M339W/389W Financial Mathematics for Actuarial Applications University of Texas at Austin Sample In-Term Exam 3 Instructor: Milica Cudinaˇ

Notes: This is a closed book and closed notes exam. Time: 50 minutes

MULTIPLE CHOICE TRUE/FALSE 1 (5) a b c d e

1 (2) TRUE FALSE 2 (5) a b c d e

2 (2) TRUE FALSE 3 (5) a b c d e

3 (2) TRUE FALSE 4 (5) a b c d e

4 (2) TRUE FALSE 5 (5) a b c d e

5 (2) TRUE FALSE 6 (5) a b c d e

FOR GRADER’S USE ONLY: T/F 1. 2. M.C. Σ 2

3.1. CAPM. Problem 3.1. (10 points) State the assumptions of the Capital Asset Pricing Model. Solution: I. The market is competitive, i.e., the securities are bought and sold at the same price. There are no taxes or transaction costs. Both borrowing and lending are at the -free interest rate. II. hold only efficient portfolios. III. Homogeneous expectations: The investors have the same beliefs about the expected values, volatilities, and correlations of returns of securities.

3.2. TRUE/FALSE QUESTIONS. Please note your answers on the front page.

Problem 3.2. (2 points) Assume the assumptions of CAPM. Then, the line (CML) is the tangent line of the feasible set going through the market . True or false? Solution: TRUE

Problem 3.3. (2 points) Assume the Capital Asset Pricing Model holds. You are given the following information about X and the market: • The annual effective risk-free rate is 4%. • The market’s is 0.08 and the market’s is 0.25. • The expected return of stock X is 6% and its volatility is 0.4. • The correlation between the returns of stock X and the market is 0.2. Then, the should invest in stock X. True or false? Solution: TRUE The β for the stock X equals 0.4(−0.2) β = = −0.32. X 0.25 So, the stock X has a required return equal to

rX = rf + βX (E[Rm] − rf ) = 0.04 + (−0.32)(0.08 − 0.04) = 0.0272. Since the expected return exceeds the required return, one should invest in stock X.

Problem 3.4. (2 points) Under the CAPM, the expected return and the required return of the are equal. True or false? Solution: TRUE

Problem 3.5. (2 points) Familiarity bias can result in a systematic trading bias and result in a deviation of prices from their fundamental values. True or false? 3

Solution: FALSE

Problem 3.6. (2 points) Overconfidence bias is a tendency to hold onto losers and sell winners. True or false?

Solution: FALSE Overconfidence bias results from uninformed individuals overestimating the precision of their knowledge.

Problem 3.7. (2 points) Disposition effect can result in a systematic trading bias which can shift the asset prices from their fundamental values. True or false?

Solution: TRUE

Problem 3.8. (2 points) Herd behavior is when investors imitate each others actions. True or false?

Solution: TRUE

Problem 3.9. (2 points) Assume the CAPM holds. Investors are only allowed to invest in stock A and stock B. One investor invests $2,000 in stock A and $3,000 in stock B. Another investor has $5,000 invested in stock A. Then, necessarily, he has $7,500 invested in stock B. True or false?

Solution: TRUE

Problem 3.10. (2 points) Assume the CAPM holds. The risk premium of a zero- stock equals the risk premium of the market portfolio. True or false?

Solution: FALSE It actually equals zero.

Problem 3.11. (2 points) The (α) measures the distance the stock’s average return is away from the (SML). True or false?

Solution: TRUE 4

3.3. FREE-RESPONSE PROBLEMS. Please, explain carefully all your statements and as- sumptions. Numerical results or single-word answers without an explanation (even if they’re correct) are worth 0 points.

Problem 3.12. (10 points) Consider a portfolio of four stocks as displayed in the following table: Stock Weight Beta 1 0.1 1.2 2 0.2 1.4 3 0.5 1.0 4 0.2 β4 The expected return of the portfolio is 0.12, the annual effective risk-free rate is 0.04, and the premium is 0.06. Assuming the Capital Asset Pricing Model, calculate β4. Solution: From the conditions on the first three stocks, we get

r1 = 0.04 + 1.2(0.06) = 0.112,

r2 = 0.04 + (1.4)(0.06) = 0.124,

r3 = 0.04 + 1.0(0.06) = 0.1. Therefore, from the condition on the portfolio’s expected return, we get

0.1(0.112) + 0.2(0.124) + 0.5(0.1) + 0.2r4 = 0.12 ⇒ r4 = 0.17 Finally, we have 0.17 − 0.04 13 β = = ≈ 2.17. 4 0.06 6

Problem 3.13. (10 points) In a market, the risk-free interest rate is given to be 0.04. Consider an investment I in this market, whose is 0.4. You construct a portfolio where you put a third of your wealth in the investment I and the remainder of your wealth in the risk-free asset. The expected return of this portfolio is 0.08. You decide to rebalance your portfolio so that four fifths of your wealth gets invested in the investment I and the remainder is invested in the risk-free asset. What is the volatility of this portfolio?

Solution: Let’s denote the volatility of investment I by σI and the volatility of the new portfolio by σP 0 . Then, σP 0 = 0.8σI . The expected return of the old portfolio is 1 2 [R ] = [R ] + r . E P 3E I 3 f So,

E[RI ] = 3E[RP ] − 2rf = 3(0.08) − 2(0.04) = 0.24 − 0.08 = 0.16. 5

We are given the Sharpe ratio of the investment I, and so we can calculate 0.16 − 0.04 12 σ = = = 0.3. I 0.4 40 Finally, the new portfolio’s volatility is

σP 0 = 0.8(0.3) = 0.24.

