Credit Value Adjustment (Payo-at-Maturity contracts, Equity Swaps, and Interest Rate Swaps)

Transcription

1 Credit Value Adjustment (Payo-at-Maturity contracts, Equity Swaps, and Interest Rate Swaps) Dr. Yuri Yashkir Dr. Olga Yashkir July 30, 2013 Abstract Credit Value Adjustment estimators for several nancial derivatives were developed, and typical features of CVAs are numerically investigated. Keywords: Credit Value Adjustment, Equity Swap, Interest Rate Swap YASHKIR CONSULTING YASHKIR CONSULTING 1

2 Contents 1 Credit Value Adjustment (CVA) denition 3 2 Transition matrix 3 3 Risk-neutral Probabilities of Default 3 4 CVA for Payo-at-Maturity contracts CVA calculation The CVA Calculation Examples CVA for Equity Swaps The Equity Swap Expected Exposure calculation Price tree Migration probability tree Building the s th Monte Carlo scenario Backward Induction Expected Exposure The Equity Swap Rate calculation Test Results The Credit value adjusted Swap Rate Fair Swap rate for defaultable counterparties Swap rate examples Source Codes 12 8 References 12 A Credit Spread Data 12 B Transition Matrix 14 C Yield Curve 14 2

3 1 Credit Value Adjustment (CVA) denition The Credit Value Adjustment is based on the assumption that the value of a nancial contract with a defaultable counterparty is reduced by the value of the expected default losses. Simple CVA estimators were developed for Payo-at-Maturity contracts, for equity swaps ( CONSULT- ING, 2013), and for interest rate swaps. General formula for CVA (Gregory, 2010) is as follows: CV A = (1 ρ) n EE i P D i (1) where ρ is the recovery rate, EE i is the Expected Exposure (EE) at time t = 0 due to a cash ow at t i, and P D i is the risk-neutral Probability of Default (PD) between t i 1 and t i. Time points are chosen at cash ow dates. The values of EE are calculated based on specics of given contract type in following sections. Default probabilities for a counterparty with the known credit rating R are calculated using credit rating migration matrix T and credit spread curves S(R, t). 2 Transition matrix Calculation of a transition matrix T for a given time period T is necessary if available transition matrix M corresponds to a dierent time period T 0. If T = K T 0 with K = 1, 2,... then i=1 T = M M (K times) M K (2) If K is not an integer (typically, it can be 1 12, 1 2, 1 1 2, etc.) then the matrix T = MK according to (Israel et al., 2001) is calculated as follows: Using an obvious formula we calculate the matrix-generator Q log M: n Q = ( 1) M = e log M (3) k+1 Dk where D = M I (here I is the identity matrix). The upper limit n of the expansion is dened by the criterion e Q M ϵ (5) where ϵ is the required accuracy. The required power of the matrix M is then calculated as T = e KQ using the following series n (K Q) k T = I + (6) k! 3 Risk-neutral Probabilities of Default We start from calculation of the transition matrices T(i, i+1) for time periods (t i, t i+1 ). Assuming that the historical transition matrix T corresponds to a time period = t i+1 t i we calculate the following two matrices: { T(0, i) = T i for time period: (0, t i ) (7) T(0, i + 1) = T i+1 for time period: (0, t i+1 ) k (4) 3

4 From the credit spread curve data S(R, t) we obtain (by proper interpolation, if necessary) credit spreads s k,i corresponding to credit ratings k [1 : k m ] [AAA, AA, A, BBB, BB, B, C] for the time period of (0, t i ). Implied default probabilities are then calculated as δ k,i = 1 e s k,it i 1 ρ Next step is to modify transition matrices (7) as follows: (8) Replace default probabilities (the last column) with implied default probabilities δ k,i Rescale matrix elements (except δ k,i ) to make each row sum to be a unit Finally, marginal transition matrices for (t i, t i+1 ) time periods are calculated as: T(i) = T(0, i + 1) T(0, i) 1 (9) Using the set of transition matrices (9) we can build the probability map for rating migration of the counterparty: { p k,0 = 1 p k,i = p k,i 1 T(i) for k corresponding to the initial rating R, zero otherwise i = 1 : n (10) The probability map element p k,i is the probability of the counterparty to have a rating k at time t i. Therefore, probabilities of default are: 4 CVA for Payo-at-Maturity contracts 4.1 CVA calculation P D i = p km,i (11) The Credit value adjustment calculation is straightforward in case of derivatives with payo at maturity (vanilla options, forward rate agreements, etc.). The present value (PV) of the contract is calculated using appropriate pricer (the trinomial trees, Monte Carlo simulations, etc.). Given credit rating R of the counterparty we calculate the probability of default P D as follows. From the credit spread curve data S(R, t) we obtain (by proper interpolation, if necessary) the credit spread s corresponding to the credit rating R for the time period of (0, T ). Implied default probability is then calculated as P D = 1 T e s (12) 1 ρ Finally, we obtain the CVA value 4.2 The CVA Calculation Examples CV A = (1 ρ) P V P D (13) Let P V = 100 and the recovery rate ρ = 0.5. The credit spread data were taken as of December 2000 (see Appendix A). Results of the CV A calculation for dierent counterparty ratings are presented in Table 1 and in Figures (1) and (2). 4

