Department of Economics ECO 204 Microeconomic Theory for Commerce (Ajaz) Test 2 Solutions

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1 Department of Economics ECO 204 Microeconomic Theory for Commerce (Ajaz) Test 2 Solutions YOU MAY USE A EITHER A PEN OR A PENCIL TO ANSWER QUESTIONS PLEASE ENTER THE FOLLOWING INFORMATION LAST Name (example: Trudeau) FIRST Name (example: Justin) STUDENT ID NUMBER (example: ) PUT AN X BELOW THE SECTION WHICH YOU RE REGISTERED IN Tue 11 am 1 pm Tue 2 pm - 4 pm Wed 11 am 1 pm Wed 2 pm - 4 pm IMPORTANT NOTES Proceed with this exam only after getting the go-ahead from the Instructor or the proctor Do not leave during the first hour of the exam or the last 15 minutes of the exam You are not allowed to leave the exam room unattended. If you need to go to the washroom, please raise your hand and a proctor will accompany you to the washroom. You are allowed to go to the washroom ONLY. You are required to stop writing and turn your exam face down when asked to stop by the instructor or proctor at the end of the exam Please note that proctors will take down your name for academic offenses, which will be treated in accordance with the policies as published by the Faculty of Arts and Sciences. EXAM DETAILS Duration: 2 hours Total number of questions: 4 Total number of pages: 21 (including title page) Total number of points: Bonus Points Please answer all questions. To earn credit you must show all calculations. This is a closed note and closed book exam. You may use a non-programmable calculator. Sharing is not allowed.. KEEP YOUR ANSWERS AS BRIEF AS POSSIBLE AND SHOW ALL NECESSARY CALCULATIONS GOOD LUCK! Page 1 of 20

2 QUESTION 1 [TOTAL 20 Points] (a) [5 Points] The following table contains some data on Delta Airline s and Boeing s monthly stock price = P, monthly returns = RET, and monthly returns without dividends = RETX : DATE DELTA P DELTA RET DELTA RETX BOEING P BOEING RET BOEING RETX 10/30/2015 $ $ /30/2015 $ $ /31/2015 $ $ Source: CRSP through U of Toronto Calculate Delta and Boeing s dividend per share (if any) on 11/30/2015 and 12/31/2015. State all necessary assumptions, show all essential steps, and state the final answer up to two decimal places. From the definition of returns: Or: Returns = Capital Gains + Dividend Yield RET = RETX + Dividend Yield Dividend Yield = RET RETX Dividend at time t Price at time t 1 = RET t RETX t Dividend Payment at time t = (RET t RETX t ) Price at time t 1 Dividend Payment at time t = (Dividend Yield at time t) Price at time t 1 To see if there was a dividend payment on 11/30/2015 we would calculate: Dividend Payment on 11/30/2015 = (RET 11/30/2015 RETX 11/30/2015 ) Price on 10/30/2015 Using the same procedure to see if a dividend was issued on 12/31/2015 we obtain: DATE DELTA P DELTA RET DELTA RETX DELTA DIV YIELD DELTA DIV BOEING P BOEING RET BOEING RETX BOEING DIV YIELD BOEING DIV 10/30/2015 $ $ - $ $ - 11/30/2015 $ $ 0.14 $ $ /31/2015 $ $ - $ $ - On 11/30/2015, Delta issued a dividend of $0.14/share while Boeing issued a dividend of $0.91/share. Neither company issued a dividend on 12/30/2015. Page 2 of 20

3 (b) [5 Points] An investor has collected data from 6/29/2007 through 12/31/2015 on the monthly returns of Delta Airlines, Boeing, and the S&P 500 Index, as well as data on the monthly rates of US T-Bills issued on the first day of the next month (notice the missing value below): DATE DELTA RET BOEING RET SP500 RET Monthly rate of T- Bill on 1st day of next month 6/29/ /31/ /30/ /31/ Missing: Monthly rate of T-Bill issued on 1/1/2016 Source: CRSP through U of Toronto and FRED The annualized rate of a one-month T-Bill issued on 1/1/2016 was 0.260%. What was the monthly rate of a T-Bill issued on 1/1/2016? Show all essential steps, state the final answer up to six decimal places, and use this value in parts below. Since 0.260% is in annualized terms, we divide by 12 to get the monthly rate in % terms: = %. Next, we 12 divide by a 100 to express this in terms of basis points: = Thus, we get: 100 DATE DELTA RET BOEING RET SP500 RET Monthly rate of T- Bill on 1st day of next month 6/29/ /31/ /30/ /31/ Page 3 of 20

