An investor has initial wealth wo that she must divide between two risky assets. Denote the...
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An investor has initial wealth wo that she must divide between two risky assets. Denote the total investment into asset i by w; for i E {1, 2}. There are two possible states H and L for the economy. Assume that the economy will be in state H with probability p and in L with probability (1-p).The first asset pays a gross return RH if the state of the economy is H and a gross return 0 if the state is L. I.e. in state H, an investment of w₁ in asset 1 results in a return w₁RH and in state L, the return is 0. The second asset pays R in both states. Denote the fraction of initial wealth that is invested in asset i by a for i = {1,2} so that w; = a,wo and a₁ + a₂ = 1. (a) For an arbitrary investment portfolio {a₁, a2), determine the final wealth of the investor. (b) Assume the investor has a Bernoulli utility function u (w) where w denotes the final wealth. Assume that u (w) > 0 and u" (w) < 0. Write down the expected utility resulting from an arbitrary portfolio {01, 02} and characterize the first order conditions for the optimal portfolio. Argue also that second order conditions are also satisfied at the point that satisfies the first order conditions. (c) Consider now a third asset that pays 0 in H and R per unit invested in L. Assume further that the utility function takes the form u (w) = In w.For what values of p, RH, R¹, R will the in- vestor invest in assets 1 and 3 only? An investor has initial wealth wo that she must divide between two risky assets. Denote the total investment into asset i by w; for i E {1, 2}. There are two possible states H and L for the economy. Assume that the economy will be in state H with probability p and in L with probability (1-p).The first asset pays a gross return RH if the state of the economy is H and a gross return 0 if the state is L. I.e. in state H, an investment of w₁ in asset 1 results in a return w₁RH and in state L, the return is 0. The second asset pays R in both states. Denote the fraction of initial wealth that is invested in asset i by a for i = {1,2} so that w; = a,wo and a₁ + a₂ = 1. (a) For an arbitrary investment portfolio {a₁, a2), determine the final wealth of the investor. (b) Assume the investor has a Bernoulli utility function u (w) where w denotes the final wealth. Assume that u (w) > 0 and u" (w) < 0. Write down the expected utility resulting from an arbitrary portfolio {01, 02} and characterize the first order conditions for the optimal portfolio. Argue also that second order conditions are also satisfied at the point that satisfies the first order conditions. (c) Consider now a third asset that pays 0 in H and R per unit invested in L. Assume further that the utility function takes the form u (w) = In w.For what values of p, RH, R¹, R will the in- vestor invest in assets 1 and 3 only?
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Answer rating: 100% (QA)
a The final wealth of the investor is given by final wealth pw1R1 w2R2 1 pw1R1 w2R2 w1pR1 1 pR1 w2pR... View the full answer
Related Book For
Income Tax Fundamentals 2013
ISBN: 9781285586618
31st Edition
Authors: Gerald E. Whittenburg, Martha Altus Buller, Steven L Gill
Posted Date:
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