FIN8340 - Final Exam Fixed Income Securities Exam time is: 60 hours. Total points for this exam is: 600 points, corresponding to 60% of your nal grade. 0.0.1 Instructions Read carefully the questions. Summarize you answers in a Report in the rst work sheet of the excel le. The report in the rst work sheet should be formatted in such a way that when I press "print" I will be able to get all of your results, plots, and explanations. Your grade will be based on the report. It is advisable that you preview and print the report before you submit your exam for a last check. You should execute your work in the work sheets assigned for each problem in an orderly fashion so that I can check it, if necessary. However, if a results is not reported in the report, do not assume that I will be able to nd it in some place of the workbook, and that you will get credit for it. If you nd a question unclear, explain your approach, make any assumption that you need to make, and provide the best answer you can. Given the structure of the test, and for fairness, I will not reply to any queries. The workbook contains the data that you need for your work. 0.0.2 Con dentiality Agreement. By taking this exam you acknowledge that you have agreed to the terms of the con dentiality agreement reproduced below for you convenience. You agree that the contents of the nal take home exam are con dential and that the disclosure of that information could compromise the integrity of the student evaluation process. The exam is available to you solely to test your knowledge of the exam subject matter. You are expressly prohibited from disclosing, publishing, reproducing, transmitting, discussing, or accessing any previous exam and 1
any related information including, without limitation, questions, answers, worksheets, computations, drawings, diagrams, length or number of exam segments or questions, or any communication, including oral communication regarding or related to the exam, in whole or in part, in any form or by any means, oral or written, electronic or mechanical, for any purpose, without the prior express written permission of the instructor. Mid term exams and in class labs are in the public domain, and thus are excluded from con dentiality. However, while taking the exam, you are expressly prohibited from any communication, including oral communication, regarding or related to the mid term exams and in class labs, in whole or in part, in any form or by any means, oral or written, for any purpose, without the prior express written permission of the instructor. The instructor determines, in its sole discretion that you have undertaken or participated in any action that compromises the integrity and con dentiality of the examination. Failure to comply with the terms of this agreement before the nal grade is issued will be handled through the established procedures of the KSU University Judiciary Program, which includes either an informal resolution by a faculty resulting in a F grade, or a formal hearing procedure, which may subject a student to the Code of Conduct s minimum one semester suspension requirement. The term of this Agreement is perpetual. You waive any right to challenge the validity of this agreement on the grounds that it was transmitted and entered into electronically. You agree that entering into the Agreement electronically is equivalent to signing the Agreement. 2
1 Problem 1 a) Write the price function and plot the price of a 15 year bond, nominal value 1000, coupon 5.5% with annual payment, as a function of the yield to maturity. Use a range of yields from 0 to 20% with increments of 2%. b) Compute the $duration analytically and numerically when the yield is 7%. Use i) 1 basis point increment for the numerical approximation ii) use a smaller increment equal to :0000001. Do you notice any di erence in the two numerical approximations? c) Explain what does this number measure. d) Plot the linear approximation of the bond function about the 6% yield. e) What is the error from the approximation when the yield is 6%. f) What is the error when the yield is 5.9999%. g) What is the error when the yield is 12% e) Comment on e, f, and g. P (y) = X15 55 + 1000 (1+y) t (1+y) 15 y 1400 1200 1000 800 600 400 200 0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 x P (:07+:0001) P (:07) :0001 = 8239: 68 = 3
