ECMC49F Midterm Instructor: Travis NG Date: Oct 26, 2005 Duration: 1 hour 50 mins Total Marks: 100 [1] [25 marks] Decision-making under certainty (a) [5 marks] Graphically demonstrate the Fisher Separation Theorem for the case where an individual ends up borrowing in the capital market. Label the following 3 points on your graph. i. Initial endowment (Y 1, Y 2 ) ii. Optimal production point (P 1, P 2 ) iii. Optimal consumption point (C 1, C 2 ). (b) [5 marks] List the mathematical equilibrium conditions for getting (P 1, P 2 ) and (C 1, C 2 ). What are the intuitions of each of the conditions? (c) [5 marks] What is the net present value of (i) her initial wealth, (ii) her wealth after she optimizes? How much is she borrowing? How much is she going to repay in the second period? (d) [5 marks] In words, describe the relationship between NPV rule as a project selection rule for a firm and the Fisher Separation Theorem. (e) [5 marks] In words, describe the role of capital market in a world of certainty. [2] [25 marks] Expected Utility Theory (a) [9 marks] State the minimum set of necessary conditions needed to obtain risk-return indifference curves as those used in portfolio theory. (No explanation needed) (b) [8 marks] Suppose an individual s utility function depends on her wealth only. Which one of the following 3 utility functions exhibits reasonable behavioral predictions? [hints: Consider what are the reasonable behavioral predictions under risk. Show your answers in mathematical details. i. U(W) = -(1/W) ii. U(W) = aw- bw 2 iii. U(W) = ln(w) (c) [8 marks] Suppose the same individual in fact has utility function U(W)=ln(W). Her current wealth is $1,000. If she is faced with 50/50 chance of losing and winning $50, how much will she pay to avoid such uncertain situation? [You can use either Markowitz or Arrow-Pratt approach to calculate] [3] [25 marks] Portfolio Theory Let R a and R b be the returns from two risky assets a and b respectively. Assume each of the two returns are normally distributed with expected returns and standard deviations as, E(R a ) = 5% and E(R b ) = 20%, σ(r a ) = 20% and σ(r b ) = 40%. Assume these 2 risky assets are the only investments vehicles available. (a) [2 marks] If the two risky assets have returns that are perfectly positively correlated, plot the feasible mean-variance investment opportunity set in an expected return-standard deviation space. 1
(b) [10 marks] If the two risky assets have returns that are perfectly negatively correlated, plot the feasible mean-variance investment opportunity set on the same graph. i. Point out the precise point where you will construct a portfolio with the lowest possible risk. ii. iii. What is the risk level in (i)? Evaluate the following claim, Every risk-averse individual will invest in this minimum variance portfolio. (c) [5 marks] If the two risky assets have returns that are not perfectly correlated, plot the feasible mean-variance investment opportunity set again using a new graph. Indicate where the efficient frontier/set lies. (d) [3 marks] On the same graph in part (c), suppose everyone can borrow and lend unlimited amount of money at risk-free rate of 5%, which portfolios are now mean-variance efficient? (e) [5 marks] With the aid of graphs, describe the role of capital market under the world of uncertainty. [4] [25 marks] Capital asset pricing model (a) [5 marks] List the assumptions for CAPM. (No explanation needed) (b) [5 marks] Suppose the market is expected to pay out 15% with a standard deviation of 8%. Borrowing rate is 5%. Write down the precise expression for the Capital Market Line and for the Security Market Line. (c) [5 marks] If investor A invests by borrowing an additional amount equal to 50% of her initial amount of investment money she sets aside, what is her portfolio s expected return and standard deviation? If part of her portfolio consists of a mutual fund that is expected to pay 10%, can we tell how much is the mutual fund return s standard deviation? (d) [5 marks] Suppose you work for an investment company and currently are working on an IPO (i.e., initial public offering) project. What are the necessary steps for evaluating the optimal price of that IPO? (e) [5 marks] Many stocks had their prices fell dramatically in 2001. Such observation in itself is evidence that the CAPM is false. Evaluate this statement. 2
