final_1 2023 - sol1

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FIN 206 - Spring 2023 FINAL EXAM FIN 206 - INVESTMENTS TIME ALLOWED: 2 HOURS Students should attempt to answer: 1. All the five questions in Section A , 2. Three out of the four questions in Section B Make sure that you enter your name and student ID on each booklet . THIS IS A CLOSED-BOOK EXAMINATION PLEASE TURN OVER

FIN 206 - Spring 2023 SECTION A Instructions: You should answer all 5 questions in this section. Each question consists of a short statement, for which you must decide whether it is true or false , and explain your answer . All questions in this section carry equal weight of 8% (4% for the correct true or false assessment, the remaining 4% for the quality of the discussion), and together they make 40% of the overall grade. Question 1: Suppose that you are now 30 and would like $2 million at age 65 for your retirement. You would like to save each year an amount that grows by 5% each year. Assuming that r=8%, you should start saving now $6, 472.97. ANSWER: TRUE, it is immediate from the growing annuity formula ࠵?࠵? = ! "#$ ((1 + ࠵?) % − (1 + ࠵?) % ) = ! &.&(#&.&) ((1 + 0.08) *) − (1 + 0.05) *) ) = $2࠵?࠵?. That 2ml/308.977=6, 472.97. (8 %) Question 2: A trader who wants to short a stock is betting on the price of the stock to drop in the near future. Thus, in a market where prices reflect true fundamentals no one would short. ANSWER: FALSE, one may short to hedge a risk that it is exposed to, or for many other reasons. Thus, even if the stock is fairly priced and you do not have any private information on it being overpriced, you might want to short. Think for example about sellers of call options, who accumulate risk on the underlying assets. (8 %) Question 3: Consider a 4-year T-note with face value $100 and 7% coupon, selling at $103.50 and yielding 6%. If the T-note pays semi-annual coupons, then the duration is 3.565 years

FIN 206 - Spring 2023 ANSWER: TRUE, (8 %) Question 4: Suppose that the annual volatility of the market is 25%, and we consider an asset with volatility 40% and a CAPM beta of 1.2. Assuming that the CAPM holds, the percentage of the total variance of the asset attributable to non-systematic risk is 26.46%. (hint: recall that according to the CAPM: ࠵? + 3 − ࠵? , = ࠵? - 5࠵? . 6 − ࠵? , 7 + ϵ / ) ANSWER: TRUE,

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FIN 206 - Spring 2023 So, in our case we have (0.4) ! = (1.2) ! (0.25) ! + ࠵?࠵?࠵?[࠵?] , so we get 0.2646. (8 %) Question 5: In a binomial model, in order to correctly price an option, one needs to know the probability with which the underlying asset is going to appreciate or depreciate. ANSWER: FALSE, the premium can be obtained through risk-neutral pricing as ࠵? = 01 ! 2(4#0)1 " 42" , where r is the risk-free rate and ࠵? = 42"#6 7#6 . (8 %) END OF SECTION A (Total Points for Section A: 40%) SECTION B Instructions: You should answer THREE of the 4 questions in this section. This section counts for 60% of your overall grade, each question has weight 20% . Question 6:

FIN 206 - Spring 2023 You are a hedge-fund manager, who considers buying put options on Google. Google is anticipated not to pay dividends. The current price of one share is $100. The stock can either increase in value to $120 or drop to $90. The probability of it appreciating is 0.5. a) What is the fair premium of a 1-period lived At-the-Money (ATM) put option? b) What is the fair premium of a 2-period lived At-the-Money (ATM) put option? c) Now, suppose that Google will pay dividends at a rate of 5% of its stock price at the end of period 1. What is the fair premium of a 2-period lived At-the-Money (ATM) American call option? ANSWER: a) First, observe that ࠵? = 48& 4&& = 1.2 while ࠵? = 9& 4&& = 0.9 . Because ࠵? = 0 , we have that ࠵? = 4 * . The option is ATM, so K=100. As a result, ࠵? = 8& * . b) In the second period, stock values can be either 100 ∗ 1.2 8 = 144 , with risk-neutral probability ࠵? 8 = 4 9 or 100 ∗ 0.9 8 = 81 , with risk-neutral probability (1 − ࠵?) 8 = : 9 or finally 100 ∗ 1.2 ∗ 0.9 = 108 , with risk-neutral probability 2࠵?(1 − ࠵?) = 2 4 * 8 * = : 9 . Thus, the option is only exercised in case the stock is worth 81 and we get ࠵? = 4 9 (100 − 81) = 8.4. Note that at the first period ࠵? 6 = max D100 − 90 = 10, 8 * (100 − 81) = 12.6G = 12.6 , so even if one could exercise the put early that would not be optimal. c) Now, if one does not exercise the option the stock valuer at date one drops to 120 ∗ 0.95 = 114 in the up state, and 90 ∗ 0.95 = 85.5 in the down state. Thus, the final stock prices are 114 ∗ 1.2 = 136.8 , 114 ∗ 0.9 = 102.6 = 85.5 ∗ 1.2 , and 85.5 ∗ 0.9 = 76.95 . Now, if one only exercises the option at the end, then it would be exercised always, except from the case in which the stock price is 76.95, which happens with probability (1 − ࠵?) 8 = : 9 . At the interim stage we have ࠵? 7 = max D20, 8 * (102.6 − 100) + 4 * (136.8 − 100) = 14G = 20, ࠵? 6 = 1 3 (102.6 − 100) = 0.87 ࠵? = 1 3 20 + 2 3 0.87 = 7.25

