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Fixed Income Securities: Valuation, Risk, And Risk Management, 1st Edition Test Bank

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    Test bank to accompany the textbook Fixed Income Securities: Valuation, Risk, and Risk Management by Pietro Veronesi Preliminary Version Author: Javier Francisco Madrid 1

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    Chapter 1 1. Is the following an arbitrage opportunity? A gift that makes me feel good just by having it. Ans. This is an arbitrage opportunity because it doesn’t cost anything at initi- ation and it generates a positive profit by a certain date in the future (i.e. ’you feel good’). 2. Is the following an arbitrage opportunity? A bond that cost nothing but will payoff zero with certainty in the future. Ans. This is not an arbitrage opportunity, since it doesn’t give a positive payoff in the future. 3. Is the following an arbitrage opportunity? A free car that if I repair well, I won’t have to spend money on gasoline or maintenance costs (i.e. repairs) ever. Ans. This is not an arbitrage opportunity, since I have to pay money (to repair the car) in order to be free of future costs. 4. Is the following an arbitrage opportunity? Suppose you are in the desert and are given a bag of ice with a penny inside. Assume that the ice will melt instantly and the cost of disposing of the bag is zero. Ans. This is an arbitrage opportunity because even though I can’t take advan- tage of the ice, I gain the penny for free. 5. Is the following an arbitrage opportunity? A security that cost zero and might pay a dollar in the future, but pays zero otherwise. Ans. This is an arbitrage opportunity because I get for free the chance of getting a dollar in the future. 6. What steps would you follow in order to take advantage of the following arbitrage opportunity (if there is one)? Security A costs $3 and pays $5 in 2 years, while security B costs $3 and pays $4 in 2 years. Ans. You borrow security B and sell it, which means you receive $3, with these proceeds you buy security A. In 2 years you receive $5 and have to pay $4. You make a $1 profit. 7. What steps would you follow in order to take advantage of the following arbitrage opportunity (if there is one)? Security A costs $100 and pays $120 in 3 years. Security B costs $100 and pays $110 in one year. Your friend tells you that he would like you to lend him $110 in a year and that he would give $130 the following year. Finally you know that in two years, with $130, you can invest in a security that will pay you either $140 or $121 (with equal probability) after a year. 2

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    Ans. You borrow security A and sell it, with the proceeds you buy security B. After a year you lend the money to your friend. The next year when he pays back, you invest in the risky security. After the third year, this will give you either $140 or $121, while you have to pay $120. So you either have a profit of $20 or $1. 8. What steps would you follow in order to take advantage of the following arbitrage opportunity (if there is one)? Security A costs $100 and pays $110 in 2 years. Security B costs $100 and pays $109 in one year. You know that in a year with $109 you can invest in a security that pays $120 or $109 (with equal probability) the following year. Ans. This is not an arbitrage opportunity. 9. Intuitively, is the Federal Funds rate generally higher, lower or the same as LIBOR? Why? Ans. The LIBOR rate should be higher than the Federal Funds rate, since it should include a higher probability of default by the banks trading LIBOR. 10. Intuitively, is LIBOR generally higher, lower or the same as the repo rate? Why? Ans. LIBOR should be higher than the repo rate since it is not collateralized with another security, as it occurs with the repo rate. 11. What are the steps to take a long position on a given U.S. security via the repo market? Ans. The trader must take the following steps (repo): At time t : i. Buy bond at P t and deliver the bond to the repo dealer. ii. The repo dealer will pay P t − haircut to the trader, which is made whole (minus the haircut) for the cost of the bond. At time T : iii. The trader gets the bond from the repo dealer and sells it for P T . iv. With the proceeds P T the trader pays back ( P t − haircut) × (1+Repo) to the repo dealer. 12. What are the steps to take a short position on a given U.S. security via the repo market? Ans. The trader must take the following steps (reverse repo): At time t : 3

