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An investor measures the utility of her wealth using the utility function Uw) = In(w) for w > 0. (i) Derive the absolute and relative risk aversions for this investor's utility function, and the first derivative of each. [4] (ii) Comment on what this tells us about the proportion of her assets that this investor will invest in risky assets. [2] The investor has $100 available to invest in two possible assets, Asset A and Asset B. The future value of Asset A depends on an uncertain future event. . Every El invested in Asset A will be worth El.30 with probability 0.75 and 10.40 with probability 0.25. Asset B is risk-free, so every El invested in Asset B will always be worth $1. The investor does not discount future asset values when making investment decisions She decides to invest a proportion a of her wealth in Asset A and the remaining proportion 1 - a in Asset R. (iii) Express her expected utility of wealth in terms of a. [2] (iv) Determine the amount that she should invest in each of Asset A and B to maximise her expected utility, using your result from part (iii). [5] [Total 13] Consider an asset whose return follows the probability density function /(x). (i) Write down a formula for the variance of the return on the asact, defining any additional notation you use. [1] (ii) Write down a formula for the shortfall probability for the return on the asset below a level L. [1] The relurns on an asset follow a Normal distribution with mean u = 6% per annum and variance o" = 23% per annum. An investor buys (500 of the asset. (iii) Determine the shortfall probability for the value of the asset in one year's time below a value of 6480. 121 (iv) Explain what can be deduced about an investor's utility function if the investor makes decisions based on: (a) the variance of returns. (b) the shortfall probability of returns. [2] [Total 6]