# Fins2624 Assignment 2013

FINS2624MOCKTERMS1, 2016INSTRUCTIONS1) Write your name, student ID number and signature in the assigned space on the front page.2) To make sure that we can identify your exam if your student ID number is hard to read,please tick the boxes corresponding to your student ID number on the front page. Do this bymarking the box corresponding to each of the digits in your seven digit student ID number,going from left to right. For example, my ID number is z3352704, so I would fill in the boxesas below.112233445566778899003) Fill in the details requested by the generalized answer sheet.4) This paper has two sections:Section A: 30 multiple choice questions (worth a total of 30 marks)Section B: 5 free-format problem (worth a total of 42 marks)5) Mark your answers to the multiple choice questions of section A in the generalized answersheet using a 2B pencil. Write down the solution of the free format problem in the answerssheets in the back. All answers and solutions must be written in ink. Pencils may not beused.6) Notation and terminology in this exam are as defined in the lectures.7) Assumptions that have been made throughout the lectures may be assumed throughout theexam unless otherwise stated. For instance, unless explicitly relaxed, you may assume thatthere are no transaction costs, that bonds have no default risk, that investors are rational,that there are no restrictions on short positions etc.8) Total number of marks for this paper is 72.9) This is a closed book exam. Except calculators according to the UNSW guidelines, no othermeans are allowed.10) This question sheet and the generalized answer sheet must bereturned in full.Do nottear the pages.11) You must prove your identity using your student ID card when turning in the exam.Page 2 of 26Please see over

FINS2624 S2 2012 Mockterm FAMILY NAME OTHER NAMES STUDENT ID rm SIGNATURE THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF BANKING AND FINANCE M oc kt e FINS2624 PORTFOLIO MANAGEMENT MOCK MID-TERM EXAMINATION - SEMESTER 2, 2012 TIME ALLOWED: N/A To make sure that we can identify your exam if your student ID number is hard to read, please tick the boxes below to fill in your seven digit student ID number. Detailed instructions are on the next page. 1 2 3 4 5 6 7 8 9 0 Page 1 of 21 Please see over 1 2 3 4 5 6 7 8 9 0 FINS2624 S2 2012 Mockterm INSTRUCTIONS 1) Write your name, student ID number and signature in the assigned space on the front page. 2) To make sure that we can identify your exam if your student ID number is hard to read, please tick the boxes corresponding to your student ID number on the front page. Do this by marking the box corresponding to each of the digits in your seven digit student ID number, going from left to right. For example, my ID number is z3352704, so I would fill in the boxes as below. 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 3) Fill in the details requested by the generalized answer sheet. 4) This paper has two sections: Section A: 30 multiple choice questions (worth a total of 30 marks) Section B: 5 free-format problem (worth a total of 42 marks) 5) Mark your answers to the multiple choice questions of section A in the generalized answer sheet using a 2B pencil. Write down the solution of the free format problem in the answers sheets in the back. All answers and solutions must be written in ink. Pencils may not be used. 6) Notation and terminology in this exam are as defined in the lectures. 7) Assumptions that have been made throughout the lectures may be assumed throughout the exam unless otherwise stated. For instance, unless explicitly relaxed, you may assume that there are no transaction costs, that bonds have no default risk, that investors are rational, that there are no restrictions on short positions etc. 8) Total number of marks for this paper is 72. 9) This is a closed book exam. Except calculators according to the UNSW guidelines, no other means are allowed. 10) This question sheet and the generalized answer sheet must be returned in full. Do not tear the pages. 