MSc. Finance/CLEFIN 2016/2017 Edition

20135 Theory of Finance – Part I September 2017 Exam – Time Available 40 minutes

INSTRUCTIONS: Please answer according to the contents and goals of the first part of the course only. Give your answers in good faith (see the pledge) and note that “None of the above is likely or appropriate.”, refers to the claims (A) – (D), not to the text of the question (i.e., this not the space to show that you do not like a question). Only one of the answers/ options is correct. Write only in the spaces provided ON THE FIRST PAGE and in question 9 at the end of the test. No other work will be marked and referring to any other answers or material in any format, represents a violation of the honor code. NAME: _________________________________________________ ID NUMBER: ________________________

Question Answer

1 D

2 A

3 D

4 E

5 C

6 B

7 A

8 C

9 D

NOTE: Question 9 requires you to provide one open answer worth 0.5 points.

Question 1 (0.75 points) Consider the mean-variance (of portfolio returns) diagram below. In the plot, A’, A’’ and A’’’ represent three distinct portfolios and Rf is the risk-free rate.

Which of the following statements is LEAST LIKELY: (A) On the capital market line characterized by ray Rf-A’’, the two-fund separation theorem holds. (B) All the rays coming out of Rf and going through A’, A’’, and A’’’ are instances of capital transformation lines; however, only the ray Rf-A’’ represents the unique capital market line, that maximizes the Sharpe ratio achievable by the investor. (C) Given the diagram, portfolios A’ and A’’’ must be characterized by a correlation that is strictly less than one. (D) The capital market line corresponding to the ray Rf-A’’ is characterized by one specific portfolio with the highest Sharpe ratio, the tangency portfolio, that therefore dominates all other portfolios. (E) All of the above are likely and appropriate.

Answer D

Debriefing.

Question 2 (1 point) After a long search, Mary has recently found a new job at the Bitterchocolate Factory, that will pay her a high and relatively stable (i.e., scarcely variable over time) annual income for her managerial services. You know that Mary had invested her wealth in just one equity ETF and riskless cash (through a bank account) before she had found her new job. Consider the problem of predicting whether, after finding her new job, Mary will change the percentage weight assigned to stocks and cash in her portfolio. Which of the following statements is MOST LIKELY: (A) Without additional information on the correlation between Mary’s labor income and the returns on the equity ETF returns, nothing precise can be said. (B) A few precise predictions might be computed only if we knew that the sign of the correlation between Mary’s future labor income at the Bitterchocolate Factory and the returns on the equity ETF is positive. (C) Without additional information exclusively concerning the variances of the returns on the equity ETF and the mean rate of labor income, nothing precise can be said. (D) A few precise predictions might be computed only if we knew that the sign of the covariance between Mary’s future labor income at the Bitterchocolate Factory and the returns on the equity ETF is negative. (E) None of the above is likely/appropriate.

Answer A

Debriefing.

Question 3 (1 point) With reference to the expected utility theorem, which of the following statements is MOST LIKELY: (A) The theorem states that alternative lotteries (gambles) may be ranked either by using the primitive preferences of individuals, or by sorting the expected value of the cardinal, Von Neumann-Morgenster felicity that they yield, and that two rankings will be identical when a specific set of conditions (axioms) hold; the expectation is to be computed on the basis of the beliefs of the individual that therefore affect her decision under uncertainty. (B) Among the required axioms of choice under uncertainty, we have the axioms of completeness, continuity, transitivity, risk aversion under ambiguity, independence of more relevant alternatives, and the ability to always compute a certainty equivalent of a lottery. (C) Among the required axioms of choice under uncertainty, we have the axioms of completeness, continuity, transitivity, consistency in lottery reduction, independence of irrelevant alternatives, and the ability to always compute a certainty equivalent of a lottery. (D) A and C are correct. (E) B and C are correct. (F) All of the above are likely or appropriate.

Answer D

Debriefing.

