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Topic 4. Measuring Credit Risk (Individual Loan)

4.1 Components of credit risk

4.2 Usefulness of credit risk measurement

4.3 The return of a loan

4.4 Default risk models

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4.1 Components of credit risk Credit risk consists of

1. Default risk – a possibility that a counterparty in a financial contract will not fulfill a contract commitment to meet his/her obligations stated in the contract.

E.g. A borrower fails to repay the loan to a lender.

2. Spread risk – the reduction in market value of the financial contract due to the decline of the credit quality of the debtor or counterparty.

E.g. A fall in the price of the bond as a result of credit rating downgrade of the bond’s issuer.

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4.2 Usefulness of credit risk measurement Supporting origination decision

Credit risk measurement could assist a FI to determine whether a certain credit-linked asset is worth to take or not and to determine the price of this asset. For example, by assessing the creditworthiness of a borrower, a bank can decide whether to grant the loan or not or determine the level of loan interest rate to be charged.

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4.2 Usefulness of credit risk measurement Supporting portfolio optimization

Credit risk management could assist FIs to understand more about the risk-return characteristics of the portfolio of credit-linked assets (e.g. loans or bonds). Then, the optimized portfolio, in the sense that a portfolio with minimum risk for a given level of return, can be established.

Supporting capital management

Credit risk measurement could provide the information to FI in order to set appropriate reserve (economic capital) to maintain the FI’s target credit rating.

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4.2 Usefulness of credit risk measurement

(John Hull (2007), “Risk Management and Financial Institutions”, Pearson International Edition)

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4.3 The return of a loan The annual loan interest rate (in short, loan rate) is

made up of1. Base lending rate (BR): reflects the FI’s weighted-

average cost of capital or its marginal cost of funds, such as the federal funds rate, or LIBOR (London Interbank Offered Rate).

2. Credit risk premium or margin (m): the spread to be charged over the BR for compensating the credit risk of the borrower.

Loan rate (per annum) = BR + m (4.1)

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4.3 The return of a loan The annual gross return (effective yield), k, per dollar

lent on a loan is given by

Other than loan interest, the direct and indirect fees and charges relating to a loan include

• Loan origination fee (of): a % of a loan to be charged to the borrower for processing the application.

• Compensating balance (b): a % of a loan that a borrower cannot actively use for expenditures. Instead, these balance must be kept on deposit at the FI.

• Reserve requirement (RR): a % of the FI’s compensating balance to be charged by the Federal Reserve.

(4.2) borrowedamount Actual

chargesother interest loan Annual k

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4.3 The return of a loanAssumptions:

1. No default

2. Neglecting the present value effect.

Suppose the contractual loan amount is $D.

Loan interest = D(BR + m)

Other charges = Dof

Actual amount borrowed = D(1 – b) + DbRR

= D(1 – b(1 RR))

By Eq. (4.2),

(4.3) 11

RRb

ofmBRk

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4.3 The return of a loan By taking the present value effect into consideration

(origination fee (of) is usually an up-front payment and the loan interest is paid at the year end), Eq. (4.3) is modified to

where d is the annual discount rate.

(4.4) 11

1

1

RRb

ofd

mBRk

10

4.3 The return of a loan Example 4.1

Contractual loan amount (D): $1 million

Term of the loan: 1 year

Loan rate (BR + m): 14% per annum

Loan origination fee (of) : 0.125%

Compensating balance (b): 10%

Reserve requirement (RR): 10%

Discount rate (d): 10% per annum

From Eq. (4.4),

rateLoan 14.12%

%101%101

%125.0%101

1%14

k

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4.3 The return of a loanExpected return Taking the default risk into account, the annual gross

return, r, per actual dollar lent of the loan will be less than k and r is a random variable.

The effective amount of loan receive at the end of the year (random)=D*×(1 + r),

where D* = D(1 – b(1 RR)). We assume the lender can recover R (0 R 1) of D*

when default occurs.

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4.3 The return of a loan Let the random variable be the default time of a

borrower. Denote q as the probability that no default in one year.

More precisely, q is expressed as

q = Pr( > 1)

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4.3 The return of a loan

So,

The E(r) in Eq. (4.5) is defined as the annual expected return per actual dollar lent of the loan.