Some of the Mathematical formulas we have covered: [1] Suppose your utility has the following structure U(C 0,C 1 ) = U(C 0 ) + U(C 1 )/(1+ ρ) To derive the slope of the corresponding indifference curve, we take total derivative: U'(C 0 )dc 0 + [1/(1+ ρ)]u'(c 1 )]dc 1 = 0 Rearranging, we can the slope as: dc 1 /dc 0 = -(1+ ρ) [U'(C 0 )/ U'(C 1 )] [2] Multi-period Present value: PV = C 0 + C 1 /(1+R) + C 2 /(1+R) 2 +. Where C t = cash flow at time t, can be positive or negative, and assuming a constant discount rate R Perpetuity: PV = C/R Annuity: PV = C[1/R 1/(R(1+R) t )], where t = the period the annuity ends [3] Arrow-Pratt Risk Premium: π(w,z) = (1/2)σ 2 z(-u (W)/U (W)) Where z is the random payoff from gamble. Degree of Absolute risk aversion: ARA = (-U (W)/U (W)) Degree of Relative risk aversion: RRA = W(-U (W)/U (W)) [4] Asset j s return in State s: r js = (W s W 0 ) / W 0 Expected return on asset j: E(r j ) = s α s r js Asset j s variance: σ 2 j = s α s [r js - E(r j )] 2 Asset j s standard deviation: σ j = σ 2 j Covariance of asset i s return & j s return: Cov(r i, r j ) = E[(r is - E(r i )) (r js - E(r j ))]= s α s [r is - E(r i )] [r js - E(r j )] Correlation of asset i s return & j s return: ρ ij = Cov(r i, r j ) / (σ i σ j ), where -1 ρ ij 1 Properties concerning mean and variance: E(ũ+a) = a + E(ũ) E(aũ) = ae(ũ) Var(ũ+a) = Var(ũ) Var(aũ) = a2var(ũ) Variance of 2 random variables: Var(aũ+bẽ)=aVar(ũ) + bvar(ẽ) + 2abCov(ũ,ẽ), where a & b are constants and ũ and ẽ are random variables. [5] Cross-sectional regression equation of CAPM [R it R ft ] = γ 0 + γ 1 β i + γ x Χ i + ε it, where Χ is a vector of variables potentially explain the excess return of asset i over risk-free rate. If CAPM is true, of cause the vector s estimated coefficients γ x will be zero. 3
Midterm solution for ECMC49F Travis NG Date: Oct 26, 2005
Question 1 (a) Labeling the diagram is all you need to get the marks. C 2 Capital Market line (P 1, P 2 ) * (C 1, C 2 ) * (Y 1, Y 2 ) * U c&p U 0 C 1 (b) To get (P 1, P 2 ), set MRT = (1+R). Intuition is to invest and invest until your marginal return from the marginal investment is just the same as the cost of capital. To get (C 1, C 2 ), set MRS = (1+R). Intuition is to borrow or lend until your subjective rate of time preference is the same as the cost of borrowing/lending. (c) PV of initial wealth = Y 1 + Y 2 /(1+R) PV of optimized wealth = C 1 + C 2 /(1+R) She borrows (C 1 P 1 ) She repays (C 1 P 1 ) (1+R) which is the principal amount plus the interest charge. (d) I know most of you are taking the commerce program. In management accounting, or finance, you usually deal with the calculation of NPV, or the evaluation of whether to take on a project. NPV rule is a well-known rule in practice. Managers know they have to take the projects which generate positive NPV in order to maximize their shareholder s wealth. But most of them don t know why. This question proves that you truly appreciate such rule based on solid theoretical background. You learn to calculate NPV in other course, and you learn the underlying theory in economics. The most I want you to get out of this course is help you make sense out of what you eventually will be doing in managing businesses that involves finance, or simply your personal finance. That doesn t necessary mean you make more 2
money, but it is about fun. Knowing what you are doing is way more fun than having no idea about what you are doing but are still obligated to do it. When you eventually become a manager or own your business, you know exactly why you need to calculate the NPV. Again, NPV rule refers to a firm taking all the projects that have positive NPV. From the above graph, we know that the optimal point to invest is when marginal investment return equals the cost of capital. This is exactly what is meant by NPV rule. All those projects with positive NPV represents points on the POS to the left of (P1, P2). Any point to the right of (P1, P2) means those projects having negative NPV. (e) Another big piece of why study financial economics is here. Why do we need a capital market? The easy answer is: because having it, everyone is better off. Economists like you know the answer, and are able to draw the diagram in part (a) to prove your point to your parents, your girlfriend, your friends and pets. Different individuals or firms are exposed to different opportunities. Some are constrained to have limited lucrative production