P (:07+:0000001) P (:07) :0000001 = 8245:0 = X15 X15 t55 + 151000 = 8245: 01 (1+:07) t+1 (1+:07) 16 t55 (1+:07) t+1 + 151000 (1+:07) 16 = 8245: 01 Linear approximation L(y) = P (:06) 8245: 01(y :06) L(:06) P (:06) = 0:0 L(:059999) P (:059999) = 0:001 15 L(:12) P (:12) = 100: 556 2 Problem 2 Assume that you manage a portfolio that includes several dozens of di erent bonds. The characteristics of the portfolio are Value Yield Modi ed Duration Convexity $103 768 :81 5:5% 7: 438 073 5 70: 949 696 Assume that in the market there are three bonds with the following speci cations. 1) Bond H 1 Coupon 6%, paid annually. maturity = 20 year, par value =100, yield =5:75%. 2) Bond H 2 Coupon 5%, paid annually maturity =5 year par value = 100, yield = 5:25%. 3) Bond H 3 4
Coupon 7%, paid annually maturity =7 year par value = 100, yield = 5:65%. a) Compute the price, $duration and $Convexity of all the bonds. For H 1 we have: Price function H 1 (y) = Price X20 $Dur(H 1 ) = X20 6 + 100 (1+y) t (1+y) 20 6 + 100 = 102: 927 (1:0575) t (1:0575) 20 X20 $Convexity(H 1 ) = For H 2 we have: t(6) 20(100) = 1194: 66 (1:0575) t+1 (1:0575) 21 X20 Price function H 2 (y) = Price H 2 (:0525) = $Dur(H 2 ) = 5X $Convexity(H 2 ) = For H 3 we have: 5X t(t+1)6 + (20)(21)100 = 19516: 237 (1:0575) t+2 (1:0575) 22 5X 5 (1+y) t + 100 (1+y) 5 5 + 100 = 98: 925 07 (1:0525) t (1:0525) 5 t(5) 5(100) = 427: 012 41 (1:0525) t+1 (1:0525) 6 5X Price function H 3 (y) = H 3 (:0565) = 7X t(t+1)5 + (5)(6)100 = 2354: 732 8 (1:0525) t+2 (1:0525) 7 7X 7 + 100 (1+y) t (1+y) 7 7 + 100 = 107: 630 87 (1:0565) t (1:0565) 7 5
$Dur(H 3 ) = 7X $Convexity(H 3 ) = t7 7(100) = 600: 557 21 (1:0565) t (1:0565) 8 7X t(t+1)7 + (7)(8)100 = 4196: 22 (1:0565) t+2 (1:0565) 9 b) Use the rst bond above to hedge your portfolio to the rst order change in y. In other words, form a portfolio that is duration neutral. How many unit of the bond should you buy or sell? Report the number q. Since the value of the portfolio is a function of the yield we write P (y) We denote the change in P (y) as dp (y). by For a small change in the yield we know that dp (y) can be approximated @P (y) dy: @y So we write dh(y) = $Dur(H)dy dp (y) = $Dur(P )dy dp (y) @P (y) dy = $Dur(P )dy @y dp (y) + qdh(y) = 0 implies (q$dur(h) + $Dur(P ))dy = 0 $Dur(P ) q$dur(h) = 0 $Dur(P ) + q$dur(h) = 0 q = $Dur(P ) $Dur(H) The total dollar duration of the portfolio P (y) is equal to modi ed duration times the price of the portfolio, i.e. P (y)md = $Dur(P ) 7: 438 073 5(103 768 :81) = 771840 to nd q we need q = $Dur(P ) $Dur(H 1 ) : The quantity q to be SHORTED is then equal to q = 771840 1194: 66 6 = 646: 075
c)what is the value of your hedged portfolio? Check what happens to your unhedged portfolio when the yield moves upward of 10 basis points? What happens when the yields goes up 300 bp s. Comment. The rst order approximation of P (y) = 103768: 81 771840(y :055) Answer Around x 0 f(x) = f(x 0 ) + f 0 (x 0 )(x x 0 ) + error Unhedged P (:055 + :001) P (:055) = 771: 84 P (:055 + :03) P (:055) = 23155: 2 Let s now form the hedged portfolio. What is the value of the portfolio? The value is: P (y) qh 1 (y) = P (:055) 646: 075H 1 (:0575) 37270: 5 P (:055) 646: 075H 1 (:0575) = 37270: 5 Let s see what happens if the yield goes to up 10 bp s The new value of the hedged portfolio is: P (:055 + 0:001) 646: 075H 1 (:0575 + :001) 37264: 2 P (:055 + 0:001) = 102997 103768: 81 771840(:055 + 0:001 :055) = 102997 H 1 (:0575 + :001) = 101: 742 646: 075H 1 (:0575 + :001) = 65732: 7 P (:055 + 0:001) 646: 075H 1 (:0575 + :001) = 37264: 2 102997 65732: 7 = 37264: 3 7
The new value of the hedged portfolio is: The loss is 38036: 1 37270: 5 = 765: 6 Let s see what happens if the yield increases 300 bps The new value of the hedged portfolio is: P (:055 + :03) 646: 075H 1 (:0575 + :03) = The new value is P (:055 + :03) = 80613: 6 H 1 (:0575 + :03) = 74: 442 8 P (:055 + :03) 646: 075H 1 (:0575 + :03) = 32518: 0 and the loss is 32518: 0 37270: 5 = 4752: 5 d) Use bonds H 1 and H 2 and H 3 above to hedge your portfolio to the second order changes in y, while keeping the value of portfolio unchanged. In other words, form a portfolio that is duration neutral and convexity neutral. This means that you want to nd 3 quantities q 1 ; q 2 ; and q 3 such that the duration and convexity of your portfolio are both equal to zero, and the total investment necessary to hold the position is unchanged. Report the system that you have set up and the solution, i.e. q 1 ; q 2 ; and q 3 : Answer Or we can write the system in the form 0 = q 1 H 1 (y) + q 2 H 2 (y) + q 3 H 3 (y) P 0 (y) = q 1 H 0 1(y) + q 2 H 0 2(y) + q 3 H 0 3(y) P 00 (y) = q 1 H 00 1 (y) + q 2 H 00 2 (y) + q 3 H 00 3 (y) We can now use the information about our bonds to set the hedge 8
