Unformatted text preview: University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain ECO 204 Summer 2009 S. Ajaz Hussain Practice Problems 24 Solutions Please help improve the course by sending me an email about typos or suggestions for improvements For these problems, it'll be helpful to remember these formulas for expected value, expected variance, expected covariance, expectation of a sum and variance of a sum: E[Rp] = (1 )RF + E[RM] 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) Always express returns and standard deviation in % terms. Question 1 Suppose you invest $1,000 in a mutual fund: the price of a share today is $100. After extensive research, you believe that a year from now the share price will be: State of the Economy Boom Normal Recession Probability 0.25 0.50 0.25 Stock Price $140 $110 $80 You expect the dividend to be $4 (in finance, this would correspond to a dividend yield of 4% = 1 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain $4/$100, where dividend yield is defined as dividend income per dollar invested in the stock at the start of the period). (a) What is the expected return of the stock? Hint: Calculate the return for each state of the economy and compute EV. Assume there is zero inflation. Answer: Recall that (nominal) returns are defined as: capital gains plus dividend yields or: Return between time t 1 and t = (Pt Pt1)/Pt1 + Dt/Pt1 Since inflation is zero, real returns equal nominal returns and for each state of the economy are: State of the Economy Boom Normal Recession Probability 0.25 0.50 0.25 Returns ($140 $100)/$100 + $4/$100 = 0.44, 44% ($110 $100)/$100 + $4/$100 = 0.14, 14% ($80 $100)/$100 + $4/$100 = 0.16, 16% The expected return is = 0.25(44%) + 0.5(14%) + 0.25(16%) = 14% (b) What is the risk of this stock's return? Answer: The expected variance is: E[2] = 0.25(44% 14%)2 + 0.5(14% 14%)2 + 0.25(16% 14%)2 = 450%2 ( i.e. in %2 units) The standard deviation, "risk", is E = {E[2]}1/2 = (450 %2)1/2 = 21.2% If you've taken ECO 220, you can utilize the standard deviation to say what % of the time expected returns will be within +/ 1 standard deviation of the expected returns. (c) In your calculation of risk, are a boom and recession economy treated "equally"? Why might this be troubling for some investors? Answer: 2 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain Some investors may be troubled more by the potential downside risk of a 16% return than the potential upside of a 44% return. Note how the expected standard deviation a measure of risk does not distinguish between the 44% upside and the 16% downside. Both events are equally likely and equally spaced from the EV of 14%. Question 2 After graduating from U of T, you are hired by a nonprofit organization The "Save the Economists" foundation. The foundation has an endowment one half of which is in "Goolab Jammins" stocks (Goolab Jammins makes hip Indian fusion desserts) and the other half is in 3 month Canadian bonds. The price of Goolab Jammins stocks is sensitive to the price of sugar: at times, when the Caribbean sugar crop fails, the price of sugar spikes up and Goolab Jammins suffers significant losses. The "Save the Economists" board of directors gives you the following table with the Goolab Jammins' rate of return Probability Rate of Return (a) What is the expected rate of return on Goolab Jammins stock? Answer: The expected return is: EV(GJ) = 0.5(25%) + 0.3(10%) + 0.2(25%) = 10.5% (b) What is the expected risk of Goolab Jammins stock? Answer: Expected risk is measured by expected standard deviation, which is the square root of expected variance: E[2]GJ = 0.5(25% 10.5%)2 + 0.3(10% 10.5%)2 + 0.2(25% 10.5%)2 3 Normal Year Bullish Stock Market 0.5 25% Bearish Stock Market 0.3 10% Abnormal Year Sugar Crisis 0.2 25% University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain E[2]GJ = 357.25%2 EGJ = {E[2]GJ}1/2 = 18.9% Suppose the return on 3month Central Bank of Canada bonds is 5%. (c) What is the expected return of "Save the Economists" foundation's portfolio? Answer: Since the endowment is split equally between GJ stocks and government bonds, you can use this formula for the expected return on the portfolio: E[Rp] = (1 )RF + E[RM] E[Rp] = (10.5%) + (5%) = 7.75% (d) What is the expected risk of "Save the Economists" foundation's portfolio? Answer: Since expected risk is