Credit Risk
Financial firms want to measure credit quality of a loan or
bond to decide where to invest. Credit quality analysis is
becoming more important for at least four reasons:
1. Low-risk firms now borrow directly in financial markets
using commercial paper, leaving poorer quality firms as the
prime customers of financial firms such as banks.
2. The competition for consumer loans (credit cards etc.) is
increasing, reducing the rates financial firms’ can charge.
3. The default rates on consumer loans has been increasing
during the last 10 years.
4. The amount of high-risk (junk) bonds and their rate of
default has been increasing during the last few years.
Calculating a Loan Return
Before we consider how to estimate credit quality, consider
the how to measure the promised return on a loan.
Factors that influence a loan’s return
1. Loan interest rate - k
2. Base lending rate - L - Prime Rate or Fed Funds rate
3. Risk premium - m
4. Compensating balances - b
5. Reserve requirements - R
6. Loan fees - f
All of these are measured as an amount per dollar or loan.
Example
For each dollar loaned we are promised 1 + k dollars:
1 + k = 1 + [f + L + m]/[1 - b(1 - R)]
Problem: Suppose that the Prime Rate is 12%. For a
particular loan, Husky requires a risk premium of 2%, a loan
origination fee of 1/8%, compensating balances of 10%, and
its reserve requirement is 10%. What is the loan’s promised
return?
1 + k = 1 + [.00125 + .12 + .02]/[1 - .1(1 - .1)] = 1.1552
=> k = 15.52%
Expected vs. Promised Return
Of course, Husky only earns 15.52 percent if all the loan
payments are made, i.e., the borrower does not default.
Husky is more interested in its expected return (E[r]) for the
loan. Assume that the probability of full repayment of a loan
or bond investment is p. If there is some collateral backing
the loan, then even in default, Husky will get a proportion 
of principal and interest back with probability (1 - p). Thus
E[r] = p(1 + k) + (1 - p)(1+k)
If a default leads to a total loss because there is no collateral
or corporate assets backing the loan or bond, then
E[r] = p(1 + k)
Examples of Expected Return
Example 1: Suppose for the previous problem, Husky
expects 95% probability of repayment there are no assets
backing the loan in default, what is its expected return?
E[r] = .95(1 + .1552) = 1.0974
Example 2: Altman and Kishor’s (1998) data show that about
55% of the value of defaulted loans are recovered. If Husky
expects to recover 55%, what is its expected return?
E[r] = .95(1 + .1552) + (1 - .95).55(1 + .1552)= 1.1292
This shows that the greater the collateral or corporate assets
backing a loan or bond, the larger the expected return.
Collateral vs. Default Risk
Assuming risk neutral pricing of loans and bonds, rates on
various loans or bonds should follow:
E[r] = p(1 + k) + (1 - p)(1+k) = (1 + i)
where i is the risk-free rate.
Question: Why should this hold, particularly when p or  (or
both) are equal to 1 or close to 1?
We can rearrange the equation above to get:
[p +  - p](1 + k) = 1 + i
This shows that if we can increase either p or  then we can
offer a lower, more competitive loan rate to our borrowers.
Estimating Repayment
Probability
Default Risk Models are used to estimate the probability
of default - repayment probability (used in previous model)
is just one minus this probability. Models include:
Borrower-specific factors such as:
• Reputation - borrowing and repayment history
• Leverage - debt/equity
• Earnings Volatility - operating leverage, industry
• Collateral - pledged asset as security
• Market-specific factors such as:
• monetary policy, GNP growth,
business cycle, ratings upgrade/downgrades.
Credit Scoring Models
Credit Scoring Models use data on observed borrower
characteristics to calculate default probabilities or default risk
classes. Three general types:
1. Linear Probability Models
• Regression to predict default probability (Zi), e.g.
Zi = (1 - p) = b1(Debt/Equity) + b2(Sales/Assets) + e
A. Get data on many loans. For borrowers that defaulted
Zi = 1, otherwise, Zi = 0. For each borrower, get their
Debt/Equity ratio and Sales/Assets ratio.