Question 4 (1 point) In Homework 1, you have obtained the following diagram (it is magnified for emphasis):

The two minimum-variance frontiers differ because of the inclusion of real estate in the largest of the two asset menus. Which of the following statements is MOST LIKELY: (A) The diagram shows that as an asset class, real estate (the fifth, additional asset) expands/improves by little the available investment opportunities, which must be a result of real estate being strongly correlated with other asset classes or a portfolio thereof; regardless of real estate, the minimum-variance frontier almost touches the vertical axis because the asset menu includes the almost riskless 1-month T-bills. (B) The two mean-variance frontiers differ because one has been computed using data on the returns of five risky assets and the other, using data on the returns of four assets. (C) The diagram shows that as an asset class, real estate expands/improves the available investment opportunities only for investors with a low degree of aversion to risk, whose mean-variance indifference curves are likely to be tangent to the mean-variance frontier on the upper, rightmost portion of the frontier. (D) The diagram shows that the global minimum-variance portfolio yields a positive expected return in spite of the fact that the resulting variance is almost nil. (E) All of the above are likely or appropriate.

Answer E

Debriefing. Rather obvious for those who had spent some time before pondering the findings of Homework 1.

0.00%

0.20%

0.40%

0.60%

0.80%

1.00%

0.00% 0.05% 0.10% 0.15% 0.20%

E[r]

Variance

Minimum Variance Frontier

Five Assets Four Assets

Question 5 (1 point) You know that for small variations and risks, John is characterized by the following property: if John’s wealth increases by a small percentage amount, then John will decrease by some percentage amount his overall investment in the available risky assets in his choice menu; if John’s wealth decreases by a small percentage amount, then John will increase by some percentage amount his investment in all available risky assets in his choice menu Which of the following statements is MOST LIKELY: (A) John is characterized by increasing relative risk aversion; therefore, at least locally (for small risks and variations) his coefficient of absolute risk aversion must be either constant or increasing. (B) John is characterized by decreasing relative risk aversion; however, from this fact, we cannot simply infer that John is characterized by decreasing absolute risk aversion, because decreasing absolute risk aversion is compatible with constant relative risk aversion. (C) John is characterized by increasing relative risk aversion; however, from this fact, we cannot simply infer that John is characterized by constant or increasing absolute risk aversion, because decreasing absolute risk aversion is compatible with increasing relative risk aversion. (D) John is characterized by decreasing relative risk tolerance; therefore, at least locally (for small risks and variations) his coefficient of absolute risk tolerance must be either constant or increasing. (E) None of the above is likely/appropriate.

Answer C

Debriefing.

Also note that under decreasing ARA(W), when wealth increases the investment in risky assets increases; however, there is no guarantee that this will occur at a rate sufficient to ensure that the elasticity of the weight w.r.t. is equal to or exceeds 1; that will depend on the utility function. This is why option A above is incorrect (better not the MOST LIKELY or correct, given the presence of option C).

Question 6 (1.25 points) Mr. Tollgates, a finance researcher, has found a highly paid job with a well-known top 5 university. We know that before securing his new post, Mr. Tollgates was investing 20% in a single risky portfolio that is characterized by a yearly risk premium of 8%; the standard deviation of excess returns is not known. You are then informed that one year after he has started earning his newly found, hefty researcher’s income, the percentage he invests in the risky asset has doubled. Mr. Tollgates maximizes a mean-variance objective with (CARA, risk aversion) coefficient 𝜅𝜅 that is not affected by her new occupational status. Which of the following statements is MOST LIKELY: (A) Mr. Tollgates must be characterized by a CARA coefficient 𝜅𝜅 = 15, while his new job will return wages characterized by a negative covariance with risky returns of -0.2; however, the correlation between his future wages and risky returns cannot be established. (B) Mr. Tollgates must be characterized by a product between his CARA coefficient (𝜅𝜅) and the variance of the risky portfolio of 0.4; this can be used to deduce that his new job will return wages characterized by a negative ratio between covariance with risky returns and their variance of -0.2; however, the correlation between his future wages and risky returns, his coefficient of risk aversion, and the variance of the risky asset returns cannot be computed. (C) We do not have enough information to compute either Mr. Tollgates’ CARA coefficient (𝜅𝜅) or the covariance between wages and risky returns. (D) Mr. Tollgates must be characterized by a CARA coefficient 𝜅𝜅 = 15, while his new job will returns wages characterized by a negative covariance with risky returns of -1; however, the correlation between his future wages and risky returns cannot be computed. (E) None of the above is likely or appropriate.