5 Table 1: Basic CVA vs Rating Rating T = 1m T = 10y AAA AA A BBB BB B C AAA AA A BBB BB B C AAA AA A BBB BB B C cva cva Rating Rating Figure 1: Maturity 1 m Figure 2: Maturity 10y 5 CVA for Equity Swaps 5.1 The Equity Swap Expected Exposure calculation Terminology: 5

6 N T type P osition t k m r 0 dt r q M Notional Maturity Notional: xed or oat long or short Swap period Number of swap periods xed payment rate time step Interest rate Dividend rate Number of Monte Carlo scenarios (14) Price tree The price of an underlying equity can be simulated using the trinomial tree algorithm. Time axis is presented with discreet time points t j = (j 1)dt, (T = (n + 1)dt is the option maturity, j = 1 (n + 1)). Equity prices at time points t j and tree nodes i are S ji = S 0 u i j (15) j = 1 (n + 1) i = 1 (2j 1) The scale factors for the price moving up by u, or moving down by u 1 are: u = e σ 2dt (16) Probabilities for price movement up (p u ), down (p d ) or staying the same (p m ) are: u e (r q)dt/2 1 p u = (17) (u 1) 2 u e (r q)dt/2 u p d = (18) (u 1) 2 p m = 1 p u p d (19) The Equity option contract (long/short position ±) cash ow (oat leg) at a payment date t k = t k (the tree node (j, i)) for a given Monte Carlo scenario s is: ( ) Where N (s) k V (s) ki is the notional value (N (s) k = ±N (s) k S ji S j 1,i (s) j 1 1 N for xed notional) S (s) j 1,i is the equity price at previous time point t j 1 at a previous tree node (i (s) j 1 ) j 1 6 (20)

7 In case of the oating notional it is reset at time t j 1 to be used at t j. Reset formula is as follows: N k = N k 1 S ji S j 1,i (s) j 1 In the following two sections below (5.1.2, 5.1.3) the Monte Carlo process is described in details. As a result, at each payment time t k for all tree nodes (i) the array of prices V (s) ki is obtained (due to number of Monte Carlo paths through these nodes). Finally, averaging by s leads to cash ow values to be used in the tree pricing procedure Migration probability tree (21) w ki = V (s) ki (22) For each scenario the path starts at the tree root (1, 1) and goes through the tree nodes switching either up (i (i + 2)), down (i i) or staying the same (i (i + 1)) according to probabilities p u, p d, and p m. A probability P ji to reach a node (j, i) is calculated as follows: P 11 = 1, P 22 = p m P j,i 2 p u, if i = 2j + 1; j = 1, n P j,i 2 p u + P j,i 1 p m, if i = 2j; j = 3, n P j+1,i = P ji p d + P j,i 1 p m + P j,i 2 p u, if i = 3, 2j 3; j = 2, n P j,i 1 p m + P j,i p d, if i = 2; j = 3, n P j,i p d, if i = 1; j = 1, n (23) Building the s th Monte Carlo scenario We start from the tree root node (1, 1). From each node (j, i) the path goes up or down according to the random value of ϵ ϵ < p d downward (i i) p d < ϵ < p d + p m same (i i + 1) (24) ϵ > p d + p m upward (i i + 2) where ϵ is a uniformly distributed random number (0 < ϵ < 1). For each M.C. scenario s we add V (s) ki to the sequence of prices for nodes (i) at payment times t k along the Monte Carlo path. After all Monte Carlo scenarios are done we take average of s-sequences and record it as w ki as in (22). Attention! In case of not sucient number of Monte Carlo scenarios some nodes are never reached. In these cases w ki = Backward Induction Working back from (n + 1, i) to the tree root (1, 1): At maturity we have node values Q n+1,i : Q n+1,i = w km,i (25) 7

