Solution Guide to Exercises for Chapter 4 Decision making under uncertainty

THE ECONOMICS OF FINANCIAL MARKETS R. E. BAILEY Solution Guide to Exercises for Chapter 4 Decision making under uncertainty 1. Consider an investor who makes decisions according to a mean-variance objective. (a) Sketch indifference curves for the investor (with expected wealth on the vertical axis and the standard deviation of wealth on the horizontal axis). In this answer, wealth is replaced by portfolio rate of return. This particular answer is not affected, though it is not always permissible to make the replacement. µ P Figure 1: Indifference curves in µ P, space. (b) What do you consider the main properties which the indifference curves should be assumed to have? How would you justify these assumptions? It is typically assumed that µ P is a good in the sense that greater expected return is preferred to less, and that is a bad, in the sense that greater risk (standard deviation of return) is worse than less. This imples that higher indifference curves are preferred and that indifference curves are increasing in µ P, space. In addition it is usually assumed that indifference curves are convex from below (as drawn in figure 1). The assumptions made so far do not necessarily imply this. Convexity is commonly justified by introspection, namely that it seems reasonable for the curves to have this shape. Why should that be? The argument is that for low levels of risk (small ) only a small increase to expected return, µ P, is needed to compensate for an increment in risk (i.e. the indifference curve is relatively flat for low levels of risk). But at high levels of risk, the same increment in must be compensated by a larger increase in expected return for the investor to remain on the same indifference curve. Whether you find this argument convincing is entirely up to you. Another justification for convexity is to ask how an investor who did not have convex indifference curves would behave. Suppose, for instance, that the indifference curves are concave from below (see figure 2). 1

In this case, implausible predictions are generated. Wby? Assume that the investor faces an upward sloping budget constraint given by the line 0B (this will be justified in chapter 5). The investor would then choose one of two strategies: either (i) invest as much as possible in risky assets or (ii) invest nothing in risky assets. The investor would be a plunger, i.e. would never adopt a policy between the extremes. If there is no upper limit on the riskyness of the portfolio, then the investor s portfolio may be undefined: if a non-zero level of risk is chosen, then the investor would seek to bear unbounded risk. More reasonably, there may be an upper-limit on the investor s risk, say at σp (imposed by banks or the government). The plunger would then invest up to the permitted limit. Thus, if investors have concave indifference curves they must be plungers. The evidence does not support this prediction. Hence, the assumption is implausible. (Note that it is possible but not inevitable that an investor with convex indifference curves may choose to invest all or zero of his/her wealth in risky assets may not must.) µ P B σ P 0 Figure 2: Indifference curves for a plunger. (c) Compare the indifference curves for two investors one of whom is more risk averse than the other. The more risk averse investor (left panel of figure 3) requires a greater increase in expected return to compensate for a given increment in risk than a less risk investor (right panel). Not that the indifference curves for both investors are positively sloped and convex from below. 2. Explain and assess the role of the Expected Utility Hypothesis (EUH) in the theory of portfolio selection. Guidelines: This is a typical past examination question for which there exist many correct answers at various levels of performance. There is no single right (or wrong) answer. Hence, there follows advice about how to answer the question. (a) The two main concepts here are the Expected Utility Hypothesis (EUH) and portfolio selection. 2

µ P More risk averse µ P Less risk averse Figure 3: Indifference curves in µ P, space. The latter is fairly obvious but your answer could begin by outlining the investor s portfolio selection problem, that is, selection of portfolio proportions, a 1, a 2,..., a n of initial wealth, A to attain some specified objective. This question focuses on the EUH but it is permissible to note that there are other objectives and that an assessment of the role of the EUH should be aware (at least) of these alternatives. (b) The question asks you to explain the EUH, i.e. to present its assumptions and implications. The assumptions (or axioms) of the EUH are conditions imposed on individual actions. Answers have some discretion about how much detail is provided about the assumptions of the EUH. (For a third class mark it is not essential to include much about the assumptions; a lower-second class answer would include some brief remarks; an uppersecond class answer would typically list the axioms and a first class answer might offer some comments note that there are normally several different ways of achieving any mark. Some answers might receive a high mark because they are exceptionally good on some other part of the question.) (c) The main assumptions underlying the EUH are summarised as follows: i. Consider an event (a set of one or more states) and compare actions that differ in their consequences for states in the event but have consequences that are identical with one another for states not in the event. The first assumption states that the decisionmaker s ordering of the actions is independent of the common consequences for states not in the event. That is, roughly speaking, if action A is preferred over action B when the common consequences for states outside the event are favourable, then action A remains preferred over action B when favourable is replaced by more favourable, or less favourable, or whatever. ii. Consider the consequences in any particular state (i.e. in one row of the payoff array). The second assumption asserts that preferences over consequences for the given state are independent of the state in which they occur. Less formally, the decision-maker cares only about the consequence not the label of, or index of (say, a subscript k ), the state in which it is received. iii. The third assumption asserts, again rather imprecisely, that the decision-maker s degree of belief about whether a state will occur is independent of the consequences in the state. 3