the square root of expected variance, use the formula: Var(a x + b y) = a2 Var(x) + b2 Var(y) + 2 a b cov(x,y) Var(Rp) = Var( GJ + Bonds) = ()2 Var(GJ) + ()2 Var(Bonds) + 2 Cov(GJ, Bonds) But since bonds are riskless: Var(Bonds) = 0 and the Cov(GJ, Bonds) = 0. Thus: Var(Rp) = ()2 357.25 + ()2 0 + 2 0 Var(Rp) = 89.3125%2 Thus, the risk of the portfolio is: E[ P] = {E[2]P}1/2 = 9.45% Thanks to ECO 204, you're an expert in the economics and finance of portfolios. You notice that in years of a sugar crisis in the Caribbean, a Hawaiian company "Brown Sugar" reaps massive profits. You investigate further and discover that "Brown Sugar" returns are: 4 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain Probability Rate of Return Normal Year Bullish Stock Market 0.5 1% Bearish Stock Market 0.3 5% Abnormal Year Sugar Crisis 0.2 35% (e) What is the expected return for "Brown Sugar"? Answer: EV(BS) = 0.5(1%) + 0.3(5%) + 0.2(35%) = 6% (f) What is the expected risk for "Brown Sugar"? Answer: E[2]BS = 0.5(1% 6%)2 + 0.3(5% 6%)2 + 0.2(35% 6%)2 E[2]BS = 217%2 EBS = {E[2]BS}1/2 = 14.73% (g) Is "Brown Sugar" a candidate investment for diversifying "Save the Economists" portfolio? Why? Answer: Yes. This is because BS stock has negative covariance with GJ stock, a portfolio of these two stocks will serve to reduce risk through diversification. You can see this from the formula: Var(a x + b y) = a2 Var(x) + b2 Var(y) + 2 a b cov(x,y) (a and b are constants) Which implies that a portfolio with "a" fraction in GJ stocks and "b" fraction in BS stocks will have variance: Var(a GJ + b BS) = a2 Var(GJ) + b2 Var(BS) + 2 a b cov(GJ,BS) (a and b are constants) Variance is always positive (why?). Observing that cov(GJ, BS) < 0 tells us that the portfolio variance will be lower with a GJ, BS portfolio than with a GJ and a positively correlated stock. 5 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain (h) Consider the following 3 portfolios: (1) 100% of portfolio in Goolab Jammins stock (2) 50% in bonds,50% in Goolab Jammins (3) 50% in Goolab Jammins, 50% in Brown Sugar Do you notice anything interesting about these portfolios? Hint: Examine the expected return and risk of each portfolio. Answer: From parts (a) and (b), observe that portfolio 1 has return, risk of 10.5%, 18.9% respectively. From parts (c) and (d), observe that portfolio 2 has return, risk of 7.75%, 9.45% respectively. How do these compare to portfolio 3? Let's see. The expected return of this portfolio is a weighted average of the expected returns on GJ and BS stocks: E[Rp] = (1 )E[RGJ]+ E[RBS] E[Rp] = (10.5%) + (6%) = 8.25% The expected variance comes from the formula: Var(a GJ + b BS) = a2 Var(GJ) + b2 Var(BS) + 2 a b cov(GJ,BS) (a and b are constants) Since of the portfolio is in GJ stocks and the other half in BS stocks: Var(Rp) = Variance( GJ, BS) = ()2 Var(GJ) + ()2 Var(BS) + 2 Cov(GJ, BS) Var(Rp) = Variance( GJ, BS) = ()2 Var(GJ) + ()2 Var(BS) + 2 Cov(GJ, BS) From above, we know Var(GJ) and Var(BS). Thus we need to know Cov(GJ, BS). You know the formula for cov between x and y is: cov(X,Y) = i=1:n (Probability of X & Y) (X EV of X) (Y EV of Y) Thus: 6 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain Cov(GJ, BS) = 0.5(25% 10.5%)(1% 6%) + 0.3 (10% 10.5%)( 5% 6%) + 0.2(25% 10.5%)(35% 6%) Cov(GJ, BS) = 240.5 The negative covariance shows that BS can be used to hedge against GJ. Returning to: Var(Rp) = Variance( GJ, BS) = ()2 Var(GJ) + ()2 Var(BS) + 2 Cov(GJ, BS) Variance( GJ, BS) = ()2 (18.9%)2 + ()2 (14.73%)2 + 2 (240.5) Variance( GJ, BS) = 23.3%2 Risk( GJ, TBills) = {Variance( GJ, BS)}1/2 = 4.83% In summary: Expected Risk Portfolio Allocation Expected Returns 1 2 3 100% GJ 50% GJ, 50% Bonds 50% GJ, 50% BS 10.5% 7.75% 8.25% 18.9% 9.45% 4.83% What is interesting is that the risk of a portfolio with GJ, BS (two risky assets) is lower than the risk of a portfolio of GJ, TBills (one risky, one risk free asset). This is because the negative covariance between GJ, BS reduces risk of the portfolio even though both stocks are riskier than TBills. Question 3 (Actual CFA Level 1 question). You manage an equity fund with an expected risk premium of 10% and an expected standard deviation of 14%. The rate of TBills is 6%. Your client chooses to invest $60,000 of portfolio in your equity fund and $40,000 in a TBill money market fund. What is the expected return and standard deviation of your client's portfolio? By the way, the "risk premium" in finance is the difference between the returns of a risky and a risk free asset. 