Consider the following portfolio choice problem. The investor has initial wealth w and utility u(x) = . There is a safe asset (such as a US government bond) that has net real return of zero. There is also a risky asset with a random net return that has only two possible returns, R₁ with probability 1- q and Ro with probability 9. We assume R₁ <0, Ro> 0. Let A be the amount invested in the risky asset, so that w - A is invested in the safe asset. 1) What are risk preferences of this investor, are they risk-averse, risk neutral or risk-loving?

Consider the following portfolio choice problem. The investor has initial wealth w and utility u(x) = . There is a safe asset (such as a US government bond) that has net real return of zero. There is also a risky asset with a random net return that has only two possible returns, R₁ with probability 1- q and Ro with probability 9. We assume R₁ <0, Ro> 0. Let A be the amount invested in the risky asset, so that w - A is invested in the safe asset. 1) What are risk preferences of this investor, are they risk-averse, risk neutral or risk-loving?

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter7: Uncertainty

Section: Chapter Questions

Problem 7.9P

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ANSWER E PLEASE ONLY Consider the following portfolio choice problem. The investor has initial wealth w andutility u(x) = (x^n) / n. There is a safe asset (such as a US government bond) that has netreal return of zero. There is also a risky asset with a random net return that has onlytwo possible returns, R1 with probability 1 − q and R0 with probability q. We assumeR1 < 0, R0 > 0. Let A be the amount invested in the risky asset, so that w − A isinvested in the safe asset.a) What are risk preferences of this investor, are they risk-averse, riskneutral or risk-loving?b) Find A as a function of w. c) Does the investor put more or less of his portfolio into the risky assetas his wealth increases? d) Now find the share of wealth, α, invested in the risky asset. How doesα change with wealth? e) Calculate relative risk aversion for this investor. How does relativerisk aversion depend on wealth?
Consider the following portfolio choice problem. The investor has initial wealth w andutility u(x) = (x^n) /n. There is a safe asset (such as a US government bond) that has netreal return of zero. There is also a risky asset with a random net return that has onlytwo possible returns, R1 with probability 1 − q and R0 with probability q. We assumeR1 < 0, R0 > 0. Let A be the amount invested in the risky asset, so that w − A isinvested in the safe asset.a) What are risk preferences of this investor, are they risk-averse, riskneutral or risk-loving?b) Find A as a function of w.
Consider the following portfolio choice problem. The investor has initial wealth w andutility u(x) = (x^n) /n. There is a safe asset (such as a US government bond) that has netreal return of zero. There is also a risky asset with a random net return that has onlytwo possible returns, R1 with probability 1 − q and R0 with probability q. We assumeR1 < 0, R0 > 0. Let A be the amount invested in the risky asset, so that w − A isinvested in the safe asset. Calculate relative risk aversion for this investor. How does relative risk aversion depend on wealth?
Consider the following portfolio choice problem. The investor has initial wealth w andutility u(x) = (x^n) /n. There is a safe asset (such as a US government bond) that has netreal return of zero. There is also a risky asset with a random net return that has onlytwo possible returns, R1 with probability 1 − q and R0 with probability q. We assumeR1 < 0, R0 > 0. Let A be the amount invested in the risky asset, so that w − A isinvested in the safe asset.1) What are risk preferences of this investor, are they risk-averse, riskneutral or risk-loving?2) Find A as a function of w.
ANSWER C AND D PLEASE ONLY Consider the following portfolio choice problem. The investor has initial wealth w andutility u(x) = (x^n) / n. There is a safe asset (such as a US government bond) that has netreal return of zero. There is also a risky asset with a random net return that has onlytwo possible returns, R1 with probability 1 − q and R0 with probability q. We assumeR1 < 0, R0 > 0. Let A be the amount invested in the risky asset, so that w − A isinvested in the safe asset.a) What are risk preferences of this investor, are they risk-averse, riskneutral or risk-loving?b) Find A as a function of w. c) Does the investor put more or less of his portfolio into the risky assetas his wealth increases? d) Now find the share of wealth, α, invested in the risky asset. How doesα change with wealth?
