Credit RiskFinancial 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 ReturnBefore 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.

ExampleFor 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 ReturnOf 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 ReturnExample 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 RiskAssuming 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 ProbabilityDefault 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 ModelsCredit 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) = (Debt/Equity) + (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 and , e.g.,

Zi = (Debt/Equity) + (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). Theregression 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.

ExampleFor 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 issueValue 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 BondAaa, Aa, A 0.00% 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00Baa 0.04 0.00 0.00 0.00 0.00 0.00

0.04 0.00 0.04 0.00 0.04 0.00Ba 0.17 0.00 0.60 0.38 0.60 2.30

0.17 0.00 0.77 0.38 1.36 2.67B 2.30 0.81 1.86 1.97 2.59 4.99

2.30 0.81 4.11 2.76 6.60 7.61Caa 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 RatesOne 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 ExpectationsUsing 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 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 + - 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 RatesThe 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 Firms1. 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.