11) You must prove your identity using your student ID card when turning in the exam. Page 2 of 21 Please see over FINS2624 S2 2012 Mockterm FOMRULA SHEET The price of a bond, P , is given by: c 1 FV P = 1− + T y (1 + y) (1 + y)T where c is the dollar coupon, y is the yield-to-maturity, T is the time to maturity and F V is the face value. If X and Y are two stochastic variables and α and β are two constants then: E(X + Y ) = E(X) + E(Y ) E(αX) = αE(X) V ar(X) = Cov(X, X) Cov(X, Y ) = Cov(Y, X) Cov(X, Y ) = ρXY σX σY Cov(αX, βY ) = αβCov(X, Y ) The solution to the quadratic equation ax2 + bx + c = 0 is x1/2 = −b ± √ b2 − 4ac 2a Page 3 of 21 Please see over FINS2624 S2 2012 Mockterm SECTION A OF THE MOCKTERM EXAM 30 MARKS IN TOTAL THIRTY MULTIPLE CHOICE QUESTIONS • Mark the answers in the generalized answer sheet using a 2B pencil. Or don’t, since this is a mockterm. • Each question is worth one mark and there is no negative marking. 1. Which of the following statements about bonds is true? a) A bond with a higher coupon rate must be a better investment than a bond with a lower coupon rate. b) A bond that trades at a discount must be a better investment than a bond that trades at a premium. c) A bond with a higher yield-to-maturity is a better investment than a bond with a lower yield to maturity. d) Several of the above statements are true. e) None of the above statements are true. E is correct. C is false since the yield-to-maturity, y is the constant hypothetical interest rate that solves c 1 FV P = 1− + T y (1 + y) (1 + y)T It says nothing about whether the bond is correctly priced (which we assume it is unless we have some specific reason to believe otherwise) or about how good an investment it is. Similar arguments go for A and B, e.g. the statements have no relevant pricing implications. 2. Which of the following statements about the efficient frontier is true? a) In practice, only the market portfolio is on the efficient frontier. b) The portfolios on the efficient frontier are only dominated by other portfolios on the efficient frontier. c) The efficient frontier tends to be concave. d) Several of the above statements are true. e) None of the above statements are true. The efficient frontier is a set of portfolios that minimize the standard deviation for their respective expected returns., so A is false. The portfolios on the efficient frontier are not dominated by any portfolios, so B is false. C is correct. Page 4 of 21 Please see over FINS2624 S2 2012 Mockterm 3. Which of the following statements about the yield-to-maturity is true? a) Discounting all cash flows of a bond with the bond’s yield-to-maturity only gives us the correct price if we have a flat term structure of interest rates. b) The yield-to-maturity is upwards sloping. c) The yield-to-maturity is always a spot rate. d) Several of the above statements are true. e) None of the above statements are true. E is correct. The yield-to-maturity, y is the constant hypothetical interest rate that solves 1 FV c 1− + P = T y (1 + y) (1 + y)T The concept does not rely on the actual existence of such an interest rate, so A and C are false. By definition, it is a constant so B is false. 4. Which of the following statements about bonds is true? a) A zero-coupon bond always trades at a discount as long as its yield-to-maturity is positive. b) A zero-coupon bond tends to have a lower yield-to-maturity than a coupon bond with the same time to maturity. c) In reality, zero-coupon bonds have higher default risk than coupon bonds. d) Several of the above statements are true. e) None of the above statements are true. A is correct. If the bond’s yield is positive the face value will be discounted and there are no coupons to balance that effect out. B is incorrect as the coupon rate does not have any direct effect on the yield. If anything, the zero-coupon bond would tend to have a higher yield as it has a higher duration and the term-structure of interest rates is typically (but not necessarily) upwards sloping. C is false as the coupon rate says nothing about default