B. Run the regression to get coefficients b1 and b2, e.g.,
Zi = .50(Debt/Equity) + .10(Sales/Assets)
C. Use the regression to estimate the default probability of
a prospective borrower. For example, assume
D/E = .3 and S/A = 2 for a new borrower then:
Zi = (1 - p) = .50(.3) + .1(2) = .35
=> p = .65 which can be used to get expected return.
2. Logit Models - are similar to the linear regression
approach except that the predicted values are
statistically forced to be in the interval (0,1). The
regression model can give estimates outside this
interval.
3. Discriminant Models - groups borrowers into high and low
default risk classes contingent on their characteristics.
For example, Altman estimates the following model.
Z = 1.2(WC/TA) + 1.4(RE/TA) + 3.3(EBIT/TA) +
0.6(ME/LTD) + 1.0(Sales/TA)
where Z = borrower’s score, WC = working capital,
TA = total assets, EBIT = earnings before interest and
taxes, ME = market equity, and LTD = long-term
debt book value.
The larger the Z, the lower the default risk. The average Z
values for a group of old defaulted loans and non-defaulted
loans is used to decide whether a new loan is made.
Example
For Altman’s model, the average Z for defaulted loans was
1.61 and for non-defaulted loans was 2.01. The Z value
between these two groups (1.81) is used as the cutoff value to
decide whether to make a loan.
Problem: Suppose that a borrower has the following ratios:
WC/TA=.2, RE/TA=0, EBIT/TA=-.2, ME/LTD=.1
and Sales/TA=2. If the cutoff Z-score is 1.81, should
we make them a loan?
Z = 1.2(.2) + 1.4(0) + 3.3(-.2) + 0.6(.1) + 1.0(2) = 1.64
Because 1.64 < 1.81, we would reject the borrower.
The discriminant model with different variables is used to
judge Sovereign (country-specific) risk - Chapter 16.
Mortality Rate Derivation
• One way to get the default probability of a loan (bond) is to
look at historic default rates for loans (bonds) of similar
credit quality and age.
• This historic data can be obtained from a vendor/consultant
of gathered internally if enough data on loans (bonds) is
available at the financial firm.
• To calculate a mortality (default) rate, one must first group
loans (bonds) by rating or risk category.
• Next, for each risk category and each year, calculate rates:
(1 - p) = Value of loans defaulting in year 1 after issue
Value of all loans outstanding in year 1
Altman and Suggitt Default
Rates for 1991-1996 period.
First entry is a marginal and the second a cumulative rate.
Rating
Age
One Year
Two Years
Three Years
Loan Bond Loan Bond Loan Bond
Aaa, Aa, A
0.00% 0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00
Baa
0.04 0.00 0.00 0.00 0.00 0.00
0.04 0.00 0.04 0.00 0.04 0.00
Ba
0.17 0.00 0.60 0.38 0.60 2.30
0.17 0.00 0.77 0.38 1.36 2.67
B
2.30 0.81 1.86 1.97 2.59 4.99
2.30 0.81 4.11 2.76 6.60 7.61
Caa
15.24 2.65 7.44 3.09 13.03 4.55
15.24 2.65 21.55 5.66 31.77 9.95
• Of course, the results you get will depend upon how good
your data is and large financial firms with good data have a
competitive advantage.
• Note that the loans default rate typically exceeds the bonds
default rate. Some financial firms use default rates calculated
from bonds to set loan rates because there is less data on loan
rates. This may seriously under-estimate default.
• Data on some syndicated loan prices and fees is published
daily in the Wall Street Journal.
Using Calculated Default
Rates to Set Loan Rates
One way to use the data in the previous table is to either set
minimum loan rates or as a comparison to decide loan rates
based upon your own opinions of future default rate.
Example 1: Suppose that a borrower you rate as B quality
asks for a 3-year loan. Based on the table data, assuming a
risk-free rate of 7% and collateral such that 55% of the loan
would be recouped in case of default, what should the
minimum loan rate be? Use the previous equation:
[p +  - p](1 + k) = 1 + i so
k = [1 + i]/[p +  - p] - 1
From the table (1 - p) = .066 so p = .934
k = [1 + .07]/[.934 + .55 - .55(.934)] - 1
=> k =10.3%
Example 2: Now suppose that you know that during the
1991-1996 period covered by the data in the chart, the
economy was poorer than it will be in the future. Then you
can simply assume that the default rate will fall to, say 5%.