iv. Technical assumptions (essentially continuity) to ensure that a preference ordering can be represented by a utility function. (d) The first, most basic, of the implications of the assumptions is that the investor s preferences can be expressed in the form of expected utility: U = U(W 1, W 2,..., W l ) = π 1 u(w 1 ) + π 2 u(w 2 ) + + π l u(w l ) (1) where π k is the probability that the individual assigns to state s k. The function u( ) is called the von Neumann-Morgenstern utility function after John von Neumann and Oscar Morgenstern (though strictly they were not the first to use the concept). Notice that the u( ) is the same for all states, though its argument, W k, is generally not. Both the probabilities and the von Neumann-Morgenstern utility function are allowed to differ across investors. It is assumed, however, that u (W ) > 0 for all relevant levels of W, i.e. individuals prefer more wealth to less. You should emphasize in your answers that the EUH implies a separation of beliefs (about which state of the world will occur) from preferences for wealth. If an individual behaves according to the EUH, that individual acts as if he/she assigns probabilities to states and values wealth independently of which state actually occurs (what matters is the amount of the wealth, not in which state that amount is received). (e) Your answer should now move on to discuss the EUH in the context of portfolio selection. Here the most important implication is the Fundamental Valuation Relationship (FVR). Again, answers will differ in how much depth they go into on the FVR. It is not expected that answers would present as much detail as in question (3), below. See the guidelines for question (3) for an outline of the topics to cover in your answer. Answers should, at least, include (i) a brief statement of the FVR, (ii) note that it is a necessary condition for portfolio optimisation, (iii) note that it cannot be solved for the optimal portfolio without more information about the von Neumann-Morgenstern utility function, and (iv) note that mean-variance analysis can be considered as an important special case. (f) Moving on to the assessment of the EUH, answers should note that it is central to finance, via the FVR. You may also comment that the FVR can be extended to include a broader set of behaviour such as intertemporal optimisation. It is important to make these points, though there is not much in quantity to write about them. (g) Answers should comment on the weaknesses of the EUH. The basic weakness of the EUH is that there is evidence mainly from experimental studies that individual s behaviour differs from the predictions of the EUH. By implication one or more of the assumptions of the EUH is violated. It is permissible for answers to include an overview of behavioural finance, which is mostly an attack on the EUH. (But you should probably not devote too much of the available time in covering behavioural finance in answer to this question it is relevant but not essential.) (h) Finally, you may wish to conclude your answer with comments about alternatives to the EUH. That is, consider the following: if the EUH is open to criticism (as it is), are there any better approaches to portfolio selection? You could mention one or two other theories such as mean-variance analysis (which can be interpreted as a special case of the EUH) or behavioural finance (a collection of different supposedly more realistic theories). There is no single right conclusion. Your answer will be assessed according to how well you present the argument, not merely on what the argument is. You may conclude that the EUH is rather weak because it does not make very many predictions and those 4