risk. 5. Which of the following statements about investors are true? a) A mean-variance investor will always choose the asset with the highest return. b) A mean-variance investor will always choose the asset with the lowest risk. c) A mean-variance investor will trade off expected returns and risk. d) Several of the above statements are true. e) None of the above statements are true. C is correct. The investor will balance expected returns and risk against each other. Page 5 of 21 Please see over FINS2624 S2 2012 Mockterm 6. Which of the following statements about duration are true? a) The duration of a liability is always negative. b) The duration of a bond is always positive. c) The duration of a portfolio is a weighted average of the duration of the bonds that are included in that portfolio. d) Several of the above statements are true. e) None of the above statements are true. D is correct. A is false and B is true, since the duration of a liability or a bond is always positive. This is obvious if you think about duration as an adjusted time-to-maturity measure. C is true. See lecture notes 2 for a sketch of the proof. 7. Which of the following statements about yield-to-maturity are true? a) A bond with a yield-to-maturity of 7% is never more sensitive to changes in the yield than a bond with a yield to maturity of 5%. b) A bond with a yield-to-maturity of 7% is always more sensitive to changes in the yield than a bond with a yield to maturity of 5%. c) A bond with a yield-to-maturity of 5% is never more sensitive to changes in the yield than a bond with a yield to maturity of 7%. d) A bond with a yield-to-maturity of 5% is always more sensitive to changes in the yield than a bond with a yield to maturity of 7%. e) None of the above statements are true. E is correct. The statements are not true, since the sensitivity depends on duration. 8. Which of the following statements about diversification are true? a) The diversification benefits in a portfolio of two assets are smaller when the correlation between the returns of the assets is larger. b) The diversification benefits in a portfolio of two assets are larger when the correlation between the returns of the assets is larger. c) The diversification benefits of combining three perfectly correlated assets is typically larger than the diversification benefits of combining two perfectly correlated assets. d) Several of the above statements are true. e) None of the above statements are true. A is correct. A is true and B is false as the less correlated two assets are, the more of their risk will ”cancel out”. C is false as there are no diversification benefits of combining perfectly correlated assets. Page 6 of 21 Please see over FINS2624 S2 2012 Mockterm 9. Which statement is true regarding the market portfolio? a) It includes all publicly traded financial assets. b) It lies on the efficient frontier. c) All securities in the market portfolio are held in proportion to their market values. d) It is in principle unobservable. e) All of the above statements are true. E is correct. 10. Which of the following statements is false about the capital allocation line (CAL)? a) It includes all publicly traded financial assets. b) Investors prefer assets further to the right on the CAL to assets further to the left on the CAL. c) The risk-free asset plots on the CAL. d) More than one of the above statements are false. e) None of the above statements is false. D is correct. A is false as assets generally carry some unsystematic (and hence unpriced) risk. They will therefore plot below the CAL. B is false (as a general statement) because the investors’ choice of assets on the CAL depend on their risk aversion. 