ECMC49S Midterm. Instructor: Travis NG Date: Feb 27, 2007 Duration: From 3:05pm to 5:00pm Total Marks: 100

Transcription

1 ECMC49S Midterm Instructor: Travis NG Date: Feb 27, 2007 Duration: From 3:05pm to 5:00pm Total Marks: 100 [1] [25 marks] Decision-making under certainty (a) [10 marks] (i) State the Fisher Separation Theorem and explain what it means in words (no graph). (ii) Which project selection rule it suggests for firms to adopt? (iii) State at least 1 advantage and 1 disadvantage of using this project selection rule. (i) Fisher Separation theorem given perfect and complete capital markets, the production/investment decision is solely governed by objective market criteria (say, interest rate) and therefore is independent of the individual s subjective preference. In other words, while consumption decision is subjectively determined, the investment decision is objectively determined. (ii) The net present value rule of selecting projects that have +ve NPV. (iii) Advantage: Bill Gates do not have to survey his shareholders in order to decide what project to take. Of course, there are many other advantages. Disadvantage: Ignore principal-agency problem (corporate governance), ignore real options, sometimes it is hard to compute NPV without a clear-cut interest rate to use (especially when dealing with risky projects), and it depends on the accuracy of estimations of future cashflows. Any reasonable disadvantage will be given marks. (b) [10 marks] The inter-temporal consumption model attempts to explain people s motives to save and invest. (i) In your own words, briefly explain why sometimes people prefer to invest than to put money into the bank? (ii) What exactly does investing mean? (iii) Why would some people invest huge amount this period and simultaneously borrow money from the bank to spend? (i) Whether to invest or to save, the ultimate goal is to maximize utility given the constraints. While an individual s utility inevitably depends not only on his consumption today but also his future consumption, the task is to try to optimize consumption across periods by means of investing or saving. Whether saving or investing then depends on which activity would achieve higher utility. (ii) Investing means putting money into any productive activity, expecting to generate a return in the future. From an economics point of view, investing does not confine to investing in the capital market. Investing in my own education, hoping that the accumulated human capital will generate decent return in my future is also a form of investment. (iii) This is nothing wrong according to Fisher Separation Theorem. As long as the projects are worthwhile to take (which yields +ve NPV), the individual should take it, irrespective of his own subjective rate of time preference. Such act maximizes his own current wealth (in present value term), and then he can decide how he s going to spend his person s wealth in the way that suit his preference the most, which may or may not include borrowing money from the bank. (c) [5 marks] Suppose in the 3-period risk-free world, Sherwin has $10,000 this period and nothing in the subsequent two periods. There are two projects available for him to invest: Project A requires $15,000 of initial capital this period and generates $20,000 in the next period; Project B requires $8,000 of initial capital that generates $5,000 in the next period and $6,000 in the third period. The bank pays 15% interest per period. Briefly describe what and how he should do? 1

2 y0 y1 y2 market interest rate $10,000 $0 $0 15%/period 1st 2 nd 3 rd NPV project A -$15,000 $20, project B -$8,000 $5,000 $6, NPV = CF 0 +CF 1 /(1+R f )+CF 2 /[(1+R f )^2], where CF t = Cashflow in period t. NPV for both projects are positive; therefore Sherwin should take both projects. (It is not ok to use IRR rule to select projects as it yields the same project selection results only if we are in a 2-period model.) He does not have that much money, but he can always borrow money from the bank to finance his projects. He then decides how to consume across the 3 periods based on his own subjective rate of time preference, which may or may not include lending or borrowing from the bank again. [2] [25 marks] Expected Utility Theory (a) [5 marks] I hate to be described as risk-averse because I am borrowing OSAP to finance my college education and by the time I am done, I will be so deeply in debt. There is nothing riskier than messing up my finance in the next few years for the sake of a degree without any guarantee of a decent job. I am therefore risk-loving. Evaluate this statement. This is just the other way of asking if you truly understand the simple concept, that being risk-averse doesn t mean you don t take risk. (b) [5 marks] It seems so reasonable to decide which risky project to take based on which gives the highest expected return. What is wrong with that? This is also another way to ask if you truly understand happiness = expected utility in the world of uncertainty. The game in the St. Peterborough s paradox gives unlimited expected return, but you never pay large sum of money to play the game. This is because you care more than expected return. You care about the risk involved in getting the expected return in question. In the world of uncertainty, there is nothing more dangerous than behaving based on a single dimension, namely, expected return. (c) [10 marks] Parminder s utility function is U(W)=(W) 1/2, where W denotes her year-end wealth. Parminder is a professional soccer player and she is thinking of whether to buy insurance to cover her risk of injury. Suppose she will make $60,000 in the next year and an injury will cause her miss games and will therefore only get paid $10,000 instead from her club. She estimates that her current wealth is $100,000 and her expected total expense this year is around $35,000 i. Suppose the insurance company charges a price of $5,000. While. If her odds of getting injured are 10%, would she buy the insurance? odds (p) year end wealth U(W) px(u(w)) E(U(W) state initial wealth expense income not injured $100, $35, $60, $125, injured $100, $35, $10, $75, insured $100, $40, $60, $120, So, she should insure herself and pay that $5,000 insurance premium. 2