which it does make are not always supported empirically. On the other hand you may conclude that there are few reliable alternatives to the EUH; in the absence of a generally acceptable alternative, the EUH may be the most appropriate model for the study of portfolio selection. 3. What is meant by the Fundamental Valuation Relationship? Discuss its role in the theory of portfolio decision making. Guidelines: This is a typical examination question for which there are many correct answers. There is no single right or wrong answer. Here are some points that correct answers would include: (a) The Fundamental Valuation Relationship, FVR, expresses the necessary, first-order, condition for maximising Expected Utility in portfolio theory. Hence, it is relevant when the investor behaves according to the EUH, not otherwise. (b) In the context of the EUH, the portfolio selection decision solves the following optimisation problem: choose portfolio proportions, a 1, a 2,..., a n to maximize expected utility, E[u(W )], where: W = (1 + r 1 )a 1 A + (1 + r 2 )a 2 A + + (1 + r n )a n A, where r j is the (random) rate of return on asset j, A is initial wealth, and a 1 + a 2 + + a n = 1. The function u( ) is the investor s von Neumann-Morgenstern utility function, with positive but diminishing marginal utility, u (W ) > 0, u (W ) < 0. (c) For the above optimisation problem, the FVR can be expressed as: E[(1 + r j )u (W )] = λ, j = 1, 2,..., n, (2) where λ > 0 is the same for all j. To understand why the FVR is necessary for maximisation of EUH, start from a maximum and suppose that the investor receives an additional small, one unit, of initial wealth. If this is invested in asset j, an additional 1 + r j units of wealth is obtained. The individual values wealth in utility terms; hence, the increment in utility equals (1 + r j )u (W ) (this is correct because the change in initial wealth was small, so that a linear approximation is permissible). But increment in utility will depend on the state (the future is uncertain) which state will occur is not known when the investment decision is made. Hence, the investor values the utility in expected terms, i.e. the increment to expected utility is E[(1 + r j )u (W )]. Now this must be equal for all assets, otherwise the starting point could not have been a maximum of utility. Note that the assumption of diminishing marginal utility, u (W ) < 0, implies that a solution of the FVR is indeed a maximum of expected utility. (d) Dividing through by λ gives the general form of the FVR: where H u (W )/λ. E[(1 + r j )H] = 1, j = 1, 2,..., n, (3) (e) If a risk-free asset is available, the FVR must hold for that too: E[(1 + r 0 )H] = 1, (1 + r 0 )E[H] = 1, E[H] = 1/(1 + r 0 ). (4) 5

Also, note that in the presence of a risk-free asset it is often convenient to express the FVR as: E[(r j r 0 )H] = 0, j = 1, 2,..., n, (f) The role of the FVR in the theory of portfolio selection is that it establishes a condition that must be satisfied for every investor who behaves according to the assumptions of the EUH. In principle, though rarely in practice, the FVR could be solved to obtain the optimum portfolio composition. The only case for which the portfolio proportions are easily computable is when the von Neumann-Morgenstern Utility function is quadratic, that is, when the investor behaves according to a mean-variance criterion. More generally, the FVR plays a very important role because the implications of the FVR can be used in empirical applications, i.e. testable predictions can be obtained. 4. Consider a world in which there are exactly two assets (labelled A and B) and two states of nature (labelled 1 and 2). The payoff of the assets in the two states is as follows: Assets A B State 1 3 9 State 2 6 0 Denote by x A and x B the number of units of assets A and B purchased by an investor. (a) The investor is assumed to have preferences which depend on the total amount received in each state, not whether the return comes from one asset or the other. Apart from this, the investor s preferences are unspecified. If the preferences are represented by an objective function, what are the arguments of the function? The investor s preferences are represented by U = U(W 1, W 2 ) where W k denotes the level of wealth in state k: W 1 = 3x A + 9x B (5) W 2 = 6x A (6) (b) Suppose that the investor behaves according to the expected utility hypothesis and believes that the probability of state 1 is 1 3 and the probability of state 2 is 2 3. How would you express the investor s objective function (ie the function that the investor seeks to maximize)? Write out the investor s objective function in as much detail as you can, given the information which is provided. Now preferences are represented by: U = U(W 1, W 2 ) = 1 3 u(w 1) + 2 3 u(w 2) (7) where u( ) denotes the investor s von Neumann Morgenstern utility function (as a function of wealth in state k, W k ). (c) Now suppose that the investor behaves according to a mean-variance objective. How would you express the investor s preferences? Write out the investor s objective function in as much detail as you can, given the information which is provided. [Hint: calculate the expected value of wealth and the variance of wealth, as functions of x A and x B.] 6

In this case preferences are represented by a function of expected wealth and standard deviation of wealth, say G(µ W, σw 2 ), where: E[W ] µ W = x A E[V A ] + x B E[V B ] = x A { 1 3 3 + 2 3 6} + x B{ 1 3 9 + 2 3 0} = 5x A + 3x B (8) var(w ) σw 2 = E{W E[W ]} 2 = 1 3 {W 1 E[W ]} 2 + 2 3 {W 2 E[W ]} 2 = 1 3 {3x A + 9x B (5x A + 3x B )} 2 + 2 3 {6x A + 0x B (5x A + 3x B )} 2 = 1 3 { 2x A + 6x B } 2 + 2 3 {x A 3x B } 2 = 1 3 {4x2 A + 36x2 B 24x Ax B } + 2 3 {x2 A + 9x2 B 6x Ax B } = 2x 2 A 12x A x B + 18x 2 B ***** 7