11. Which of the following statements is false about the CAPM? a) The CAPM describes the required return of any risky asset. b) The CAPM assumes that investors agree on the statistical properties of all assets. c) The CAPM assumes that all investors have the same preferences. d) More than one of the above statements are false. e) None of the above statements is false. C is correct. Investors typically have different risk preferences, and the CAPM allows for this. 12. The risk free rate is 4% and the expected return of the market portfolio is 12%. If a stock has a CAPM β of 1.5, a standard deviation of returns of 20% and a current price of $100, what is the expected price one year from now? a) $100 b) $116 c) $120 d) There’s not enough information to say. e) None of the above. First find the expected return through the CAPM equation: E(r) = 0.04 + 1.5(0.12 − 0.04) = 0.16 Since the price today is $100, the expected price in one year is 100(1 + 0.16) = 116. B is correct. Page 7 of 21 Please see over FINS2624 S2 2012 Mockterm 13. If the standard deviation of a stock’s return increases, what happens to its required return? a) It decreases. b) It increases. c) It stays the same. d) There’s not enough information to tell. e) None of the above. D is correct, because the standard deviation has a systematic and an unsystematic component. If the standard deviation increases as a result of an increase in unsystematic risk alone, the required return is not affected. 14. Suppose a stock that was traded on the market magically disappears. What happens to the CAL? a) Its slope decreases. b) Its slope increases. c) Its slope stays the same. d) There’s not enough information to tell. e) None of the above. Since the stock was traded, it was held in the optimal risky portfolio. By definition, the optimal risky portfolio maximizes the Sharpe ratio. If one stock disappears, the portfolio has to be adjusted, and it can only be adjusted to a worse position. A is correct. For the next 3 questions, assume that there is some risky portfolio Q, which has an expected return of 15%, and a standard deviation of 12%. The risk-free rate is 5%. There is also some asset P that is not part of Q, which has an expected return of 10%, and a standard deviation of 7%. 15. What is the Sharpe ratio of Q? a) 0.71 b) 1 c) 1.34 d) 0.83 e) There’s not enough information to tell D is correct. From the definition of the Sharpe ratio: SQ = 0.15 − 0.05 ≈ 0.83 0.12 Page 8 of 21 Please see over FINS2624 S2 2012 Mockterm 16. What is the Sharpe ratio of the market portfolio? a) Between 0.71 and 1.34 b) Larger than or equal to 0.83 c) Exactly 1 d) Lower than or equal to 0.83 e) There’s not enough information to tell We cannot calculate the Sharpe ratio of the market, but we know that it must be the best possible Sharpe ratio given the investment universe. Since SP = 0.1 − 0.05 ≈ 0.71 0.07 SQ = 0.15 − 0.05 ≈ 0.83 0.12 The market portfolio must have a Sharpe ratio that is larger or equal to 0.83. B is correct. 17. You want to invest in some combination of the Q and the risk-free asset to achieve an expected return of 40%. What is the standard deviation of your returns? a) 42% b) 32% c) 16% d) 40% e) There’s not enough information to tell Let wQ be your investment in Q. It must be chosen so that (1 − wQ )0.05 + wQ 0.15 = 0.4 0.05 + wQ (0.15 − 0.05) = 0.4 wQ = 0.4 − 0.05 = 3.5 0.15 − 0.05 As we saw in lecture 5, it follows that the standard deviation is: σ = wQ σQ = 3.5 · 0.12 = 0.42 A is correct. For the next 4 questions, consider a market where the risk free rate is 3% and the expected return on the market portfolio is 10%. Page 9 of 21 Please see over FINS2624 S2 2012 Mockterm 18. What is the required rate of return of an investment with a beta of 0.5? a) 10% b) 5% c) 3% d) There’s not enough information to tell. e) None of the above. According to the CAPM, the required rate of return is E(r) = 0.03+0.5(0.1−0.03) = 0.065 E is correct. 19. What is the standard deviation of the market portfolio returns? a) 1 b) 20% c) 22.59% d) There’s not enough information to tell. e) None of the above. D is correct. 20. An investment has a beta of 0.8 and an expected return of 8%. Should you undertake the investment? a) Yes. b) No. c) It doesn’t matter. d) There’s not enough information to tell. e) None of the above. The required rate of return is E(r) = 0.03 + 0.8(0.1 − 0.03) = 0.086 > 0.08. Since the required return is larger than the expected return, B is correct. 