Consider the Arrow's portfolio model with one risky asset and one risk-free asset. The von Newmann-Morgenstern utility functions of an investor is: u(w) = In w, where w represents wealth, and In natural log. Denote as a the amount invested in the risky asset. Let the initial wealth be $10,000, the interest rate of the risk-free asset 5%, and the probability distribution of the return of the risky asset X = (1%, 5%; 0.55, 0.45). 1.1 Write the equation of the expected utility of final wealth 1.2 Take the first derivative of the expected utility with respect to a. 1.3 Now evaluate the first derivative, that you found in 1.2 above, at a = 0. What can you conclude about the optimal value of a? Why is that so?

Consider the Arrow's portfolio model with one risky asset and one risk-free asset. The von Newmann-Morgenstern utility functions of an investor is: u(w) = In w, where w represents wealth, and In natural log. Denote as a the amount invested in the risky asset. Let the initial wealth be $10,000, the interest rate of the risk-free asset 5%, and the probability distribution of the return of the risky asset X = (1%, 5%; 0.55, 0.45). 1.1 Write the equation of the expected utility of final wealth 1.2 Take the first derivative of the expected utility with respect to a. 1.3 Now evaluate the first derivative, that you found in 1.2 above, at a = 0. What can you conclude about the optimal value of a? Why is that so?

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

See similar textbooks

Related questions

Q: Simon has current wealth of 836, including $20 in cash. With probabil- ity Simon's money will be…

A: The expected value of contingent consumption in economic theory is a concept that relates to…

Q: Suppose that a decision maker is an expected utility maximiser with an increasing and continuous…

A: A risk-averse person does not want to take risks from economic activities. An increase in risk level…

Q: Stock X has a 9.5% expected return, a beta coefficient of 0.8, and a 30% standard deviation of…

A: The required rate of return (RRR) is the minimum return an investor will accept for owning a…

Q: Question 14 Sioned has a utility function of the formu(W) = ln(W). What is her coefficient of…

A: The Arrow-Pratt absolute risk aversion coefficient is given as: a = -u''(w)u'(w)Where u''(w) is the…

Q: probability

A: Decision-making refers back to the system of selecting a path of action among numerous alternatives…

Q: Consider the following portfolio choice problem. The investor has initial wealth w and utility u(x)…

A: Note:- since you have posted multiple subparts, as per guidelines we can solve only the first 3…

Q: Consider the following portfolio choice problem. The investor has initial wealth w and utility u(x)…

A: In economics, a portfolio problem is a problem in time finance, where an investor has to choose the…

Q: Suppose you are the money manager of an investment fund. The fund consists of four stocks with the…

A: Beta coefficient of portfolio = (Investment in Stock A / Total investment) * Beta of Stock A +…

Q: You have a log utility, U = In(W), where W is your wealth. Currently, you own $1,000. You are given…

A: Utility:The utility is want satisfying power of a commodity. It can be expressed in cardinal and…

Q: The risk free rate is 3%. The optimal risky portfolio has an expected return of 9% and standard…

A: The expected return is a term used to estimate the average return an investor can anticipate from an…

Q: Consider an individual who has Dollars to invest in two risky assets. The means and variances of the…

A: Given: Individual has A dollars in 2 risky assets. The means of rate of returns are μ1 < μ2 The…

Q: Priyanka has an income of £90,000 and is a von Neumann-Morgenstern expected utility maximiser with…

A: The Von Neumann-Morgenstern (VNM) utility index is an index that measures an individual's preference…

Q: Consider the following portfolio choice problem. The investor has initial wealth w and utility u(x)…

A: Risk aversion is a term used to describe an individual's preference for avoiding risk or uncertainty…

Q: a. Suppose you forecast that the standard deviation of the market return will be 20% in the coming…

A: a. The market risk aversion can be calculated as follows:

Q: 8. Risk and return Suppose Frances is choosing how to allocate her portfolio between two asset…

A: An investment is an asset or item obtained determined to create income or appreciation. Appreciation…

Q: The following table shows the betas and actual expected returns of 4 mutual funds in the market:…

A: Capital Asset Pricing Model is a model in finance that calculates the expected return rate for an…

Q: A risk-averse expected-utility maximizer has initial wealth w0 and utility function u. She facesa…

A: The objective of the question is to determine the expected utility of a risk-averse individual who…

Q: Consider the following portfolio choice problem. The investor has initial wealth w and utility u(x)…

A: Utility function, u(x) = x^n/n Initial wealth = w There is a safe asset (such as a US government…

