Problem 4: Recall the following portfolio allocation problem: an individualwith initial wealth wo has to choose an allocation between a safe asset (with azero rate of return) and a risky asset with a random rate of return & € [−1, 1],where E[8] > 0. If the individual invests & dollars in the risky asset, then finalwealth is (wo-x)+x(1+5) = w₁ +x6. The individual chooses to maximizeexpected utility denoted by E[u(wo +xd)].1. Suppose the Bernoulli utility function is u(w)solute risk aversion is a constant.= -e. Show that ab-2. Let x* (wo) denote the optimal amount of investment in the risky asset.Show that for the utility function of Part a., da = 0, i.e. the amount ofinvestment in the risky asset is independent of initial wealth. Explain thisresult.dwo3. Now suppose that the utility function is u (w) = aw - bw², where a, b > 0.The parameters a and b are such that marginal utility of wealth is positivefor all w. What can we say about in this case? Explain your result.

The utility function of the consumer shows the satisfaction received by the consumer is based on the…

Problem 4: Recall the following portfolio allocation problem: an individual with initial wealth wo has to choose an allocation between a safe asset (with a zero rate of return) and a risky asset with a random rate of return = [−1,1], where E[6] > 0. If the individual invests x dollars in the risky asset, then final wealth is (wo-x) + x (1 + d) = w₁ +xd. The individual chooses x to maximize expected utility denoted by E[u(w₁ + xd)]. 1. Suppose the Bernoulli utility function is u(w) solute risk aversion is a constant. dwo ew. Show that ab- 2. Let x*(wo) denote the optimal amount of investment in the risky asset. Show that for the utility function of Part a., da = 0, i.e. the amount of investment in the risky asset is independent of initial wealth. Explain this result. 3. Now suppose that the utility function is u (w) = aw – ½bw², where a, b > 0. The parameters a and b are such that marginal utility of wealth is positive for all w. What can we say about in this case? Explain your result.

Problem 4: Recall the following portfolio allocation problem: an individual with initial wealth wo has to choose an allocation between a safe asset (with a zero rate of return) and a risky asset with a random rate of return = [−1,1], where E[6] > 0. If the individual invests x dollars in the risky asset, then final wealth is (wo-x) + x (1 + d) = w₁ +xd. The individual chooses x to maximize expected utility denoted by E[u(w₁ + xd)]. 1. Suppose the Bernoulli utility function is u(w) solute risk aversion is a constant. dwo ew. Show that ab- 2. Let x*(wo) denote the optimal amount of investment in the risky asset. Show that for the utility function of Part a., da = 0, i.e. the amount of investment in the risky asset is independent of initial wealth. Explain this result. 3. Now suppose that the utility function is u (w) = aw – ½bw², where a, b > 0. The parameters a and b are such that marginal utility of wealth is positive for all w. What can we say about in this case? Explain your result.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter7: Uncertainty

Section: Chapter Questions

Problem 7.7P

See similar textbooks

Similar questions

9) A risk-neutral individual with current wealth w has already decided investing all his wealth in a project that has two possible net wealth outcomes wn and we (with probabilities pr and pe, respectively) where wr > w > we > 0, PrPe E (0,1), and pr + Pe = 1. Before he invests, he realises that there is a source of information that tells the individual which outcome will be realised with truth, what is the value of this information for the individual? a) pe(@ – w;), b) (Pn – Pe)w, c) Phwn + PeWe – w, d) Pr(w – wn).
3) A risk-loving individual has $1000 to invest. The individual maximizes his/her expected utility and has a monotonic utility function. Show that he/she will never choose a diversified portfolio - that is, show that he/she will either keep the entire $1000 in a safe, or invest the entire $1000 in a risky assesst, for which each $1 invested yields $] with probability p, and SB with probability (1-p), where $B<$1<$J.
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?
1. Tanner is choosing between two investment options. He can invest $500 now and get (guaranteed) $550 in one year, or invest $500 now and get (guaranteed) $531.40 back later today. The risk-free rate is 3.5%. Which investment should Tanner prefer? A) $531.40 later today, since $1 today is worth more than $1 in one year. B) $550 in one year, since it is $50 more than he invested rather than $31.40 more than he invested. C) Neither - both investments have a negative NPV. D) Tanner should be indifferent between the two investments, since both are equivalent to the same amount of cash today.
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?
A maximizing investor with preferences u(u, o) = 0.2u – 0.50^2 will allocate a portfolio worth 4000 between a risk free asset with a return of 4 percent and the market asset with a return of 20 percent and risk of 4 percent. How many dollars should be invested in the market asset? %3D
5. Find the expected value assuming the risk factor is 30 % and the interest rate is 12% , if you will receive $20,000 one year from today.
Suppose Caroline is choosing how to allocate her portfolio between two asset classes: risk-free government bonds and a risky group of diversified stocks. The following table shows the risk and return associated with different combinations of stocks and bonds.CombinationFraction of Portfolio in Diversified StocksAverage Annual ReturnStandard Deviation of Portfolio Return (Risk)(Percent)(Percent)(Percent)A 0 1.50 0B 25 3.00 5C 50 4.50 10D 75 6.00 15E 100 7.50 20There is a relationship between the risk of Caroline's portfolio and its average annual return.Suppose Caroline currently allocates 75% of her portfolio to a diversified group of stocks and 25% of her portfolio to risk-free bonds; that is, she chooses combination D. She wants to reduce the level of risk associated with her portfolio from a standard deviation of 15 to a standard deviation of 5. In order to do so, she must do which of the following? Check all that apply. Sell some of her stocks and use the proceeds to purchase…
Buying and selling prices for risky investments obviously are related to certain equivalents. This problem, however, shows that the prices depend on exactly what is owned in the first place. Suppose that your utility for wealth (A) can be represented by the utility function u(A) = In [(A)] You currently have R1000 in cash. A business deal of interest to you yields a reward of R100 with probability 0,5 and RO with probability 0,5. 2.1 If you own this business deal in addition to the R1000, what is the smallest amount for which you would sell the deal? 2.2 Suppose you do not own the deal. Formulate an appropriate equation and solve with algebra to find the largest amount you would be willing to pay for the deal. 2.3 Explain why the amounts in 2.1 and 2.2 are slightly different.
Please explain in detail about expected utility to get a positive upvote. An individual has a utility function U = W¼, where W is her total wealth. She has one safe asset worth Rs 5,000, and another risky asset whose value can be either Rs 5,000 or Rs 1,400 with equal probabilities. What is her expected utility? (a) Rs 11,400 (b) Rs 100 aw lo boeoqmoo vmonoos to on g cubire cou s o iva alagos ad a adWnooni lanou lo OAuti (c) Rs 2,580 (d) Rs 90