Financial Markets and Investments
10th January 2011
Closed-book exam. During the exam you can only check the official formulas’ sheet.
Answer directly on the exam sheet. Duration: 2.5h
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Name: Number:
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GROUP I ( 45 points)
There is only one correct answer per question. Mark your choices with circles. Each correct answer is
worth 5 points and for each incorrect answer there is a deduction of 1.5 points.
1. Diversification among assets improves the opportunities faced by all risk-averse investors,
a) Only if correlations are not larger than 0;
b) Only if the assets have similar variances;
c) Irrespective of the correlation coefficients (provided not all assets are perfectly and positively
correlated);
d) None of the above.
2. In a world where there is a risk free asset and where you can both apply or borrow money at
the same rate and without limits, which of the following statements is true?
a) The only efficient combination of risky assets does not involve shortselling;
b) No investor will choose to invest only on the risk free asset;
c) All the efficient portfolios will have the same Sharpe Ratio;
d) None is true.
3. If, without shortselling restrictions, the tangent porfolio (many assets) involves shortselling of
asset i and a long position in asset j; then, if shortselling would be forbidden, in the new tangent
portfolio:
a) There would still be for sure a long position on asset j, but no investment in asset i;
b) It is not certain that it would still be efficient to buy asset j, but surely there would be no
investment in asset i;
c) It could be that buying both assets, i and j, would become efficient;
d) None of the above.
4. As the number of assets increase, the variance of homogeneous portfolios converges to
a) Zero;
b) The average of all individual asset variances in the market;
c) The average of all return covariances in the market;
d) None of the above
5. Suppose and investor with utility function given by U (w) = ae−bW , with a and b constants.
What should be the sign of a and b so that the investor prefers more to less and is risk averse?
a) a > 0; b > 0
b) a > 0; b < 0
c) a < 0; b > 0
d) a < 0; b < 0
6. Whenever and investment A dominates another investment B, in terms of first order stochastic
dominance,
a) Then A also dominates B in terms of second and third order stochastic dominance;
b) A is the best investment also in terms of the “Safety First” criteria;
c) Despite this, risk lover investors amy prefer B to A.
d) None of the above.
7. Suppose there is a risk free asset with return equal to Rf and consider as valid all CAPM
assumptions. In this context, the beta on any efficient portfolio:
a) Must be 1, because all efficient portfolios are perfectly correlated with the market portfolio;
b) It is proportional to 1, depending on how much it is invested in the market portfolio;
c) It is equivalent to the portfolio’s volatility, since risk of efficient portfolios is measured by
volatility;
d) It must always be less than 1.
8. When portfolios are actively managed
a) It is common to use strategies of “market timing” and “market selection”;
b) Managers try to replicate a market index;
c) The resulting portfolios will always have negligible systematic risk (as they are very well
diversified);
d) None of the above.
9. There is theoretical ground to invest according to technical analysis strategies,
a) If one believes markets are weak efficient;
b) If one believes markets are semi-strong efficient;
c) If one believes markets are strong efficient;
d) None of the above.
GROUP II ( 30 points)
Answer briefly to the two questions of this group (without exceeded the available space). Each correct
answer is worth 15 points.
1. Discuss the differences between absolute and relative risk aversion.
Answer:
2. Choose ONE of the following statements and discuss whether it is true or false:
I. To an investor who does not verify the Von-Neuman-Morgensten axioms, one should rec-
ommend safe portfolios according to criteria such as Roy, Kataoka or Telser.
II. If we assume there is a risk-free asset and consider all risky financial assets worldwide,
then there is only one combination of risky assets that is efficient, irrespective of investors’
nationality.
Answer:
GROUP III
This group has 3 numerical problems.
Problem 1 (40 points)
For this problem it is not necessary to present computations, values are enough. Each correct answer
is worth 2 points.
Consider two independent investment funds.
One of the funds performance depends on the stock market - Market Fund (MF). The other fund
performance depends on the results of the Portuguese team on the Soccer World Cup - Soccer Fund
(SF). The table bellow presents estimated returns and probabilities for different scenarios:
Market Fund (MF) Soccer Fund (SF)
Probability Return Probability Return
0.25 -10% 1/3 -20%
0.5 8% 1/3 5%
0.25 20% 1/3 42%
Based upon the above information fill the blank spaces “. . . . . . ” with the appropriate values (present
values rounded to two decimal digits) or with abbreviations “MF” or “SF” to designate one of the
funds.
