### A Risk Neutral Framework For The Pricing Of Credit Derivatives

.. A or B (6) The probabilities of transition from period 2 to period 3 are obtained as: {RN}02 {RN}23 = {RN}03 Where {RN}ij is the risk-neutral transition matrix from period i to period j Thus, {RN}23 = {RN}-102 {RN}03 Table -6 shows the risk neutral probabilities of transition from period 2 to period 3. From this table, it can be seen that P (F / EA ) = 0.074 and P (F / EB ) = 0.176. In addition, we know that P(EA) = 0.181 and P(EB) = 0.530 (refer Table -5a). Thus, the risk-neutral probability that Rs.

100 is received in period 3 is 0.074 x 0.181 + 0.176 x 0.530 = 0.107 The value of the derivative is obtained as 20.76 Table-6 Risk neutral probabilities of transition from period 2 to period 3 after 3 periods after 2 periods A B C D A 0.742 0.169 0.074 0.015 B 0.145 0.625 0.176 0.053 C 0.069 0.180 0.603 0.148 D 0.000 0.000 0.000 1.000 6. VALUATION OF A BOND 6.1 A Plain Bond The risk-neutral transition matrices developed for the valuation of a credit derivative can be used to value a bond. Let us consider a 3-period zero-coupon bond rated B. The pay-offs from the bond are the same as that from the following portfolio: 1. A credit risk free zero-coupon bond with a maturity of 3 periods.

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2. A credit derivative which requires payment of Rs. 60 if the rating of a bond rated B currently touches D anytime during the first three periods. The value of the credit-risk free bond is 100 / (1+ 0.1358)3 = 68.25 The value of the credit derivative calculated as in the previous section turns out to be Rs. 6.96. This amount is arrived as follows: RN(3)BD * 60 / (1 + r03 )3 = 6.96 Thus, the value of the bond is obtained as the difference between the credit risk free bond and the credit derivative, i.e. Rs. 68.25 – Rs.

6.96 = Rs. 61.29. In this case again, the interest rate tree is not required for the valuation of the bond. This gives a powerful framework to split up the loan price into its building blocks – a time value of money (represented by the risk-free rate), and a risk premium (for bearing credit risk). 6.2 Coupon Bond with Prepayment Option So far, interest rate risk has not been relevant to the valuation of any of the loan or derivative products.

However, in case of a loan with an embedded prepayment option, the interest rate volatility needs to be considered. It is here that our model departs from presently available models in giving a framework to price prepayment options embedded in loans. To understand the justification for using the interest rate tree, let us examine the drivers that lead the borrower to prepay the loan: Fall in interest rates: With a fall in interest rates, a borrower can obtain cheaper funding in the market, and hence would be motivated to prepay the current (fixed rate) loan. Improvement in credit rating: As spreads narrow with enhancement in credit rating, borrowers with rating upgrades are driven to opt for the prepayment because they too can obtain refinance at cheaper rates now. However, the decision to prepay is not independently determined by interest rate decline and credit rating upgradation. In some situations, the changes in interest rate and in credit ratings could exert opposite influences on the decision to prepay.

Hence, we need to consider the simultaneous impact of changes in interest rates and credit ratings on the valuation of the bond. We shall value the loan (& the embedded option) using a backward recursive method for computing the expected present value of the loan in a risk-neutral world. In order to move into a risk-neutral framework with respect to interest rates, we have used a recombining binomial interest rate tree (refer Chart-1). Chart-1 The Risk-Neutral Interest Rate Tree Consider a 12% coupon bond rated B with a maturity of 3-periods. Let us suppose the issuer has the option to prepay starting from the first period. The prepayment amount is fixed at Rs. 98.

The steps in the valuation of the bond and the embedded option are: Step-1: List all possible interest rate paths from period 0 to period 3. All these paths are equally likely. Period Path 0-1 1-2 2-3 1 15.0% 16.0% 14.4% 2 15.0% 16.0% 12.0% 3 15.0% 11.5% 12.0% 4 15.0% 11.5% 10.0% Step-2: For each interest rate path (the following calculation illustrates the valuation procedure for the second interest rate path, i.e. 15.0%, 16.0% and 12.0%), obtain the value of the bond as described under: Let Vij be the value of the bond in the ith period if its rating is j. At maturity, the bonds value is Rs.

112 if its rating is A, B or C and Rs. 40 if the rating is D. For valuing the bond in period 2, we use the risk-neutral transition matrix from period-2 to period-3. So, V2A is obtained as (7) Here 12.0% is the interest rate for period 3 in the first interest rate path. Similarly other V2j are obtained.