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    i. Borrows the bond from the repo dealer and sells it at P t . ii. Receives P t which he posts as collateral with the repo dealer. At time T : iii. The trader buys the bond for P T and gives it back to the repo dealer. iv. The repo dealer pays P t × (1+Repo). The dealer makes a profit if this is larger than P T . 13. What’s the return on capital for a trader who entered into a one-month repo where P t = 98 . 5, P T = 99 . 01, Repo= 5% and haircut= 0 . 8? Ans. The return on capital is 0.13. 14. What’s the profit for a trader who entered into a one-week reverse repo where P t = 99 . 40, P T = 99 . 48 and Repo= 6%? Ans. The profit is 0.0347. 15. You find a bond that has a repo rate substantially lower than the GCR. Is this, for certain, an arbitrage opportunity? Ans. No, it might be that the bond is short on supply (hard to find), which means that the trader might be thinking that the bond is overpriced and is speculating that the price will fall. By entering in a reverse repo, the trader might make a substantial profit, even willing to forgo part of the rate to be in the transaction. 16. You are told that there is an ample supply for the bond mentioned in question 13. Does this affect your previous answer? Ans. Yes, since it shows that the lower rate is not due to scarcity of a specific type of bond. 17. What are the gains from trade of entering into a swap for these two firms? Firm A Firm B Fixed Rate 13% 16% Floating Rate LIBOR + 3% LIBOR + 6% Ans. Gains from trade are zero. 18. What are the gains from trade of entering into a swap for these two firms? Firm A Firm B Fixed Rate 10% 15.5% Floating Rate LIBOR + 2% LIBOR + 4% Ans. Gains from trade are 3.5%. 4

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    19. What are the gains from trade of entering into a swap for these two firms? Firm A Firm B Fixed Rate 9% 5% Floating Rate LIBOR + 7% LIBOR + 4% Ans. Gains from trade are 1%. 5

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    Chapter 2 1. Do discount factors depend on compounding frequency? Why? Ans. No, discount factors do not depend on compounding frequency, since they are terms of exchange (prices) between having money at time t versus having at a later date T . If they changed with compounding frequency, there would be an arbitrage opportunity. 2. What effect does inflation have on discount factors? Ans. Higher inflation makes less appealing money in the future, so discount factors will go down. 3. Can a bond be quoted in more than one interest rate? Ans. Yes. It can be quoted in various compounding frequencies. 4. From the following data obtain the discount curve: a. A zero coupon bond P z (0 , 0 . 5) = 99 . 20. b. A coupon bond paying 3% quarterly P (0 , 0 . 25) = 100 . 5485. c. A coupon bond paying 6% quarterly P (0 , 0 . 75) = 100 . 1655. d. A coupon bond paying 5% semiannually P (0 , 1) = 103 . 0325. Ans. The discount factors are the following: t Z (0 , t ) 0.25 0.9980 0.50 0.9920 0.75 0.9870 1.00 0.9810 5. Using the previous discount curve price the following: A zero coupon bond expiring at t = 0 . 75. Ans. The price of the bond is 98.70. 6. Using the previous discount curve price the following: A 1-year coupon bond paying 4% quarterly. Ans. The price of the bond is 102.0580. 7. Using the previous discount curve price the following: A 6-month coupon bond paying 7% semiannually. Ans. The price of the bond is 102.6720. 8. Using the previous discount curve price the following: A 9-month coupon bond paying 5% semiannually. 6

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    Ans. The price of the bond is 103.6625. 9. For the following scenario, check if there is a mispriced security: a. A zero coupon bond P z (0 , 0 . 5) = 99 . 00. b. A coupon bond paying 6% quarterly P (0 , 0 . 25) = 101 . 1955. c. A coupon bond paying 4% quarterly P (0 , 0 . 50) = 102 . 0830. d. A coupon bond paying 7% semiannually P (0 , 0 . 75) = 105 . 8440. Ans. The mispriced bond is [a.] the zero coupon bond. 10. For the following scenario, check if there is a mispriced security: a. A zero coupon bond P z (0 , 0 . 25) = 99 . 40. b. A zero coupon bond P z (0 , 0 . 50) = 98 . 00. c. A coupon bond paying 3% quarterly P (0 , 0 . 50) = 100 . 4880. Ans. The mispriced security is [b.]. 11. For the following scenario, check if there is a mispriced security: a. A coupon bond paying 1% quarterly P (0 , 0 . 25) = 100 . 6498. b. A coupon bond paying 4% semiannually P (0 , 0 . 25) = 101 . 8980. c. A coupon bond paying 3% quarterly P (0 , 0 . 50) = 101 . 2978. d. A coupon bond paying 5% quarterly P (0 , 0 . 75) = 103 . 4425. e. A coupon bond paying 4% semiannually P (0 , 1 . 00) = 103 . 5880. Ans. The mispriced bond is [a.]. 12. For the following scenario, check if there is a mispriced security: a. A zero coupon bond P z (0 , 0 . 5) = 99 . 50. b. A coupon bond paying 3% quarterly P (0 , 0 . 50) = 100 . 9948. c. A coupon bond paying 5% quarterly P (0 , 0 . 75) = 102 . 7288. d. A coupon bond paying 2% semiannually P (0 , 1 . 25) = 102 . 8720. e. A zero coupon bond P z (0 , 1 . 25) = 98 . 4. Ans. The mispriced bond is [d.]. 13. For the following scenario, check if there is a mispriced security: a. A zero coupon bond P z (0 , 0 . 25) = 99 . 30. b. A zero coupon bond P z (0 , 0 . 50) = 98 . 70. c. A coupon bond paying 3% semiannually P (0 , 0 . 50) = 100 . 1850. d. A coupon bond paying 2% semiannually P (0 , 0 . 75) = 101 . 4880. 7