The total duration of the portfolio P (y) is equal to the number of certi - cates times the $duration of each certi cate. $Convexity = 103 768 :81(70: 949 696) = 7: 362 37 10 6 Hence 2 4 0 771840 7362366 0 = q 1 103 + q 2 99 + q 3 108 771840 = q 1 1195 + q 2 427 + q 3 601 7362365: 5 = q 1 19516 + q 2 2355 + q 3 4196 3 2 102: 926 60 98: 925 07 3 2 107: 630 87 5 = 4 1194: 66 427: 012 41 600: 557 215 4 19516: 237 2354: 732 8 4196: 22 q 1 q 2 q 3 3 5 So we have 2 102: 926 60 98: 925 07 3 107: 630 87 4 1194: 66 427: 012 41 600: 557 215 19516: 2 3 237 2354: 732 8 4196: 22 q 1 = 4q 2 5 q 3 3 2 0 4 771840: 5 = 4 7362366: 1 2 3 237: 960 7357: 535 6989: 97 These are the q s we are looking for. f) Consider now the hedged portfolio. Compute its value at the time the hedge is set up. Then check what happens to your hedged portfolio when the yield moves upward of 10 basis points? What happens when the yields goes down 300 bp s. (Hint, do not forget to de ne a QUADRATIC price function for the portfolio). Comment. Answer. As expected the hedged portfolio is worth at the time of the hedge P (:055) 237: 960H 1 (:0575) 7357: 53H 2 (:0525) + 6989: 97H 3 (:0565) = 103769 Rede ne the portfolio quadratic function P (y) = 103768: 81 771840(y :055) + :5 7362366(y :055) 2 Let s apply a 10 basis points change to the 3 bonds 9
P (:055+:001) 237: 960H 1 (:0575+:001) 7357: 53H 2 (:0525+:001)+6989: 97H 3 (:0565 + :001) = 102285: The loss is 103648 103 768 :81 = 120: 81 Let s apply a 300 basis points change to the 3 bonds P (:055 + :03) 237: 960H 1 (:0575 + :03) 7357: 53H 2 (:0525 + :03) + 6989: 97H 3 (:0565 + :03) = 62255: 6 Now the loss is 104756 103 768 :81 = 987: 19 3 Problem 3 Nelson and Siegel propose to t the yield curve with a function of the following form.!! R(0; ) = 0 + 1 1 e + 2 1 e Here ; is the maturity, and is a scaling parameter. a) Brie y, explain the meaning of the parameters 0 ; 1 ; and 2. b) Estimate the 0 ; 1 ; and 2 and ; for the U.S. government securities constant maturities on 1/2/2004, using the data in the data sheet. Convert to continuos compounding. Report the parameters estimates and the value of the minimized loss function, i.e. the sum of the squared di erences of the rates. 6:0217% 5:3243% 0:1437% 3:33 loss function = 4:69407E 06 c) Use the curve that you have estimated to price a 15 year U.S. government year bond, with annual coupon 5%. and face value equal to 1000. Use continuos compounding for the discount factor. e 10
Price 1023:5235 d) Estimate the $ sensitivities to changes in 0 ; 1 ; and 2 of a portfolio made of 1000 certi cates. Numerically, estimate all sensitivities with an increment of :0001: Assume that you expect a 1% decrease of 1 and you have no view with respect to 0 ; and 2 : Assume in addition that you want the face value of the portfolio unchanged, regardless of the strategy you take and that you can use the 1-year, 5-year and 10-year bonds (recall that you have the data for the zeros) to immunize your portfolio. Report the system that you need to solve. You don t need to solve it. 4 Problem 5 Merton (1973) suggests that the payo of a limited liability company s stock is akin to that of an option. For instance, consider a rm with market value V, and debt with face value K on date T. The value of the equity in the rm, S(T ), at maturity of the debt, T, is the price of a hypothetical option on the rm value. a) What type options is it? What is its payo function? b) Draw the payo function S T (V T ) assuming that the strike price K, which in this case represents the value of the rms debt, is equal to 8M c) Consider the fundamental balance sheet identity. What type of options does the value of debt resembles? What is the payo function of the bond B? Draw it. Comment on the economic meaning of the two pictures. B = V T S T = V T Max(V T K; 0) VT if V = T < 6 6 if x > 6 The two pictures show that the equity payo is similar to that of a call option. It start to pay only after the debt has been fully repaid. On the 11