4 (c) [3 Points] The following table gives the summary statistics of the data set in part (b): DELTA RET BOEING RET SP500 RET Monthly rate on T- Bill on 1st day of next month Mean Variance Source: CRSP through U of Toronto and FRED. Data range from 6/29/2007 through 12/31/2015 Suppose that on Jan 1 st, 2016, an investor invests $1m in a risk free asset and a (i.e. one) risky asset. True or false: T-Bills are a risk free asset because of the four assets listed above, T-Bills has the smallest variance (and therefore risk )? False. While T-Bills have the lowest variance (and thus lowest standard deviation, a measure of risk), the reason they are risk free is because a T-Bill purchased on 1/1/2016 has zero risk provided the investor holds on to the T-Bill for one month (its duration) and cashes the payment in US$. Page 4 of 20

5 (d) [7 Points] The following table gives the summary statistics of the data set in part (b): DELTA RET BOEING RET SP500 RET Monthly rate on T- Bill on 1st day of next month Mean Variance Source: CRSP through U of Toronto and FRED. Data range from 6/29/2007 through 12/31/2015 Suppose that on Jan 1 st, 2016, an investor invests $1m in a risk free asset and a (i.e. one) risky asset. Of the three risky assets (Delta, Boeing, S&P 500) which one should the investor select as the risky asset in her portfolio (remember in this part the portfolio consists of a risk free and a risky asset)? Provide a short explanation backed by graphs/numbers (if necessary). The investor can select either Delta, or Boeing, or SP 500 (actually, an ETF tracking the SP 500) as the risky asset -- but which one should he pick? Let s use the same logic we used to choose the optimal combination of two risky assets to construct a synthetic risky asset. The following graph shows the two combinations ( 1 and 2 ) of two risky assets ( A and B ): R t B E 2 1 E F A R We argued that combination 2 was better than combination 1 because it has a higher Sharpe ratio ( price of risk ) based on this argument, we constructed the synthetic asset by looking for the combination that gave us the highest price of risk. We can use the same logic here pick the asset with the highest price of risk. The following table shows the price of risk calculated by: Price of Risk = Risk Premium of risky asset Risk of risky asset = Return of risky asset Risk free rate Risk of risky asset Page 5 of 20

6 DELTA BOEING SP500 Risk Premium = Risk = Price of Risk (Sharpe s Ratio) = Delta has the highest price of risk. Thus, if the investor wishes to invest $1m (for one month in January 2016) in a risk free asset and one risky asset, that risky asset should be Delta stocks. Page 6 of 20

7 QUESTION 2 [TOTAL 10 Points] (a) [5 Points] Suppose an investor has a mean-variance utility function (with utility parameter c > 0). Does this investor have monotone (aka more is better ) preferences? What about convex (aka taste for variety ) preferences? The mean-variance utility function is: U = E[ ] c 2 Here E[ ] = Expected returns, 2 = Variance of returns (risk squared), c > 0 is a parameter. Now, the marginal utilities are: U = c < 0 (for a risky asset and risk aversion (i. e. c > 0)) U E[ ] = 1 > 0 Risk is a bad good and, as such, the investor does not have monotone preferences. Now, the indifference curves look like (one needs to show that the indifference look like this by, amongst other things, computing MRS = c and noting that the slope steepens with risk): Expected Return Risk The investor does have convex preferences. For example, notice that the investor prefers bundle C to all combinations of bundles A and B : Page 7 of 20

8 C B Expected Return A Risk Page 8 of 20

9 (b) [5 Points] Consider an investor with a mean-variance utility function with the parameter c = 0.5. Suppose that initially, the investor has purchased an asset with a (historical) average return of 0.872% and risk of standard deviations. Now suppose that the risk of this asset increases by 1%. What must happen to this asset s expected return in order for this investor to continue holding this asset? In general, will the extra returns required to compensate the investor for higher risk rise, fall, or stay the same with the level of risk? State all necessary assumption, show all essential steps, and state the final answer up to six decimal places. The mean-variance utility function is: Initially: U = E[ ] c 2 = E[ ] E[ ] = estimated by historical average returns = Risk = U = E[ ] c 2 = ( ) 2 = Now, the new level of risk is 1.01( ) = standard deviations. Following the greater risk, for the investor to continue holding on to the asset, the investor must be indifferent to the initial situation. For this to happen we see that the asset has to offer higher returns: U after higher risk = New Expected Returns 0.5 ( ) 2 = Initial U = New Expected Returns = ( ) 2 = This makes intuitive sense: the investor demands a higher return to absorb the greater risk. To see the relationship between the extra returns required to compensate the investor for higher risk and the level of risk we note from the previous part that: MRS = c That is, the slope of the indifference curves is proportional to the level risk. As the asset becomes riskier, the investor demands ever higher compensating returns. Page 9 of 20

