 | 0.222 | 0.22 | E | 0.40 | 0.60 | 0.216 | 0.18 | F | 0.50 | 0.50 | 0.210 | 0.15 | G | 0.60 | 0.40 | 0.204 | 0.13 | H | 0.70 | 0.30 | 0.198 | 0.13 | I | 0.80 | 0.20 | 0.192 | 0.14 | J | 0.90 | 0.10 | 0.186 | 0.17 | K | 1.00 | 0.00 | 0.180 | 0.20 | | | | | | Corr(D.E) = -1.00 | Portfolio | wD | wE | E(rP) | σP | A | 0.00 | 1.00 | 0.240 | 0.35 | B | 0.10 | 0.90 | 0.234 | 0.30 | C | 0.20 | 0.80 | 0.228 | 0.24 | D | 0.30 | 0.70 | 0.222 | 0.19 | E | 0.40 | 0.60 | 0.216 | 0.13 | F | 0.50 | 0.50 | 0.210 | 0.08 | G | 0.60 | 0.40 | 0.204 | 0.02 | H | 0.70 | 0.30 | 0.198 | 0.04 | I | 0.80 | 0.20 | 0.192 | 0.09 | J | 0.90 | 0.10 | 0.186 | 0.15 | K | 1.00 | 0.00 | 0.180 | 0.20 |

3. The correlation coefficients between pairs of stocks are as follows:

Corr(A,B) = 0.85
Corr(A,C) = 0.60
Corr(A,D) = 0.45

Each stock has an expected return of 8% and a standard deviation of 20%.

a. If your entire portfolio is comprised of stock A and you can add some of only one other stock to your portfolio, which stock, B, C, or D, would you choose? Explain your choice.

Answer:

The correct choice is Stock D. Intuitively, we note that since all stocks have the same expected rate of return and standard deviation, we choose the stock that will result in lowest risk. This is the stock that has the lowest correlation with Stock A.
b. Would your answer to part (b) change for more risk-averse or more risk-tolerant investors? Explain.

Answer:

No, the answer to part (b) would not change, at least as long as investors are not risk lovers. Risk neutral investors would not care which portfolio they held since all portfolios have an expected return of 8%.

c. Suppose that in addition to investing in one more stock, you can invest in T-bills as well. Would you change your answer to parts (a) and (b) if the T-bill rate is 8%? Explain.

Answer:

No, the answers to parts (a) and (b) would not change. The efficient frontier of risky assets is horizontal at 8% since each stock has the same expected return but different standard deviations. Therefore, the new efficient frontier runs from the risk-free rate through this linear frontier. The best portfolio is again, the one with the lowest variance. The optimal complete portfolio (the one that is formed by investing in both the risk-free asset and the best risky portfolio) depends on the individual investor’s risk aversion.

4. Suppose that the market can be described by the following three sources of systematic risk with associated risk premiums:

Risk Factor | Risk Premium (%) | Risk Factor Beta | Industrial production (I) | 1.24 | 1.2 | Interest rates (R) | 0.91 | 0.5 | Consumer confidence (C) | 1.03 | 0.3 |

Calculate the equilibrium rate of return on this stock using the APT. The current market estimate for the stock’s expected return is 10% and the T-bill rate is 6%. Is the stock under- or over-priced? Explain.

Answer:

APT required return = r = 6 + (1.24 x 1.2) + (0.91 x 0.5) + (1.03 x 0.3) = 8.252 %

Since the stock’s expected return, 10%, is higher than its required return, 8.252%, this stock is currently underpriced. The demand for the stock will increase, pushing its price upward and its expected return downward, until the expected return is just equal to the required return.

