I need the answer of question number 5 and 6.

I need specific process of the solutions.

Homework 3 UNIVERSITY OF MINNESOTA ECON 4751 - Financial Economics Due Thursday, February 25, 2016 Instructions: Please answer all questions clearly and concisely. Show all work. This can include screenshots of an appropriately set-up Excel spreadsheet. If Excel is used, please indicate which formulas were used in which cells. Illegible homework will not be graded and will receive an automatic 0. You may handwrite or type up problem sets. 20 points total. 1 point for the Problem 1, 3 points for problem 2, 3 points for problems 3, 6 points for problem 4, 3 points for problem 5, and 4 points for problem 6 . Problem 1: Dene the following terms: Risk Neutral Fair Game Capital Allocation Line Systematic Risk Portfolio Opportunity Set Problem 2: Suppose there are many stocks in the security market and that the characteristics of stocks A and B are given by E(rA ) = 0.1, E(rB ) = 0.15, (rA ) = 0.05, (rB ) = 0.1 and Corr(rA , rB ) = 1. Here represents standard deviation. Investors are not constrained to invest their wealth in only one security but they can invest it accross dierent securities. Is it possible in this economy to invest at a risk-free rate? If yes, what will be the risk free rate? Explain. Problem 3: This problem will get you familiar to the idea that investors with dierent risk aversion coecients will make dierent choices when facing the same assets. It will also give you practice with drawing and labeling indierence curves passing through assets on an expected return - standard deviation graph. 1 Suppose Owen is an investor with risk aversion coecient 1, Theo is an investor with risk aversion coecient 2, and Tony is an investor with risk aversion coecient 3. They all have utility functions given by 1 U (r) = E(r) A V ar(r). 2 There are four assets available in the nancial markets. The rst asset is risk-free and has expected return given by r = 0.03. Asset A is risky and has an expected return of 0.07 and a variance of 0.03. Asset B is risky and has an expected return of 0.08 and a variance of 0.04. Asset C is risky and has an expected return of 0.09 and a variance of 0.06. A. If Owen is choosing one of the four assets to invest all his wealth in, which one would he choose? Why? How about Theo and Tony? B. Draw and label each asset on the graph that has expected return on the vertical axis and standard deviation on the horizontal axis. C. Draw indierence curves that pass through each asset for Owen. Please pay attention to the intersection of these indierence curves with the vertical axis. Problem 4: Suppose there are two assets available to an investor. One is risk-free and has a return of 3 percent. The other is risky and has an expected return of 8 percent and a variance of 0.05. The investor's utility is given by 1 2 U (r) = E(r) AV ar(r) 3 2 and his risk aversion coecient is 3. A. The investor is trying to decide what fraction of his wealth he will invest in the risky asset. Write down the investor's maximization problem. B. Take the rst order conditions for the investor's problem. Find a formula for y , the optimal fraction of wealth that the investor will invest in the risky asset. C. Find the value of y given the characteristics of both assets and the risk-aversion of the investor. Suppose now that the investor's utility was given by 1 U (r) = E(r) A(r) 2 where A is his risk aversion coecient and is the standard deviation of returns. Suppose investor cannot borrow at any risk-free rate. D. Fix a level of utility for this investor and use it to draw an indierence curve on a graph that has expected return on the vertical axis and standard deviation on the horizontal axis. 2 E. Write down a rule about how the investor will choose what fraction of wealth he will invest in the risky asset and which fraction in the risk-free asset. F. Apply the rule you wrote above to nd the value of y given the characteristics of both assets and the risk-aversion of the investor. Problem 5: George works as a nancial advisor in Wall Street. He typically invests in a collection of 50 equities drawn from several dierent industries. During a meeting with his clients, one client told him the following: \"I trust your stock-picking ability and I would like to hear some advice on how I should invest my money in your ve best ideas. Why invest in 50 companies when you obviously have strong opinions on a few of them?