Credit Risk. Chapter 22. Credit Ratings. In the S&P rating system, AAA is the best rating. After that comes AA, A, BBB, BB, B, and CCC The corresponding Moody’s ratings are Aaa, Aa, A, Baa, Ba, B, and Caa Bonds with ratings of BBB (or Baa) and above are considered to be “investment grade”.
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Historical data provided by rating agencies are also used to estimate the probability of default
Define V(t) as cumulative probability of the company surviving to time t.
The recovery rate for a bond is usually defined as the price of the bond immediately after default as a percent of its face value
Average default intensity(h) over life of bond is approximately
where s is the spread of the bond’s yield over the risk-free rate and R is the recovery rate
The calculation of default intensities using historical data are based on equation(22.1) and table 22.1. From equation (22.1),we have
The calculations using bond prices are based on equation(22.2)and bond yields published by Merrill Lynch. The results shown are averages between December 1996 and Oct. 2007.The recovery rate is assumed to be 40% and, the risk-free interest rate is assumed to be the 7-year swap rate minus 10 basis points. Foe example, for A-rated bonds the average Merrill Lynch yield was 5.993%. The average swap rate was 5.398%, so that the average risk-free rate was 5.298%. This gives the average 7-year default probability as
max(VT -D, 0)
where VT is the value of the firm and Dis the debt repayment required
An option pricing model enables the value of the firm’s equity today, E0, to be related to the value of its assets today, V0, and the volatility of its assets, sV
This equation together with the option pricing relationship enables V0 and sV to be determined from E0 and sE
Denote Qi(T) as the probability that company i will default between time zero and time T, and Pij(T) as the probability that both i and j will default. The default correlation measure is
Qi(ti) = N(xi ) or xi = N-1[Qi (ti)]
where N is the cumulative normal distribution function.
(default probability distributions from market data)
(estimated by asset correlation)
Choice of copula
(Gaussian / normal copula)
Fully defined multivariate distribution of the probability of defaulting within time T
The measures can be calculated from each other
Suppose that we wish to simulate the defaults for n companies . For each company the cumulative probabilities of default during the next 1, 2, 3, 4, and 5 years are 1%, 3%, 6%, 10%, and 15%, respectively
N-1(0.01) = -2.33, N-1(0.03) = -1.88,
N-1(0.06) = -1.55, N-1(0.10) = -1.28,
N-1(0.15) = -1.04
-2.33, the company defaults in the first year