21. If the market expects a return of 20.5% from a certain stock, what is its beta? a) 21 b) 2.5 c) 2.45 d) There’s not enough information to tell. e) None of the above. The CAPM states that E(ri ) = rf + β(E[rM ] − rf ) β= 0.205 − 0.03 E(ri ) − rf = = 2.5 E(rM ) − rf 0.1 − 0.03 B is correct. Page 10 of 21 Please see over FINS2624 S2 2012 Mockterm 22. According to the CAPM a well-diversified portfolio’s rate of return is a function of a) market risk. b) unsystematic risk. c) unique risk. d) reinvestment risk. e) liquidity risk. A is correct. 23. Which of the following statements is true about the security market line, SML? a) It is a different name for the CAL. b) It contains all efficient portfolios. c) It is convex for risk averse investors. d) More than one of the above statements are true. e) None of the above statements are true. B is correct. 24. The total value of the market portfolio is $100,000. Stock A is trading at $10. The company has issued 1,000 stocks. There are 5 other companies on the market. What is the weight of company A in the market portfolio? a) 20% b) 0.01% c) 10% d) 1% e) 25% The weight of an asset i in a market with N assets is wi = Vi N P Vj j=1 where Vj is the market value of asset j. In this case we have wA = 10 · 1000 = 0.1 100000 C is correct. Page 11 of 21 Please see over FINS2624 S2 2012 Mockterm 25. The shape of the term structure of interest rates is determined by a) Expectations on future spot rates b) The risk aversion of investors c) Mismatches between the investment horizon of investors in bonds and issuers of bonds d) A and C e) All of the above The expectations hypothesis predicts A and the liquidity preference theory predicts B and C. Since both effects are likely at work, E is correct. 26. The convexity of a bond... a) Leads to an undesirable asymmetry in the price response to yield changes b) Leads to a desirable asymmetry in the price response yield changes c) Is a measure of the bond’s time to maturity d) A and C e) B and C B is correct. Please see lecture slides 3 for details. 27. If all spot rates decreased by 1%, the duration of a 10-year 5%-coupon bond would... a) Increase b) Decrease c) Be unaffected d) Change in a direction that depends on its convexity e) There’s not enough information to tell If all spot rates decreased, the yield of all bonds would decrease too. The present value of cash flows further away in time would increase relatively more, so the duration would increase. A is correct. 28. Duration... a) Is a measure of the time to maturity of a bond’s cash flows b) Is a measure of the sensitivity of a bond’s price to changes in the yield c) Can be used to approximate the price of a bond at other yields d) A and B e) All of the above E is correct. Page 12 of 21 Please see over FINS2624 S2 2012 Mockterm 29. According to the expectations hypothesis a) The term structure of interest rates is upwards sloping b) Investors are risk-neutral c) Future spot rates are known today d) All of the above e) None of the above E is correct. The expectations hypothesis states that E(s rt ) =s ft , which does not imply any of the statements. 30. A synthetic instrument... a) Is typically a portfolio of other instruments b) Is typically used to set up an arbitrage trade c) Cannot be traded d) A and B e) All of the above D is correct. Page 13 of 21 Please see over FINS2624 S2 2012 Mockterm SECTION B OF THE MOCKTERM EXAM 42 MARKS IN TOTAL FIVE FREE-FORMAT QUESTIONS 1. Assume that the interest rate is flat, with spot rates of 11% for all maturities. Consider a market with the following two bonds: • Bond A is a one-year zero-coupon bond with a face value of $100. It trades for $90.09. • Bond B is a three-year zero-oupon bond with a face value of $100. It trades for $73.12. (a) (1 mark) Which of the two bonds is more sensitive to changes in the yield and why? Bond B is more sensitive, since it has a larger duration. (b) (5 marks) Suppose you have a liability of $1,000 maturing in 4 years. Set up a portfolio of bonds that immunizes the liability. Your