10 QUESTION 3 [TOTAL 45 Points + 5 Bonus Points] Consider a financial portfolio with a fraction (1 β) consisting of a risk free asset and a fraction β consisting of a risky asset. (a) [10 Points] [This part is independent of other parts in this question] Show that there is a linear relationship between the portfolio s expected return and the portfolio risk. What are the intercept and slope of this linear function? Please show all steps. The expected return of the portfolio is: The variance of portfolio returns is: E[ p ] = E[(1 β) + β ] = (1 β)e[ ] + βe[ ] = + β(e[ ] ) Var[ p ] = p 2 = Var[(1 β) + β ] Var[ p ] = 2 p = (1 β) 2 Var[ ] + β 2 Var[ ] + 2β(1 β) cov[, ] 0 σ r 0 Here we used the fact that a risk free asset s returns have zero variance and zero covariance with the risky asset. Next: Thus: Var[ p ] = p 2 = β 2 2 β = p = Portfolio Risk Risky asset Risk E[ p ] = + β(e[ ] ) = + p (E[ ] ) = + (E[ ] ) p We see that there is a linear relationship between portfolio returns and risk: E[ p ] = + (E[ ] ) p = Intercept + Slope p Here the intercept is the risk free rate and the slope is the risky asset s price of risk. Page 10 of 20

11 (b) [5 Points] Suppose an investor with a mean-variance utility function (with c > 0) constructs a financial portfolio with a fraction (1 β) consisting of a risk free asset and a fraction β consisting of a risky asset. Derive the expression for the optimal fraction of the portfolio invested in the risk asset, i.e. β. Show all steps. The investor chooses β to: max β U = E[ p] c p 2 = + β{e[ ] } cβ 2 2 Fraction of Portfolio in Risky Asset du dβ = + E[ ] 2cβ 2 = 0 = β = (E[ ] ) 2c 2 = Risk Premium of Risky Asset 2 c (Risky Asset Risk) 2 Page 11 of 20

12 (c) [10 Points] [This part is independent of other parts in this question] Consider the investor in part (b) and suppose that her optimal portfolio is a leveraged portfolio where she has borrowed 10% of the value of her own funds at the risk free rate and invested her funds and the borrowed funds in a risky asset whose price of risk is What is the risk of this risky asset if her utility parameter c = ? Show all essential steps and express the final answer up to six decimal places The investor chooses β to: At the optimal solution: Fraction of Portfolio in Risky Asset max β U = E[ p] c p 2 = + β{e[ ] } cβ 2 2 = β = (E[ ] ) 2c 2 = Risk Premium of Risky Asset 2 c (Risky Asset Risk) 2 In this case, she has borrowed 10% of the value of her own funds at the risk free rate and invested everything in a risky asset, so that: Fraction of Portfolio in Risky Asset = β = (E[ ] ) 2c 2 = Risk Premium of Risky Asset 2 c (Risky Asset Risk) 2 = 1.1 Notice that: Thus: β = (E[ ] ) 2c 2 = 1 2c (E[ ] ) = 1 2c (Risky asset s price of risk) = 1.1 = 1 2(1.1)c (Risky asset s price of risk) = 1 ( ) (1.1)( ) Page 12 of 20

13 (d) [10 Points] [This part is independent of other parts in this question] Consider the investor in part (b) and suppose that she has constructed the optimal portfolio. True or false: all else equal, at the optimal solution, which has the greater impact on optimal utility: a small increase in the risky asset s return or the same increase in the risk free asset s return? The investor chooses β to: At the optimal solution: Fraction of Portfolio in Risky Asset max β U = E[ p] c p 2 = + β{e[ ] } cβ 2 2 = β = (E[ ] ) 2c 2 = Risk Premium of Risky Asset 2 c (Risky Asset Risk) 2 We re being asked for the impact on optimal utility from a small increase in the risky asset return vs. an equivalent increase in the risk free rate. We could substitute β into the expression for utility, derive U, and then compare the marginal utility of higher returns on the risky asset vs. the risk free asset. However, this would be complicated. A faster approach is to use the envelope theorem. The original utility function is: U = + β{e[ ] } cβ 2 2 To see the impact on optimal utility from a small increase in the risky asset s returns -- by the envelope theorem we differentiate utility with respect to E[ ]: du de[ ] = β Next we evaluate this derivative at the optimal values: du de[ ] = β To see the impact on optimal utility from a small increase in the risky free asset s returns we -- by the envelope theorem differentiate utility with respect to : du d = 1 β Next we evaluate this derivative at the optimal values: du d = 1 β Which is better: a small increase in the risky asset return or an equivalent increase in the risk free rate? That is, which expression is larger: du de[ ] = β v. du d = 1 β The answer is simple: if at the initial optimal solution, over 50% of the portfolio is in the risk asset then a small increase in the risky asset s returns will have a larger impact on optimal utility than the equivalent increase in the risk free asset s returns. Page 13 of 20

