1.1 Implied probability of default and credit yield curves

Transcription

1 Risk Management Topic One Credit yield curves and credit derivatives 1.1 Implied probability of default and credit yield curves 1.2 Credit default swaps 1.3 Credit spread and bond price based pricing 1.4 Pricing of credit derivatives 1

2 1.1 Implied probability of default and credit yield curves The price of a corporate bond must reflect not only the spot rates for default-free bonds but also a risk premium to reflect default risk and any options embedded in the issue. Credit spreads: compensate investor for the risk of default on the underlying securities 2

3 The spread increases as the rating declines. In general, it also increases with maturity (for BBB-rating or above). The spread tends to increase faster with maturity for low credit ratings than for high credit ratings. 3

4 Term structures of forward probabilities of default Year Cumulative default probability (%) Forward default probability in year (%) % + ( %) % = % % + ( %) % = % P [τ def 2] = cumulative default probability up to Year 2 = %; P [τ def 2 τ def > 1] = forward default probability of default in Year 2 = %. 4

5 Probability of default assuming no recovery Define y(t ) : y (T ) : Q(T ) : τ : Yield on a T -year corporate zero-coupon bond Yield on a T -year risk-free zero-coupon bond Probability that corporation will default between time zero and time T Random time of default The value of a T -year risk-free zero-coupon bond with a principal of 100 is 100e y (T )T while the value of a similar corporate bond is 100e y(t )T. 5

6 Assuming zero recovery upon default, there is a probability Q(T ) that the corporate bond will be worth zero at maturity and a probability 1 Q(T ) that it will be worth 100. The value of the risky bond is {Q(T ) 0 + [1 Q(T )] 100}e y (T )T = 100[1 Q(T )]e y (T )T. Since the yield on the risky bond is y(t ), so 100e y(t )T = 100[1 Q(T )]e y (T )T. The T -year survival probability is given by S(T ) = 1 Q(T ) = e [y(t ) y (T )]T. Note that the probability Q(T ) is the risk neutral probability since it is inferred from prices of traded securities. 6

7 As a summary, assuming zero recovery upon default, the survival probability as implied from the bond prices is seen to be 100e y(t )T S(T ) = price of defaultable bond = 100e y (T )T price of default free bond = e credit spread T, where credit spread = y(t ) y (T ). Here, the T -year credit spread is the difference in the yield of the risky zero-coupon and its riskfree counterpart, both with maturity T. Alternative proof Assuming zero recovery and independence of the interest rate process and default event, and letting τ be the random default time, we then have price of risky bond = E[100e T 0 r u du 1 {τ>t } ] (zero recovery) = E[100e T 0 r u du ]E[1 {τ>t } ] (independence) = price of riskfree bond S(T ). 7

8 Example Suppose that the spreads over the risk-free rate for 5-year and a 10- year BBB-rated zero-coupon bonds are 130 and 170 basis points, respectively, and there is no recovery in the event of default. The default probabilities can be inferred from the term structure of credit spreads as follows: P [τ 5] = Q(5) = 1 e = P [τ 10] = Q(10) = 1 e = The probability of default between five years and ten years is Q(5; 10) where Q(10) = Q(5) + [(1 Q(5)]Q(5; 10) or P [τ 10 τ > 5] = Q(5; 10) =

9 Credit spreads and default intensities (hazard rates) The default intensity (hazard rate) at time t is defined so that λ(t) t is the probability of default between time t and t + t conditional on no earlier default. If S(t) is the cumulative probability of the company surviving to time t (no default by time t), then probability of default occurring within (t, t + t] = S(t) S(t + t) = S(t)λ(t) t. Taking the limit t 0, we obtain so that ds(t) S(t) = λ(t) dt with S(0) = 1, S(t) = e t 0 λ(u) du = e λ(t)t = 1 Q(t), where Q(t) is the probability of default by time t and λ(t) is the average default intensity between time 0 and time t. 9

10 The average default intensity λ(t) can be visualized as the credit spread over (0, t) since S(t) = e [y(t) y (t)]t = e t 0 λ(u) du = e λ(t)t, t [0, T ]. The unconditional default probability density q(t) is defined so that q(t) t gives the probability of default that occurs within (t, t + t). Let F (t) be the distribution function of the random default time τ, where we then have q(t) = F (t). F (t) = P [τ t], Recall that q(t) t = S(t)λ(t) t so that q(t) = e t 0 λ(u) du λ(t) = S(t)λ(t), t 0, where S(t) = 1 F (t). Also, the probability of surviving until time t, conditional on survival up to s, where s t, is given by P [τ > t τ > s] = S(t) S(s) = e t 0 λ(u) du t e s = e s λ(u) du. 0 λ(u) du 10

11 Recovery rates Amounts recovered on corporate bonds as a percent of par value from Moody s Investor s Service are shown in the table below. Class Mean (%) Standard derivation (%) Senior secured Senior unsecured Senior subordinated Subordinated Junior subordinated The amount recovered is estimated as the market value of the bond one month after default. Seniority of the bond among outstanding bonds issued by the same issuer is an important determinant of the recovery rate of that bond. Bonds that are newly issued by an issuer must have seniority below that of existing bonds issued earlier by the same issuer. 11