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    Ans. The mispriced bond is [d.], since it requires the discount factor Z (0 , 0 . 75) to be larger than the previous ones. 14. What is the price on a 4.5-year floating rate bond that pays a semiannual coupon (no spread)? Ans. The price of the coupon is 100. 15. What is the price on a 5.75-year floating rate bond that pays a semiannual coupon (no spread)? We know the following: a. There is a coupon bond paying 3% quarterly P (0 , 0 . 25) = 100 . 0448. b. Last quarter the semiannually compounded rate was 3%. Ans. The price of the floating rate bond is 100.7895. 16. What is the price of a 0.5-year floating rate bond that pays a quarterly coupon equal to floating rate plus a 1% spread? We know the following: a. There is a zero coupon bond P z (0 , 0 . 25) = 99 . 80. b. There is a coupon bond paying 2% quarterly P (0 , 0 . 5) = 100 . 3960. Ans. The price of the floating rate bond is 100.4980. 17. What is the price of a 0.75-year floating rate bond that pays a semiannual coupon equal to floating rate plus 2% spread? We know the following: a. There is a zero coupon bond P z (0 , 0 . 25) = 99 . 70. b. There is a zero coupon bond P z (0 , 0 . 50) = 99 . 20. c. There is a coupon bond paying 3% quarterly P (0 , 0 . 75) = 101 . 7380. Ans. The price is 102.3145. 18. You have two coupon bonds with same maturity, one pays 5% semiannu- ally and the other 5% quarterly. Which one has a higher yield? Ans. The second bond has a higher yield because it compounds more frequently than the first one, increasing the expected return. 19. A Treasury dealer quotes the following 182-day bill at a 3.956% discount. What is the price of the security? Ans. The price of the Treasury is 98.00. 20. A Treasury dealer quotes the following 91-day bill at a 3.956% discount. What is the price of the security? Ans. The price of the Treasury is 99.00. 8

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    Chapter 3 Use the following discount factors when needed. t Z (0 , t ) 0.25 0.9840 0.50 0.9680 0.75 0.9520 1.00 0.9360 1.25 0.9190 1.50 0.9040 1.75 0.8880 2.00 0.8730 2.25 0.8587 2.50 0.8445 2.75 0.8308 3.00 0.8175 3.25 0.8047 3.50 0.7924 3.75 0.7806 4.00 0.7691 1. Calculate the duration of the following security: 5-year zero coupon bond. Ans. The duration of the security is 5.00. 2. Calculate the duration of the following security: 2-year fixed coupon pay- ing 5% quarterly. Ans. The duration of the security is 1.9138. 3. Calculate the duration of the following security: 1.25-year floating coupon paying float + 50 bps semiannually. You know that last quarter the semi- annual rate was 6.4%. Ans. The duration of the security is 0.2534. 4. Calculate the duration of the following portfolio: i. 5 units of a 2-year fixed rate bond paying 6% quarterly. ii. 2 units of a 1.75-year floating rate bond paying float + 80 bps semi- annually. You know that the reference rate was 6.5% three months ago. iii. 6 units of a 1-year zero coupon bond. iv. 5 units of a 1.5-year floating rate bond with no spread paid semian- nually. Ans. The duration of the portfolio is 0.8805. 5. Calculate the duration of the following portfolio: 9

