INVESTMENTS Modeling Default Risk The Corporate Bond Market 25 20 15 10 5 0 -5 Mortgage Rates (Home Loan Mortgage Corporation) FED Fund Apr-01 Apr-99 Apr-97 Apr-95 Apr-93 Apr-91 Apr-89 Apr-87 Apr-85 Apr-83 Apr-81 Apr-79 Apr-77 Apr-75 Apr-73 Apr-71 -10 Spread Figure 1: Mortgage and FED rates, Source : www.federalreserve.gov/releses/hr 20 18 16 14 12 10 8 6 4 2 AAA BAA Spread Figure 2: Corporate rating spreads, Source : www.federalreserve.gov/releses/hr Jan-99 Jan-94 Jan-89 Jan-84 Jan-79 Jan-74 Jan-69 Jan-64 Jan-59 Jan-54 Jan-49 Jan-44 Jan-39 Jan-34 Jan-29 Jan-24 Jan-19 - 16% 14% 12% 10% 8% 6% 4% 1 Year Maturity 30 Years Maturity Figure 3: Corporate rating spreads, Source : Moody’s 100% 90% 80% 70% 60% 50% 40% 30% 20% BBB AAA AA- Figure 4: Corporate rating migration for industry-sector, Source : Standard Poor’s. D CCC B B B+ BB BB BB+ BBB BBB BBB+ A A A+ AA- AA AAA 0% AA+ 10% Caa B3 B2 B1 Ba3 Ba2 Ba1 Baa3 Baa2 Baa1 A3 A2 A1 Aa3 Aa2 Aa1 0% Aaa 2% Bond Valuation with Default Risk 100 Cashflow Conditioning on Survival c/2 0.5 1 1.5 2 2.5 3 3.5 4 τi Non-Default Default RandomDefaultTime τ% Figure 5: Chart cash ﬂow Conditioning on survival. Assuming no default risk, P0 = 8 � er·ti + 100 · er·4 i=1 How does the default risk aﬀect the bond price? (1) Modelling Default Risk Modelling default risk is central to the pricing and hedging of credit sen sitive instruments. Two approaches to modelling default risk: • Structural approach, ”ﬁrst-passage”: default happens when the to tal asset value of the ﬁrm falls below a threshold value (for example, the ﬁrm’s book liability) for the ﬁrst time. • Reduced-form, ”intensity-based”: the random default time τ� is gov erned by an intensity process λ. For pricing purpose, the reduced-form approach is adequate, and will be the focus of this class. Modelling Random Default Times The probability of survival up to time t: (2) P rob(τ� ≥ t) The probability of default? before time t: P rob (τ� < 0) = 1 − P rob (τ� ≥ t) (3) We assume that T� is exponentially distributed with constant default intensity λ: 1 Survival Probability: ( ) Prob τ% ≥ t = e- λt 0 0 100 t (year) Figure 6: Survival Probability. Default Probability and Credit Quality One-Year default probability = 1 − eλ Default intensity λ =? D CCC B B B+ BB BB BB+ BBB BBB BBB+ AA A+ AAAA AA+ AAA 0% 10% AAA BB 20% AA+ BB Figure 7: Survival Probability. AA B+ 30% AAB 40% 50% A+ B A CCC 60% A D 70% BBB+ 80% BBB 90% BBB 100% BB+ AAA AA+ AA AAA+ A ABBB+ BBB BBBBB+ BB BBB+ B BCCC D AAA 91.95% 2.31% 0.62% 0.00% 0.00% 0.08% 0.14% 0.00% 0.07% 0.05% 0.17% 0.00% 0.00% 0.00% 0.00% 0.00% 0.19% 0.00% AA+ 4.11% 84.71% 1.36% 0.15% 0.03% 0.06% 0.04% 0.00% 0.03% 0.00% 0.00% 0.00% 0.00% 0.03% 0.00% 0.00% 0.00% 0.00% AA 2.86% 8.75% 85.42% 3.44% 0.83% 0.49% 0.11% 0.08% 0.07% 0.11% 0.00% 0.12% 0.00% 0.00% 0.07% 0.00% 0.00% 0.00% AA0.48% 2.88% 7.24% 83.67% 4.47% 0.66% 0.35% 0.13% 0.17% 0.21% 0.08% 0.06% 0.05% 0.10% 0.00% 0.00% 0.00% 0.00% A+ 0.16% 0.19% 2.60% 8.61% 82.27% 5.25% 1.13% 0.59% 0.45% 0.11% 0.08% 0.06% 0.09% 0.00% 0.00% 0.18% 0.19% 0.00% A 0.20% 0.48% 1.49% 3.02% 8.08% 82.50% 8.58% 2.26% 0.93% 0.69% 0.51% 0.37% 0.05% 0.03% 0.14% 0.00% 0.00% 0.00% A0.12% 0.10% 0.25% 0.50% 2.75% 5.44% 77.39% 8.32% 2.24% 0.59% 0.34% 0.18% 0.28% 0.23% 0.21% 0.00% 0.19% 0.00% BBB+ 0.04% 0.00% 0.50% 0.23% 0.46% 3.18% 7.21% 75.24% 7.83% 2.67% 0.67% 0.31% 0.33% 0.10% 0.00% 0.36% 0.19% 0.00% Figure 8: Survival Probability, Migration table, Source: RiskMetrics T M . 