(4.5) )1()1(

)1(1

111 ***

RRkq

RqkqrE

RDqkDqrDE

q

1 q

no default

default

D*·(1+k)

D*R

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4.3 The return of a loan Two ways for increasing E(r):

1. To increase k by increasing m and/or b and/or of.

2. To increase q by screening and/or credit rationing (refusing to make the loan or limiting the size of the loan to the borrowers with unsatisfactory credit quality).

In general, you cannot increase k and q simultaneously since the high fees and base rates may reduce the probability of repayment.

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4.4 Default risk modelsQualitative model (Expert system) Under the qualitative model, the FI manager would

base on two categories of factors namely borrower-specific factors and market-specific factors to assess the probability of default of the borrower and to price the loan subjectively.

Because of the judgment is based on the FI manager experience, so this model is also called expert system.

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4.4 Default risk models Borrower-specific factors:

1. Reputation

2. Leverage (Debt/Equity)

3. Volatility of earnings

4. Collateral

Market-specific factors:1. Business cycle

2. Level of interest rates

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4.4 Default risk modelsCredit scoring models Credit scoring model is a quantitative model. It uses data on observed borrower economic and

financial characteristics either to calculate the probability of default or to sort borrowers into different risk classes.

The examples of the observed borrower characteristics include

• Consumer debt: income, assets, age and occupation.• Commercial debt: cash flow information and financial

ratios.

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4.4 Default risk models The statistical techniques, such as regression and

discriminant analysis, are involved in building the models.

In contrast to qualitative model, the credit scoring model is more cost effective and consistent.

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4.4 Default risk modelsDiscriminant Score The discriminant score attempts to classify the

customers into two groups – Default and not default. It does this by assigning a score to each borrower. The discriminant score of the borrower i (DSi) is the weighted sum of the borrower data:

where

Xij, j = 1, …, n, is set of casual variables that reflect quantitative information about the borrower i.

j is a constant that reflects the importance of the jth casual variable (Xij ) .

(4.6) 1

n

jijji XDS

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4.4 Default risk models The most famous discriminant score is Altman’s Z-score,

which is developed by Edward Altman in 1968. It uses accounting ratios to predict default.

The Z-score, Z, is calculated as

where

X1: Working capital (= current assets – current liabilities) /Total assets

X2: Retained earnings/Total assets

X3: Earning before interest and taxes/Total assets

X4: Market value of equity/Book value of long-term debt

X5: Sales/Total assets

(4.7) 0.16.03.34.12.1 54321 XXXXXZ

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4.4 Default risk models According to the value of Z as predicted in Eq. (4.7),

we have the following cases: Z > 2.99 low default risk firm 1.81 Z 2.99 indeterminant default risk firm Z < 1.81 high default risk firm

Example 4.2

Suppose the financial ratios of a potential borrowing firms are:

X1 = 0.2; X2 = 0; X3 = –0.2; X4 = 0.1; X5 = 2.0.

From Eq. (4.7), Z = 1.64 < 1.81 The FI should not make a loan to this borrower.

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4.4 Default risk models Weakness:

Only considers two extreme cases (default/no default). The cases in between, such as delay of interest or principal payment, are not considered.

Weights need not be stationary over time. Ignores hard to quantify factors such as business cycle

effects.

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4.4 Default risk modelsLogit Model The logit model is to relate the discriminant score (DS)

to the probability of default (Pc).

The logit model transform the DS into Pc as follow

where c is a constant. The parameter c needs to be determined through the

statistical methods such as maximum likelihood estimation.

When DS , Pc 0.

When DS , Pc 1.

(4.8) .1

1DScc e

P

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4.4 Default risk models

DS

Pc

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4.4 Default risk models Example 4.3

Assume c = –3. From Example 4.2, the estimated probability of default under logit model is

7958.01

164.13

ePc

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4.4 Default risk modelsMarginal and cumulative default probabilities Marginal default probability over the period (s, t] (denote as

p(s,t)):

The probability that a borrower will default over the period (s, t], given that there was no default until s.

p(s,t) = Pr (s < t | > s) (4.9) Cumulative default probability at the time t (denote as

cp(t)):

The probability that a borrower will default on or before time t.

cp(t) = Pr ( t) (4.10) When s = 0 in (4.9), we have

p(0,t) = Pr (0 < t | > 0) = Pr( t)= cp(t)

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4.4 Default risk models For n 2, we let 0 = t0 < t1 < t2 < … < tn. The marginal

and cumulative default probabilities are related by

where .