Since there is a prepayment option we compare the value of the bond with the prepayment amount Rs. 98. If the value of the bond V2j is greater than the prepayment amount, the issuer is likely to prepay. V2A, V2B and V2C are 99.03, 96.67 and 90.42 respectively. Since V2A is greater than Rs. 98 (the prepayment amount), the issuer will exercise the prepayment option and the value of the bond will become Rs. 98.

Since default is an absorbing state, the risk-neutral transition matrix indicates that the probability of migrating to any other state from a state of default is zero. Hence, the value of the bond at the default node is simply the present value of Rs. 40 received at the end of the original maturity period, the interest rate tree providing the discount rates. At this point, the coupon payment will be received, and the value of the bond will be incremented by Rs. 12, except in case of default, where no cash flow occurs.

V2A, V2B V2C and V2D can then be used to compute V1A, V1B V1C and V1D in a similar fashion, except that the interest rate applicable will be r12 along the chosen interest rate path. (8) This procedure is repeated till the current period is reached. At this point, the credit rating applicable is known (B in the present illustration), and thus the bond value can be ascertained. A schematic representation of the backward recursion of the bond cash flows is shown below for the second interest rate path, i.e. r01 = 15%, r12 = 16%, and r23 = 12%, yielding the bond value (net of the value of the embedded prepayment option) to be Rs. 84.47.

Strike price / Prepayment amount: 98 Period 0 1 2 3 A 104.83 110.00 112 B 84.47 101.47 108.67 112 C 94.10 102.42 112 D 30.79 35.71 40 The value at each node includes the coupon received at that node. Step-3: Since we are operating in a risk neutral framework and each of the interest rate paths is equally likely, the value of the bond is the simple average of values obtained for each of the interest rate paths. Step-4: A similar exercise can be carried out to price a plain bond, i.e. where there are no embedded options. In this case, at each node, there would be no need to compare the amount arrived at by the backward recursion with any other amount, because there is no option available to exercise.

Step-5: The difference between the two values (Step-3 and Step-4) represents the value of the embedded option. Table-7 contains an analysis of the price of the bond with a prepayment option into the price of a plain bond, and that of a prepayment option. Table-7 Components of the price of a bond with an embedded prepayment option r01 r12 r23 Bond value with embedded option Value of plain bond Value of embedded prepayment option 15.0% 16.0% 14.4% 83.21 83.21 0.00 15.0% 16.0% 12.0% 84.47 84.61 0.14 15.0% 11.5% 12.0% 87.48 87.63 0.15 15.0% 11.5% 10.0% 88.31 88.89 0.58 Price of instrument 85.87 86.09 0.22 7. CONCLUSION A loan or a bond can be viewed as a portfolio of a risk-free instrument and a credit derivative that pays an amount equal to the loss in the event of default. This gives a powerful framework to split up the loan price into its building blocks – a time value of money (represented by the risk-free rate), and a risk premium (for bearing credit risk).

Ascertaining the risk-neutral transition matrices is critical in this framework, where the payoff depends on the credit rating of a certain party. These transition matrices can be obtained from empirically observed or real world transition matrices, observed bond prices and observed interest rates. However, when it comes to the pricing of bonds with embedded prepayment options, the demarcation between credit rating upgradation as a trigger for the exercise of the option and interest rate decline as the trigger for the exercise of the option is not very clear. In this scenario, the best way to combine interest rate and credit risk is to use both, the risk neutral transition matrix, and the risk neutral interest rate tree. The valuation procedure involves backward recursion of the bond cash flows (including the embedded option) starting out with the terminal period, separately for each interest rate path.

The price of the bond is the mean of the values arrived at in each path. It is noteworthy that the mean can be used only in a risk-neutral setting. The option value is the difference between the bond value (alongwith the embedded option) and the value of a plain bond. Bibliography Aguais, Scott D.; Dr. Forest, Lawrence Jr.; Krishnamoorthy, Suresh and Mueller Tim, Creating Value from Both loan structure and price 2. Das, Satyajit, Credit Derivatives – Trading and Management of Credit & Default Risk 3. Das, Sanjiv R.; An Overview of Credit Derivatives, Harvard Business Review, July 1997 4.

Belkin Barry; Dr. Forest, Lawrence Jr. and Dr. Suchower, Stephen, Measures of Credit Risk and Loan Value in KPMGs LAS, Financial Services Consulting -Risk Solutions KPMG Peat Marwick LLP. Business Essays.