from period 2 to period 3 are obtained as:{RN}02 {RN}23 = {RN}03Where {RN}ij is the risk-neutral transition matrix from period i to period jThus, {RN}23 = {RN}-102 {RN}03Table -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 = 0.107The value of the derivative is obtained as 20.76Table-6Risk neutral probabilities of transition from period 2 to period 3Rating after 3 periodsRating after 2 periodsABCDA0.7420.1690.0740.015B0.1450.6250.1760.053C0.0690.1800.6030.148D0.0000.0000.0001.0006.VALUATION OF A BOND6.1A Plain BondThe 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.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 = 68.25The 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 = 6.96Thus, 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).6.2Coupon B...
