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Unformatted text preview: 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). Answer : 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.- 6 P P Utility increases Utility increases 6 Figure 1: Indifference curves in P , P space. (b) What do you consider the main properties which the indifference curves should be as- sumed to have? How would you justify these assumptions? Answer : It is typically assumed that P is a good in the sense that greater expected return is preferred to less, and that P 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 P , 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 P ) only a small increase to expected return, P , is needed to com- pensate 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 P 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 B (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 investors 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 investors risk, say at * P (imposed by banks or the government). The plunger would then invest up to the permitted limit....
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- Spring '11