eco204_summer_2009_HW_12

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Unformatted text preview: University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain ECO 204 Summer 2009 S. Ajaz Hussain HW 12 Please help improve the course by sending me an email about typos or suggestions for improvements For this HW, it's helpful to have the following expressions handy: E[2] = i=1:n pi(xi EV[x])2 E[cov(x,y)] = i=1:n pi (xi EV[x])(yi EV[y]) E[a x + b y] = a E[x] + b E[y] (a and b are constants) Var(a x + b y) = a2 Var(x) + b2 Var(y) + 2 a b cov(x,y) (a and b are constants) Question 1 We've seen that if an investor wants to put a proportion of the portfolio into every risky asset (the "market") and the remainder proportion 1 into a single riskless asset, then: = Desired portfolio risk/Market risk = p / m We've also seen that it is possible to mix two risk assets to have a completely riskless portfolio (which by definition will give you the risk free rate because otherwise there'd be arbitrage). For your convenience, here is one such example from the practice problems: 1 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain 20072008 Final Exam Question: The following table gives the expected return and standard deviation for returns of stocks A and B: Stocks A B Expected Return (R) 10% 15% Standard Deviation () 5% 10% The covariance of stock A and B returns is 50. What is the risk free rate? You may find following formulas helpful: ax2 + bx + c = 0 x = - b b 2 - 4ac 2a Rp = RX + (1 )RY 2p = 22X + (1 )22Y + 2 (1 ) cov(RX,RY). Answer: A risk free asset has zero risk. Let us combine stocks A and B to obtain a portfolio of zero risk. By definition, the return of that portfolio will be the risk free return. That is, suppose fraction is in stock A and fraction 1 is in stock B. Then if this portfolio has 0 risk it must be that: E[Rp] = E[RA]+ (1 )E[RB] = RF Now, how do we choose to construct a portfolio of zero risk from stocks A and B? Simple: we want the expected risk of this portfolio to be 0 so that: E[2p]= 22A + (1 )22B + 2 (1 ) cov(RA,RB) = 0 E[2p] = 2(5)2 + (1 )2(10)2 2 (1 )(50) = 0 E[2p] = 252 + (1 2 + 2)(100) (1 ) (100) = 0 E[2p]= 252 + 100 200 + 1002 100 + 1002 = 0 E[2p] = (25 + 100 + 100) 2 + 100 (200 + 100) = 0 E[2p] = 225 2 300 + 100 = 0 This is a quadratic equation with a = 225, b = 300, c = 100. The solution is: = 300 90,000 - 4(225)(100) 2(225) = 300/450 = 0.67 That is: = 0.67. Given that = 0.67 (i.e. 67% of the portfolio is in Stock A), we can compute the risk free return. With this allocation, since the portfolio has 0 risk, the expected return of the portfolio be equal to the risk free return: 2 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain E[Rp] = E[RA]+ (1 )E[RB] = RF E[Rp] = 0.67 (10%) + 0.33 (15%) = 11.65% Hence: RF = 11.65% In this question you will explore whether combining the following two risky assets can diversify risk completely, i.e. generate a portfolio with zero risk. Consider the following table of returns and probabilities for two risky assets A and B: State Strong Normal Weak Rate of Return A 50% 10% 30% Rate of Return B 30% 10% 50% Probability 0.20 0.60 0.20 (a) Calculate the expected return and risk of asset A. (b) Calculate the expected return and risk of asset B. (c) Calculate the expected covariance of asset A and asset B returns. (d) Is it possible to construct a riskless portfolio with proportion of the portfolio in asset A and proportion 1 in asset B? Show numerical calculations for these particular assets and algebraically for any two assets. Hint: use the technique in the example given at the beginning of the question. (e) Instead of constructing a riskless portfolio of risky assets A and B, calculate the (proportion of portfolio in asset A) which minimizes risk of a portfolio of assets A and B. Show numerical calculations for these particular assets and algebraically for any two assets. Hint: write down the expression for portfolio variance and differentiate with respect to . Question 2 (Summer 2008 Final Exam Question) Ajax sat at his desk and thought intently about a proposal on his desk. As the CEO of Brr! Yani foods, he had recently been exploring opportunities for growth in South Korea. The Executive Summary of the proposal contained this table: 3 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain Total Market Size (annual units) Anticipated Market Capture (%) Annual Fixed Costs Average Variable Cost South Korea Market 500,000 40% $2m $30 (a) What is Brr! Yani's annual cost in millions of dollars if it is the only firm to enter the South Korean market? Show all steps and calculations. Ajax learns that a Japanese rival company VLM is also considering entering the South Korean market. Neither company knows the other company's actions. Wall and Bay Street analysts predict VLM and Brr! Yani's fixed cost and average variable to be identical. Ajax calculates that either firm, if it were the only entrant, could capture 40% of the total market size at a price of $55 per unit. If both companies were to enter the market, each firm is expected to sell 30% of the total market size at a price of $45. (b) Fill the payoffs (in `000s of $) for South Korea entry game below. Show all steps and calculations. VLM Brr! Yani Enter Enter Don't Enter (c) Copy the payoffs from the table in part (b) below. Calculate the pure strategy Nash equilibriums for the simultaneous game below. Show all calculations and payoffs. 4 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain Brr! Yani Enter Don't Enter Enter VLM Don't Enter (d) Calculate the Sub Perfect Nash Equilibrium if VLM enters first. Question 3 (20072008 Final Question) In a business deal, Gussy has been identified as sometimes being straightforward and at other times bluffing, while Jen has been identified as being sometimes trusting and at other times skeptical. The payoffs from a oneshot simultaneous game between Gussy and Jen are: Gussy Straightforward Bluffing Jen Trusting 20, 20 50, 10 Skeptical 10, 10 0, 0 (a) What is the pure strategy Nash equilibrium? (b) What is the mixed strategy Nash equilibrium? Show your calculations below, clearly stating any assumptions you've made: Question 4 (20072008 Final Exam Question) Courtney and Alma play a game: Courtney Left Right Alma Up 1, 1 0, 5 Down 5, 0 4, 4 (a) What is the pure strategy Nash equilibrium in a oneshot game? Show your work in the matrix below: Courtney 5 Left Right Alma Up 1, 1 0, 5 Down 5, 0 4, 4 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain (b) Suppose Courtney and Alma play the game for three rounds. Predict the outcome of the game in each round. Show all calculations and steps below: (c) Suppose Courtney and Alma play the game repeatedly forever. Predict the possible outcomes of the game. State any assumptions and show all calculations clearly. 6 ...
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This note was uploaded on 05/02/2011 for the course ECO 204 taught by Professor Hussein during the Fall '08 term at University of Toronto.

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