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    i. 3 units of a 0.75-year fixed rate bond paying 6% quarterly. ii. 4 units of a 2-year fixed rate bond paying 3% semiannually. iii. 7 units of a 1.75-year zero coupon bond. iv. 1 unit of a 2-year floating rate bond with no spread paid semiannually. Ans. The duration of the portfolio is 1.4617. 6. You have two bond coupon with the same maturity, one has a 9% coupon paid semiannually and the other a 8% coupon paid semiannually. Which one has a higher duration? Ans. The one with the lower coupon has a higher duration. Higher coupons mean that a higher proportion of the total cash flows from the bond will be paid more quickly. These cash flows will also be less sensitive to interest rates as cash flows paid later are more exposed to interest rate variation. 7. Calculate the MacCaulay Duration for the following security: 1-year fixed rate coupon bond paying 6% semiannually. You know that the yield of the bond is 6.72%. Ans. The MacCaulay Duration for the bond is 0.9854. 8. Calculate the Modified Duration for the same security. Ans. The Modified Duration for the bond is 0.9533. 9. What is the duration of the following portfolio? i. Long a 1.5-year zero coupon bond. ii. Short a 2-year fixed coupon bond paying 1% quarterly. Ans. Trying to compute duration from this long-short portfolio brings many problems since we can’t adequately weigh the securities within the port- folio. 10. What is the dollar duration of the following portfolio? i. Long a 1.5-year zero coupon bond. ii. Short a 2-year fixed coupon bond paying 1% quarterly. Ans. Dollar duration for the portfolio is -41.0462. 11. What is the dollar duration of the following portfolio: i. Long a 1-year fixed coupon bond paying 4% quarterly. ii. Long a 1.75-year floating rate bond paying float plus 80 bps semian- nually. You know that the reference rate was set at 6% six months ago. iii. Short a 2-year zero coupon bond. 10

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    Ans. The dollar duration of the portfolio is -51.8169. 12. What is the dollar duration of the following portfolio: i. Long a 2-year fixed coupon bond paying 7% quarterly. ii. Short three 1.25-year floating rate bonds paying float plus 80 bps semiannually. You know that the reference rate was set at 7% six months ago. iii. Short two 0.5-year zero coupon bonds. Ans. The dollar duration is 13.2098. 13. What is the PV01 of the following portfolio? i. Long a 1-year fixed coupon bond paying 4% quarterly. ii. Long a 1.75-year floating rate bond paying float plus 80 bps semian- nually. You know that the reference rate was set at 6% six months ago. iii. Short a 2-year zero coupon bond. Ans. The PV01 is -0.0052. 14. What is the PV01 of the following portfolio? i. Long a 2-year fixed coupon bond paying 7% quarterly. ii. Short three 1.25-year floating rate bonds paying float plus 80 bps semiannually. You know that the reference rate was set at 7% six months ago. iii. Short two 0.5-year zero coupon bonds. Ans. The PV01 is 0.0013. 15. Compute the 95% VaR for the following portfolio: i. A 1.5-year fixed rate bond paying 2% quarterly. ii. A 0.75-year floating rate bond paying float plus 80 basis points semi- annually. You know that the reference rate was set to 6% six months ago. iii. A 0.25 zero coupon bond. Additionally you know that μ dr = 0 and σ dr = 0 . 4233. Ans. The 95% V aR = 1 . 3116. 16. Suppose that you calculate VaR from Duration. In your many results you find that: i. using historical data (of whatever length) or a normal distribution does not affect the result; 11

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    ii. you find that kurtosis between historical data and the normal distri- bution is almost identical; iii. You find the expected change in the portfolio μ P = 0, with very small standard errors. Given the above, can you say that this Duration based VaR is an appro- priate approach to measure risk? Ans. No, it is not. It is still internally inconsistent: Duration measures small changes, but VaR uses extreme values (large changes). 17. Mr. Brown wants to invest $100,000 for the next five years. He purchases an annuity from a financial institution. Currently the term structure is flat at 10% (yearly compounded). i. If the payments are made yearly, what is the amount that the finan- cial institution will agree to pay Mr. Brown? ii. Assume that there is a 5-year fixed coupon bond that pays 12% coupon every year. What is the price and duration of the bond? iii. How much must the financial institution invest in the long-term bond in order to hedge the position? What should it do with the remainder of the money? Ans. The results for the immunization exercise are: i. The financial institution would pay a yearly amount of $26,379.75. ii. The price of the security is $107.58 with 4.074 duration. iii. The financial institution should invest 93.05% of the proceeds from the investment in the long-term bond and the rest in the money market account. 12

