Credit Risk

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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.
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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
loans 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.
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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 loans promised
return? 1 k 1 .00125 .12 .02/1 -
.1(1 - .1) 1.1552 gt k 15.52
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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 (Er) 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 Er p(1 k) (1 - p)?(1k)
If a default leads to a total loss because there
is no collateral or corporate assets backing the
loan or bond, then Er p(1 k)
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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? Er .95(1 .1552)
1.0974 Example 2 Altman and Kishors (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? Er
.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.
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Collateral vs. Default Risk
Assuming risk neutral pricing of loans and bonds,
rates on various loans or bonds should
follow Er p(1 k) (1 - p)?(1k) (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.
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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.

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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)

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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 gt 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.
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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 borrowers 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.
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Example
For Altmans 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/TA0,
EBIT/TA-.2, ME/LTD.1 and Sales/TA2. 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 lt 1.81, we would
reject the borrower. The discriminant model with
different variables is used to judge Sovereign
(country-specific) risk - Chapter 16.
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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

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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 1.3
6 2.67 B 2.30 0.81 1.86 1.97 2.59 4.99 2.30 0.8
1 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
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• 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.

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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 gt k 10.3
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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 gt k 9.5 Of
course, if you believe you can get a higher rate
from a borrower then you might charge more.
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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
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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 gt
p 0.969 so default rate 0.031 This is the
markets 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
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• Note that the two-year effect, 0.972, is just
  the one-year effect (0.986) squared. This is
  because we assumed the one-year and two-year
  rates were the same.
• Assuming that the recovery proportion is .55, the
  implied cumulative two-year default rate is just
• p ? - ?p2 p .55 - .55p2 0.972
• gt 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.

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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 ? -
?p2 p .55 - .55p2 0.964 gt 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.
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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.
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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
companys 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.
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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.