• The expected return of the MF fund is . . . . . . % and its volatility is . . . . . . %
• The expected return of the SF fund is . . . . . . % and its volatility is . . . . . . %.
• The correlation between the returns of the two funds is . . . . . .
• Consider combinations of the two funds.
– The homogeneous portfolio has expected return equal to . . . . . . % and . . . . . . % volatility.
– The minimum variance portfolio has . . . . . . % volatility, . . . . . . % expected return and the
proportion invested in the fund MF is . . . . . . %.
– Assuming shortselling is not allowed, the maximum risk portfolio requires investing . . . . . . %
in SF.
• Consider now the two fund as alternative investments.
– For an RL = 0% , the best fund according to the Roy criteria is . . . . . .
– For an α = 0.35, the best fund according to the Kataoka criteria is . . . . . .
– For RL = 0% and α = 0.35, the best fund according to the Telser criteria is . . . . . .
1
– An investor with utility function U (W ) = 10W − 10 W 2 chooses the fund . . . . . . , because
the expected utility of investing in MF is . . . . . . while it is . . . . . . for investing in SF.
• Finally, suppose there is a risk free asset in this market with Rf = 5% and that you are only
interested in combining the risk free asset with fund MF.
– To achieve the minimum risk combination one much deposit . . . . . . % of the investment.
– Investor with an optimal risk level of 12% put . . . . . . % of their investment in MF and expect
. . . . . . % of return.
Problem 2 (20 points)
Supose that the CAPM is valid in a given market and that the equilibrium security market line is
given by
R̄ei = 0.05 + 0.09βi.
The firm XYZ has just paid a dividend of 1 euro per stock. The next year dividend is expected to
be of 1.2 euros per stock. From then on analyist think the dividend growth rate will be equal to the
average dividend market growth which is 6%.
Besides the above information we know the beta of the XYZ firm’s stocks is 1.11.
a) Find out the equilibrium expected return of the XYZ stocks. . . . . . . . . . . . . . . . . . . . . . . . . . . . [0.25p]
b) What is the fair value for the stocks of the XYZ firm? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [1p]
c) Assuming the market price of the stocks under analysis is 17 euros, what would you recommend?
Explain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [0.25p]
d) Enumerate the assumptions underlying the computations above. . . . . . . . . . . . . . . . . . . . . . . . .[0.5p]
Solution:
Problem 3 (65 points)
In the context of the Single Index Model, the table bellow summarises information concerning to
individual assets and teh market index:
Assets α(%) β σ Index
A 2% 1.75 14% R̄M 11%
B 3% 0.75 6% σM 18.35%
We also know that, in this market, the returns of all risky assets are normally distributed and that
there exists a risk free asset with Rf = 3.75%.
1. Find the expected return and volatility of each individual asset and the covariance between the
two assets’ returns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .[10p]
2. Assume there is no restrictions to shortselling and that the Rf rate is the same both for deposit
and borrowing.
a) What is the efficient way of combining A and B (portfolio T ). . . . . . . . . . . . . . . . . . . . [12.5p]
b) What is the risk and expected return of that portfolio T ? . . . . . . . . . . . . . . . . . . . . . . . . . .[7.5p]
c) Write down the equation of the efficient frontier in this market. . . . . . . . . . . . . . . . . . . . .[2.5p]
d) Would you change your answer if shortselling would be forbidden? Explain . . . . . . . .[2.5p]
3. A given investor of this market has a risk profile compatible with indifference curves of the
following type R̄p = exp {0.488 σp } + K for any K ∈ R.
[If you did not answer the previous question consider the following R̄p = 0.0375+0.5208σp for the efficient
frontier.]
a) What is the optimal risk level for this investor? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [10p]
b) What is the expected return and composition of the optimal portfolio? . . . . . . . . . . . . [7.5p]
c) Would the optimal portfolio change if the rate for borrowing money would differ from the
deposit rate and be equal to 7%? Explain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [2.5p]
4. The wife of the above mentioned investor dislikes investments whose probability of negative
returns is higher than 25%. Does the optimum portfolio of her husband verify this safety
requirement? Explain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [10p]
[If you did not find the optimal portfolio in the previous question, assume R̄∗ = 10.69% and σ∗ = 13.32%]
[For z ∼ N (0, 1) we have Pr(z ≤ −0.67449) = 0.25]
Solution:
Extra space for answers (if needed):