16% 14% 12% 10% 8% 6% 4% 2% 0% Aaa Aa1 Aa2 Aa3 A1 A2 A3 Baa1 Baa2 Baa3 Ba1 Ba2 Ba3 B1 B2 Figure 9: One-Year Default Rates by Modiﬁed Ratings, 1983-1995, Source: Moodys (1996). B3 BBB 0.04% 0.38% 0.22% 0.15% 0.40% 1.11% 3.00% 8.36% 77.76% 9.46% 4.21% 1.59% 0.52% 0.13% 0.14% 0.00% 0.56% 0.00% Pricing A Defaultable Bond For simplicity, let’s ﬁrst assume that the riskfree interest rate r is a constant. Consider a τ -year zero-coupon bond issued by a ﬁrm with default intensity λ: P0 = $100 · e−r·τ · P rob(τ� ≥ τ ) (4) P0 = $100 · e−r·τ · e−λ·τ (5) P0 = $100 · e−(r+λ)·τ (6) where we assume that conditioning on a default, the recovery value of the bond is 0 (we have also assumed risk-neutral pricing). The yield on the defaultable bond is r + λ, resulting in a credit spread of λ. Time Variation of Default Probability 10 8 6 4 2 0 -2 GDP Growth / Chain QQQ% Linear (GDP Growth / Chain QQQ%) Figure 10: Chart Annual GDP Growth Rate, source: Bureau of Economic Analysis Stochastic Dec-01 Dec-00 Dec-99 Dec-98 Dec-97 Dec-96 Dec-95 Dec-94 Dec-93 Dec-92 Dec-91 Dec-90 Dec-89 Dec-88 Dec-87 Dec-86 -4 Default Intensity In general, the credit quality of a ﬁrm changes over time. A more realistic model is to treat the arrival intensity as a random process. Suppose that intensities are updated with new information at the begin ning of each year, and are constant during the year. Then the probability of survival for t years is For example, � � E e−λ0 +λ1 +···+λt−1 (7) � � ¯ − λt + εt+1 λt+1 − λt = k λ (8) Can you calculate the probability of survival for τ years? What is the price of a τ -year zero-coupon bond? What if the riskfree interest rate is also stochastic? Example: A portfolio consists of two long assets $100 each. The prob ability of default over the next year is 10% for the ﬁrst asset, 20% for the second asset, and the joint probability of default is 3%. What is the expected loss on this portfolio due to credit risk over the next year assuming 40% recovery rate for both assets. Probabilities: 0.1 · (1 − 0.2) − def ault probability of A (9) 0.2 · (1 − 0.1) − def ault probability of B (10) 0.03 − joint def ault probability (11) Expected losses: 0.1 · (1 − 0.2) · 100 · (1 − 0.4) = 4.8 (12) 0.2 · (1 − 0.1) · 100 · (1 − 0.4) = 10.8 (13) 0.03 · 200 · (1 − 0.4) = 3.6 4.8 + 10.8 + 3.6 = $19.2 mio. (14) (15) Example: Assume a 1-year US Treasury yield is 5.5% and a Eurodollar deposit rate is 6%. What is the probability of the Eurodollar deposit to default assuming zero recovery rate)? 1 1−π = 1.06 1.055 π = 0.5% (16) (17) Example: Assume a 1-year US Treasury yield is 5.5% and a and a default probability of a one year CP is 1%. What should be the yield on the CP assuming 50% recovery rate? 1 1−π 0.5π = + 1+x 1.055 1.055 = 6% (18) (19)