opportunities, while others are well-endowed with very profitable ideas. It may be because those others are just brilliant, or more knowledgeable, or relatively creative, or having good networking. Usually, these others have good ideas and to execute all those ideas, they need to raise capital, either from friends, from banks, from venture capitalists or from issuing equity. In short, those constrained by the amount of money available to them can get needed finance from those abundant with their money. This is what we called financial intermediation. The transfer of surplus fund to finance-deficit individuals or firms to carry out lucrative projects makes everyone better off. And capital market creates a centralized place to co-ordinate such activities of financial intermediation. In your simple graph drawn in part (a), you get a simplified idea that proves everyone is better off with the introduction of capital market. 3
Question 2 (a) The minimum set of necessary conditions needed is as follows: Expected Return E(r) Indifference Curve - Represents individual s willingness to trade-off return and risk - Assumptions: 5 Axioms of choices under uncertainty Prefer more to less (Greedy) Risk aversion Assets jointly normally distributed Increasing Utility Standard Deviation σ(r) (b) By exhibiting reasonable behavioral predictions, I mean (1) risk-aversion, (2) you are less risk-averse if you get richer (in absolute term) and (3) you are about as riskaverse no matter how rich or poor (in relative term). (1) risk-averse means U (W)>0 and U (W)<0 or the utility is concave. (2) means ARA is going to decrease as W increases. (3) means RRA is going to be constant and is independent of changes of W. Thus, this question boils down to checking whether those 3 utility functions satisfy the above 3 conditions. I am glad that most of you scored nearly perfect in this question. (c) Most of you nailed this question too. Assuming U(W) = ln(w), we employ the Markowitz approach first: E[U(W)] = 0.5ln(1050) + 0.5ln(950) = 6.9065 (round up to 4 decimal places) To find the certainty equivalent, set: E[U(W)] = U(CE) 6.907 = ln(ce) CE= e 6.907 = 998.745 Thus risk premium = $1,000 - $998.745 which is about $1.255. (It doesn t matter if you get a higher or lower value. All that is needed is how you set up the question) Alternatively, apply the Arrow-Pratt approach by using the formula given. 4
π(w,z) = (1/2)σ 2 z(-u (W)/U (W)) dln(w)/dw = (1/W) d(1/w)/dw = -1/W 2 σ 2 z = 0.5(50 2 ) + 0.5(50 2 ) = 2500 RP = ½ (2500)(1/1000) = $1.25, which is about the same as the Markowitz premium. Question 3 (a) Since we are already given the expected returns (i.e., means) and standard deviations of the 2 risky assets. If they are perfectly positively correlated, we can draw the following graph. E(r p ) 20% 5% Min-var. opp. set 20% 40% σ p 5
(b) (i) If they are perfectly negatively correlated, we can instead plot the following graph. E(r p ) 20% Min-var. opp. set 5% 20% 40% (ii) As you can see from the above graph, you can perfectly hedge yourself against risk by choosing the risk-free portfolio with a appropriate weight of investment to be put on a and b. At that portfolio, you are risk-free. Risk-level is minimized at zero. (iii) Of course you can choose the risk-free portfolio. But not everyone is going to do that. It all depends on the risk preference of individual investors. For example, I know I can choose to invest in the risk-free portfolio. But if I find the risk-return trade-off in favour of my taking some more risk, I will be willing to invest another portfolio in order to anticipate some more expected return, with the fact that doing so I also expose myself to higher risk. σ p 6
(d) If the two assets have returns that are not perfectly correlated, we get the following graph instead. The blue colored portion of the min-variance opportunity set is the efficient frontier/set. E(r p ) 20% Min-var. opp. set 5% 20% 40% σ p (e) Suppose you can lend or borrow money at the capital market, and suppose the borrowing rate equals the lending rate, then we get the following graph instead. E(r p ) Capital Market Line 20% Min-var. opp. set 5% 20% 40% Remember the capital market line passes through the risk-free point (i.e., the y- intercept at 5%) and the market portfolio (i.e., the tangency point). σ p 7