Problem 3.14. (15 points) According to your model, the economy over the next year could be good or bad. You are a pessimist and believe that the economy is twice as likely to be bad than good. Consider two assets, X and Y , existing in this market. If the economy is good the return on asset X is 0.12, and the return on asset Y is 0.11. If the economy is bad the return on asset X is −0.03 and the return on asset Y is −0.01. You construct a portfolio P using assets X and Y so that the portfolio’s expected return equals 0.025. Calculate the volatility of this portfolio’s return.

Solution: First, we need to figure out the weight wX the asset X is given in portfolio P . The expected returns of assets X and Y are 1 2 0.12 − 0.06 [R ] = (0.12) + (−0.03) = = 0.02, E X 3 3 3 1 2 0.11 − 0.02 [R ] = (0.11) + (−0.01) = = 0.03. E Y 3 3 3

So, the portfolio P must be equally weighted, i.e., wX = 0.5. Hence, the distribution of the portfolio’s return can be described as ( 0.115, with probability 1/3 R ∼ P −0.02, with probability 2/3 The second moment of the portfolio’s return is 1 2 [R2 ] = (0.115) × + (−0.02)2 × = 0.004675. E P 3 3 The variance of the portfolio’s return is 2 V ar[RP ] = 0.004675 − 0.025 = 0.00405. √ Finally, its volatility equals σ = 0.00405 = 0.0636396.

Problem 3.15. (10 points) Assume the CAPM holds. Let the risk-free interest rate be 0.04 and let the expected return of a market portfolio be equal to 0.08. Suppose that stock X has βX = 2 and that stock Y has βY = 0.8. Using the risk-free asset, stock X, and stock Y , you create a portfolio such that the weight given to X equals the weight given to Y while the weight of the risk-free asset is 0.2. What is the expected return of this portfolio? 6

Solution: The β of the risk-free asset is zero. Hence, the β of the portfolio is

βP = 0.4βX + 0.4βY = 0.4(2.8) = 1.12. So, realizing that the expected return of the portfolio equals its required return, we get

E[RP ] = rf + βP (rm − rf ) = 0.04 + 1.12(0.04) = 2.12(0.04) = 0.0848.

Problem 3.16. (10 points) Suppose that your market consists exactly of the five different stocks whose information is given in the following table: Stock Price per share Number of (in 106) 1 20 12 2 20 14 3 30 10 4 35 4 5 30 4 What are the portfolio weights in the market portfolio? Solution: The total market value of the shares of the five stocks are (in millions of $):

MV1 = 20 ∗ 12 = 240,

MV2 = 20 ∗ 14 = 280,

MV3 = 30 ∗ 10 = 300,

MV4 = 35 ∗ 4 = 140,

MV5 = 30 ∗ 4 = 120. The total market value of all the shares in the market is (in millions of $): MV = 240 + 280 + 300 + 140 + 120 = 1080. The weights of the five stocks in the market portfolio are

w1 = 240/1080 = 2/9, w2 = 280/1080 = 7/27, w3 = 300/1080 = 5/18,

w4 = 140/1080 = 7/54, w5 = 120/1080 = 1/9.

3.4. MULTIPLE CHOICE QUESTIONS. Problem 3.17. (5 points) Consider a portfolio of four stocks as displayed in the following table: Stock Weight Beta 1 0.2 1.2 2 0.2 1.4 3 0.3 1.0 4 0.3 -0.4 What is the portfolio’s beta? (a) 0.6 (b) 0.7 7

(c) 0.8 (d) Not enough information is provided. (e) None of the above. Solution: (b)

βP = 0.2(1.2) + 0.2(1.4) + 0.3(1.0) + 0.3(−0.4) = 0.7.

Problem 3.18. (5 points) For a certain stock, you are given that its expected return equals 0.12 and that its β equals 1.2. For another stock, you are given that its expected return equals 0.07 and that its β equals 0.4. Both stocks lie on the Security Market Line (SML). What is the risk-free interest rate rf ? (a) 0.04 (b) 0.045 (c) 0.0625 (d) 0.1075 (e) None of the above. Solution: (b) Denote the expected return of the market portfolio by rm. Then,

0.12 = rf + 1.2(rm − rf ),

0.07 = rf + 0.4(rm − rf ). Subtracting the second equation from the first one, we get 0.05 0.05 = 0.8(r − r ) ⇒ r − r = = 0.0625. m f m f 0.8 Substituting the obtained risk premium of the market portfolio into the first equation above, we obtain

rf = 0.12 − 1.2(0.0625) = 0.045.

Problem 3.19. (5 points) Assume the Capital Asset Pricing Model holds. You are given the following information about stock X, stock Y, and the market: • The required return and volatility for the market portfolio are 0.08 and 0.25, respectively. • The required return and volatility for the stock X are 0.06 and 0.4, respectively. • The correlation between the returns of stock X and the market is 0.25. • The volatility of stock Y is 0.3. • The correlation between the returns of stock Y and the market is 0.2. Calculate the required return for stock Y. (a) 0.0489 (b) 0.0542 (c) 0.0691 (d) 0.0734 (e) None of the above. 8

Solution: (c) The βs of stocks X and Y are 0.4(−0.25) β = = −0.4, X 0.25 0.3(0.2) β = = 0.24. Y 0.25 So, the required return of stock X must satisfy

0.06 = rX = rf + (−0.4)(0.08 − rf ) ⇒ 0.06 = rf − 0.032 + 0.4rf

⇒ 1.4rf = 0.092 ⇒ rf = 0.0657. Finally, the required return of stock Y equals

rY = 0.0657 + 0.24(0.08 − 0.0657) = 0.0691.