Answer B

Debriefing.

The answer derives from

𝜔𝜔�𝑡𝑡𝑠𝑠𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡 = 0.2 =𝐸𝐸𝑡𝑡[(𝑟𝑟𝑡𝑡+1 − 𝑟𝑟𝑓𝑓)]

𝜅𝜅𝜎𝜎2 ⟹ 𝜅𝜅𝜎𝜎2 =0.080.2

= 0.4,

so that 𝜔𝜔�𝑡𝑡𝑠𝑠𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑠𝑠𝑠𝑠 = 0.4 = 𝐸𝐸𝑡𝑡��𝑟𝑟𝑡𝑡+1−𝑟𝑟

𝑓𝑓��𝜅𝜅𝜎𝜎2 −

𝐶𝐶𝐶𝐶𝐶𝐶𝑡𝑡[𝑌𝑌𝑡𝑡+1,𝑟𝑟𝑡𝑡+1]𝜎𝜎2

⟹𝐶𝐶𝐶𝐶𝐶𝐶𝑡𝑡[𝑌𝑌𝑡𝑡+1, 𝑟𝑟𝑡𝑡+1]

𝜎𝜎2= 𝐸𝐸𝑡𝑡[(𝑟𝑟𝑡𝑡+1 − 𝑟𝑟𝑓𝑓)]

𝜅𝜅𝜎𝜎2− 0.4 =

0.080.4

− 0.4 = −0.2

However, the separate correlations between the covariance of labor income and risky asset returns, the coefficient of risk aversion, and the variance of excess returns cannot be computed because we have not been given the standard deviation of wage growth and/or the covariance between labor income and risky asset returns.

Question 7 (1.25 point) John has a felicity of monetary wealth function of power type, 𝑈𝑈𝐽𝐽𝑒𝑒ℎ𝑠𝑠(𝑊𝑊) = 100−3/(2𝑊𝑊2) with some initial wealth W0. You know that for some small bet defined in absolute, monetary values as h = 1 euro, the minimum odds that John requires to enter the bet equals 0.55. The bet is small enough to allow you to apply the approximations based on the (R/A)RA coefficients that were covered during the lectures. Which of the following statements is MOST LIKELY: (A) John’s initial wealth must be 15 euros and at this initial wealth, his absolute risk aversion is 0.2. (B) John’s initial wealth must be 15 euros and at this initial wealth, his absolute risk aversion is 5. (C) John’s initial wealth must be 10 euros and at this initial wealth, his absolute risk aversion is 1/3. (D) John’s initial wealth cannot be determined because power utility implies constant relative risk aversion that as such fails to depend on his wealth. (E) None of the above is likely/appropriate.

Answer A

Debriefing.

Because we know the formula for the minimum odds, we know from the question that

0.55 ≅12

+14𝐴𝐴𝐴𝐴𝐴𝐴(𝑊𝑊) × 1 =

12

+14

×𝛾𝛾𝑊𝑊0

× 1 =12

+14

×3𝑊𝑊0

× 1,

where the equality comes from the fact that we have emphasized that “The bet is small enough to allow you to apply the approximations based on the (R/A)RA coefficients that were covered during the lectures.” As seen in the lectures, in the case of power utility, ARA(W) = 𝛾𝛾/𝑊𝑊. At this point, note that

0.55𝑊𝑊0 −12𝑊𝑊0 = 0.05𝑊𝑊0 =

34⟹𝑊𝑊0 =

0.750.05

= 15 𝑒𝑒𝑒𝑒𝑟𝑟𝐶𝐶𝑒𝑒.

Moreover, ARA(W) = 𝛾𝛾/𝑊𝑊 = 3/15 = 0.2. Question 8 (1.25 points) Sue is characterized by the following map of indifference curves in the classical mean-variance space.