14 (e) [5 Points] [This part is independent of other parts in this question] Suppose you wish to create a synthetic asset consisting of two risky assets A and B. True or false: regardless of their attitude towards risk, each investor will create a synthetic asset that is identical to that of every other investor. Give a short explanation. True. One creates a synthetic asset by maximizing the price of risk (aka Sharpe s ratio) which is independent of utility: Price of Risk = (E[ ] ) = Risk Premium of Synthetic Asset Risk of Synthetic Asset Page 14 of 20

15 ** Please use the following information for all remaining parts ** Consider the following information based on data from 6/29/2007 through 12/31/2015 (source CRSP at U Toronto and FRED): Covariance Table DELTA RET BOEING RET SP500 RET DELTA RET BOEING RET SP500 RET DELTA RET BOEING RET SP500 RET Mean Variance Std. Dev The one month rate on a T-Bill issued on 1/1/2016 was An investor wishes to create a financial portfolio (for the month of January) consisting of T-Bills and a synthetic asset consisting of Delta stocks, Boeing stocks, and stocks of a S&P 500 tracking ETF. Label Delta, Boeing, and the SP 500 tracking ETF assets 1, 2, and 3 respectively. The optimal synthetic asset consists of: w 1 = Fraction of Delta Shares in Synthetic Asset 0.54 w 2 = Fraction of Boeing Shares in Synthetic Asset 0.68 w 3 = Fraction of S&P 500 Tracking ETF in Synthetic Asset The synthetic asset s price of risk is Suppose that the investor s mean-variance utility function parameter c = 1. (f) [10 Points + Bonus 5 Points] Interpret the negative weight on the S&P 500 tracking ETF. What fraction of the portfolio consisting of the risk free asset and the synthetic asset consists of the synthetic asset above? Bonus 5 points if you derive the synthetic asset s risk by using the figures in the covariance table. The investor chooses β to: max β U = E[ p] c p 2 = + β{e[ ] } cβ 2 2 Where the risky asset in fact is the synthetic asset consisting of Delta, Boeing, and a SP 500 tracking ETF stocks. At the optimal solution: Fraction of Portfolio in Risky Asset = β = (E[ ] ) 2c 2 = Risk Premium of Risky Asset 2 c (Risky Asset Risk) 2 We need to calculate the synthetic risky asset s risk premium and risk. First the risk premium: In turn, Risk Premium = Returns Risk Free Rate Page 15 of 20

16 Returns = w w w 3 3 = 0.54(0.0201) ( ) 0.22( ) = Risk Premium = Returns Risk Free Rate = = Next, the synthetic asset s risk the easy way is as follows: Price of Risk = Risk Premium of Synthetic Asset Risk of Synthetic Asset = Risk of Synthetic Asset = Risk of Synthetic Asset = = Bonus answer: Var(Synthetic Asset Returns) = w w w w 1 w 2 cov w 1 w 3 cov w 2 w 3 cov 23 Var(Synthetic Asset Returns) = (0.025) ( ) + ( 0.22) 2 ( ) + 2(0.54)(0.68)( ) + 2(0.54)( 0.22)( ) + 2(0.68)( 0.22)( ) = Synthetic Asset Risk = Var(Synthetic Asset Returns) = = Thus: Fraction of Portfolio in Risky Asset = β = Risk Premium of Risky Asset c (Risky Asset Risk) 2 = % Page 16 of 20