(4.11) ,11)(1

1

n

iiin ttptcp

n

n

ii xxxxx

3211

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4.4 Default risk modelsProof of (4.11)

We prove (4.11) by using Mathematical Induction (MI).

When n = 2,

),(1,011

,01,01),(1,0

,01),(,0

(why?) Pr|Pr,0

PrPr

Pr)(

211

11211

1211

11211

211

22

ttptp

tptpttptp

tpttptp

tttttp

ttt

ttcp

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4.4 Default risk modelsAssume (4.11) is true when n = k,

Let n = k + 1,

By substituting (A) into cp(tk), (4.11) can be proved to be true for all n 2 by MI.

(A) ,11)(1

1

k

iiik ttptcp

11

1

1

1

11

,,1)(

Pr1|PrPr

Pr|PrPr

PrPr

Pr)(

kkkkk

kkkkk

kkkkk

kkk

kk

ttpttptcp

ttttt

ttttt

ttt

ttcp

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4.4 Default risk models The marginal and cumulative default probabilities can

be derived from the• historical data about the default experience of the bonds

(Mortality Rate Model).• current market information of the defaultable and non-

defaultable bonds

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4.4 Default risk modelsMortality rate model The probability of defaults is estimated from the past

financial market data on defaults. So, it is a kind of market-based model.

The marginal mortality rate in year i (MMRi) for a particular rating class, say B, is estimated as

(4.12)

year prior in the defaultsfor adjusted issue of year in

goutstandin bonds B grade of valueTotal

issue of year in

defaulting bonds B grade of valueTotal

i

iMMRi

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4.4 Default risk models Actually, MMRi = Pr(i – 1< i | > i – 1) = p(i – 1, i).

Weakness: Backward-looking. Sensitive to the period chosen to calculate the MMRs.

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4.4 Default risk models

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4.4 Default risk modelsCurrent market information

Notation:B(0, N): Price of N-year non-defaultable zero coupon bond

with the face value of $1.

Bd (0, N) : Price of N-year defaultable zero coupon bond with the face value of $1.

y : Bond yield of N-year non-defaultable zero coupon bond

yd : Bond yield of N-year defaultable zero coupon bond

R : The fraction of the face value can recover when the defaultable bond defaults (0 R 1).

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4.4 Default risk models Assume simple compounding and the compounding

frequency is 1. We have

The credit risk yield spread (credit risk premium) is defined as yd – y.

To compensate the investor for the default risk in Bd(0,N), yd > y or Bd(0,N) < B(0,N).

N

d

d

N

yNB

yNB

1

1,0

1

1),0(

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4.4 Default risk models Suppose $1 is invested into the N-year defaultable zero

coupon bond. Assume R = 0,

where r is the annual return of the defaultable bond.

1 cp(N)no default

default

(1+yd)N

0cp(N)

NB

NcprE

yNcprE

d

N

Nd

N

,0

11

111

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4.4 Default risk models To make the investor to be indifferent between the

defaultable and non-defaultable N-year bonds, we must have E((1 + r)N) = (1 + y)N. So,

(4.13) ),0(

,0)(1

,0

1

,0

1

,0

11

NB

NBNcp

NB

Ncp

NB

NB

Ncpy

d

d

d

N

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4.4 Default risk models Example 4.4

Given

Assume R = 0. Find cp(1) and cp(2).

Maturity (Year)

Price per $100 face value ($)

US treasury zero coupon bond 1 90.91

US treasury zero coupon bond 2 81.16

B-rated US corporate zero coupon bond 1 86.36

B-rated US corporate zero coupon bond 2 71.82

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4.4 Default risk modelsFrom Eq. (4.14),

1151.0

16.81

82.711

)2,0(

2,01)2(

1,005.091.90

36.861

)1,0(

1,01)1(

B

Bcp

pB

Bcp

d

d

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4.4 Default risk modelsRAROC model The RAROC (risk-adjusted return on capital) model

evaluates and price credit risk based on market data. It was first pioneered by Bankers Trust and has now

been adopted by most of the large banks, although with some proprietary differences between them.