12 Finite recovery rate In the event of a default, the bondholder receives a proportion R of the bond s no-default value. If there is no default, then the bondholder receives 100. The bond s no-default value is 100e y (T )T and the probability of a default is Q(T ). The value of the bond is so that [1 Q(T )]100e y (T )T + Q(T )100Re y (T )T 100e y(t )T = [1 Q(T )]100e y (T )T + Q(T )100Re y (T )T. The implied probability of default in terms of yields and recovery rate is given by Q(T ) = 1 e [y(t ) y (T )]T 1 R. 12

13 Numerical example on the impact of different assumptions of recovery rates on default probability estimation Suppose the 1-year default free bond price is $100 and the 1-year defaultable XY Z corporate bond price is $80. (i) Assuming R = 0, the probability of default of XY Z as implied by the two bond prices is Q 0 (1) = = 20%. (ii) Assuming R = 0.6, we obtain Q R (1) = = 20% 0.4 = 50%. The ratio of Q 0 (1) : Q R (1) = 1 : 1 1 R. 13

14 Calculation of default intensity with non-zero recovery rate Consider a 5-year risky corporate bond that pays a coupon of 6% per annum (paid semiannually) Yield on the corporate bond is 7% per annum (with continuous compounding) Yield on a similar risk-free bond is 5% per annum (with continuous compounding) The yields imply that (i) price of the riskfree bond = 3e e e e = (ii) price of the risky bond = 3e e e e = The present value of expected loss from default over the 5-year life of the bond = =

15 Let Q denote the constant unconditional probability of default per year. Assuming that defaults can happen at times 0.5, 1.5, 2.5, 3.5 and 4.5 year (immediately before coupon payment dates), we can calculate the expected loss from default in terms of Q. Calculation of loss from default on a bond in terms of the default probability per year, Q. Notional principal = $100. Time Default Recovery Risk-free Loss given Discount PV of expected (years) probability amount($) value($) default($) factor loss($) 0.5 Q Q 1.5 Q Q 2.5 Q Q 3.5 Q Q 4.5 Q Q Total Q 15

16 Consider the 3.5 year row in the table. The expected value of the riskfree bond at Year 3.5 (time to expiry is 1.5 years) is 3 + 3e e e = The amount recovered if there is a default is 40, so the loss given default is = The present value of this loss = e Q = Q = 54.01Q. The total expected loss is Q. Setting this equal to 8.75, we obtain Q = 3.03%. 16

17 Generalization - term structure of default probabilities Suppose we have bonds maturing in 3, 5, 7, and 10 years, we could use the first bond to estimate a default probability per year for the first 3 years, the second bond to estimate default probability per year for years 4 and 5, the third bond for years 6 and 7, and the last bond for years 8, 9 and 10. For example, suppose λ [0,3] is the default intensity in the first 3 years, which has been obtained from an earlier calculation based on 3-year risky and riskfree bonds. We compute λ [3,5] using 5-year bonds by following the sample calculations as shown in the above, except that the default intensity at times 0.5, 1.5 and 2.5 are set to be the known quantity λ [0,3]. The default intensity at times 3.5 and 4.5 are set to be λ [3,5], a quantity to be determined. 17

18 Construction of a credit risk adjusted yield curve is hindered by 1. The general absence in money markets of liquid traded instruments on credit spread. In recent years, for some liquidly traded corporate bonds, we may have good liquidity on trading of credit default swaps whose underlying is the credit spread. 2. The absence of a complete term structure of credit spreads as implied from traded corporate bonds. At best we only have infrequent data points. 18

19 The default probabilities estimated from historical data are much less than those derived from bond prices For example, from historical data published by Moody s, an A-rated company has average cumulative default rate Q(7) of = 0.91%. The average 7-year default intensity λ(7) is determined by so that S(7) = = = e λ(7) 7 λ(7) = 1 7 ln = = 0.13%. On the other hand, based on bond yields published by Merrill Lynch, the average Merrill Lynch yield for A-rated bonds was 6.274%. The average riskfree rate was estimated to be 5.505%. As an approximation, the average 7-year default intensity is = = 1.28% Here, the recovery rate is assumed to be

20 Seven-year average default intensities (% per annum). Rating Historical default Default intensity Ratio Difference intensity from bonds Aaa Aa A Baa Ba B Caa Corporate bonds are relatively illiquid and bond traders demand an extra return to compensate for this. Bonds do not default independently of each other. This gives rise to risk that cannot be diversified away, so bond traders should require an expected excess return for bearing the risk. 20

21 Implied default probabilities (equity-based versus credit-based) Recovery rate has a significant impact on the defaultable bond prices. The forward probability of default as implied from the defaultable and default free bond prices requires estimation of the expected recovery rate (an almost impossible job). The industrial code mkm V estimates default probability using stock price dynamics equity-based implied default probability. For example, the JAL stock price dropped to 1 in early Obviously, the equity-based default probability over one year horizon is close to 100% (stock holders receive almost nothing upon JAL s default). However, the credit-based default probability as implied by the JAL bond prices is less than 30% since the bond par payments are somewhat partially guaranteed even in the event of default. 21

22 1.2 Credit default swaps The protection seller receives fixed periodic payments from the protection buyer in return for making a single contingent payment covering losses on a reference asset following a default. protection seller 140 bp per annum Credit event payment (100% recovery rate) only if credit event occurs protection buyer holding a risky bond 22

23 Protection seller earns premium income with no funding cost gains customized, synthetic access to the risky bond Protection buyer hedges the default risk on the reference asset 1. Very often, the bond tenor is longer than the swap tenor. In this way, the protection seller does not have exposure to the full period of the bond. 2. Basket default swap gain additional premium by selling default protection on several assets. 23