final answer should specify what position in each bond you should take, e.g. ”Buy 4 of bond A and sell 1 of bond B”. First, we need to specify the durations of the instruments we need to work with. Since all are zero-coupon bonds, the durations are DA = 1, DB = 3 and DL = 4. To immunize our liability we need to take an offsetting (long) position in a portfolio consisting of bonds A and B. Let wA be the weight of bond A in that portfolio (and (1 − wA ) by implication the weight of bond B. Since the duration of a portfolio is the weighted average of the durations of the bonds that it consists of, we have that: DP = wA DA + (1 − WA )DB We choose WA so that this duration equals that of the liability: DP = wA DA + (1 − WA )DB = DL wA = 4−3 DL − DB = = −0.5 DA − DB 1−3 Since this is negative, we must short sell bond A. The resulting weight in bond B is wB = 1 − (−0.5) = 1.5. To immunize our liability, we need to make an investment of the same present value. To find these values we first need to price the bonds: PA = 100 · 1.11−1 ≈ 90.09 PB = 100 · 1.11−3 ≈ 73.12 PL = 1000 · 1.11−4 ≈ 658.73 The resulting investments in bond A, NA , and bond B, NB are: Page 14 of 21 Please see over FINS2624 S2 2012 Mockterm −0.5 · 658.73 wA PL ≈ ≈ −3.66 PA 90.09 w B PL 1.5 · 658.73 NB = ≈ ≈ 13.51 PB 73.12 NA = (c) (2 marks) What practical problems can you forsee with this immunization? • There may be non-parallel shifts in the yield curve. • If the yield shifts the liability would no longer be immunized and rebalancing the bond portfolio may be costly or difficult. • When time passes the liability would no longer be immunized and rebalancing the bond portfolio may be costly or difficult. • In practice there may be transaction costs associated with the trades, such as bid-ask spreads and short-selling costs (or short-selling restrictions). 2. You find the following term structure of interest rates in the market (where rt denotes the spot rate for time t): r1 r2 r3 r4 r5 = 6% = 8% = 8.5% = 9% = 10% (a) Assume that this term structure is explained by the liquidity theory of the term structure of interest rates. What does this tell you about each of the following (i) (1 mark) the market expectations on future spot rates It tells you nothing. (ii) (1 mark) the risk preferences of investors Investors are risk averse. (iii) (1 mark) the relative investment horizons of issuers of bonds and investors in bonds Issuers of bonds have longer investment horizons than investors (as the term structure is upwards sloping. (b) (5 marks) What is the arbitrage free forward rate for an investment that begins at time 2 and ends at time 5, 2 f5 ? This should be motivated with a no-arbitrage argument. First note that investing one (the trade can obviously be scaled by an arbitrary factor without point deductions) dollar at the forward rate yields an amount (1 +2 f5 )3 at time 5. The replicating trades include the following: Page 15 of 21 Please see over FINS2624 S2 2012 Mockterm 1 = 1.08−2 ≈ 0.86 dollars at time zero. (1 + r2 )2 1 • Reinvest the = 1.08−2 ≈ 0.86 dollars for five years (until time 5). (1 + r2 )2 • Repay 1 dollar at time 2. (1 + r5 )5 1.15 • Withdraw = ≈ 1.38 dollars at time 5. (1 + r2 )2 1.082 • Borrow Since the cash flows at all times but time 5 are equal for the forward trade and the replicating trades, the time 5 cash-flows must also be equal in the absense of arbitrage. It follows that: 1.15 = (1 +2 f5 )3 1.082 1/3 1.15 − 1 ≈ 11.35% 2 f5 = 1.082 3. (5 marks) You observe the following bonds in the market: • Bond A is a three-year bond with a face value of $100 that pays an annual coupon of $10. It trades for $117.02. • Bond B is a two-year zero-coupon bond with a face value of $200. It trades for $188.52. • Bond C is a two-year zero-coupon bond with a face value of $1000. It trades for $961.17. Show that there is an arbitrage opportunity in this market by describing a set of trades that would result in an arbitrage profit. Your final answer should specify what position in each bond you should take, e.g. ”Buy 4 of bond A and sell 1 of bond