Q: An investor has a power utility function with a coefficient of relative risk aversion of 3. Compare…

A: Utility measures the satisfaction, happiness, or well-being that a person derives from consuming…

Q: Using the Utility Function in Portfolio Management, where the utility function is the constant…

A: The certainty equivalent is a return that is assured and somebody would prefer to receive now over…

Q: Consider the following portfolio choice problem. The investor has initial wealth w and utility u(x)…

A: A key factor to understand investors and venture capitalist's decision making behavior is to study…

Q: Consider the following portfolio choice problem. The investor has initial wealth w and utility u(x)…

A: The level of risk aversion can vary from person to person, and is often influenced by factors such…

Question

1. **Consider the Arrow’s portfolio model with one risky asset and one risk-free asset. The von Neumann-Morgenstern utility functions of an investor is:**

$ u(w) = \ln w $,

where $ w $ represents wealth, and $ \ln $ is the natural log. Denote as $ \alpha $ the amount invested in the risky asset. Let the initial wealth be $10,000, the interest rate of the risk-free asset 5%, and the probability distribution of the return of the risky asset

$ \tilde{X} = (1\%, 5\%; 0.55, 0.45) $.

**1.1 Write the equation of the expected utility of final wealth.**

**1.2 Take the first derivative of the expected utility with respect to $ \alpha $.**

**1.3 Now evaluate the first derivative, that you found in 1.2 above, at $ \alpha = 0 $. What can you conclude about the optimal value of $ \alpha $? Why is that so?**

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Trending now

This is a popular solution!

Step by step

Solved in 4 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, economics and related others by exploring similar questions and additional content below.

Similar questions

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
10. Which one of the following measures may be used to measure the risk of an investment on its own? a) Expected return of the investment. b) Expected utility of the investment for an investor. c) Standard deviation of the possible outcomes of the investment. d) The Bernoullian utility function's value of a good investment outcome.
The value of a successful project is $420,000; the probabilities of success are 1/2 with good supervision and 1/4 without. The manager is risk neutral, not risk averse as in the text, so his expected utility equals his expected income minus his disutility of effort. He can get other jobs paying $90,000, and his disutility for exerting the extra effort for good supervision on your project is $100,000. (a) Show that inducing high effort would require the firm to offer a compensation scheme with a negative base salary; that is, if the project fails, the manager pays the firm an amount stipulated in the scheme. (b) How might a negative base salary be implemented in reality? (c) Show that if a negative base salary is not feasible, then the firm does better to settle for the low-pay, low-effort situation.
Expected utility in portfolio theory
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.
I need help with question d
You plan to invest $1,000 in a corporate bond fund or in a common stock fund. The following table represents the annual return (per $1,000) of each of these investments under various economic conditions and the probability that each of those economic conditions will occur. Compute the expected return for the corporate bond and for the common stock fund. Show your calculations on excel for expected returns. Compute the standard deviation for the corporate bond fund and for the common stock fund. Would you invest in the corporate bond fund or the common stock fund? Explain. If choose to invest in the common stock fund and in (c), what do you think about the possibility of losing $999 of every $1,000 invested if there is depression. Explain.
(a) Calculate the risk-premium on this portfolio and provide a brief interpretation of it (b) Calculate the minimum sale price of the capital assets for the average investor.
advanced microeconomics, uncertainty
A widely used utility function in the economics literature is the constant rate of risk aversion utility function. It is given by: utc) = c*(1-n) (1-n) Assume that an agent lives tor three periods (t-0,1,2) and discounts future utity at rate B (per period), The agent is born with asset level ag and his/her labour market income is yo and y, for periods O and 1 respectively, the agent retire in the last period (no labour income). The interest rate in this economy is Please answer the following questions based on the information displayed here. Choose the best option avallable. Select one: O a. The budget constrain in period te2 is given by: C2 + ay = az(1+r) O b. The budget constrain in period te2 is given by: c2= az{1+1) O G. The budget constrain in period t=2 is glven by: ute, )+ ag = a;(1+1) +ya d. The budget constrain in period t=2 is given by: C2= az(1+r) +y3 e. The budget constrain in period te2 is given by: c, + ag = az(1+) +ya
Use the utility function u = E (r,) – 0.5 A o, where A is the risk aversion parameter and A = 3.5. Use the average annual return on the S&P 500 as E(r,). Use the average annual interest rate on the US Government 10-year treasury bond over the past decade as r Solve for the investor's optimal allocation between the risky and the risk-free asset.