24 A bank lends 10mm to a corporate client at L + 65bps. The bank also buys 10mm default protection on the corporate loan for 50bps. Objective achieved by the Bank through the default swap: maintain relationship with the corporate borrower reduce credit risk on the new loan Risk Transfer Corporate Borrower Interest and Principal Bank Default Swap Premium If Credit Event: par amount If Credit Event: obligation (loan) Financial House Default swap settlement following Credit Event of Corporate Borrower 24

25 Settlement of compensation payment 1. Physical settlement: The defaultable bond is put to the Protection Seller in return for the par value of the bond. 2. Cash compensation: An independent third party determines the loss upon default at the end of the settlement period (say, 3 months after the occurrence of the credit event). Compensation amount = (1 recovery rate) bond par. 25

26 Selling protection To receive credit exposure for a fee (simple credit default swaps) or in exchange for credit exposure to better diversify the credit portfolio (exchange credit default swaps). Buying protection To reduce either individual credit exposures or credit concentrations in portfolios. Synthetically to take a short position in an asset which are not desired to sell outright, perhaps for relationship or tax reasons. 26

27 Funding cost arbitrage Should the Protection Buyer look for a Protection Seller who has a higher/lower credit rating than himself? A-rated institution as Protection Seller Lender to the A-rated Institution 50bps annual premium funding cost of LIBOR + 50bps AAA-rated institution as Protection Buyer BBB risky reference asset coupon = LIBOR + 90bps LIBOR-15bps as funding cost Lender to the AAA-rated Institution 27

28 The combined risk faced by the Protection Buyer: default of the BBB-rated bond default of the Protection Seller on the contingent payment Consider the S&P s Ratings for jointly supported obligations (the two credit assets are uncorrelated) A + A A BBB + BBB A + AA + AA + AA + AA AA A AA + AA AA AA AA The AAA-rated Protection Buyer creates a synthetic AA asset with a coupon rate of LIBOR + 90bps 50bps = LIBOR + 40bps. This is better than LIBOR + 30bps, which is the coupon rate of a AA asset (net gains of 10bps). 28

29 For the A-rated Protection Seller, it gains synthetic access to a BBB-rated asset with earning of net spread of Funding cost of the A-rated Protection Seller = LIBOR + 50bps Coupon from the underlying BBB bond = LIBOR + 90bps Credit swap premium earned = 50bps 29

30 In order that the credit arbitrage works, the funding cost of the default protection seller must be higher than that of the default protection buyer. Example Suppose the A-rated institution is the Protection Buyer, and assume that it has to pay 60bps for the credit default swap premium (higher premium since the AAA-rated institution has lower counterparty risk). spread earned from holding the risky bond = coupon from bond funding cost = (LIBOR + 90bps) (LIBOR + 50bps) = 40bps which is lower than the credit swap premium of 60bps paid for hedging the credit exposure. No deal is done! 30

31 Counterparty risk in CDS Before the Fall 1997 crisis, several Korean banks were willing to offer credit default protection on other Korean firms. US commercial bank LIBOR + 70bp Hyundai (not rated) 40 bp Korea exchange bank Higher geographical risks lead to higher default correlations. Higher geographic risks lead to higher default correlations. Advice: Go for a European bank to buy the protection. 31

32 How does the inter-dependent default risk structure between the Protection Seller and the Reference Obligor affect the credit swap premium rate? 1. Replacement cost (Seller defaults earlier) If the Protection Seller defaults prior to the Reference Entity, then the Protection Buyer renews the CDS with a new counterparty. Supposing that the default risks of the Protection Seller and Reference Entity are positively correlated, then there will be an increase in the swap rate of the new CDS. 2. Settlement risk (Reference Entity defaults earlier) The Protection Seller defaults during the settlement period after the default of the Reference Entity. 32

33 Hedge strategy using fixed-coupon bonds Portfolio 1 One defaultable coupon bond C; coupon c, maturity t N. One CDS on this bond, with CDS spread s The portfolio is unwound after a default. Portfolio 2 One default-free coupon bond C: with the same payment dates as the defaultable coupon bond and coupon size c s. The default free bond is sold after default of the defaultable counterpart. 33

34 Comparison of cash flows of the two portfolios 1. In survival, the cash flows of both portfolio are identical. Portfolio 1 Portfolio 2 t = 0 C(0) C(0) t = t i c s c s t = t N 1 + c s 1 + c s 2. At default, portfolio 1 s value = par = 1 (full compensation by the CDS); that of portfolio 2 is C(τ), τ is the time of default. The price difference at default = 1 C(τ). This difference is very small when the default-free bond is a par bond. Remark The issuer can choose c to make the bond be a par bond such that the initial value of the bond is at par. 34

35 This is an approximate replication. Recall that the value of the CDS at time 0 is zero. Let B(0, t N ) denote the price of a zero-coupon default-free bond. Neglecting the difference in the values of the two portfolios at default, the no-arbitrage principle dictates C(0) = C(0) = B(0, t N ) + ca(0) sa(0). Here, (c s)a(0) is the sum of present value of the coupon payments at the fixed coupon rate c s. The equilibrium CDS rate s can be solved: s = B(0, t N) + ca(0) C(0). A(0) B(0, t N ) + ca(0) is the time-0 price of a default free coupon bond paying coupon at the rate of c. 35