B”. This should be motivated with a no-arbitrage argument. Bond A is a red herring. We start by getting the implied two year spot rates from bonds B and C: r2A r F VA −1= PA r 200 − 1 = 3% 188.52 r F VB −1= PB r 1000 − 1 = 2% 961.17 = r2B = Since these are not equal there must be an arbitrage opportunity. We would like to invest in the high-yield bond, i.e. bond A, and hedge our cash flows with the low-yield bond, i.e. bond B. Suppose we buy NA = 1 bond A. The resulting cash flows are: CF0 = −188.52NA CF2 = +200NA To cancel out the time two cash flow, we sell NB of bond B. The resulting cash flows are: Page 16 of 21 Please see over FINS2624 S2 2012 Mockterm CF0 = +961.17NB CF2 = −1000NB In order to have zero net cash flows at time 2, it must hold that: 200NA = −1000NB NB = − 1 200 ·1=− 1000 5 So we end up buying one bond A and selling 1 5 bond B. The resulting net cash flows are: CF0 = 15 · 961.17 − 188.52 = 3.714 CF2 = 0 Obviously, any trades where NA = 5NB are acceptable. 4. Consider a market with the following three bonds (note the non-standard face value of bond B): Bond A B L YTM 4% 5.93% ? Price 96.15 ? 90.7 Cash flows t=1 t=2 100 0 5 5 0 100 t=3 0 100 0 a) (1 mark) What is the price of bond B? This is found either by (a version of) the bond pricing formula or discounting each cash flow and adding them all together: PB = 5 5 5 100 (1 − 1.0593−3 ) + 95 · 1.0593−3 = + + = 93.30445 2 0.0593 1.0593 1.0593 1.05933 b) (2 marks) Determine all spot rates. r1 and r2 can be read immediately from the zero-coupon bonds: r1 = 4% r2 = 5% We can get r3 by bootstrapping: 5 5 100 + + 2 1.04 1.05 (1 + r3 )3 1/3 5 5 =⇒ r3 = 100 93.30445 − − − 1 = 6% 1.04 1.052 93.30445 = Page 17 of 21 Please see over FINS2624 S2 2012 Mockterm c) (2 marks) Calculate the duration of all bonds. DA and DL can be deduced directly as they’re zero-coupon bonds: DA = 1 DL = 2 DB can be calculated using the duration formula: DB = 5/1.0593 5/1.05932 100/1.05933 ·1+ ·2+ · 3 ≈ 2.85 93.30445 93.30445 93.30445 d) (4 marks) Suppose that bond L is a liability that your company has. Form a portfolio consisting of bonds A and B that immunizes that liability. How much do you invest in bonds A and B respectively? Please show all the steps leading up to your answer. Set up a portfolio with an appropriate weight in asset A to achieve the target duration: wA DA + (1 − wA )DB = DL wA + (1 − wA )2.85 ≈ 2 wA ≈ 0.46 wB ≈ 1 − 0.46 = 0.54 Since the total investment must be equal to the present value of the liability your investments in each asset would be: IA ≈ 0.46 · 90.7 ≈ 41.70 IB ≈ 0.54 · 90.7 ≈ 49.00 e) (4 marks) Determine the forward rate between t = 1 and t = 3, 1f 3, through a noarbitrage argument. That is, you must describe the potential arbitrage trades that motivate your answer. The cash-flow consequences of investing at the forward rate are: CF1 : −1 CF3 : +(1 +1 f3 )2 We set up a replicating portfolio. First we borrow the present value of CF1 above. The cash flow consequences of this are: CF0 : +1/(1 + r1 ) = 1/1.04 CF1 : −1 We then reinvest this amount until time 3, with cash flow consequences: CF0 : −1/(1 + r1 ) = −1/1.04 CF3 : +(1 + r3 )3 /(1 + r1 ) = 1.063 /1.04 Page 18 of 21 Please see over FINS2624 S2 2012 Mockterm Our net cash flow consequences are: CF0 : 1/(1 + r1 ) − 1/(1 + r1 ) = 0 CF1 : −1 CF3 : +(1 + r3 )3 /(1 + r1 ) = 1.063 /1.04 Since CF0 and CF1 are the same as in the strategy we’re replicating, CF3 must also be the same in the absence of arbitrage opportunities: CF3 : +(1 + r3 )3 /(1 + r1 ) = 1.063 /1.04 = (1 +1 f3 )2 We solve this equation for 1 f3 : 1 f3 = p 1.063 /1.04 − 1 = 0.07 5. Suppose there are three bonds trading in the market with a face value of $100 and maturing in two years. The bonds all pay annual coupons. Their coupon rates are 10%, 20% and 30% respectively and they trade for $97.40, $115.08 and $132.48 respectively. a) (5 marks) Show that there is an arbitrage opportunity. The first step is to figure out if there is indeed a mispricing in the market that allows for an arbitrage trade. We know that in an arbitrage-free