1920.1050.048B0.1600.5850.1560.100C0.0880.1560.5760.181D0.0000.0000.0001.000Table-4bReal world probabilities of transition over 3 periodsRating after 3 periodsCurrent RatingABCDA0.5480.2330.1370.081B0.1940.4730.1850.148C0.1140.1850.4520.248D0.0000.0000.0001.000Table-4cReal world probabilities of transition over 4 periodsRating after 4 periodsCurrent RatingABCDA0.4690.2540.1590.118B0.2120.3970.1980.194C0.1330.1980.3650.305D0.0000.0000.0001.000In order to obtain the 2-period risk-neutral transition probabilities, p(2)Aj , we require the price V(2)A of a two-period zero coupon bond currently rated A. The expected pay-off in the risk neutral framework is p(2)AA x100 + p(2)AB x100 + p(2)AC x100 + p(2)AD x40. Thus, the following relationship must hold :V(2)A = (5)Since the expected pay-off is calculated using risk-neutral probabilities, it is free from credit risk. Further, since the pay-off is independent of the interest rate, using the 2-period risk free rate for discounting is justified.As in the 1-period case, we assume that the risk-neutral probabilities of transition from state A to other states are proportional to real-world probabilities. The R.H.S of Eqn.(5) reduces to a function of p(2)A. Thus, the risk neutral transition probabilities p(2)Aj can be obtained. The same methodology may be used to obtain p(2)Bj and p(2)Cj and thus the 2-period risk neutral transition matrix is obtained. Tables 5a-c show the 2,3 and 4-period risk neutral transition matrices respectively.Note that the M-period risk-neutral transition matrix is not the Mth power of the 1-period matrix. Table-5aRisk neutral probabilities of transition over 2 periodsRating after 2 periodsCurrent RatingABCDA0.6270.2070.1130.052B0.1810.5300.1760.112C0.1020.1830.5030.212D0.0000.0000.0001.000Table-5bRisk neutral probabilities of transition over 3 periodsRating after 3 periodsCurrent RatingABCDA0.5030.2560.1510.089B0.2230.3940.2130.170C0.1370.2220.3430.298D0.0000.0000.0001.0...
