Credit Risk Yiling Lai 2008/10/3

Outline Introduction Credit Ratings Historical Default Probabilities Recovery Rates Estimating Default Probability from Bond Prices Comparison of Default Probability Estimates

Introduction Credit risk raises from the possibility that borrowers and counterparties in derivatives transactions may default.

Credit Rating Credit Rating assesses the creditworthiness of corporate bonds. Default Risk S&P Moody’s 中華信評 Low AAA Aaa twAAA AA+ , AA, AA- Aa1, Aa2, Aa3 twAA+, twAA, twAA- A+, A , A- A1, A2, A3 twA+, twA, twA- BBB+, BBB, BBB- Baa1, Baa2, Baa3 twBBB+, twBBB, twBBB- BB+, BB, BB- Ba1, Ba2, Ba3 twBB+, twBB, twBB- B+, B, B- B1, B2, B3 twB+, twB, twB- High CCC… Caa… twCCC… Investment grade bond Non-investment grade bond (high yield bond, speculative grade bond or junk bond)

Historical Default Probabilities Average cumulative default rates(%), 1970-2006. Source: Moody’s From this table, we can calculate unconditional default probability and conditional default probability. Term 1 2 3 4 5 7 10 15 20 Aaa 0.000 0.026 0.099 0.251 0.521 0.992 1.191 Aa 0.008 0.019 0.042 0.106 0.177 0.343 0.522 1.111 1.929 A 0.021 0.095 0.220 0.344 0.472 0.759 1.287 2.364 4.238 Baa 0.181 0.506 0.930 1.434 1.938 3.959 4.637 8.244 11.362 Ba 1.205 3.219 5.568 7.985 10.215 14.005 19.118 23.380 35.093 B 5.236 11.296 17.043 22.054 26.794 34.771 43.343 52.175 54.421 Caa-C 19.476 30.494 39.717 46.904 52.622 59.938 69.178 70.870

Historical Default Probabilities Unconditional default probability The probability of a bond defaulting during a particular year as seen at time 0. Term 1 2 3 4 5 7 Aaa 0.026 0.073 … Aa 0.008 0.011 0.023 0.064 0.071 A 0.021 0.074 0.125 0.124 0.128 Baa 0.181 0.325 0.424 0.504 Ba 1.205 2.014 2.349 2.417 2.23 B 5.236 6.06 5.747 5.011 4.74 Caa-C 19.476 11.018 9.223 7.187 5.718 Increasing 0.352 =0.506-0.181 Decreasing

Historical Default Probabilities Conditional default probability (default intensity or hazard rate) The probability that the bond will default during a particular year conditional on no earlier default. Average cumulative default rates Term 1 2 3 4 Caa-C 19.476 30.494 39.717 … Unconditional default probability Term 1 2 3 4 Caa-C 19.476 11.018 9.223 … The probability that the bond will survive until the end of year 2 is 100-30.949=69.506% Conditional default probability is 9.223/69.506 = 13.27% (for 1-year time period)

Default Intensities

Default Intensities Q(t): the probability of default by time t.

Average recovery rate (%) Recovery Rates The recovery rate for a bond is normally defined as the bond’s market value immediately after a default, as a percent of its face value. Recovery rates on corporate bonds as a percentage of face value. 1982-2003 (Source: Moody’s) Class Average recovery rate (%) Senior secured 54.44 Senior unsecured 38.39 Senior Subordinated 32.85 Subordinated 31.61 Junior subordinated 24.47

Recovery Rates Recovery rates are significantly negatively correlated with default rates. Recovery rate = 59.1 - 8.356 x Default rate The recovery rate: the average recovery rate on senior unsecured bonds in a year measured as % The default rate: the corporate default rate in the year measured as % A bad year for the default rate is usually double bad because it is accompanied by a low recovery rate. Default rate = 0.1% => Recovery rate = 58.3% Default rate = 3% => Recovery rate = 34.0%

Estimating Default Probability from Bond Prices Assumption: The only reason a corporate bond sells for less than a similar risk-free bond is the possibility default. An approximate calculation: 這個假設並不完美，因為公司債的價格會受到“流動性”的影響。liquidity S = 200 base points and R = 40% => h=0.02/(1-0.4)=3.33%

A More Exact Calculation The corporate bond: Period: 5 years Coupon: 6% per annum (paid semiannually) Yield: 7% per annum (with continuous compounding) Price: 95.34 A similar risk-free bond Yield: 5% per annum (with continuous compounding) Price: 104.09 The expected loss from default over the 5-year life of the bond is 104.9-95.34=$8.75

A More Exact Calculation Assumption: defaults can happen at times 0.5, 1.5, 2.5, 3.5, and 4.5 years (immediately before coupon payment dates). Notional Principal=$100 Risk-free rates: 5% (with continuous compounding) Recovery rate = 40% => 100*40%=$40

A More Exact Calculation Consider the 3.5 years: The expected value of the risk-free bond at time 3.5 years: The loss given default is 104.34-40=64.34 The PV of this loss is 1 2 3 4 5 years Cash flow 103

