AMS Portfolio Theory and Capital Markets I

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1 ams-q01-sol-05-m.nb 1 AMS Portfolio Theory and Capital Markets I Solutions 5 - Utility Theory and Portfolio Choice Robert J Frey, Research Professor Stony Brook University, Applied Mathematics and Statistics frey@ams.sunysb.edu Solutions to exercises for Class 5. Chapters refer to Luenberer s Investment Science. February 23, 2005 Chapter 9, Problem 2 Let f(x) be the pdf of the payoff x. Note that the terminal wealth iven x is W w + x. The expected utility of terminal wealth is, therefore, - -a HW -w+xl f HxL x = - -a W -a Hx-wL f HxL x = - -a W -a Hx-wL f HxL x Thus, if we are comparin two investments with price and payoff {w 1, x 1 } and {w 2, x 2 }, then their relative rankin will be independent of of W. Chapter 9, Problem 6 We'll start out with the HARA expression itself. Linear hara = 1 - ÅÅÅÅÅÅÅÅÅÅÅ i j a x ÅÅÅÅÅÅÅÅÅÅÅ k by { z ; If we set a = 1 and then take the limit as Ø 1, then Limit@hara ê. a Ø 1, Ø 1D x

2 ams-q01-sol-05-m.nb 2 Quadratic If we look at HARAs with = 2, then we et the quadratic polynomial ExpandA 1 - ÅÅÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅÅ b2 2 + a b x - a2 x ÅÅÅÅÅÅÅÅÅÅÅÅ 2 2 z i j ÅÅÅÅÅÅÅÅÅÅÅ a x k by { ê. Ø 2E An affine transformation puts U(x) into the form we want with c = a / 2b. SimplifyA% 1 ÅÅÅÅÅÅÅÅ a b + ÅÅÅÅÅÅÅÅ b 2 a E x - a x2 ÅÅÅÅÅÅÅÅÅ 2 b Exponential If we set b =1 and then take the limit as Ø -,then Limit@hara ê. b Ø 1, Ø -D - -a x Power For simplicity we set a =1 and then take the limit as b Ø 0. Limit@hara ê. a Ø 1, b Ø 0D x H ÅÅÅÅÅÅÅ x 1- ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ L-1+ We can further simplify this to H1 - L 1- ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ and then make the substitution that c = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ H1-L1- to brin it into its final form. Loarithmic The text suests lookin at the function below. We'll take its limit as Ø 0 x

3 ams-q01-sol-05-m.nb 3 H1 - L1- LimitA ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Hx - 1L, Ø 0E Lo@xD Note that for 0 this function is equivalent to a power law utility (with an affine transformation of its standard form). Thus, the lo utility function is the dividin line between power laws with positive and neative exponents. Arrow-Pratt The Arrow-Pratt risk aversion coefficient is shown below. Note that this can easily be restated in form desired. D@#, 8x, 2<D ahara = SimplifyA- ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ E &@harad D@#, xd a - a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ b + a x - b We now make the substituton c = a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ a - a and d = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ b - b a - a SimplifyAaHara ã True 1 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ c x + d ê. 9c Ø a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ a - a, d Ø ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ b - b a - a =E Chapter 9, Problem 14 Let p = 0.25 be the probability of winnin the race. For a $1 bet we have w = 5 as the payoff on a win and l = 0 as the payoff on a loss. The individual's utility function is U(x) = è!!! x. The initial wealth is W. Our objective is to determine the optimal bet b. The expected utility is E@UHxLD = H1 - pl è!!!!!!!!!!! W - b + p è!!!!!!!!!!!!!!!! W + bhw!!!!!!!!! - 1L We differentiate with respect to b and solve for the root.

4 ams-q01-sol-05-m.nb 4 sol = SolveA b IH1 - pl è!!!!!!!!! W - b + p è!!!!!!!!!!!!!!!!!!!!!!!!! W + b Hw - 1L M ã 0, bep1, 1T b Ø H p - 2 p2 w + p 2 w 2 L W ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ H-1 + wl H1-2 p + p 2 wl This is the eneral solution. We can now substitute the actual parameters into the rule and then recover the actual value of b. b ê. sol ê. 8p Ø 0.25, w Ø 5< W Thus, Gavin ouht to bet rouhly 13.5% of his wealth on the bet. A plot of expect utility illustrates the result. PlotA IH1 - pl è!!!!!!!!! W - b + p è!!!!!!!!!!!!!!!!!!!!!!!!! W + b Hw - 1L M ê. 8W Ø 1, p Ø 0.25, w Ø 5<, 8b, 0, 0.3<E; As an additional illustration, we'll repeat the analysis with lo utility. Note that this ives a more conservative result.

5 ams-q01-sol-05-m.nb 5 sol = Solve@ b HH1 - pl Lo@W - bd + p Lo@W + b Hw - 1LDL ã 0, bdp1, 1T b ê. sol ê. 8p Ø 0.25, w Ø 5< Plot@HH1 - pl Lo@W - bd + p Lo@W + b Hw - 1LDL ê. 8W Ø 1, p Ø 0.25, w Ø 5<, 8b, 0, 0.15<D; b Ø ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ -W + p w ÅÅÅÅÅÅÅ W -1 + w W H5-1L H-1L 0.25