36 Cash-and-carry arbitrage with par floater A par floater C is a defaultable bond with a floating-rate coupon of c i = L i 1 + s par, where the par spread s par is adjusted such that at issuance the par floater is valued at par. Portfolio 1 One defaultable par floater C with spread s par over LIBOR. One CDS on this bond: CDS spread is s. The portfolio is unwound after default. 36

37 Portfolio 2 One default-free floating-coupon bond C : with the same payment dates as the defaultable par floater and coupon at LIBOR, c i = L i 1. The bond is sold after default. Time Portfolio 1 Portfolio 2 t = t = t i L i 1 + s par s L i 1 t = t N 1 + L N 1 + s par s 1 + L N 1 τ (default) 1 C (τ) = 1 + L i (τ t i ) The hedge error in the payoff at default is caused by accrued interest L i (τ t i ), accumulated from the last coupon payment date t i to the default time τ. If we neglect the small hedge error at default, then s par = s. 37

38 Remarks The non-defaultable bond becomes a par bond (with initial value equals the par value) when it pays the floating rate equals LI- BOR. The extra coupon s par paid by the defaultable par floater represents the credit spread demanded by the investor due to the potential credit risk. The above result shows that the credit spread s par is just equal to the CDS spread s. The above analysis neglects the counterparty risk of the Protection Seller of the CDS. Due to potential counterparty risk, the actual CDS spread will be lower. 38

39 Valuation of Credit Default Swap Suppose that the probability of a reference entity defaulting during a year conditional on no earlier default is 2%. That is, the default intensity is assumed to be the constant 2%. Table 1 shows the survival probabilities and forward default probabilities (i.e., default probabilities as seen at time zero) for each of the 5 years. The probability of a default during the first year is 0.02 and the probability that the reference entity will survive until the end of the first year is The forward probability of a default during the second year is = and the probability of survival until the end of the second year is =

40 Table 1 Forward default probabilities and survival probabilities Time (years) Default probability Survival probability = = = = P [3 < τ 4] = forward default probability of default during the fourth year (as seen at current time) = P [τ > 3] P [3 < τ 4 τ > 3] = survival probability until end of Year 3 conditional probability of default in Year 4 = = =

41 Assumptions on default and recovery rate We will assume the defaults always happen halfway through a year and that payments on the credit default swap are made once a year, at the end of each year. We also assume that the risk-free (LIBOR) interest rate is 5% per annum with continuous compounding and the recovery rate is 40%. Expected present value of CDS premium payments Table 2 shows the calculation of the expected present value of the payments made on the CDS assuming that payments are made at the rate of s per year and the notional principal is $1. For example, there is a probability that the third payment of s is made. The expected payment is therefore s and its present value is se = s. The total present value of the expected payments is s. 41

42 Table 2 Calculation of the present value of expected payments. Payment = s per annum. Time (years) Probability of survival Expected payment Discount factor PV of expected payment s s s s s s s s s s Total s 42

43 Table 3 Calculation of the present value of expected payoff. Notional principal = $1. Time (years) Probability of default Recovery rate Expected payoff ($) Discount factor Total PV of expected payoff ($) For example, there is a probability of a payoff halfway through the third year. Given that the recovery rate is 40%, the expected payoff at this time is = The present value of the expected payoff is e = The total present value of the expected payoffs is $

44 When default occurs in mid-year, the Protection Buyer has to pay the premium accrued half year (between the last premium payment date and default time). Table 4 Calculation of the present value of accrual payment. Time (years) Probability of default Expected accrual payment Discount factor PV of expected accrual payment s s s s s s s s s s Total s 44

45 As a final step we evaluate in Table 4 the accrual payment made in the event of a default. There is a probability that there will be a final accrual payment halfway through the third year. The accrual payment is 0.5s. The expected accrual payment at this time is therefore s = s. Its present value is se = s. The total present value of the expected accrual payments is s. From Tables 2 and 4, the present value of the expected payment is s s = s. 45

46 Equating expected CDS premium payments and expected compensation payment From Table 3, the present value of the expected payoff is Equating the two, we obtain the CDS spread for a new CDS as s = or s = The mid-market spread should be times the principal or 124 basis points per year. In practice, we are likely to find that calculations are more extensive than those in Tables 2 to 4 because (a) payments are often made more frequently than once a year (b) we might want to assume that defaults can happen more frequently than once a year. 46

47 Impact of expected recovery rate R on credit swap premium s Recall that the expected compensation payment paid by the Protection Seller is (1 R) notional. Therefore, the Protection Seller charges a higher s if her estimation of the recovery rate R is lower. Let s R denote the credit swap premium when the recovery rate is R. We deduce that Remark s 10 s 50 = (100 10)% (100 50)% = 90% 50% = 1.8. A binary credit default swap pays the full notional upon default of the reference asset. The credit swap premium of a binary swap depends only on the estimated default probability but not on the recovery rate. 47

48 Marking-to-market a CDS At the time it is negotiated, a CDS, like most swaps, is worth zero. Later, it may have a positive or negative value. Suppose, for example the credit default swap in our example had been negotiated some time ago for a spread of 150 basis points, the present value of the payments by the buyer would be = and the present value of the payoff would be The value of swap to the seller would therefore be , or times the principal. Similarly the mark-to-market value of the swap to the buyer of protection would be times the principal. 48