This will allow you to set the loan rate at
k = [1 + .07]/[.95 + .55 - .55(.95)] - 1
=> k =9.5%
Of course, if you believe you can get a higher rate from a
borrower then you might charge more.
Using the Yield Curve Market Expectations
Using historical loan defaults and recovery rates has the
drawback that the future may differ significantly from the
past. Also, we may not have enough data for accurate
estimates. Another approach is to use market rates to get the
implied combined effects of default and partial recovery:
[p +  - p](1 + k) = 1 + i so [p +  - p] = (1 + i)/(1 + k)
Example 1: If the one-year risk-free rate is 7% and the Baa
rated one-year bond yields 8.5%, what is the implied
combined default and recovery effect?
(1 + i)/(1 + k) = (1 + .07)/(1 + .085) = 0.986
Example cont.
Suppose that you believe that the recovery proportion is .55,
what is the implied default rate?
[p +  - p] = [p + .55 - .55p] = 0.986
=> p = 0.969 so default rate = 0.031
This is the market’s forward-looking expectation of the
default rate assuming that the recovery rate is .55.
Example 2: If the two-year risk-free rate is 7% and the
Baa rated two-year bond yields 8.5%, what is the
implied two-year combined default and recovery
effect?
(1 + i)2/(1 + k)2 = (1 + .07)2/(1 + .085)2 = 0.972
• Note that the two-year effect, 0.972, is just the one-year
effect (0.986) squared. This is because we assumed the oneyear and two-year rates were the same.
Assuming that the recovery proportion is .55, the implied
cumulative two-year default rate is just
[p +  - p]2 = [p + .55 - .55p]2 = 0.972
=> p = 0.969 so the one-year default rate stays the same
at (1 - p) = 0.031 but the cumulative two-year default rate is
(1 + 0.031)2 - 1 = 0.063.
Example 3: If the two-year risk-free rate is 7.5% and the
Baa rated two-year bond yields 9.5%, what is the
implied two-year combined default and recovery
effect?
(1 + i)2/(1 + k)2 = (1 + .075)2/(1 + .095)2 = 0.964
Here we assume that the yields change and the spread
between them changes (from 1.5% to 2%). This leads to a
smaller implied two-year combined effect, i.e., a larger
cumulative default probability.
[p +  - p]2 = [p + .55 - .55p]2 = 0.964
=> p = 0.96 so the one-year default rate is (1 - p) = 0.04. The
cumulative two-year default rate is (1 + 0.04)2 - 1 = 0.082.
Using Market Data on Bond
Rates
The simplest way to set a minimum loan rate is to just look at
the rate charged for bonds of a similar quality. Using the
chart below, if you have a Baa quality borrower you might
charge 8.5%. If you think rates will continue to increase as
they have done recently then you can charge 8.75% or more.
Other More Complex Models
Used By Financial Firms
1. Risk-Adjusted Return on Capital (RAROC) - combines
duration and potential changes in risk premiums.
2. Options Model - we can get the value of a loan as a put
option, similar to what we did to value guarantees. The
problem here is that there few traded options on bonds. The
options model can be used to help identify future risk
changes, however. If a potential borrower has traded option,
we can look at the implied volatility to see if the market
expects the company’s risk to increase.
3. CreditMetrics - similar to RiskMetrics which was covered
previously. See www.riskmetrics.com.
4. Credit Risk+ - uses Poisson distribution and the average
number of defaults to calculate a many-default probability.
Portfolio Credit Risk Diversification
• Group loans (bonds) in your portfolio by industry or group
(e.g., technology) and compare to the average of competitors.
•Follow bond ratings changes by industry or group to avoid
making loans to deteriorating groups or to increase rates
charged to them.
• Set limits on the amount of loans or bonds allocated to one
borrower or group. Given a maximum loss percentage and
historical or projected loss rate:
Concentration Limit = (Max. Loss Percentage) (1/Loss Rate)
• If expected returns, return variances and correlations are
available for a portfolio of loans or bonds, select holdings to
maximize returns given a desired expected return.