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    Chapter 4 Use the following discount factors when needed. T Z (0 , T ) 0.25 0.9840 0.50 0.9680 0.75 0.9520 1.00 0.9360 1.25 0.9190 1.50 0.9040 1.75 0.8880 2.00 0.8730 2.25 0.8587 2.50 0.8445 2.75 0.8308 3.00 0.8175 3.25 0.8047 3.50 0.7924 3.75 0.7806 4.00 0.7691 1. Calculate the convexity of the following security: a 5-year zero coupon bond. Ans. The convexity of the security is 25. 2. Calculate the convexity of the following security: a 3-year fixed rate bond paying 4% coupon on a semiannual basis. Ans. The convexity of the security is 8.3780. 3. Calculate the convexity of the following security: a 3-year floating rate bond with no spread paid quarterly. Ans. The convexity of the security is 0. 4. Calculate the convexity of the following portfolio: i. 1 unit of a 2-year fixed coupon bond paying 10% coupon quarterly. ii. 1 unit of a 2-year fixed coupon bond paying 1% coupon semiannually. iii. 1 unit of a 2-year zero coupon bond. Ans. The convexity of the portfolio is 3.8230. 5. Calculate the convexity of the following portfolio: i. 2 units of a 1.5-year fixed rate bond paying 6% quarterly. ii. 4 units of a 1.75-year floating rate bond paying float + 80 bps semi- annually. You know that the reference rate was 7% three months ago. 13

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    iii. 6 units of a 2-year zero coupon bond. iv. 1 units of a 1.5-year floating rate bond with no spread paid semian- nually. Ans. The convexity of the portfolio is 2.0655. 6. Calculate the convexity of the following portfolio: i. 4 units of a 1.5-year fixed rate bond paying 4% quarterly. ii. 5 units of a 1.5-year fixed rate bond paying 5% semiannually. iii. 10 units of a 1.5-year zero coupon bond. iv. 3 units of a 1.5-year floating rate bond with no spread paid semian- nually. Ans. The convexity of the portfolio is 1.8916. 7. Calculate annualized expected returns (including convexity) for a 5-year zero coupon bond, when E [ dr ] = 0 and E [ dr 2 ] = 6 × 10 − 07 (on a daily basis). Ans. Annualized expected return on the bond is: 0.189%. 8. Calculate annualized expected returns (including convexity) for a 30-year zero coupon bond, when E [ dr ] = 0 and E [ dr 2 ] = 7 × 10 − 07 (on a daily basis). Ans. Annualized expected return on the bond is: 7.938%. 9. Calculate annualized expected returns (including convexity) for a 3-year fixed rate bond paying 2% coupon semiannually, when E [ dr ] = 0 and E [ dr 2 ] = 7 . 5 × 10 − 07 (on a daily basis). Ans. Annualized expected return on the bond is 0.0815%. 10. Suppose you hold a bond and interest rates suddenly fall. Duration says that bond prices will raise a given amount. If Convexity is included in this estimate, will bond prices go above or below what Duration predicts? Ans. Prices will go up even more than what Duration predicts. 11. Suppose you hold a bond and interest rates suddenly rise. Duration says that bond prices will fall a given amount. If Convexity is included in this estimate, will bond prices go above or below what Duration predicts? Ans. Prices will go up, partially countering the decrease predicted by Duration. 12. Compute the Term Spread and the Butterfly Spread for the following data. What shape does the yield curve have? 1-month yield 5-year yield 10-year yield 5.0% 8.0% 10.0% 14

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    Ans. The term spread is 5% and the Butterfly Spread is 1%. The shape of the yield cure is increasing. 13. Compute the Term Spread and the Butterfly Spread for the following data. What shape does the yield curve have? 1-month yield 5-year yield 10-year yield 6.0% 4.5% 4.0% Ans. The term spread is -2% and the Butterfly Spread is -1%. The shape of the yield cure is decreasing. 14. Compute the Term Spread and the Butterfly Spread for the following data. What shape does the yield curve have? 1-month yield 5-year yield 10-year yield 3.0% 4.0% 3.0% Ans. The term spread is 0% and the Butterfly Spread is 2%. The shape of the yield cure is a hump. 15. Compute the Term Spread and the Butterfly Spread for the following data. What shape does the yield curve have? 1-month yield 5-year yield 10-year yield 4.0% 3.0% 5.0% Ans. The term spread is 1% and the Butterfly Spread is -3%. The shape of the yield cure is an inverted hump. 16. You currently hold a 7-year fixed rate bond paying 5% annually. You would like to hedge against changes in the level and the slope of the yield curve and you plan to use a 1-year zero coupon bond and a 7-year zero coupon bond. Use the following table to compute the adequate positions in the hedging instruments. maturity β 1 β 2 Z ( t, T ) 1.00 1.1150 -0.2540 0.9800 2.00 0.9940 -0.3010 0.9600 3.00 0.9640 -0.1470 0.9300 4.00 0.9330 0.0080 0.8900 5.00 0.9300 0.1620 0.8500 6.00 0.9260 0.3160 0.8100 7.00 0.9270 0.4230 0.7700 8.00 0.9270 0.5300 0.7300 Ans. The position in the short term bond should be -0.4651, while the position in the long term bond should be -1.1231. 15

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