other side, debt resembles short position in a put contract. If the value of the of the rm is below those of debt, bond holders only get the value of the rm. If it is higher than that of the bond, bond holders get the full principal back. d) Let S be the stock value, V T the value of the rm asset, N(d) the normal standard probability of a V T being less or equal to d. The other parameters are the years to maturity T of the bond B, the risk free rate r; and the strike price K and the volatility of equity S. The stock of this rm is currently traded at $3 and there are 2; 000; 000 common stock certi cates held by investors. The bond and the stock are the only entries in the right hand side of the rm s balance sheet. The parameters are as follows: Bond F acev alue = 8 Risk F ree Rate = 0:05 Y ears to Maturity = 1 Stock V olatility = 0:60 Using the Merton model answer the following question: d) What is the value of the Firm V computed with the model? e) What is the value of Firm Volatility V computed with the model? Equity V alue (V ) 13:5954 F irm V olatility V 0:2677 f) What is the probability of default? Probability of Default = 0.0210 g) What is the Corporate Bond Value? B = S = Ke rt N(d 2 ) + V T [1 N(d 1 )] Corporate Bond Value =7.5954 h) What is the value of the risk free bond? B = Ke rt Risk Free Bond Value =7.6098 In addition de ne and provide the estimate of the following quantities i) What is the Expected Credit Loss ECL? 12
ECL = B F B Expected Credit Loss =0.0145 j) What is the Loss Given Default LGD? LGD = ECL 1 N(d 2 ) Loss Given Default =0.6897 k) What is the recovery rate? RR = K LDG LDG or 1 K K Recovery Rate = 0.9138 l) Brie y comment on whether these results make sense to you. 5 Short questions 5.1 Question 1-25 points Provide a de nition of yield to maturity and show how the de nition that you provided applies to the case of a bond with 10 year maturity, semiannual payment, coupon 9%, and 100 par value, that is traded at 93: 768 9. Answer It is the rate of discount the makes the present value of the future bond cash ows equal to the price of the bond. Answer y is the solution to X20 93: 768 9 = y=2 = :05 y = 10% 4:5 + 100 (1+y=2) t (1+:y=2) 20 13
5.2 Question 2-25 points You have been hired by an investment bank. The CFO of your new client company says that he has invented a bond with special characteristics that will make it very successful in the market. He wants to call it The Happy New Year Bond R. This bond has a par value of 100, the maturity is 50 years. The bond is to be oated on the rst trading day of January 2008. The reason he thought about the name is that he wants the coupon payment to be 5 dollars on July 1 of and 10 dollars on January 1 of each year. Assume that the yield curve is at at 15%. How can you (easily) compute the price of The Happy New Year Bond R? How much is it? Answer. This bond is just the combination of two bonds with par value equal to 50. One pays $5 dollar coupon semiannually on July and January, the other pays $5 annually, on January. X50 100 X 5 + 50 = 33: 348 7 (1+:15) t (1+:15) 50 5 (1+:15=2) t + 50 (1+:15=2) 100 = 66: 654 6 33: 348 7 + 66: 654 6 = 100: 003 5.3 5.4 Question 3-25 points a) What are the stylized fact of interest rates? Consider the following model for the dynamics of the interest rate. dr t = dt + dw t a) Which of the stylized facts does it t? Do you see any problem with this model? Consider now this other model for the dynamics of the interest rate. 14
dr t = ( r t )dt + p r t dw t Can this be considered an improvement over the previous one? Why? b) What is the interpretation of ; ; and? Answer Interest rate changes randomly. It is not mean reverting. It can become negative. is the speed of mean reversion. is the level at which the rate tends to revert. is the level of volatility 5.5 Question 4-25 points Consider two securities (nominal $100). One year pure discount bond selling at $95 and a two year 8% bond selling at $99 a) Compute the one-year spot rate b) What is the zero coupon rate for the 2 year time horizon? c) How is this method of extracting zero rates called? a) 95 = 100 1+R(0;1) 1 + R(0; 1) = 100 95 R(0; 1) = 5:263% = 1: 052 63 8 b) 99 = + 108 1+R(0;1) (1+R(0;2)) 2 Use the R(0; 1) that you compute in a) 99 = 8 + 108 1: 052 63 (1+R(0;2)) 2 Solve for R(0; 2) (1 + R(0; 2)) 2 = 1: 181 62 R(0; 2) = 1: 181 62 1=2 1 = 8: 702 35% c) Bootstrapping 15