market, all bonds are priced using the same term structure of interest rates. We’ll back out the implied term-structure from two bonds and check if the price of the last bond is consistent with it. We could take any two bonds, but let’s pick the first two (which we’ll denote bond 1 and bond 2). These price of these bonds should be the sum of the present value of their futre cash flows. We’ll get the following two pricing equations: 110 10 + P1 = 97.4 = 1 + r1 (1 + r2 )2 P2 = 115.08 = 20 120 + 1 + r1 (1 + r2 )2 Since we have two unknowns, we’ll use the first equation to express (1 + r2 )2 in terms of r1 . The algebra gets a bit messier here than you’d probably find in a real exam, but conceptually there’s nothing strange going on. This example also serves to illustrate one of the online quiz questions that seemed to cause some confusion: 10 110 + 1 + r1 (1 + r2 )2 2 (1 + r2 ) = 110 97.4 − 97.4 = 10 1 + r1 We’ll use this to solve for r1 in the second equation: 115.08 = 20 120 + 1 + r1 (1 + r2 )2 Page 19 of 21 Please see over FINS2624 S2 2012 Mockterm 115.08 = 20 + 120 1 + r1 , 110 97.4 − 10 1 + r1 20 120 · 97.4 120 10 + − · 1 + r1 110 110 1 + r1 120 · 10 20 − 110 − 1 ≈ 3% r1 = 120 · 97.4 115.08 − 110 115.08 = It’s now straight forward to find r2 : 10 2 (1 + r2 ) = 110 97.4 − 1.03 s 10 r2 = 110 97.4 − − 1 ≈ 12% 1.03 Now that we have the implied term structure, let’s see if the last bond (let’s call it bond 3) has a price consistent with it. We calculate the implied price: P3IM P LIED = 30 130 · ≈ 132.76 > 132.48 = P3OBSERV ED 1.03 1.122 Since this price is ”too low”, there is an arbitrage opportunity in the market. b) (3 marks) Now assume that there is also a zero-coupon bond maturing in one year with a face value of $100, a price of $97.09 and a yield of 3%. Show how you can exploit the mispricing you found in (a). It is actually not necessary to introduce this new bond, but it makes the solution less messy. Since we already showed that the implied value of r1 is 3%, we know that the price of the new bond is consistent with bonds 1 and 2. We also know that the price of bond 3 is ”too low”, so we want to buy the bond to exploit the mispricing. This leads to the following cash flow consequences: CF0 : −132.48 CF1 : +30 CF2 : +130 We’ll need to take offsetting positions in the other bonds, e.g. we need to make sure that the net cash flows at times 1 and 2 are both zero, in order to set up the arbitrage trade. Let’s offset the time 2 cash flow with bond 1 (although we could just as well have used bond 2). Suppose we buy N1 of bond 1. We want to choose N1 such that the resulting cash flows at time 2 are 130 (exactly enough to offset our cash flow from our long position in one bond 3). It must be that: CF2 = 110N1 = −130 N1 = − 130 110 Page 20 of 21 Please see over FINS2624 S2 2012 Mockterm This is a negative number, which makes sense as we are trying to offset a positive future cash flow. We interpret this as a necessity to short sell 130/110 of bond 1. We also need to replicate the cash flow at time 1. We could do this by adding either bond 2 or the new zero-coupon bond into the mix. It is much easier to use the new bond (or we would have had to solve another system of equations). So let’s assume that we buy NN EW of the new bond. We must choose NN EW so that the cash flow of 30 at time 1 from our long position in one bond 3 gets cancelled out. We must remember, however, that we also have a cash flow from our short position in bond 1 to account for. NN EW must be: CF1 = − 130 10 + 100NN EW = −30 110 NN EW = 1300/110 − 30 2 =− 100 11 Our final answer is that we should buy one of bond 3; short sell 130/110 of bond 1 and short sell 2/11 of the new zero-coupon bond. To see that this actually results in an arbitrage profit, let’s write out the resulting cash flows: CF0 : −132.48 + 97.4 · 130/110 + 97.09 · 2/11 ≈ 0.28 > 0 CF1 : 30 − 10 · 130/110 − 100 · 2/11 = 0 CF2 : 130 − 110 · 130/110 = 0 We could of course make more than $0.28 by scaling up the trade. Page 21 of 21

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