49 Basket default swaps The credit event to insure against using the k th -to-default credit default swap is the event of the k th default. A premium or spread s is paid as an insurance fee until maturity or the event of the k th default, whichever comes first. If the k th default occurs before swap s maturity, the Protection Buyer puts the defaulting bond to the Protection Seller in exchange for the face value of the bond. Sum of the k th -to-default swap spreads, k = 1, 2,..., n, for n obligors in total in the basket is greater than the sum of the individual spreads of the same set of n obligors: n k=1 s k > Why? Apparently, both sides insure exactly the same set of risks: the n defaults in the basket. At the time of the first default, the left side stops paying the huge spread s 1 while on the plain-vanilla side one just stops paying the spread s i of the first default that falls on obligor i. n i=1 s i. 49

50 Bounds on the swap premiums for the first-to-default (FtD) swaps under low default correlation Assuming all 3 obligors have the same dollar exposure, we have fee on CDS on fee on FtD portfolio of worst credit swap CDSs on all credits s C s FtD s A + s B + s C With low default probabilities and low default correlation, we have s FtD s A + s B + s C. To see this, by assuming zero default correlation, the probability of at least one default is so that p = 1 (1 p A )(1 p B )(1 p C ) = p A + p B + p C (p A p B + p A p C + p B p C ) + p A p B p C p p A + p B + p C for small p A, p B and p C. 50

51 1.3 Credit spread and bond price based pricing Market s assessment of the default risk of the obligor (assuming some form of market efficiency information is aggregated in the market prices). The sources are market prices of bonds and other defaultable securities issued by the obligor prices of CDS s referencing this obligor s credit risk How to construct a clean term structure of credit spreads from observed market prices? 51

52 Based on no-arbitrage pricing principle, a model that is based upon and calibrated to the prices of traded assets is immune to simple arbitrage strategies using these traded assets. Market instruments used in bond price-based pricing At time t, the defaultable and default-free zero-coupon bond prices of all maturities T t are known. These defaultable zero-coupon bonds have no recovery at default. Information about the probability of default over all time horizons as assessed by market participants are fully reflected when market prices of default-free and defaultable bonds of all maturities are available. 52

53 Risk neutral probabilities The financial market is modeled by a filtered probability space (Ω, (F t ) t 0, F, Q), where Q is the risk neutral probability measure. All probabilities and expectations are taken under Q. Probabilities are considered as state prices. 1. For constant interest rates, the discounted Q-probability of an event A at time T is the price of a security that pays off $1 at time T if A occurs. 2. Under stochastic interest rates, the price of the contingent claim associated with A is E[β(T )1 A ], where β(t ) is the discount factor. This is based on the risk neutral valuation principle and the money market account M(T ) = 1 T β(t ) = e t r u du is used as the numeraire. 53

54 Indicator functions For A F,1 A (ω) = { 1 if ω A 0 otherwise. τ = random time of default; I(t) = survival indicator function I(t) = 1 {τ>t} = { 1 if τ > t 0 if τ t. B(t, T ) = price at time t of zero-coupon bond paying off $1 at T B(t, T ) = price of defaultable zero-coupon bond if τ > t; I(t)B(t, T ) = { B(t, T ) if τ > t 0 if τ t. 54

55 Monotonicity properties on the bond prices 1. 0 B(t, T ) < B(t, T ), t < T 2. Starting at B(t, t) = B(t, t) = 1, B(t, T 1 ) B(t, T 2 ) > 0 and B(t, T 1 ) B(t, T 2 ) 0 t < T 1 < T 2, τ > t. Independence assumption {B(t, T ) t T } and τ are independent under (Ω, F, Q) (not the true measure). 55

56 Implied probability of survival in [t, T ] based on market prices of bonds [ B(t, T ) = E e T ] [ t r u du and B(t, T ) = E e ] T t r u du I(T ). Invoking the independence between defaults and the default-free interest rates [ B(t, T ) = E e T ] t r u du E[I(T )] = B(t, T )P (t, T ) implied survival probability over [t, T ] = P (t, T ) = B(t, T ) B(t, T ). 56

57 The implied default probability over [t, T ], P def (t, T ) = 1 P (t, T ). Assuming P (t, T ) has a right-sided derivative in T, the implied density of the default time Q[τ (T, T + dt ] F t ] = P (t, T ) dt. T If prices of zero-coupon bonds for all maturities are available, then we can obtain the implied survival probabilities for all maturities (complementary distribution function of the time of default). 57

58 Properties on implied survival probabilities, P (t, T ) 1. P (t, t) = 1 and it is non-negative and decreasing in T. Also, P (t, ) = Normally P (t, T ) is continuous in its second argument, except that an important event secheduled at some time T 1 has direct influence on the survival of the obligor. 3. Viewed as a function of its first argument t, all survival probabilities for fixed maturity dates will tend to increase. If we want to focus on the default risk over a given time interval in the future, we should consider conditional survival probabilities. conditional survival probability over [T 1, T 2 ] as seen from t = P (t, T 1, T 2 ) = P (t, T 2) P (t, T 1 ), where t T 1 < T 2. 58

59 Implied hazard rate (default probabilities per unit time interval length) Discrete implied hazard rate of default over (T, T + T ] as seen from time t H(t, T, T + T ) T = so that P (t, T ) P (t, T + T ) 1 = P def(t, T, T + T ) P (t, T, T + T ), P (t, T ) = P (t, T + T )[1 + H(t, T, T + T ) T ]. In the limit of T 0, the continuous hazard rate at time T as seen at time t is given by h(t, T ) = T ln P (t, T ). 59