\". George plans to respont to his client within the context of modern portfolio theory. A. Contrast the concepts of systematic risk and rm-specic risk, and give an example of each type of risk B. Critique the client's suggestion. Discuss how both systematic and rm-specic risk change as the number of securities in a portfolio increases. Problem 6: An investor is considering investing $100,000 in three mutual funds. The rst is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a risk-free rate of 5 percent. Returns on the stock fund and the bond fund have a correlation coecient of 0.1 and the following characteristics: Description Stock fund Bond fund Expected Return 17 % 9% Standard deviation 20% 12% Solve the following exercises in the order you think appropriate, but please make sure you answer all of them. A: When picking the optimal risky-portfolio, how much will the investor invest in each of the risky mutual funds? B: What is the expected value, standard deviation and the Sharpe ratio of the optimal risky portfolio? C: If the investor has a standard utility function over returns and has a risk aversion coecient of 3, how much will the investor invest in the risk-free mutual fund? D: What is the expected value, standard deviation and the Sharpe ratio of the optimal portfolio? 3 Chapter 6 Risk Aversion and Allocation to Risky Assets Recall Micro Theory Good B Indifference Curves U2 U3 U1 Budget Line Good A Portfolio Choice: Capital Allocation $$$$ Spending (Consumption) Saving (Investment) Risk-Free Asset Risky Asset In this chapter, we study the optimal allocation of capital into risk-free and risky assets At this point, we are not focusing on the decision of which risky assets to choose. Risk Aversion Though there are some individuals who enjoy gambling, we generally assume that investors want to reduce variance in returns An investor who is risk-averse will: - Reject fair-games or worse when there is any uncertainty - Will require a higher expected return to accompany higher variance A risk-neutral investor only cares about expected return and is indifferent to variance Utility Function for Portfolios Investor's utility for a risky portfolio depends on: - Expected return, E(r) - Variance, 2 - Investor's risk aversion index, A (A>0 for risk-averse) So for risk-averse investors, higher variance decreases utility \"Certainty Equivalent\" If we use the given utility function, we can ask: for an investor with risk aversion A, and portfolio with expected return E(rc) and standard deviation what would be the riskfree rate of return to get the same utility? Suppose E(rc) = .07, Example Using the following data, find which of the investments would give each investor their highest utility. In addition, for the investor with risk aversion 1.5 please draw the indifference curves for each option. The risk free rate is 0.04 There are three investors with three different risk aversion coefficients: 1.5, 3, 7 There are three risky investments - 1: risk premium of 0.03, std dev of 0.1 - 2: risk premium of 0.09, std dev of 0.22 - 3: risk premium of 0.14, std dev of 0.48 Computing Utility For each investor, we can choose an optimal portfolio from the 4 choices given Preferred Portfolio Assume that an investor is risk-averse. By the mean-variance criterion, portfolio A dominates portfolio B if: with at least one strict inequality Graphical Representation E(r) Expectation (good) Q R Portfolio P S St. Deviation (bad!) Indifference Curves U3 E(r) Q Portfolio P Certainty Equivalent for Portfolio P U1 U2 S Capital Allocation Investor has a certain amount of wealth to invest in risk-free (F) and risky (P) assets The proportion in invested in P is denoted by \"y\" so the remaining \"1-y\" goes into F. Investor can decide further, what to do with the risky portfolio: equity, bonds, etc. - We will leave that decision for later Capital Allocation Wealth y 0.00 0.25 0.50 0.75 0.90 1.00 100,000 risky asset risk-free Amount in P Amount in F $0 $100,000 $25,000 $75,000 $50,000 $50,000 $75,000 $25,000 $90,000 $10,000 $100,000 $0 Portfolio Choice Set For