Credit Risk
Return on a Individual Loan
Factors: interest payments, fees, credit risk premium, collateral, other requirements such as compensating balances and reserve requirements.
Return = inflow/outflow
k = (f + (BR + M ))/(1-[b(1-R)])
where: k: return on the loan
f: the loan origination fee
BR: the base blending rate
M: the credit risk premium
b: the compensating balance requirement
R : the reserve requirement imposed by the Fed
Example : BR=12%, M=2%, f =0.125%, b = 10%, R=10%
k = (f + (BR + M ))/(1-[b(1-R)]) = (0.00125 +(0.12 + 0.02)/(1-0.1*(1-0.1))
=15.52%
Expected return: E(r) = p(1+k), p is the probability of repayment of the loan.
Lending Rates and Rationing
At retail: Usually a simple accept/reject decision rather than adjustments to the rate.
Credit rationing.
If accepted, customers sorted by loan quantity.
At wholesale:
Use both quantity and pricing adjustments.
Measuring Credit Risk
Qualitative models: borrower specific factors are considered as well as market or systematic factors.
Specific factors include: reputation, leverage, volatility of earnings, covenants and collateral.
Market specific factors include: business cycle and interest rate levels.
Credit scoring models
Linear probability models:
Example: Suppose the estimated linear probability model is Z = 1.1X1 + .6X2 + .5X3 + error, where X1 = 0.75 is the borrower's debt/equity ratio; X2 = 0.25 is the volatility of borrower earnings; and X3 = 0.15 is the borrower’s profit ratio.
a.What is the projected probability of repayment for the borrower?
Z = 1.1(.75) + .6(.25) + .5(.15) = 1.05%.
The expected probability of repayment is 1 0.0105 = 98.95%.
Logit models: overcome weakness of the linear probability models using a transformation (logistic function) that restricts the probabilities to the zero-one interval
Altman’s linear discriminant model:
Z=1.2X1+ 1.4X2 +3.3X3 + 0.6X4 + 1.0X5
Critical value of Z = 1.81.
X1 = Working capital/total assets.
X2 = Retained earnings/total assets.
X3 = EBIT/total assets.
X4 = Market value equity/ book value LT debt.
X5 = Sales/total assets.
Problems:
Only considers two extreme cases (default/no default).
Weights need not be stationary over time.
Ignores hard to quantify factors including business cycle effects.
Database of defaulted loans is not available to benchmark the model.
Mortality rate models
The probability of default is estimated from past data on defaults.
Marginal Mortality Rates:
MMR1 = (Value Grade B default in year 1)/ (Value Grade B outstanding yr.1)
MMR2 = (Value Grade B default in year 2)/ (Value Grade B outstanding yr.2)
RAROC models
RAROC = One year income on a loan/ Loan risk or capital at risk
Risk adjusted return on capital. This is one of the more widely used models.
Incorporates duration approach to estimate worst case loss in value of the loan:
L = -DL x L x (R/(1+R))
where R is an estimate of the worst change in credit risk premiums for the loan class over the past year.
RAROC = one-year income on loan/L
Example: A bank is planning to make a loan of $5,000,000 to a firm in the steel industry. It expects to charge an up-front fee of 1.5 percent and a servicing fee of 50 basis points. The loan has a maturity of 8 years and a duration of 7.5 years. The cost of funds (the RAROC benchmark) for the bank is 10 percent. Assume the bank has estimated the maximum change in the risk premium on the steel manufacturing sector to be approximately 4.2 percent, based on two years of historical data. The current market interest rate for loans in this sector is 12 percent.
a.Using the RAROC model, estimate whether the bank should make the loan?
RAROC = Fees and interest earned on loan/ Loan or capital risk
We ignore up-front fees in the calculation of the loan’s income.
Loan risk, or L= -DL*L*(R/(1 + R) = = -7.5 * $5m * (.042/1.12) = -$1,406,250
Expected interest= 0.12 x $5,000,000=$600,000
Servicing fees= 0.0050 x $5,000,000=$25,000
Less cost of funds= 0.10 x $5,000,000=-$500,000
Net interest and fee income=$125,000
RAROC = $125,000/1,406,250 = 8.89 percent. Since RAROC is lower than the cost of funds to the bank, the bank should not make the loan.
Credit Risk in a Loan (Asset) Portfolio
Migration analysis
Track credit rating changes within sector or pool of loans.
Rating transition matrix.
Risk grade: end of year
123Default
Risk grade: 1|.85.10.04.01
beginning2|.12.83.03.02
of year3|.03.13.80.04
Loan concentration model
Concentration limits
On loans to individual borrower
Concentration limit = Maximum loss Loss rate
Where the Maximum loss expressed as percent of capital.
Applying portfolio theory to loan portfolios
Using loans to construct the efficient frontier
Minimum risk portfolio
Low risk
Low return.
Require
Expected return on loan(measured by all-in-spread)
Loan risk
Correlation of loan default risks
KMV Portfolio Manager model
Ri = AISi - E(Li) = AISi - [EDFi × LGDi]
i = ULi = Di × LGDi
= [EDFi(1-EDFi)]½ × LGDi
ij = correlation between systematic return components of equity returns of
Loan volume-based models
Use commercial bank call reports to estimate national allocations
Shared national credit database that breaks commercial and industrial loan volume into 2-digit SIC codes
Use above as market benchmarks to measure loan allocation deviation
Loan loss ratio-based models
Estimate loan loss risk by SIC sector.
loan loss ratio in ithsector= + i(total loan loss ratio)
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