Which of the following statements is LEAST LIKELY: (A) Sue is risk-averse for all wealth levels (B) Sue is neither of DARA or IARA type, but she switches from DARA to IARA in correspondence to some specific combination of risk and mean return

(C) Sue is at first a risk-lover and then she becomes risk-averse (D) Sue will demand an increasing compensation for taking on risk, when risks are small, but then, past some level, she will demand a declining compensation for risks (E) All of the above are likely/appropriate.

Answer C

Debriefing. As seen and commented already in one open question of one 2016 exam, we know that the one in the picture is exactly the case of one investor who is always risk-averse but at first she is of IARA type and then of DARA type: her indifference curves are always increasing, but at first they are convex and then concave.

Question 9 (1.5 points, note the presence of an open question portion) Consider the following picture, computed in Excel and discussed in the lectures in correspondence to second set of solved exercises:

Which of the following statements is MOST LIKELY: (A) In the presence of human capital, the mean-variance demand for long-term government bonds (dark solid bars) is the only demand used to hedge the uncertainty caused by labor income uncertainty and that tactically switches sign as a function of whether an investor is employed or not (B) The mean-variance investor plotted in the picture must be highly risk-averse because one can observe that occasionally the demand of a few types of stock portfolios has turned negative, i.e., stocks must be sold short (C) Because the neutral equity portfolio dominates the asset allocation over time, this investor must be approximately risk-neutral (D) In the presence of predictability of asset returns, the mean-variance demand for long-term government bonds (dark solid bars) is the only demand used in large proportions to time the state of investment opportunities that tactically switches sign as a function of the value taken by the predictor(s) (E) None of the above is likely or appropriate

Answer D

-220%

-160%

-100%

-40%

20%

80%

140%

200%

260%

320%19

85Q1

1985

Q419

86Q3

1987

Q219

88Q1

1988

Q419

89Q3

1990

Q219

91Q1

1991

Q419

92Q3

1993

Q219

94Q1

1994

Q419

95Q3

1996

Q219

97Q1

1997

Q419

98Q3

1999

Q220

00Q1

2000

Q420

01Q3

2002

Q220

03Q1

2003

Q420

04Q3

2005

Q220

06Q1

2006

Q420

07Q3

2008

Q220

09Q1

2009

Q420

10Q3

2011

Q220

12Q1

Growth Portfolio Neutral Portfolio Value Portfolio3M US BILL 10Y US BOND

Briefly explain your answer clarifying, in the light of this exercise and of the related Homework 2, how one would go about testing for and exploiting predictability of asset returns in an Excel framework. Which predictors would you use for the excess returns series/assets also mentioned in the picture? Why do the time-varying weights in the picture oscillate between positive and negative values? What difficulty would be posed by the decision to constrain the portfolio weights to be non-negative and not exceed 100%? Please motivate all your claims and make sure to organize a reply based on the elements of the course only (0.5 points, 10 lines max, TEN not 10 thousand…).

Debriefing. As seen in the second exercise set and in Homework 2, one would estimate any predictability relationship and test its statistical significance, using a simple linear regression framework, which consists of regressing (excess) asset returns—say, one at the time—on a selected number of predictors known to be useful and economically valuable from the financial literature. The regressions are usually of an out-of-sample, predictive nature, i.e., they use time t information on the RHS to predict excess returns on the LHS. One would then exploit any predictability patterns—interesting this step does not require that in the first step the linear relationships had to be found to be statistically significant—by using the regression on an out-of sample basis to forecast on month t the excess return (also called, the conditional risk premium) for month t + 1. The (quite common) predictors indicated during the lectures and used in practice are the log price-dividend ratio, that when above average tends to predict lower future returns, and the term spread (often the difference between 10- and 1-year sovereign interest rates), that when above average tends to predict higher future returns. The reported mean-variance weights oscillate because as new information on the log price-dividend ratio and the term spread arrive, forecasts of subsequent excess asset returns change, and hence mean-variance portfolio weights do change. Because no short-sale constraints have been imposed, such weights can also turn negative. If we had desired to constraint the portfolio weights to be positive, then we would have set up the problem solving it not in closed-form using simple algebra, but on a numerical basis using instead Excel’s Solver tool. The difficulty is that the solver is slow and, at least when one sticks to naked Excel, the optimization has to be repeated a number of times “by hand” which is slow and often unreliable.