FIN 206 - Spring 2023 (Total points for this question: 20%) Question 7: There are 3 risky assets (A, B, C), with expected returns, respectively, of 10%, 25% and 9%, and volatilities 20%, 30% and 40%. The correlation between A and B is 0.5; the correlation between B and C is 0 and that between A and C is -0.5. a) Construct the variance-covariance matrix and derive an expression for the covariance between an individual asset (be it A, B or C) and a portfolio of the 3 assets together with weights (w1, w2, w3). ANSWER: Σ = L 0.2 8 0.5 ∗ 0.2 ∗ 0.3 −0.5 ∗ 0.2 ∗ 0.4 0.5 ∗ 0.2 ∗ 0.3 0.3 8 0 −0.5 ∗ 0.2 ∗ 0.4 0 0.4 8 M The covariance between asset A and the portfolio is: ࠵?࠵?࠵?5࠵? ; , ࠵? < 7 = ࠵? ; 0.2 8 + ࠵? = (0.5 ∗ 0.2 ∗ 0.3) + ࠵? ! (−0.5 ∗ 0.2 ∗ 0.4) b) Derive the Mean-Variance Efficient Portfolio (MVE) of the 3 assets, and provide an interpretation for the weights that you find. (hint: the inverse of the variance-covariance matrix is: Σ #4 = L 50 −16.6667 12.5 −16.6667 16.6667 −4.16667 12.5 −4.16667 9.375 M ) We have that

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FIN 206 - Spring 2023 ࠵? .>? = Σ #4 ࠵? @ 1 A Σ #4 ࠵? @ = L 50 −16.6667 12.5 −16.6667 16.6667 −4.16667 12.5 −4.16667 9.375 M L 0.05 0.2 0.04 M (1 1 1) L 50 −16.6667 12.5 −16.6667 16.6667 −4.16667 12.5 −4.16667 9.375 M L 0.05 0.2 0.4 M = L −0.33334 2.33334 0.16666 M 2.16667 = L −0.2 1.1 0.1 M c) Suppose that T-Bills pay 5% interest, and that you have mean-variance preferences with coefficient of risk-aversion of 2. What is your optimal portfolio? ANSWER: the MVE has an expected return of −0.2 ∗ 0.1 + 1.1 ∗ 0.25 + 0.1 ∗ 0.09 = 0.264 and volatility −0.2 8 ∗ 0.2 8 + 1.1 8 0.3 8 + 0.1 8 0.4 8 + 2(−0.2) ∗ 1.1 ( 0.5 ∗ 0.2 ∗ 0.3) + 2(−0.2) ∗ 0.1(−0.5 ∗ 0.2 ∗ 0.4) = 0.1 . Thus, the Sharpe Ratio of the MVE is &.8B:#&.&) &.4 # = 0.68 As a mean-variance optimizer with risk aversion of 2, the optimal portfolio weight on the MVE is &.B( :∗&.4 # = 0.54 . (Total points for this question: 20%) Question 8: After graduation, you are hired by AQR. They ask you to find arbitrage opportunities in stock markets. They provide you with a dataset where they have estimated the CAPM beta of the following assets: IBM (1.3), Microsoft (1.5) and Tesla (1.8). Expected returns are: 12% for the market portfolio; 20% for IBM, 22% for Microsoft and 30% for Tesla. The risk-free rate is 5%. You work under the assumption that the CAPM is the correct model for stock returns. You must provide a report with the following elements in it: i) First, draw a plot of the Securities Market Line, and comment on how well the securities line up in it. ANSWER: the SML is a line with intercept equal to 5% and slope equal to the excess return of the market, which is 0.12-0.05=0.07.