60 Proof We have First, we recall 1 P (t, T, T + T ) = P (t, T ) P (t, T, T + T ). h(t, T ) = lim H(t, T, T + T ) T 0 = 1 P (t, T, T + T ) lim T 0 T P (t, T, T + T ) = lim T 0 1 T [ P (t, T ) P (t, T + T ) 1 = lim 1 T 0 P (t, T + T ) = 1 P (t, T ) T P (t, T ) = ln P (t, T ). T ] P (t, T + T ) P (t, T ) T 60

61 Forward spreads and implied hazard rate of default For t T 1 < T 2, the simply compounded forward rate over the period (T 1, T 2 ] as seen from t is given by F (t, T 1, T 2 ) = B(t, T 1)/B(t, T 2 ) 1 T 2 T 1. This is the price of the forward contract with expiration date T 1 on a unit-par zero-coupon bond maturing on T 2. To prove, we consider the compounding of interest rates over successive time intervals. 1 B(t, T 2 ) }{{} compounding over [t, T 2 ] = 1 [1 + F (t, T 1, T 2 )(T 2 T 1 )] B(t, T 1 ) }{{}}{{} simply compounding over [T 1, T 2 ] compounding over [t, T 1 ] Defaultable simply compounded forward rate over [T 1, T 2 ] F (t, T 1, T 2 ) = B(t, T 1)/B(t, T 2 ) 1 T 2 T 1. 61

62 Instantaneous continuously compounded forward rates f(t, T ) = lim F (t, T, T + T ) = T 0 T f(t, T ) = lim F (t, T, T + T ) = T 0 T Implied hazard rate of default Recall P (t, T 1, T 2 ) = B(t, T 2) B(t, T 1 ) B(t, T 2 ) B(t, T 1 ) and upon expanding, we obtain ln B(t, T ) ln B(t, T ). = 1 + F (t, T 1, T 2 )(T 2 T 1 ) 1 + F (t, T 1, T 2 )(T 2 T 1 ) = 1 P def(t, T 1, T 2 ), P def (t, T 1, T 2 ) [1 + F (t, T 1, T 2 )(T 2 T 1 )] }{{} B(t,T 1 )/B(t,T 2 ) = [F (t, T 1, T 2 ) F (t, T 1, T 2 )](T 2 T 1 ). 62

63 P def (t, T 1, T 2 ) Define H(t, T 1, T 2 ) = (T 2 T 1 )P (t, T 1, T 2 ) rate of default. We then have as the discrete implied H(t, T 1, T 2 ) = B(t, T 2) [F (t, T 1, T 2 ) F (t, T 1, T 2 )] B(t, T 1 ) P (t, T 1, T 2 ) = B(t, T 2) B(t, T 1 ) [F (t, T 1, T 2 ) F (t, T 1, T 2 )]. Taking the limit T 2 T 1, then the implied hazard rate of default at time T > t as seen from time t is the spread between the forward rates: h(t, T ) = f(t, T ) f(t, T ). Alternatively, we obtain the above relation using f(t, T ) f(t, T ) = T = T ln B(t, T ) B(t, T ) ln P (t, T ) = h(t, T ). 63

64 The local default probability at time t over the next small time step t 1 t Q[τ t + t F t {τ > t}] r(t) r(t) = λ(t) where r(t) = f(t, t) is the riskfree short rate and r(t) = f(t, t) is the defaultable short rate. Recovery value View an asset with positive recovery as an asset with an additional positive payoff at default. The recovery value is the expected value of the recovery shortly after the occurrence of a default. 64

65 Payment upon default Define e(t, T, T + T ) to be the value at time t < T of a deterministic payoff of $1 paid at T + T if and only if a default happens in [T, T + T ]. e(t, T, T + T ) = E Q [β(t, T + T )[I(T ) I(T + T )] F t ]. Note that I(T ) I(T + T ) = { 1 if default occurs in [T, T + T ] 0 otherwise, E Q [β(t, T + T )I(T )] = E Q [β(t, T + T )]E Q [I(T )] = B(t, T + T )P (t, T ), E Q [β(t, T + T )I(T + T )] = B(t, T + T ), and B(t, T + T ) = B(t, T + T )/P (t, T + T ). 65

66 It is seen that e(t, T, T + T ) = B(t, T + T )P (t, T ) B(t, T + T ) [ ] P (t, T ) = B(t, T + T ) P (t, T + T ) 1 On taking the limit T 0, we obtain = T B(t, T + T )H(t, T, T + T ) e(t, T, T + T ) rate of default compensation = e(t, T ) = lim T 0 T = B(t, T )h(t, T ) = B(t, T )P (t, T )h(t, T ). The value of a security that pays π(s) if a default occurs at time s for all t < s < T is given by T T π(s)e(t, s) ds = π(s)b(t, s)h(t, s) ds. t This result holds for deterministic recovery rates. t 66

67 Random recovery value Suppose the payoff at default is not a deterministic function π(τ) but a random variable π which is drawn at the time of default τ. π is called a marked point process. Define π e (t, T ) = E Q [π F t {τ = T }]. which is the expected value of π conditional on default at T and information at t. Conditional on a default occurring at time T, the price of a security that pays π at default is B(t, T )π e (t, T ). Since the time of default is not known, we have to integrate these values over all possible default times and weight them with the respective probability of default occurring. The price at time t of a payoff of π at τ if τ [t, T ] is given by T t π e (t, s) B(t, s)p (t, s) h(t, s) ds. }{{} B(t,s) 67

