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ECO 204, 2016-2017 (AJAZ), Test 2`
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
∎ Department of Economics ∎ ECO 204 ∎ Microeconomic Theory for Commerce ∎ 2016-2017 (Ajaz)
Test 2
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EXAM DETAILS
Duration: 2 hours
Total number of questions: 4
Total number of pages: 18 (including title page)
Total number of points: 110 + 5 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!
ECO 204, 2016-2017 (AJAZ), Test 2`
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
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 $ 50.84 0.133051 0.133051 $ 148.07 0.130737 0.130737
11/30/2015 $ 46.46 -0.083497 -0.086153 $ 145.45 -0.011549 -0.017694
12/31/2015 $ 50.69 0.091046 0.091046 $ 144.59 -0.005913 -0.005913
Source: CRSP through CHASS @ 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.
ECO 204, 2016-2017 (AJAZ), Test 2`
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
(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
6/29/2007 0.034121 -0.04404 -0.017816 0.0041333
7/31/2007 -0.095432 0.075603 -0.031982 0.0036000
… … … … …
… … … … …
11/30/2015 -0.083497 -0.011549 0.000505 0.0001917
12/31/2015 0.091046 -0.005913 -0.01753 Missing: Monthly rate of T-Bill issued on 1/1/2016
Source: CRSP through CHASS @ 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.
ECO 204, 2016-2017 (AJAZ), Test 2`
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
(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 0.0201 0.00872 0.00390 0.0003842
Variance 0.0205 0.00614 0.00215 0.0000007
Source: CRSP through CHASS @ U of Toronto and FRED. Data range from 6/29/2007 through 12/31/2015
Suppose that on Jan 1st, 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”)?
ECO 204, 2016-2017 (AJAZ), Test 2`
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
(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 0.0201 0.00872 0.00390 0.0003842
Variance 0.0205 0.00614 0.00215 0.0000007
Source: CRSP through CHASS @ U of Toronto and FRED. Data range from 6/29/2007 through 12/31/2015
Suppose that on Jan 1st, 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).
ECO 204, 2016-2017 (AJAZ), Test 2`
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
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?
ECO 204, 2016-2017 (AJAZ), Test 2`
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
(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 0.07838 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.
ECO 204, 2016-2017 (AJAZ), Test 2`
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
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.
ECO 204, 2016-2017 (AJAZ), Test 2`
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
(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.
ECO 204, 2016-2017 (AJAZ), Test 2`
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
(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 0.108483. What is the risk of this risky asset if her utility parameter 𝑐 = 0.629139? Show all essential steps and express the final answer up to six decimal places
ECO 204, 2016-2017 (AJAZ), Test 2`
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
(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?
ECO 204, 2016-2017 (AJAZ), Test 2`
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
(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.
ECO 204, 2016-2017 (AJAZ), Test 2`
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
** 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 CHASS @ U Toronto and FRED):
Covariance Table
DELTA RET BOEING RET SP500 RET
DELTA RET 0.02052 0.00364 0.00167
BOEING RET 0.00364 0.00614 0.00254
SP500 RET 0.00167 0.00254 0.00215
DELTA RET BOEING RET SP500 RET
Mean 0.0201 0.00872 0.00390
Variance 0.0205 0.00614 0.00215
Std. Dev. 0.1433 0.07838 0.04632
The one month rate on a T-Bill issued on 1/1/2016 was 0.000216667. 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:
𝑤1 = Fraction of Delta Shares in Synthetic Asset 0.54
𝑤2 = Fraction of Boeing Shares in Synthetic Asset 0.68
𝑤3 = Fraction of S&P 500 Tracking ETF in Synthetic Asset -0.22
The synthetic asset’s price of risk is 0.101792. 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.
ECO 204, 2016-2017 (AJAZ), Test 2`
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
ECO 204, 2016-2017 (AJAZ), Test 2`
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
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 𝑌0 and 𝑌1 (in units of corn) respectively. Let 𝐶0 and 𝐶1 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 𝑟 > 0.
(a) [15 Points] Suppose an agent’s preferences are represented by a general differentiable utility function
𝑈(𝐶0, 𝐶1). 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.
ECO 204, 2016-2017 (AJAZ), Test 2`
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
ECO 204, 2016-2017 (AJAZ), Test 2`
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
(b) [15 Points] Suppose that an agent’s preferences over are represented by the utility function 𝑈 = min(𝛼𝐶0, 𝛽𝐶1)
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.
ECO 204, 2016-2017 (AJAZ), Test 2`
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
![Department of Economics ECO 204 Test 2 Solutions€¦ · We are told that the CMP has an interior solution so that: [ ] [ ] [ ] The FOCs are: ECO 204, 2013 - 2014, Test 2 This test](https://img.pdfslide.us/doc/110x75/601733f2c9b66e1f38386eb4/department-of-economics-eco-204-test-2-solutions-we-are-told-that-the-cmp-has-an.jpg)