Hugo has a concave ubility function of U(W)=√W. His only asset is shares in an Internet start-up company. Tomorrow he will learn the stock's value. He belleves that it is worth $225 with probability 80% and $256 with probability 20%. What is his expected utsty? What risk premium would he pay to avoid bearing this risk? The stock's expected utility (EU) is EU = (Enter a numeric response using a real number rounded to two decimal places.) han fro
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Two stocks are available. The corresponding expectedrates of return are r¯1 and r¯2; the corresponding variances and covariances areσ12, σ22, and σ12. What percentages of total investment should be invested ineach of the two stocks to minimize the total variance of the rate of return ofthe resulting portfolio? What is the mean rate of return of this portfolio?
A risk-averse expected-utility maximizer has initial wealth w0 and utility function u. She facesa risk of a financial loss of L dollars, which occurs with probability π. An insurance companyoffers to sell a policy that costs p dollars per dollar of coverage (per dollar paid back in theevent of a loss). Denote by x the number of dollars of coverage.(a) Give the formula for her expected utility V (x) as a function of x.(b) Suppose that u(z) = −e−zλ, π = 1/4, L = 100 and p = 1/3. Write V (x)using these values. There should be three variables, x, λ and w. Find the optimal value of x,as a function of λ and w, by solving the first-order condition (set the derivative of the expectedutility with respect to x equal to zero). (The second-order condition for this problem holds butyou do not need to check it.) Does the optimal amount of coverage increase or decrease in λ,where λ > 0?(c) Repeat exercise (b), but with p = 1/6.(d) You should find that for either (b) or (c), the optimal coverage…
Jen is choosing a portfolio. For this choice, she is an expected utility maximizer. We fix the following preference representation for Jen: if she earns w dollars with probability 1, her utility is √w. There are two stocks she can buy, A or B. She will choose one. Stock A will be worth 1000 with probability ¹/2, and it will be worth 2000 with probability 1/2. Stock B will be worth 250 with probability 2/3 and 5000 with probability 1/3. Which does she choose?
Consider an investor with initial wealth yo, who maximizes his expected utility from final wealth, E[u()]. This investor can invest in two a risky securities, 1 and 2, with random return ři and ř2. Those risky returns are two binomial variables, perfectly correlated. More specifically, with probability p we have ř =r and r, = r+ô, and with probability 1- p we have î = 0 and ř2 = -6, where r> 0 and ổ > 0. We assume that this investor has log preferences, that is u(y) = log(y). 1. For a given fraction, a, of the initial wealth, invested in risky security 2, what is the distribution of final wealth, g1? 2. Determine the expression of Elu(1)] as a function of a. 3. Explain why there is an upper bound and a lower bound for a, and determine those bounds. 4. Determine the optimal fraction a*. 5. When p = 0.5, describe qualitatively the optimal investment strategy. Does it make sense?
Suppose you visit with a financial adviser, and you are considering investing some of your wealth in one of three investment portfolios stocks, bonds, or commodities. Your financial adviser provides you with the following table, which gives the probabilities of possible returns from each investment To maximize your expected return, you should choose: Stocks Bonds Probability Return Probability Return 0.15 20% 0.15 16.7% 06 10% T 04 7.5% 0.25 8% 0.45 3.3% OA bonds OB stocks OC. commodities OD. All of the portfolios have the same expected return. If you are risk-averse and had to choose between the stock or the bond investments, you would choose OA the stock portfolio because there is less uncertainty over the outcome OB. the bond portfolio because there is less uncertainty over the outcome. OC. the stock portfolio because of greater expected return. OD. the bond portfolio because of greater expected return. Commodities Probability Return 02 20% 0.2 15% 0.2 8% 02 02 5% 0%