68 Building blocks for credit derivatives pricing Tenor structure δ k = T k+1 T k, 0 k K 1 Coupon and repayment dates for bonds, fixing dates for rates, payment and settlement dates for credit derivatives all fall on T k, 0 k K. 68

69 Fundamental quantities of the model Term structure of default-free interest rates F (0, T ) Term structure of implied hazard rates H(0, T ) Expected recovery rate π (rate of recovery as percentage of par) B(0, T From B(0, T i ) = i 1 ) 1 + δ i 1 F (0, T i 1 T i ), i = 1, 2,, k, and B(0, T 0) = B(0, 0) = 1, we obtain B(0, T k ) = Similarly, from P (0, T i ) = k i= δ i 1 F (0, T i 1, T i ). B(0, T k ) = B(0, T k )P (0, T k ) = B(0, T k ) e(0, T k, T k+1 ) = δ k H(0, T k, T k+1 )B(0, T k+1 ) P (0, T i 1 ), we deduce that 1 + δ i 1 H(0, T i 1, T i ) k i=1 = value of $1 at T k+1 if a default has occurred in (T k, T k+1 ] δ i 1 H(0, T i 1, T i ). 69

70 Taking the limit δ i 0, for all i = 0, 1,, k ( B(0, T k ) = exp ( B(0, T k ) = exp Tk 0 Tk 0 f(0, s) ds e(0, T k ) = h(0, T k )B(0, T k ). ) [h(0, s) + f(0, s)] ds ) Alternatively, the above relations can be obtained by integrating f(0, T ) = ln B(0, T ) with B(0, 0) = 1 T f(0, T ) = h(0, T ) + f(0, T ) = ln B(0, T ) with B(0, 0) = 1. T 70

71 Defaultable fixed coupon bond c(0) = K n=1 c n B(0, T n ) (coupon) c n = cδ n 1 + B(0, T K ) (principal) + π K k=1 e(0, T k 1, T k ) (recovery) The recovery payment can be written as π K k=1 e(0, T k 1, T k ) = K k=1 πδ k 1 H(0, T k 1, T k )B(0, T k ). The recovery payments can be considered as an additional coupon payment stream of πδ k 1 H(0, T k 1, T k ). 71

72 Defaultable floater Recall that L(T n 1, T n ) is the reference LIBOR rate applied over [T n 1, T n ] at T n 1 so that 1 + L(T n 1, T n )δ n 1 is the growth factor over [T n 1, T n ]. Application of no-arbitrage argument gives B(T n 1, T n ) = L(T n 1, T n )δ n 1. The coupon payment at T n equals LIBOR plus a spread δ n 1 [ L(Tn 1, T n ) + s par] = Consider the payment of [ 1 B(T n 1, T n ) 1 ] + s par δ n 1. 1 B(T n 1, T n ) at T n, its value at T n 1 is B(T n 1, T n ) B(T n 1, T n ) = P (T n 1, T n ). Why? We use the defaultable discount factor B(T n 1, T n ) since the coupon payment may be defaultable over [T n 1, T n ]. 72

73 Seen at t = 0, the value becomes B(0, T n 1 )P (0, T n 1, T n ) = B(0, T n 1 )P (0, T n 1 )P (0, T n 1, T n ) = B(0, T n 1 )P (0, T n ). Combining with the fixed part of the coupon payment and observing the relation [B(0, T n 1 ) B(0, T n )]P (0, T n ) = [ ] B(0, Tn 1 ) B(0, T n ) 1 B(0, T n ) = δ n 1 F (0, T n 1, T n )B(0, T n ), the model price of the defaultable floating rate bond is c(0) = K n=1 δ n 1 F (0, T n 1, T n )B(0, T n ) + s par + B(0, T K ) + π K k=1 e(0, T k 1, T k ). K n=1 δ n 1 B(0, T n ) 73

74 1.4 Pricing of credit derivatives Credit default swap revisited Fixed leg Payment of δ n 1 s at T n if no default until T n. The value of the fixed leg is s N n=1 δ n 1 B(0, T n ). Floating leg Payment of 1 π at T n if default in (T n 1, T n ] occurs. The value of the floating leg is (1 π) = (1 π) N n=1 N n=1 e(0, T n 1, T n ) δ n 1 H(0, T n 1, T n )B(0, T n ). 74

75 The market CDS spread is chosen such that the fixed leg and floating leg of the CDS have the same value. Hence s = (1 π) N n=1 δ n 1 H(0, T n 1, T n )B(0, T n ) Nn=1 δ n 1 B(0, T n ). Define the weights w n = δ n 1B(0, T n ) N k=1 δ k 1 B(0, T k ), n = 1, 2,, N, and then the fair swap premium rate is given by N n=1 w n = 1, s = (1 π) N n=1 w n H(0, T n 1, T n ). 75

76 1. s depends only on the defaultable and default free discount rates, which are given by the market bond prices. CDS is an example of a cash product. 2. It is similar to the calculation of fixed rate in the interest rate swap s = N n=1 where w n = δ n 1B(0, T n ) N k=1 δ k 1 B(0, T k ) w n F (0, T n 1, T n ), n = 1, 2,, N. 76