3 ii. Suppose her odds of getting injured is x%. What is x equal to so that she would be indifferent between buying and not buying the insurance? x=8.9635%. iii. If she is becoming more risk-averse, how would the x you calculated from (ii) change? Obviously, given the same insurance premium of $5,000, if she s more risk-averse, if the odds of getting injured are smaller, she is still willing to insure herself. Therefore, the x should be smaller. (d) [5 marks] What are the points of carefully establishing axioms for individuals' preferences when they face uncertainty? This is because we need a unified framework to scientifically study individuals' behaviors under uncertainty. To accomplish this objective, we need to develop a theory of rational decision making in the face of uncertainty which can be broadly applicable to many individuals. This task is particularly relevant for the studies of investors' behaviors because their decisions involve risk-return trade-offs. [3] [25 marks] Portfolio Theory Let R x and R y be the returns from two risky assets x and y respectively. Assume each of the two returns is normally distributed with expected returns and variances as followings: E(R x ) = 5% and E(R y ) = 9%, σ 2 (R x ) = 5.29% and σ 2 (R y ) = 7.84%. Assume these 2 risky assets are the only investment vehicles available. (a) [4 marks] If the two risky assets have returns that are perfectly negatively correlated, plot the feasible mean-variance investment opportunity set in an expected return-standard deviation space. For part a to part c, see the attached page. (b) [6 marks] Continuing from (a), what is the expected return of a completely hedged investment portfolio? (c) Assume now that we have many risky assets. a. [2 marks] Suppose everyone can borrow and lend unlimited amount of money at risk-free rate of 3%, draw the corresponding diagram in a NEW graph. b. [6 marks] Which risky portfolio from the minimum variance opportunity set is now meanvariance efficient? Why other risky portfolios on the minimum variance opportunity set are not efficient? Under what conditions would this portfolio be the market portfolio? c. [2 marks] In your diagram, show an individual who is borrowing to invest. What is his portfolio consists of? (d) [5 marks] As quoted in the Nobel Prize website, Harry Markowitz was awarded the Nobel Prize in Economics in 1990 for having developed the theory of portfolio choice. Based on what you have learnt in the lecture, suggest two points in support of his Nobel prize award. Diversification, one of the major consideration in the portfolio theory, is a very important concept. In the time Markowitz wrote his 1950 seminal paper, it revolutionalized the portfolio theory. Before his works, people think of putting some the best performing assets into a portfolio as the optimal portfolio choice, but Markowitz suggested in his paper that the important thing is the correlations among assets that should be of most concern in constructing a portfolio. Also, of course, building on his works, we have subsequent important works such as CAPM. Any point that is correct and makes sense would be awarded. 3

4 [4] [25 marks] Capital asset pricing model (a) [5 marks] List the four equilibrium conditions for CAPM. (Some explanations are needed for full marks) Please consult the notes. (b) [5 marks] In a CAPM world, suppose the market is expected to pay out 12% with a variance of 9%. Borrowing rate is 4%. If Alvin invests by borrowing an additional amount equal to 60% of his initial amount of investment money he sets aside, what is his portfolio s expected return and standard deviation? If part of his portfolio consists of a mutual fund that is expected to pay 12%, can we tell how much is the mutual fund return s standard deviation? CAPM equation is: E(R i ) = 4% + [12% - 4%]β i CML equation is: E(R p ) = 4% + [(12% - 4%)/( 9%)]σ p His portfolio s E(R p ) = (160%)R M + (-60%)R f = (160%)12% + (-60%)4% = 16.8% His portfolio s standard deviation: 16.8% = 4% + [(12% - 4%)/(30%)]σ p => σ p = 48% Even if we know that there is a mutual fund involved in Alvin s portfolio. That means the market portfolio itself has a portion of holding of that mutual fund. We know the portfolio is well-diversified, and thus individual assets may have their risks hedged by other assets inside the market portfolio. We actually have no information at all to tell how risky is the return of that specific mutual fund. (c) [5 marks] Explain one of the difficulties of testing CAPM in detail. You can pick anyone that we have talked in class. For example, 1) CAPM is an ex ante equation with expected return as the dependent variables, whereas empirical works are done using ex post equation. Whether ex post realized return can ever test an ex ante equation is controversial. 2) Joint hypotheses testing. We are not only testing to see if CAPM is the right structure behind asset pricing, we are also testing to see if the market proxy that we are using really is efficient or not. If we reject CAPM, we can either say CAPM is wrong, or simply that the market proxy we have used in the estimation is not ex post efficient. 3) There are obviously more difficulties. If the students say reasonable things, you can give them reasonable parts marks. I do hope they say something completely different than what I had mentioned in class and they all make sense. (d) [5 marks] What kind of risk is involved in an investment portfolio of any rational individual investor in a CAPM world? By rational investment, it means that unsystematic risk is absent in an investment portfolio of any rational individual investor. If the individual does not diversify away the unsystematic risk, he is not behaving rationally. Depends on the degree of risk-aversion of the individual investor, there will be different levels of systematic risk involved in their investment portfolios. If some is extremely risk-averse, may be he doesn t take any risk and his portfolio is essentially risk-free as he only holds risk-free asset. All other types of individual would have more systematic risk if he is less risk-averse. (e) [5 marks] Why is risk directly proportional to expected return as shown in CAPM? First, only risk that is not freely diversifiable will be proportional to expected return. So not all risk is proportional to expected return. Second, given that some risk is not diversifiable, to induce anyone to bear the risk needs to compensate him with higher expected return. If not, no one would have been bearing any risk at all. 4

5 E L,^l F$yt {,N r,i' i '17,i 'l"i I j< ni \ "/, 1' tl I t[, / -tit -ly'ct,-.f,y iu:,,/,.d (+)':@1'-0 4tr0sq,f tu (t-{=s' /z"kfez-f ;i * /-' -j'), a7- /" (Ril = 6,1a'1, ui'/7',/* r )ra (/-.< t1-tv/!^ Kl) L./ in,,) Et,;- J-, fj- A, l O, *2