A More Exact Calculation Suppose that the probability of default per year (assumed to be the same each year) is Q. Calculating of loss from default on a bond in terms of the default probabilities per year, Q. Time (years) Default Probabilities Recovery Amount ($) Risk-free Value($) Loss given Default($) Discount factor PV of expected loss ($) 0.5 Q 40 106.73 66.73 0.9753 65.08Q 1.5 105.97 65.97 0.9277 61.20Q 2.5 105.17 65.17 0.8825 57.52Q 3.5 104.34 64.34 0.8395 54.01Q 4.5 103.46 63.46 0.7985 50.67Q Total 288.48Q 288.48Q=8.75 => Q = 3.03%

A More Exact Calculation We can extend this calculations by changing some assumptions. Example: Defaults can take place more frequently A constant default intensity A particular pattern for the variation of default probabilities with time With several bonds we can estimate several parameters describing the term structure of default probabilities.

The Risk-Free Rate The benchmark risk-free rate that is usually used in quoting corporate bond yields is the yield on similar Treasury bonds. In fact, those derivative traders usually use LIBOR rates as short-term risk-free rates. They regard LIBOR as their opportunity cost of capital. Treasury rates are too low to be used as risk-free rates.

Asset Swaps Asset swaps = asset + interest rate swap or Asset swaps = asset + currency swap An asset swap transforms the character of an end user’s assets. Repacking an issue paying fixed rates into floating rates (or vice versa) Converting cash flow stated in one currency to another. It does not eliminate the asset from an investor’s portfolio.

Fixed Rates to Floating Rates The investor transforms the yield on its fixed rate asset into a floating rate asset.

Currency Transformation The investor converts with the swap counterparty the FFr500mm principal cost of the asset and $100mm, the investor’s base currency.

Asset Swaps Spread In practice, traders often use asset swap spreads as a way of extracting default probabilities from bond prices. Asset swap spreads provide a direct estimate of the spread of bond yields over the LIBOR/swap curve. Example: An asset swap spread for a particular bond: 150 bps The LIBOR/swap zero curve is flat at 5% The amount by which the value of the risk-free bond exceeds the value of the corporate is the present value of 150bps per year for 5years. Assuming semiannual payments

Asset Swaps Spread The sum of PV is $6.55 per $100 of principal. 288.48Q = 6.55 => Q=2.27% 1 2 3 4 5 years Cash flow 0.75 PV

Comparison of Default Probability Estimates (part I) From historical data: Based on Consider t=7: Consider an A-rated company in table 20.1: Q(7) =0.00759

Comparison of Default Probability Estimates (part I) From bond prices: Based on Recovery rate (R): 40% Risk-free rate: the 7-year swap rate minus 10 bps. For A-rated bond, the average Merrill Lynch yield was 5.993% The average swap rate was 5.398% => risk-free rate = 5.398-0.1=5.298% s = 0.05993-0.05298 = 0.00695

Comparison of Default Probability Estimates (part I) Seven-year average default intensities (% per annum) Rating Historical default intensity Default intensity from bonds Ratio Difference Aaa 0.04 0.60 16.7 0.56 Aa 0.05 0.74 14.6 0.68 A 0.11 1.16 10.5 1.04 Baa 0.43 2.13 5.0 1.71 Ba 2.16 4.67 2.2 2.54 B 6.10 7.97 1.3 1.98 Caa and lower 13.07 18.16 1.4 5.50 decline increase

Comparison of Default Probability Estimates (part II) Another way of looking at these results: excess return over the risk-free rate Expected excess return on bonds (bps) = (A)-(B)-(C) Rating Bond yield spread over Treasuries (A) Spread of risk-free rate over Treasuries (B) Spread for historical defaults (C) Excess return Aaa 78 42 2 34 Aa 87 3 A 112 7 63 Baa 170 26 102 Ba 323 129 151 B 521 366 Caa 1132 784 305 increase Historical default intensity x (1-Recovery rate)

Real-World vs. Risk-Neutral Probabilities The default probabilities implied from bond yields are risk-neutral probabilities of default. The risk-neutral valuation principle states that the price of a derivative is given by the expectation of the discounted terminal payoff under the risk neutral measure. This means that the default probability Q must be a risk-neutral probability. By contrast, the default probabilities implied from historical data are real-world probabilities of default. risk neutral valuation principle states that the price of a derivative is given by the expectation of the discounted terminal payoff under the risk neutral measure,

Why do we see such big differences between real-world and risk-neutral default probabilities? Corporate bonds are relatively illiquid. The subjective default probabilities of bond traders may be much higher than those given in Tables 20.1. Bonds do not default independently of each other because of systematic risk. Unsystematic risk: It is much more difficult to diversify risks in a bond portfolio than in an equity portfolio.

Which one is better? The answer depends on the purpose of the analysis. Risk-neutral default probabilities: To value credit derivatives To estimate the impact of default risk Real-world default probabilities: To calculate potential future losses from default when carrying out scenario analyses