now, the investor can set the allocation to risky asset, y, anywhere from 0 to 1. (If y were outside this range, what would we be doing?) Allocation to risk-free asset = 1-y We can compute expected return and standard deviation of return for any value of y between 0 and 1. Expected Return Rate of return on risky asset is rP Rate of return on risk-free asset is rf Return on complete portfolio is rC= yrP + (1-y) rf Taking expectation, we have: Can rewrite this as: Standard Deviation The risk-free asset doesn't contribute to variance Taking square root, we get So now we have, for any y, the expected return and standard deviation - we can graph it Capital Allocation Line (CAL) E(r) Expectation (good) y=0 E(rc)=rf E(rc)=rf+y(E(rp)-rf) c p y=1 E(rc)=E(rp) Portfolio for some value of y Between 0 and 1 St. Deviation (bad!) Capital Allocation Line When y=0, we have rc=rf and We can compute the slope of the line: This is the reward-to-volatility ratio (Sharpe ratio) Exercise: Draw the CAL Suppose we have risky asset P with E(rp)=0.13 and 0.25. rf=0.05. What is the expected return and standard deviation when y=0, 0.5, 1 ? What is the y-intercept and slope of the CAL? Draw the CAL. Capital Allocation and Risk Aversion We can compute the \"optimal\" level of y for different types of investors. As \"A\" varies, so does the best choice of y for maximizing utility, U. Expectation and variance are determined by the parameters (rf, E(rp), ) and y, the choice variable. Exercise Set up the portfolio allocation problem. What is the optimal fraction invested in the risky portfolio when: - Risk free=0.03, A=2, portfolio expected return=0.05, and portfolio standard deviation=0.04 - Risk free=0.02, A=1.5, portfolio expected return=0.07, and portfolio standard deviation=0.09 Utility Function The utility function is at a maximum when But make sure this is an interior solution! (strictly between 0 and 1). One of the endpoints might be the solution. How do we know that y* > 0 ? Identifying the Optimal Allocation Indifference Curve Analysis Draw CAL (as set of efficient combinations of expectation and standard deviation) Draw indifference curves Choose allocation that is on the highest-utility indifference curve and on the CAL. This is actually more time-consuming and tedious than using the derivative... Indifference Curve Analysis U3 E(r) Q U1 U2 CAL Portfolio P S Capital Market Line (CML) For risk-free asset, use the 1-month T-bills For risky asset, use a broad index of stocks. The Capital Allocation line between these two assets is called the CML Investing in a broad index is an example of a passive strategy - as opposed to an active strategy of analysis and selection of individual stocks. What is the aggregate value of A? Assumptions: - 85% of net worth is in risky assets (P) - Risk premium on P is .079 - Standard deviation of P is .208 So we have y* = .85, E(rp)-rf = .079, =.208 Substituting into equation for optimal y, A = 2.15 Exercise You estimate that a passive portfolio (e.g. an S&P 500 Index) yields an expected return of 13% with a standard deviation of 25%. You manage an active portfolio with an expected return of 18% and standard deviation 28%. The risk free rate is 8%. Draw the CML and the CAL for your fund. Which is better? Suppose a client of your fund wants to switch the 70% of his wealth invested in your fund to the S&P index, should he? Suppose he decides to leave, what fee (as a percentage of the return from your fund) could you charge to make him want to stay? What fee would you charge to another investor with a higher risk aversion coefficient? Leveraged Portfolios Can increase investment budget by borrowing to put more in the risky asset P. So y > 1. Lenders will demand an interest rate greater than the risk free rate (due to possibility of investor default). This is denoted by rfB. For values of y > 1, the return is: y*rp+(1-y)*rfB (what will this do to the CAL?) Can write this as: rC=rfB + y [rP-rfB] Leveraged Portfolio For y>1 E(rC) Leveraged portion of CAL rfB rf Unleveraged portion of CAL C More on risk aversion Often, a utility function in economics will look at utility as a function of different amounts of wealth, U(w) Utility is increasing in wealth: U'(w) > 0 Generally we see decreasing marginal utility: U''(w)