FIN 206 - Spring 2023 As IBM has a beta of 1.3, its excess return according to CAPM should be 1.3*0.07=0.091, which means that its expected return should be 0.141. This is strictly less than 20% As Microsoft has a beta of 1.5, its excess return according to CAPM should be 1.5*0.07=0.105, which means that its expected return should be 0.155. This is strictly less than 22% As Tesla has a beta of 1.8, its excess return according to CAPM should be 1.8*0.07=0.126, which means that its expected return should be 0.176. This is strictly less than 30%. ii) Second, explicitly construct a strategy that generates arbitrage profits (i.e., profits free of systematic risk) One can construct an arbitrage strategy for any of the 3 stocks. First, form a portfolio of the market and the risk-free assets such that it matches the beta of the asset. For IBM, for instance, we need a beta of 1.3. This can be achieved by a portfolio that invests 1.3 in the market, and -0.3 in the risk-free asset. This portfolio has the exact same systematic risk as IBM, but a strictly lower return. Thus, ne can generate risk-free profits by buying IBM and selling this portfolio, as many times as possible. (Total points for this question: 20%) Question 9: You work in commodity arbitrage, and you are interested trading gold. The current, spot price-per-ounce of gold is $100. He `true’ model of commodity returns is a 2-factor model with one factor being the market (MKTRF) and the other being Momentum (MOM). The expected return of gold is 15%, and its volatility is 20%. Its factor loadings are: 1.3 on MKTRF, and 2 on MOM. a) Suppose that you can trade the following securities: alpha Loading on MKTRF Loading on MOM Orange 0.2 1 0 Macrosoft 0.3 0.5 -0.5 Slate 0.25 1.5 1

FIN 206 - Spring 2023 Derive the weights of a tracking portfolio for gold. Is there an arbitrage opportunity? If yes, describe the exact trades you would execute to take advantage of it. ANSWER: It is useful to form the two pure-factor portfolio and the risk-free portfolio first. The first PFP solves: ࠵? D + ࠵? . 0.5 + (1 − ࠵? & − ࠵? . )1.5 = 1 −࠵? . 0.5 + (1 − ࠵? & − ࠵? . ) = 0 Which implies that ࠵? D 4 = 0.1 , ࠵? . 4 = 0.6 and so ࠵? E 4 = 0.3 . The expected return is 0.1 ∗ 0.2 + 0.6 ∗ 0.3 + 0.3 ∗ 0.25 = 0.275 The second PFP solves: ࠵? D + ࠵? . 0.5 + (1 − ࠵? & − ࠵? . )1.5 = 0 −࠵? . 0.5 + (1 − ࠵? & − ࠵? . ) = 1 Which implies that ࠵? D 8 = −0.9 , ࠵? . 8 = 0.6 and so ࠵? E 8 = 1.5 and the expected return of PFP2 is −0.9 ∗ 0.2 + 0.6 ∗ 0.3 + 1.5 ∗ 0.25 = 0.375 The risk-free portfolio solves: ࠵? D + ࠵? . 0.5 + (1 − ࠵? & − ࠵? . )1.5 = 0 −࠵? . 0.5 + (1 − ࠵? & − ࠵? . ) = 0 Which implies that ࠵? D & = −0.5 , ࠵? . & = 1 and so ࠵? E & = 0.5 and the expected return of PFP2 is −0.5 ∗ 0.2 + 0.3 + 0.5 ∗ 0.25 = 0.325 We need to track an asset that has loadings 1.3 on MKTRF and 2 on MOM, so we need to buy 1.3. of PFP1, 2 of PFP2 and short 2.3 of the risk-free portfolio. The alpha of the tracking portfolio so is 1.3(0.275) + 2 ∗ (0.375) − 2.3(0.325) = 0.36 This is greater than 15%, thus one should short gold and buy its tracking portfolio, to pocket the difference without carrying any systematic risk. b) Suppose you also want to track silver, which has loadings of 1.8 on MKTRF, and 0.5 on MOM. What portfolio weights track silver? ANSWER: the tracking portfolio is 1.8 of PFP1, 0.5 of PFP2 and short 1.3 of the risk-free portfolio.

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FIN 206 - Spring 2023 c) Finally, suppose that you want to derive a risk-free portfolio (that is, a portfolio free os systematic risk). How would you achieve this? Answer: already achieved in part a. (Total points for this question: 20%) END OF SECTION B (Total Marks for Section B: 60%) END OF EXAM PAPER