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MIT Sloan School of Management

J. Wang 15.407E52-456 Fall 2003

Problem Set 2: Fixed-Income SecuritiesDue: September 30, 2003

1. (i) Spot rate is the interest rate you get if you make a deposit today.

(ii) Forward rate is the interest rate of a specied term, that starts at a specied timein the future. Spot rate can be thought as an average of the forward rate.

(iii) Yield to maturity is a constant rate that if you use it to discount the bond, it willgive you the cost. YTM does not give you the whole term structure, and it is differentfor different bonds, but it is an easy-to-quote number for a rough comparison between

bonds.2. A 3-year treasury bond has a face value of $1000 and annual coupon of 8%. The 1-year

spot rate is r 1 = 2%, and the 1-year forward rates for the next two years are r 2 = 4%and r 3 = 5%

(a) P = sumofP V s = 78 .43 + 75.41 + 969.62 = $1, 123.46

(b) YTM = 3.586%

(c) 2-year spot rate = (1 + 2%)(1 + 4%) 1 = 2.995%3-year spot rate = [(1 + 2%)(1 + 4%)]

13 1 = 3.659%

(d) This is due to no-arbitrage. To lock in the payment of the second year, deposit$75.41 into the 1-year spot rate and also enter into a (1-year) forward agreementof the amount $75 .41 1.02 = $76.92. Then you receive $80 for sure at year 2. Youcan do the same to replicate the other two cashows. Since we can use the spotrate and forward rates to replicate all cashows of the T-bond, it provides sufficientinformation to price the bond.

(e) A corporate bond has default risk. So if we use the replication mechanism above,we will not capture the fact that bond payment may not be made if the companydefaults.

3. See the Excel le. The traded price of the STRIP is higher because of liquidity. Gov-ernment is not allowed to issue STRIP. Investment banks buy treasuries and resell eachpayment at different years. Since they provide liquidity to this market, they can chargea small spread, especially for longer maturities.

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4. (a) P A = 1001. 04 +100

1. 042 +11001. 043 = $1166 .51

Similarly, P B = $1110 .96 and P C = $668 .47

(b) Using the weights wA = 5122 , wB =1

122 , w C =25

122 , we can get a 1-year discountbond. The price of the bond is D 1 = $98 .279. We can use this and bond C to pricea 3-year discount bond, then a 2-year discount bond. We get D 2 = $94 .346andD 3 =$88.535

(c) Using the discount bonds, we get r 1 = 1 .751%, r 2 = 2 .951%, r 3 = 4 .143%

(d) for bond A, MD A = 100 D 1 1+100 D 2 2+1100 D 3 3100 1166 . 51 (1+4%) = 2 .650Similarly, MD B = 2 .157andMD C = 1 .503

(e) We need to short 100 M D B P BM D A P A = 77 .51 units of bond A.

(f) If rate increase by 1%:Gain from short position in A = $2,347.43

Loss from long position in B = $2,345.60Net gain = 1.82If rate decrease by 1%:Gain from short position in A = $2,431.57Loss from long position in B = $2425.10Net loss = 11.47

5. Note that the table gives the spot LIBOR today. You dont know future LIBOR sothere is no way for you to tell how much the oating bond is paying (except the rstcoupon, which is set today)

(a) London Inter-Bank Offer Rate (LIBOR) is the rate of interest at which banksborrow funds, in marketable size, in the London interbank market. Everyday theborrowing rate of a panel of banks is collected, and after removing the top andbottom 15% outliers, the average determine the LIBOR rate. Since it is determineddaily by the market, it is very widely used as a benchmark of risk-free oating rate.

(b) We invest in a 3-year oating rate bond and short a 1-year discount bond. Thenwe get paid 6-month LIBOR (semi-annually) for the next 3 years.

(c) Similarly, use a coupon bond and a discount bond to replicate the xed leg.

(d) Use the LIBOR rate as the forward rates.Floating leg: Notice that if we invest $1 and keep rolling over 6-month LIBOR,we get identical cashows as the oating bond. Therefore the bond must issue atpar. Therefore the value of the oating payment = 100 - PV(100 in 3 yeras) (Fora principle of $100) = $15.8000Fixed leg: We just need to discount all the cashows with the forward rate. Weget the value for $1 (2% annually) xed leg is $5.4625.Therefore, the swap rate = (15.8000/5.4625)*2 = 5.7849%

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6. Collateralized Debt Obligations. Consider the following risky bonds:

Bond Maturity (yrs) Face ($) Coupon (%)X 1.0 $1000 0%Y

1.0 $1000 0%These bonds are risky because there is a possibility that their respective issuers default(i.e. dont repay the entire face value of their issued bond to bondholders). Each issuerhas a 25% probability of defaulting. These default risks are independent in the sensethat the probability of both rms defaulting equals (0 .25)2. In the case of default, X doesnt repay any of its principal while Y repays half of it.

A collateralized debt obligation (CDO) pools risky debt securities (like X and Y ) into aninvestment vehicle that is sold off in recongured pieces called tranches . Cashows fromthe original securities are passed-through to the tranches according to a specic rule. Therule is generally hierarchical in the sense that there is a most senior tranch that receivesthe rst $P payments, a slightly less senior (called more junior ) tranch that receives thenext $Q payments, and so forth. In industry, this type of investment vehicle most oftenpools illiquid debt instrument like mortgages (called collateralized mortgage obligations ).

Consider a CDO made by pooling one X and one Y bond. It is split into two tranches.The senior tranch has a face value of $500 and the junior tranch has a face value of $1500. Assume the price of the X and Y bonds are $688.07 and $825.47 respectivelywhile the risk-free rate is 5%.

(a) Notice that since bond X pays at least $500, the senior tranch is risk free. So thevalue of the senior tranch is 5001. 05 = $476 .19. Since the the two tranches togetheralways give the same payments as the bond X and Y added up,Price of junior = $688.07 + $825.47 - $476.19 = $1037.35.

(b) YTM of senior = 5%, YTM of junior = 44.60%. The yields are different becausethe senior tranch is much less risky than the junior tranch.

(c) The easiest way to create the equity tranch is to use half of bond X, and subtract 5units of the credit derivative. This gives a price of $209.04 and a YTM of 70.05%.Since the two new tranches together is the same as the original junior tranch, theymust be priced the same. So the new junior tranch is priced at $743.31, with aYTM of 34.53%

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