8 Backward induction (j = n) (j = 1): Q j,i = (Q j+1,i p d + Q j+1,i+1 p m + Q j+1,i+2 p u ) e r dt i = 1,, 2j 1 (26) At each payment date t k (k = k m k = 1) we take account of cash ows: Q j,i Q j,i + w ki (27) Finally, we obtain all node values Q j,i and the Equity Swap value at t = 0: Expected Exposure Q 1,1 (28) The Expected Exposure is based on all non-negative values of Q jk at t j discounted to t = 0: EE j = 2j The Equity Swap Rate calculation max(0, Q jk )P jk e r t j (29) The Equity Swap rate r 0 can be calculated using Expected ( Exposure at ) t = 0 reduced by the CVA value based on the assumption that the present value EE (0) 1 CV A of the swap (receive oat) is oset by future xed rate payments. The CVA values are calculated using (1), (11), and (29). The result is: r 0 = EE(0) 1 CV A (30) k m t e rt k 5.3 Test Results For testing we use the credit spread data (Appendix A) and the 3m transition matrix (Appendix B). Table 2: Parameters Position Long Transition Matrix 3 m Notional 100 Recovery rate 50 % Maturity 6 m Monte Carlo Swap period 3 m Time step 1 d Volatility 25 % Interest rate 2% Dividend rate 0% If CVA is neglected then r 0 = 2.11% and EE 1 = Results of the CVA calculation are presented in Table 3. Dependencies of CVA and of the Fixed swap rates of the counterparty credit rating are presented in Figure 3 and in Figure 4. 8

9 Table 3: Equity Swap CVA and Fixed Rate Rating CV A r 0, % AAA AA A BBB BB B C AAA AA A BBB BB B C AAA AA A BBB BB B C cva Fixed rate, % Rating Rating Figure 3: CVA Figure 4: Swap rate An example of the time dependence of the CVA and of the Equity swap exposure for a C-rated counterparty is presented in Figures 5 and 6. 9

10 CVA (EQSwap), credit rating C Quantile 99%, average. Notional: fixed CVA(t) Equity Swap, credit rating C Time Time Figure 5: CVA vs time Figure 6: Equity swap exposure 6 The Credit value adjusted Swap Rate 6.1 Fair Swap rate for defaultable counterparties Consider an Interest Rate swap contract (notional N) bought by the Bank (receive oat, pay xed) with the credit rating of R B which is sold by a Counterparty (receive xed, pay oat) with the credit rating of R CP. The present values of the oat/xed receiver is n P V float = N r k e y ktk (1 pd (CP ) i ) P V fixed = N n r f e y ktk (1 pd (B) i ) where n is number of payments, is the coverage period, r k is the oat interest rate at k th payment date t k, y k is the discount rate at t k, and pd (CP ) i /pd (B) i are default probabilities of the xed/oat receiver. The fair xed rate r f can be calculated based on P V float = P V fixed which leads to (31) r f = n r k e y ktk (1 pd (CP ) i ) n e y kt k (1 pd (B) i ) (32) 6.2 Swap rate examples As an example we calculate Credit value Adjusted swap rates for a typical oat / xed Interest Rate Swap with payment frequency of 3m using transition matrix (Appendix B), credit spread data (Appendix A), and the yield curve (Appendix C). The recovery rates for both parties are 50%. In case of a 3m maturity (Table 4) the no-default swap rate is %. 10

11 Table 4: Swap Rates (%) at T = 3m Counterparty AAA AA A BBB BB B C AAA AA A BBB BB B C In case of a 5y maturity (Table 5) the no-default swap rate is 2.656% Table 5: Swap Rates (%) at T = 5y Counterparty AAA AA A BBB BB B C AAA AA A BBB BB B C An example of the swap rate dependence on the counterparty rating is presented in Figure 7 (horizontal line represent the no-default swap rate 4.214%). Swap rate (fixed rate payer rating: AAA ) Swap rate AAA AA A BBB BB B C Rating (float leg payer) Figure 7 11

12 7 Source Codes The source codes for CVA calculations can be ordered through Yashkir Consulting web site. 8 References YASHKIR CONSULTING. Equity swap price calculator (modied monte carlo trinomial tree) Jon Gregory. Counterparty Credit Risk. Wiley, Section 7. Robert B. Israel, Jerey S. Rosenthal, and Jason Z. Wei. Finding generators for markov chains via empirical transition matrices, with applications to credit ratings. Mathematical Finance, 11(2):245265, A Credit Spread Data Credit spread data used corresponds to December of 2000 Tenor AAA AA A BBB BB B C 1m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m

13 37m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m

14 99m m m m m m m m m m m m m m m m m m m m m m B Transition Matrix Table 7: 3 m Transition matrix (December 2000) AAA AA A BBB BB B C D AAA AA A BBB BB B C D C Yield Curve Table 8: Example of a yield curve tenor, y yield rate, %