77 Marked-to-market value original CDS spread = s ; new CDS spread = s Let Π = CDS old CDS new, and observe that CDS new = 0, then marked-to-market value = CDS old = Π = (s s ) N n=1 B(0, T n )δ n 1. Why? If an offsetting trade is entered at the current CDS rate s, only the fee difference (s s ) will be received over the life of the CDS. Should a default occurs, the protection payments will cancel out, and the fee difference payment will be cancelled, too. The fee difference stream is defaultable and must be discounted with B(0, T n ). CDS s are useful instruments to gain exposure against spread movements, not just against default arrival risk. 77

78 Hedge based pricing approximate hedge and replication strategies Provide hedge strategies that cover much of the risks involved in credit derivatives independent of any specific pricing model. Basic instruments 1. Default free bond C(t) = time-t price of default-free bond with fixed-coupon C B(t, T ) = time-t price of default-free zero-coupon bond 2. Defaultable bond C(t) = time-t price of defaultable bond with fixed-coupon c C (t) = time-t price of defaultable bond with floating coupon LIBOR + s par 78

79 3. Interest rate swap S(t) = swap rate at time t of a standard fixed-for-floating = B(t, t n) B(t, t N ), t t n A(t; t n, t N ) where A(t; t n, t N ) = N i=n+1 paying δ i on each date t i. δ i B(t, t i ) = value of the payment stream Proof of the swap rate formula The floating rate coupon payments can be generated by putting $1 at t n and taking away the floating interests immediately. At t N, $1 remains. The sum of the present value of the floating interests = B(t, t n ) B(t, t N ). Intuition behind cash-and-carry arbitrage pricing of CDSs A combined position of a CDS with a defaultable bond C is very well hedged against default risk. 79

80 Asset swap packages An asset swap package consists of a defaultable coupon bond C with coupon c and an interest rate swap. The bond s coupon is swapped into LIBOR plus the asset swap rate s A. Asset swap package is sold at par. Remark Asset swap transactions are driven by the desire to strip out unwanted structured features from the underlying asset. Payoff streams to the buyer of the asset swap package time defaultable bond swap net t = 0 C(0) 1 + C(0) 1 t = t i c c + L i 1 + s A L i 1 + s A + (c c) t = t N (1 + c) c + L N 1 + s A 1 + L N 1 + s A + (c c) default recovery unaffected recovery * denotes payment contingent on survival. 80

81 s(0) = fixed-for-floating swap rate (market quote) A(0) = value of an annuity paying at the $1 (calculated based on observable default free bond prices) The value of asset swap package is set at par at t = 0, so that C(0) + A(0)s(0) + A(0)s A (0) A(0)c }{{} swap arrangement = 1. The present value of the floating coupons is given by A(0)s(0). The swap continues even after default so that A(0) appears in all terms associated with the swap arrangement. 81

82 Solving for s A (0) Rearranging the terms, s A (0) = 1 [1 C(0)] + c s(0). A(0) C(0) + A(0)s A (0) = [1 A(0)s(0)] + A(0)c C(0) }{{} default-free bond where the right-hand side gives the value of a default-free bond with coupon c. Note that 1 A(0)s(0) is the present value of receiving $1 at maturity t N. We obtain s A (0) = 1 [C(0) C(0)]. A(0) 82

83 Credit spread options The terminal payoff is given by where r = riskless interest rate s = credit spread K = strike spread P sp (r, s, T ) = max(s K, 0) Discrete-time Heath-Jarrow-Morton (HJM) method Follows the HJM term structure approach that models the forward rate process and forward spread process for riskless and risky bonds. The model takes the observed term structures of riskfree forward rates and credit spreads as input information. Find the risk neutral drifts of the stochastic processes such that all discounted security prices are martingales. 83

84 Example Price a one-year put spread option on a two-year risky zero-coupon bond struck at the strike spread K = Let the current observed term structure of riskless interest rates as obtained from the spot rate curve for Treasury bonds be ( ) 0.07 r = The riskless forward rate between year one and year two is f 12 = The market one-year and two-year spot spreads are ( ) s =

85 The two-year risky rate is = The current price of a risky two-year zero coupon bond with face value $100 is B(0) = $100/(1.092) 2 = $ The discrete stochastic process for the spread under the true measure is assumed to take the form of a square-root process where the volatility depends on s(0) s( t) = s(0) + k[θ s(0)] t ± σ s(0) t where k = 0.3, θ = 0.02 and σ = 0.04, t = 1, s(0) =

86 We need to add an adjustment term γ in the drift term in order to risk-adjust the stochastic forward spread process s(t) = s(0) + k[θ s(0)] t + γ ± σ s(0) t. The adjustment term γ is determined by requiring the discounted bond prices to be martingales. Let B(1) denote the price at t = 1 of the risky bond maturing at t = 2. The forward defaultable discount factor over year one 1 and year two is, where s(1) is the forward spread 1 + f 12 + s(1) over the period. s(1) = { γ γ so that B(1) = f 12 +γ f 12 +γ+0.009, with equal probabilities for assuming the high and low values. 86

87 We determine γ such that the bond price is a martingale. B(0) = = ( γ γ ). The first term is the risky defaultable discount factor and the last term is{ the expected value of B(1). We obtain γ = so that s(1) = The current value of put spread option is [( ) + ( )]L = L, 2 where L is the notional value of the put spread option. Note that the default free discount factor 1/1.07 is used in the option value calculation. 87