1. Measuring Actuarial Default Risk
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Measuring Actuarial Default Risk
Sutee Mokkhavesa PhD
BBA FN415.13
Measuring Actuarial Default Risk
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2. Two methods of measuring Default Risk
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Overview
Default risk is the primary driver of credit risk and it is represented by the probability of default (PD)
When default occurs the actual amount of loss is determined by the exposure at default (EAD) (same as CE) and the loss given default (LGD).
Two methods of measuring Default Risk
Actuarial Methods
Market price methods
Actuarial methods measures the actual or natural probabilities of default.
Actuarial measures of default probabilities are provided by credit rating agencies.
Default risk can be measured using two approaches:( 1) actuarial methods, which provide "objective" measures of default rates, usually based on historical default data; and (2) market-price methods, which infer from traded prices of debt, equity, or credit derivatives" risk-neutral" measures of default risk.
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3. Credit rating for issuer vs credit rating for bonds
Credit Event
A credit event is a discrete state. Either it happens or does not. The issue is the definition of the event, which must be framed in legal terms.
The state of default is defined by Standard & Poor's (S&P), a credit rating agency:
The first occurrence of a payment default on any financial obligation, rated or unrated, other than a financial obligation subject o a bona fide commercial dispute; an exception occurs when an interest payment missed on the due date is made within the grace period.
Credit rating for issuer vs credit rating for bonds
For Credit Derivatives, the definition of default has to be defined more precisely and is formalised by the International Swaps and Derivatives Association (ISDA)
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5. Credit Ratings
A credit rating is an “evaluation of creditworthiness” issued by a rating agency.
The major U.S. bond rating agencies are Moody's Investors Services, Standard& Poor's (S&P), and Fitch, Inc.
Moody's define a credit rating as “opinion of the future ability, legal obligation, and willingness of a bond issuer or other obligor to make full and timely payments on principal and interest due to investors.”
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7. Classification by Credit Ratings
Ratings are broadly divided into the following:
Investment Grade, that is, at and above BBB for S&P and Baa for Moody’s
Speculative Grade, or below investment grade, for the rest.
These ratings represent objective (or actuarial) probabilities of default.
So it is possible to convert these ratings into probability of default.
The agencies use a number of criteria to decide on the credit rating, including various accounting ratios.
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8. Example: S&P’s ratings for US Industrial Corporations
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Example: S&P’s ratings for US Industrial Corporations
Note that the word capital here is used to represent total Asset
The first column (under" Leverage") shows that the ratio of total debt to total capital (debt plus book equity or assets) varies systematically across ratings. Highly rated companies have low leverage ratios, 12% for AAA firms. In contrast, BB-rated (just below investment grade) companies
Have a leverage ratio of 54%. This implies a capital-to-equity leverage ratio of 2.2 to 1
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9. Cash Flow Coverage
The right-hand panel (under "Cash Flow Coverage") also shows systematic variations in a measure of free cash flow divided by interest payments.
T his represents the number of times the cash flow can cover interest payments.
Focusing1 on earnings before interest and taxes (EBIT), AAA-rated companies have a safe cushion of 23.8, whereas BB-rated companies have coverage of only 2.5.
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10. Multiple Discriminant Analysis
Z-score model developed by Altman (1968)
MDA constructs a linear combination of accounting data that provides the best fit with the observed states of default and nondefault for the sample firms.
The variables used in the Z-score are (1) working capital over total assets, (2) retained earnings over total assets, (3) EBIT over total assets, (4) market value of equity over total liabilities, and (5) net sales over total assets. Lower scores indicate a higher likelihood of default.
Each variable enters with a positive sign, meaning that an increase in each of these variables decreases the probability of bankruptcy.
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12. Historical Default Rates
These describe the proportion of firms that default, which is a statistical estimate of the true default probability:
For example, borrowers with an initial Moody's rating of Baa experienced an average default rate of 0.34% over the next year.
Similar rates are obtained for S&P's BBB-rated credits, who experienced an average 0.36% default rate over the next year.
Firms at or below Caa have a default rate of 14.74%.
Higher ratings are associated with lower default rates.
As a result, this information could be used to derive estimates of default probability for an initial rating class.
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15. Historical Default Rates
In addition, the tables show that the default rate increases sharply with the horizon, for a given initial credit rating.
The default rate for Baa firms increases from 0.34% over one year to 7.99% over the following ten years.
Because these are cumulative default rates, the number must increase with the horizon.
For investment-grade credits, however, the increase is more than proportional with the horizon.
The ratio is 7.99/ 0.34 = 23 for Baa-rated credits, which is more than 10.
In contrast, the ratio for B-rated credits is 35.10/4.79 = 8. For speculative grade credits, the increase is less than proportional with the horizon.
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16. Historical Default Rate
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Historical Default Rate
One problem with such historical information, however, is the relative paucity of data. There are few defaults for highly rated firms over one year. In addition, the sample size decreases as the horizon lengthens. For instance, S&P reports default rates up to 15 years using data from 1981 to 2002.
The one-year default rates represent 23 years of data-that is, 1981, 1982, and so on to 2002.
There are, however, only eight years of data for 15-year default period
The sample size is much shorter. The data are also overlapping and therefore not independent.
So, omitting or adding a few borrowers can drastically alter the reported default rate.
This can lead to inconsistencies in the tables. For instance, the default rate for CCC obligors is the same, at percent, from year 11 to 13. This implies that there is no further risk of default after 11 years, which is an unrealistic implication.
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Calculating the error
Assess the accuracy of these default rates by computing their standard error.
Consider, for instance, the default rate over the first year for AA rated credits, which averaged out to X = 0.01.% in this S&P sample.
This was taken out of a total of about N = 8000 observations, which we assume to be independent
The variance of the average is, from the distribution of a binomial process
which gives a standard error of about 0.011%. This is on the same order as the average of 0.01%, indicating that there is substantial imprecision in this average default rate.
So we could not really distinguish an AA credit from an AAA credit.
The problem is made worse with lower sample sizes, which is the case in non- U.S. markets or when the true p is changing over time. For instance, if we observe a 5% default rate among 100 observations, the standard error becomes 2.2%, which is very large. Therefore, a major issue with credit risk is that estimation of default rates for low-probability events can be very imprecise.
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18. Cumulative and Marginal Default Rates
The default rates reported above are cumulative default rates for an initial credit rating - that is, they measure the total frequency of default at anytime between the starting date and year T.
It is also informative to measure the marginal default rate, which is the frequency of default during year T.
m is the number of issuers rated R at the end of year t that default in year T = t + N.
n is the number of issuers rated R at the end of year t that have not defaulted by the beginning of year t + N.
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19. Survival Rate
This is the proportion of issuers initially rated R that will not have defaulted by T:
Marginal Default Rate starting in year T
This is the proportion of issuers initially rated R that defaulted in year T, relative to the initial number in year t. For this to happen, the issuer will have survived until year t+ N- L, and then default the next year
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20. Example
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21. Sequential Default Process
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The probability of defaulting up to two years is the probability of defaulting in year one plus the probability of not defaulting in year one and defaulting in year two.
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22. Cumulative Default Rates
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