7 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain Answer: Your client's expected return of a portfolio with 60% in equity fund and 40% in TBill money fund is: EV (60% equity fund, 40% TBill money market fund) = 0.6 E[Requity]+ 0.4 E[RTBills] EV (60% equity fund, 40% TBill money market fund) = 0.6 E[Requity] + 0.4(6%) To find the return on equity fund, note how we know it's risk premium: Risk Premium on equity fund = Return on equity fund 6% = 10% Return on equity fund = 16% E[Requity] = 16% Thus: EV(60% equity fund, 40% TBill money market fund) = 0.6(16%) + 0.4*6% = 12% From this, we can derive the risk of the portfolio: Var(Rp) = Variance(60% Equity,40% TBills) Var(Rp) = (0.6)2 Var(Equity) + (0.4)2 Var(TBills) + 2 (0.6) (0.4) Cov(Equity, TBills) But Var (TBills) = 0 and Cov(Equity, TBills) = 0: Var(Rp) = (0.6)2 Var(Equity) Var(Rp) = (0.6)2 (14%)2 Risk(Rp) = { Var(Rp)}1/2 ={ (0.6)2 (14%)2}1/2 Risk(Rp) = (0.6)(14%) Risk(Rp) = 8.4% Question 4 (20072008 Final Exam Question) The following table gives the expected return and standard deviation for returns of stocks A and B: 8 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain 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: ax + bx + c = 0 2  b b 2  4ac x = 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) = 300/450 = 0.67 2(225) That is: = 0.67. 9 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain 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: E[Rp] = E[RA]+ (1 )E[RB] = RF E[Rp] = 0.67 (10%) + 0.33 (15%) = 11.65% Hence: RF = 11.65% Question 5 (20072008 Test Question) As an equity analyst for AJax Investment Bank (Motto: "We Don't Know How to Count, But We Do Know How to Invest"), you analyze the % rate of return for two stocks: "Canada Rules!" and "USA Rules!". The two charts below give the number of times rates of returns have been observed over the last 200 trading periods (Note: Xaxis = Rate of Return (%) Yaxis = # of observations) 10 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain Will purchasing these two stocks reduce AJax's risk? Show calculations clearly and give a one sentence explanation. Answer: If the two stocks have negative covariance, combining these will reduce portfolio risk. Let's check if covariance is in fact negative. The formula is: E[cov(x,y)] = i=1:npi(xi EV[x])(yi EV[y]) We need to obtain EV of Canada Rules! and EV of USA Rules! There are three events, which in decreasing order, have probabilities: 100/200 = 0.5, 60/200 = 0.3, and 40/200 = 0.2. The EV of Canada Rules! Is = 0.5(25%) + 0.3(10%) + 0.2(25%) = 10.5% The EV of USA Rules! Is = 0.5(1%) + 0.3(5%) + 0.2(35%) = 6% Note that the two stocks are negatively related: when one stock crashes (25%) the other rises (35%). Thus, these are good candidates for reducing risk since these two stocks have negative covariance: Covariance(Canada Rules!, USA Rules!) = 0.5(25% 10.5%)(1% 6%) + 0.3(10% 10.5%)( 5% 6%) + 0.2(25% 10.5%)(35% 6%) Covariance(Canada Rules!, USA Rules!) = 240.5 Question 6 (20072008 Test Question) You are an asset manager at "Etorre" Investments. Jane Murdock is your client. Recently, she borrowed money from your company and invested the borrowed amount plus her portfolio 11 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain ($100,000) in risky assets. While trying to balance the books, you discover to your horror that you forgot to note how much she borrowed. You start to panic but then think back to ECO 204: given the following data, can you figure out how much she borrowed? Rf = 5%, 2m = 100, Risk premium = 25%, Rp = 42.5% Answer: We know that: E[Rp] = (1 )Rf + E[Rm] If we can calculate , we can tell how much she borrowed. To do that, we need to calculate E[Rm]. Oberve that the risk premium is E[Rm] Rf E[Rm] = Risk premium + R f E[Rm] = 25% + 5% = 30% Substitute in: E[Rp] = (1 )Rf + E[Rm] From which: 42.5% = (1 )(5%) + (30%) (25%) + 5% = 42.5% = (42.5% 5%)/25% = 1.5 Therefore, Jane invested 1.5 times her portfolio. This implies that she borrowed 50% of her portfolio or $50,000. 12 ...
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 Economics, Microeconomics, Variance, S. Ajaz Hussain

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