Question 2 with full solutions and steps. Need to be finished in 3 hours.

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University of Waterloo Math-Business Program ACTSC 372: Corporate Finance II Assignment 2: On the Mean-Variance Portfolio Theory Spring 2014 Dr. H. Fahmy The Assignment is due on Thursday, June 12, 2014 at the beginning of class. Instructions: Late assignments are not acceptable. All your answers should be justi...ed. Final answers without any justi...cations are worth zero points. 1. [8] Suppose you have the following data on the rates of return of two assets, asset 1 and asset 2, and their corresponding probabilities in dierent states of the economy. State of Nature Expansion State Steady State Recession State re1 (%) 2 4 0 Probability 0:5 0:25 0:25 re2 (%) 0 1 3 (a) [2.5] Compute the mean, variance, and covariance of the rates of return of both assets. (b) [2.5] State the Markowitz problem (no derivation is necessary) and show that the solution of the problem, given by the equation below, is a parabola known as the set of minimum-variance frontier by plotting dierent b2p for dierent expected portfolio rates of return. In particular, use 1; 2; and 4 as targets expected rates of return in formula b2p = c d p b c 2 1 + ; c where a; b; and c are de...ned as shown in class. Sketch the parabola and show clearly on it the vertex point and the previous risk-reward combinations. (c) [0.5] On the graph drawn in part (b), show the e cient frontier. Explain the reason behind your selection. (d) [1.5] Identify the minimum variance portfolio (MVP) for theses two assets; that is, compute 2 p M VP , p M VP , and the optimal weights of the two assets in the portfolio. (e) [0.5] Find the covariance between the minimum variance portfolio, call it portfolio O, and the portfolio N , which consists of 70% of asset 1 and 30% of asset 2. Round your answer to 1 decimal place. (f) [0.5] Find the covariance between the minimum variance portfolio, O, and the portfolio M , which consists of 60% of asset 1 and 40% of asset 2. Round your answer to 1 decimal place. Compare your answer to part (e). (g) [Bonus Question [5]] Show mathematically using our class notation that the covariance between the minimum variance portfolio and any portfolio along the e cient set is a constant.1 2. [5] Consider2 the following three assets such that: 0 1 0 1 E [e r1 ] 1 @ A @ E [e r 2 A = = 2 E [e r3 ] 3 and 0 1 =@ 1 0 1 4 1 1 0 1 A: 9 1 To answer this question, read the paper: Merton, R. (1972). An Analytic Derivation of the E cient Set, Journal of Financial and Quantitative Analysis, 1851-1872. 2 Adapted from Danthine and Donaldson (2005). 1 (a) [3] Compute a; b; c; d; , and 1 : Take 1 where to be 0 A B =@ B B C C A B C D = = = = 1 C C A; D 1:346154 0:346154 0:038462 0:115385 (b) [0.5] Identify the minimum-variance portfolio; that is, ...nd its mean, variance, and the weights of the three assets forming it. 2 (c) [1.5] Draw a sketch of the minimum variance frontier in space and identify the following points on it; that is, ...nd the mean and the variance of the portfolio that corresponds to each point: (i) The point of the minimum variance portfolio; call it MVP: c = + , call it point Q: (ii) The point where W c = ; call it point N: (iii) The point where W 3. [12] Suppose3 two securities have the following joint distribution of returns, re1 and re2 : P fe r1 = and 1 and re2 = 0:15g = 0:1; P fe r1 = 0:5 and re2 = 0:15g = 0:8; P fe r1 = 0:5 and re2 = 1:65g = 0:1: (a) [2.5] Compute the means, variances, and covariance of returns for the two securities. (b) [4.5]Compute a; b; c; d; and : Plot the minimum variance frontier in the mean-standard deviation space, assuming that the two securities are the only investment vehicles available. When you plot the hyperbola, you need to identify two points on your graph: (i) the MVP point with its coordinates; that is, MVP= and (ii) point A, where p is zero. Show that p M VP ; ( p )M VP (c) (d) (e) (f) (g) the coordinates of point A are such that A = p = 0; p = 3:9 : [1] Show that the MVP is such that it has equal weights for both assets. [2] Demonstrate numerically that the e cient set is such that it starts with the MVP point on the minimum variance portfolio in the space and ends with a point, call it S, that corresponds to a portfolio with 100% asset 1 (assuming no short sales). (Hint: start from point MVP with weights (0:5; 0:5) and construct new portfolios with more weights on asset 1. It is su cient to ...nd another portfolio, call it R, so that point R corresponds to the weights (0:75; 0:25). Find the mean and standard deviation of the three portfolios, MVP; R; and S, and plot them on your hyperbola. [0.5] Show that portfolio T that has 100% weight on asset 2 is dominated by portfolio S. Plot the mean and standard deviation of portfolio T on your hyperbola ...gure. [0.5] From all your previous answers and your graph, it is obvious that security 2 is mean-variance dominated by security 1. However, it enters all e cient portfolios but one. How do you explain this? [1] Suppose the possibility of lending, but not borrowing, at 5% (without risk) is added to the previous opportunities. Draw the new set of mean-standard deviation combinations. Which portfolios are now e cient? Show the e cient set on your graph; no calculations are necessary. 3 This problem is adapted from Copeland et al (2006). The problem was originally due to Nils Hakannson, University of California, Berkeley. 2