Efficient frontier now is no longer the blue colored portion, but all the points along the capital market line. That means, if you pick ANY portfolio along the capital market line, it is mean-variance efficient. (f) The easy answer is: the introduction of capital market enhances everyone s utility. Anyone of us is made better off with the capital market. The differences between our degrees of risk aversion imply some may prefer low-risk-low-return portfolios, while others prefer high-risk-high-return portfolios. With the ability to borrow and lend, those willing to take higher risk can borrow money to do so at the market interest rate. Those willing to take lower risk can lend money out to the needed. The graphically result of utility enhancement is given in the lecture as well. And it is as follows: U U E(r U p ) Capital Market Line U-Max Point P Efficient set M Endowment Point R f σ p Without the capital market, the individual can only get U, but with the capital market, she will be borrowing money and invest in a higher-risk-higher-return portfolio that enables her to get a utility of U instead. Question 4 (a) CAPM assumes the followings: [1] Perfect market: Frictionless, and perfect information No imperfections like tax, regulations, restrictions to short selling All assets are publicly traded and perfectly divisible Perfect competition everyone is a price-taker 8
[2] Investors: Same one-period horizon Rational, and maximize expected utility over a mean-variance space Homogenous beliefs (b) The question gives the following information: E(R M ) = 15%, σ(r M ) = 8%, R f = 5% Equations for CML and SML are respectively CML: E(R p ) = R f + [(E(R M )- R f )/σ M ]σ(r p ) SML: E(R i ) = R f + [E(R M )-R f ] x β i Substitute these into the CML or SML equation to get: CML: E(R p ) = 5% + [(15%-5%)/8%]σ(R p ) SML: E(R i ) = 5% + [15%-5%] x β i (c) Investor A has income of say, Y. She is now borrowing 50% of Y at an interest rate of 5%. She invests (1+50%)Y amount of money in the market. That means her portfolio consists of -50% of risk-free asset and 150% of market portfolio according to the two-fund separation. Graphically, she will be investing along the capital market line at a point like point G where she maximizes her utility. E(r p ) 20% G Capital Market Line 15% M 5% 8% 12% σ p 9
Expected return is equal to -50%(5%) + 150%(15%) = 20% Substitute this number in the equation of CML: E(R p ) = 5% + [(15%-5%)/8%]σ(R p ) 20% = 5% + [(15%-5%)/8%]σ(R p ) σ(r p ) = 12% The above calculation involves using the CML equation. But suppose we know that there is a mutual fund involved in her 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. (d) This question requires you to link the knowledge between choosing an appropriate discount rate (as given in the supplementary notes page 11 to 14) and the CAPM. The idea is simple. But try to imagine the actual works that are needed to execute the following procedures, and you will get an idea why investment bankers are getting such a high salary. If you are working on an IPO project, you likely have all the historical data of that firm. You evaluate the historical returns of that company that is about to be listed. Calculate its historical beta. And use the CAPM equation to find out how much expected return is for that corresponding beta. Then, you forecast the future cash flows that the company is about to generate for its shareholders. Find the present value of these future expected cash flows by discounting them at the expected return that you just found from CAPM. The present value is the price you should tell your boss to sell the IPO. (e) For this part, I am flexible. I need an intelligent discussion in your answer. If you have reasonable points, you have marks. Below is what I have in mind. Stock prices falling can be due to a lot of reasons. It may be that the economy as a whole was doing really badly, which constituted the systematic risks. Or that a specific firm was plagued by some unique shocks, which constituted the unsystematic risks. As we all know, because of the Sep 11, 2001 disaster, the economy in general was slowing down. And I would not be surprised to see a lot of stocks falling in value because of the economy-wide slowdown. This is not evidence against CAPM for at least 2 reasons. First, CAPM predicts expected return, not the actual return. Second, the economy-wide slowdown leads to falling in prices of a lot of stocks actually reinforces our understanding of the effect of systematic risk. 10