5. After graduation, you start working for an investment bank and your bank’s current portfolio of TL2 million is invested as follows:

| Value | Percent of Total | Expected Annual Return | Annual Standard Deviation | Short-term bonds | TL200,000 | 10 | 4.6% | 1.6% | Domestic large-cap equities | 600,000 | 30 | 12.4% | 19.5% | Domestic small-cap equities | 1,200,000 | 60 | 16.0% | 29.9% | Total Portfolio | TL2,000,000 | 100 | 13.8% | 23.1% |

Your bank soon expects to receive an additional TL2 million and plans to invest the entire amount in an index fund that best complements the current portfolio. You are evaluating the four index funds shown in the following table for their ability to produce a portfolio that will meet two criteria relative to the current portfolio: (1) maintain or enhance expected return, and, (2) maintain or reduce volatility.

Each fund is invested in an asset class that is not substantially represented in the current portfolio.

Index Fund Characteristics | Index Fund | Expected Annual Return(%) | Expected Annual Standard Deviation(%) | Correlation of Returns with Current Portfolio | Fund A | 15 | 25 | +0.80 | Fund B | 11 | 22 | +0.60 | Fund C | 16 | 25 | +0.90 | Fund D | 14 | 22 | +0.65 |

State which fund you should recommend to your bank. Justify your choice by describing how your chosen fund best meets both of your bank’s criteria. (No calculations required)

Answer:

Fund D represents the single best addition to complement your bank's current portfolio, given the selection criteria. First, Fund D’s expected return (14.0 percent) has the potential to increase the portfolio’s return somewhat. Second, Fund D’s relatively low correlation with the current portfolio (+0.65) indicates that Fund D will provide greater diversification benefits than any of the other alternatives except Fund B. The result of adding Fund D should be a portfolio with approximately the same expected return and somewhat lower volatility compared to the original portfolio.
The other three funds have shortcomings in terms of either expected return enhancement or volatility reduction through diversification benefits. Fund A offers the potential for increasing the portfolio’s return, but is too highly correlated to provide substantial volatility reduction benefits through diversification. Fund B provides substantial volatility reduction through diversification benefits, but is expected to generate a return well below the current portfolio’s return. Fund C has the greatest potential to increase the portfolio’s return, but is too highly correlated with the current portfolio to provide substantial volatility reduction benefits through diversification.

6. Consider the following two situations:

(1) Portfolios A and B both have expected returns of 15%, but Portfolio A has a beta of 1.2 while Portfolio B has a beta of 1.0.

(2) Portfolio C has an expected return of 20% with a standard deviation of 30% while Portfolio B has an expected return of 15% with a standard deviation of 35%.

In a world in which CAPM is valid, could either, both, or neither of these situations occur when the market is in equilibrium? Explain your answer.

Answer:

(1) In a world where CAPM is correct, stocks should provide returns in association with their risks (beta coefficients). If both stocks A and B have an expected return of 15% but A has a higher risk because of its larger beta, noone would ever buy Stock A. Would you rather buy Stock B, have a return of 15% and a risk of 1.0 or would you buy Stock A, still have a return of 15% but a higher risk of 1.2? Therefore, the demand for Stock A will decrease (noone wants to buy it) and the supply of Stock A will increase (people who already own it will want to get rid of it and sell). The decreased demand and increased supply of Stock A will cause its price to decrease. With the price decrease, its expected return will start increasing and this will continue until the expected return on A is high enough to compensate for its higher risk. So, in a CAPM world, Stocks A and B cannot exist together, the above changes will have to happen. Stock A is said to be an "inefficient" stock compared to Stock B because it does not provide either "the highest return for a given level of risk" or "the lowest risk for a given level of return".

(2) This cannot happen in a CAPM world, either. These two portfolios cannot exist at the same time because noone would buy Portfolio B since it has a lower return and a higher risk than Portfolio A. Everyone would prefer to buy Portfolio A. Therefore, the demand for Portfolio B will decrease (noone wants to buy it) and the supply of Portfolio B will increase (people who already own it will want to get rid of it and sell). The decreased demand and increased supply of Portfolio B will decrease its price and increase its return until the return is large enough to compensate for its larger risk. Portfolio B is said to be an "inefficient" portfolio compared to Portfolio A because it does not provide either "the highest return for a given level of risk" or "the lowest risk for a given level of return".