17 QUESTION 4 [TOTAL 30 Points] Consider an agent who lives for exactly two periods t = 0,1 in an economy with a single good (corn). At the beginning of each period, the agent is endowed with real incomes Y 0 and Y 1 (in units of corn) respectively. Let C 0 and C 1 denote the amounts of corn consumed in t = 0 and t = 1 respectively. At t = 0, each agent can save or borrow corn at the real interest rate > 0. (a) [15 Points] Suppose an agent s preferences are represented by a general differentiable utility function U(C 0, C 1 ). Show that if the real interest rises then net savers at t = 0 are better off whereas net borrowers are worse off. State all assumptions. The inter-temporal UMP with a FV Budget Constraint is: max U(C 0, C 1 ). t. C 0 (1 + ) + C 1 = Y 0 (1 + ) + Y 1 C 0, C 1 Real FV Budget Constraint, C 0 0, C 1 0 The FOCs and KT conditions are: max U(C 0, C 1 ). t. C 0 (1 + ) + C 1 = Y 0 (1 + ) + Y 1, C 0 0, C 1 0 C 0,C 1 Total FV Income FVY max U(C 0, C 1 ). t. C 0 (1 + ) + C 1 FVY = 0, C 0 0, C 1 0 C 0,C 1 max L = U(C 0, C 1 ) λ 1 [C 0 (1 + ) + C 1 FVY] λ 2 [ C 0 ] λ 3 [ C 1 ] C 0, C 1, λ 1, λ, λ 3 max C 0, C 1, λ 1, λ, λ 3 L = U(C 0, C 1 ) λ 1 [C 0 (1 + ) + C 1 FVY] + λ 2 C 0 + λ 3 C 1 L = U λ C o C 1 (1 + ) + λ 2 = 0 0 MU 0 L = U λ C 1 C 1 + λ 3 = 0 1 MU 1 L λ 1 = [C 0 (1 + ) + C 1 FVY] = 0 C 0 (1 + ) + C 1 = FVY λ 2 0, C 0 0, λ 2 C 0 = 0 λ 3 0, C 1 0, λ 3 C 1 = 0 Noting that λ 2, λ 3 0, assuming that > 1 and that corn is a good good we have: λ 1 = MU 0 + λ > 0 λ 1 = MU 1 + λ 3 > 0 Page 17 of 20

18 At the optimal solution: L = U λ 1 [C 0 (1 + ) + C 1 FVY] + λ 2 C λ 3 C 1 0 Thus to evaluate the change in U due to a change in a parameter (in this case ) we should use the envelope theorem on the Lagrange equation. = U Thus: dl d = λ 1(C 0 Y 0 ) dl d = du d = λ 1 (C 0 Y 0 ) = λ 1 S 0 We showed that under some conditions λ 1 > 0. Thus, when increases, utility will increase if S 0, net savings at t = 0, are positive and vice versa if net savings are negative at t = 0. Page 18 of 20

19 (b) [15 Points] Suppose that an agent s preferences over are represented by the utility function U = min(αc 0, βc 1 ) where α, β > 0. Show that if the real interest rises then net savers at t = 0 are better off whereas net borrowers at t = 0 are worse off. The inter-temporal UMP with FV Budget Constraint is: max U = min(αc 0, βc 1 ). t. C 0 (1 + ) + C 1 = Y 0 (1 + ) + Y 1 C 0, C 1 However since the utility function can t be differentiated we can t setup and solve this UMP by the KT method. Instead, we note that the optimal choice must be on the corners line and the budget line: FVY αc 0 = βc 1 C 0 (1 + ) + C 1 = Y 0 (1 + ) + Y 1 FVY Solving these yields: C FVY 0 = β [β(1 + ) + α] C FVY 1 = α [β(1 + ) + α] An agent will save at t = 0 whenever: When is S 0 = Y 0 C 0 > 0? FVY S 0 = Y 0 β [β(1 + ) + α] > 0 S 0 = Y 0 β Y 0(1 + ) + Y 1 [β(1 + ) + α] > 0 α β > Y 1 Y 0 Now what is the impact on utility when increases? du d = U = min(αc 0, βc 1 ) = du αβ FVY [β(1 + ) + α] = αβ [Y 0(1 + ) + Y 1 ] [β(1 + ) + α] d = αβ Y 0 [β(1 + ) + α] FVY αβ 2 [β(1 + ) + α] 2 αβ [β(1 + ) + α] [Y 0 β FVY β(1 + ) + α ] = αβ [β(1 + ) + α] 2 [αy 0 βy 1 ] Page 19 of 20

20 When increases, utility will increase so long as αy 0 βy 1 > 0 or whenever α > Y 1. This is the same conclusion we β Y 0 reached in part (a): when increases, utility will increase if S 0, net savings at t = 0, are positive and vice versa if net savings are negative at t = 0. Page 20 of 20