Investment & Portfolio Theory

Investment & Portfolio Theory - Chapter 5 Risk Aversion...

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Chapter 5: Risk Aversion and Investment Deci- sions, Part 1 5.1 Introduction Chapters 3 and 4 provided a systematic procedure for assessing an investor’s rel- ative preference for various investment payoffs: rank them according to expected utility using a VNM utility representation constructed to reflect the investor’s preferences over random payments. The subsequent postulate of risk aversion further refined this idea: it is natural to hypothesize that the utility-of-money function entering the investor’s VNM index is concave ( U 00 ( ) < 0). Two widely used measures were introduced and interpreted each permitting us to assess an investor’s degree of risk aversion. In the setting of a zero-cost investment paying either (+ h ) or ( - h ), these measures were shown to be linked with the minimum probability of success above one half necessary for a risk averse investor to take on such a prospect willingly. They differ only as to whether ( h ) measures an absolute amount of money or a proportion of the investors’ initial wealth. In this chapter we begin to use these ideas with a view towards understanding an investor’s demand for assets of different risk classes and, in particular, his or her demand for risk-free versus risky assets. This is an essential aspect of the investor’s portfolio allocation decision. 5.2 Risk Aversion and Portfolio Allocation: Risk Free vs. Risky Assets 5.2.1 The Canonical Portfolio Problem Consider an investor with wealth level Y 0 , who is deciding what amount, a , to invest in a risky portfolio with uncertain rate of return ˜ r . We can think of the risky asset as being, in fact, the market portfolio under the “old” Capital Asset Pricing Model (CAPM), to be reviewed in Chapter 7. The alternative is to invest in a risk-free asset which pays a certain rate of return r f . The time horizon is one period. The investor’s wealth at the end of the period is given by ˜ Y 1 = (1 + r f )( Y 0 - a ) + a (1 + ˜ r ) = Y 0 (1 + r f ) + a r - r f ) The choice problem which he must solve can be expressed as max a EU ( ˜ Y 1 ) = max EU ( Y 0 (1 + r f ) + a r - r f )) , (5.1) where U ( ) is his utility-of-money function, and E the expectations operator. This formulation of the investor’s problem is fully in accord with the lessons of the prior chapter. Each choice of a leads to a different uncertain payoff distri- bution, and we want to find the choice that corresponds to the most preferred 1
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such distribution. By construction of his VNM representation, this is the payoff pattern that maximizes his expected utility. Under risk aversion ( U 00 ( ) < 0), the necessary and sufficient first order condition for problem (5.1) is given by: E [ U 0 ( Y 0 (1 + r f ) + a r - r f )) (˜ r - r f )] = 0 (5.2) Analyzing Equation (5.2) allows us to describe the relationship between the investor’s degree of risk aversion and his portfolio’s composition as per the following theorem: Theorem 5.1: Assume U 0 ( ) > 0, and U 00 ( ) < 0 and let ˆ a denote the solution to problem (5.1). Then ˆ a > 0 E ˜ r > r f ˆ a = 0 E ˜ r = r f ˆ a < 0 E ˜ r < r f Proof : Since this is a fundamental result, its worthwhile to make clear its (straightforward) justification.
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