The time horizon in the RAROC models is usually chosen as one year.

(4.14) capital Economic

loan aon incomenet year OneRAROC

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4.4 Default risk models The RAROC in Eq. (4.14) measures the one year net

loan income per unit dollar of economic capital (loan risk). A loan is approved only if RAROC is sufficiently high relative to some benchmarks for the FI such as return on equity (ROE) which measures the return stockholders require on their equity investment in the FI.

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4.4 Default risk models One year net loan income

= D ( rA – rD ) + F – OC – EL

= D (spread) + F – OC – EL (4.15)

where

D is the loan amount; rA is the loan interest rate; rD is the cost of funding; spread = rA – rD; F is other fees such as servicing fee; OC is the operating cost; EL is the expected loss over the next year.

F, OC and EL in Eq. (4.15) are measured in dollar amount. EL = E(L)

where L is the random variable to denote the loan loss over the next year.

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4.4 Default risk models Economic capital is defined as the amount of capital a

bank needs to absorb losses over a certain time horizon with a certain confidence level.

The confidence level is determined by the FI’s objective on the target rating which it would like to maintain over the next year. For a large international bank, it would like to maintain an AA credit rating (corresponding 1-year default probability is about 0.02%). This suggests that the confidence level is about 99.98%.

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4.4 Default risk models Since the expected loss (EL) is charged directly to the

borrower, so

Economic capital = L – EL (4.16)

where

is the confidence level;

L is the -th percentile of L.

If the distribution of L is continuous, L is defined through

Pr(L L ) = %.

Of course, we could modify it to cater for different types of distribution function of L. (Refer to Section 3.2 of Topic 3 for details.)

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4.4 Default risk models The loan’s yield R can be expressed as

R = BR + m. Assume the changes in BR (BR) is negligible. From

Eq. (2.10), L can be related to the -th percentile of the change of the loan credit risk premium, m , as follows

)yearnext

over the of

ondistributi theof loan) theof loan) e th

percentileth - (the lue(market va of(duration

(4.17) 1

1

m

RmLNDL LN

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4.4 Default risk models Since publicly available data on loan credit risk

premium (m) are scarce, we turn to publicly available corporate bond market data to estimate the distribution of m over the next year.

Procedures in estimating the distribution of m:1. A credit rating (AAA, AA, and so on) from certain credit

rating agency (such as S&P, Moody) is assigned to borrower.

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4.4 Default risk models Procedures in estimating the distribution of m (cont.):

2. Denote the risk premium changes of bond i over the last year as mi.

mi = (ri – rG)

where ri and rG are the yield of corporate and matched duration treasury bond respectively

The mi of all the bonds traded in that particular rating class over the last year are analyzed. This last year distribution is taken as proxy of the distribution of m over the next year.

3. The m in Eq. (4.17) equals the -th percentile of the distribution of m in step 2.

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4.4 Default risk models Example 4.5

To evaluate the credit risk of a loan with a market value of

$1 million and duration of 2.7 years to a AAA borrower.

The bank collects 400 publicly traded bonds in that class and get following distribution of m.

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4.4 Default risk models

Hypothetical Frequency Distribution of Yield Spread Changes for All AAA Bonds in 2012

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4.4 Default risk modelsTaking = 99.

From the above figure, m99 = 1.1%.

If the current average yield (R) on AAA bond is 10%, then from Eq. (4.17)

Suppose OC = 0 in Eq. (4.15). The projected (one-year) spread and fees of the loan are as follows:

Spread = 0.2% $1 million = $2,000

Fees = 0.1% $1 million = $1,000

Total loan income = $3,000

000,27$%101

%1.1million 17.2

L

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4.4 Default risk modelsIf EL = 0, then Economic capital = $27,000.

From Eq. (4.14),

If 11.1% exceeds the bank’s internal RAROC benchmark (such as ROE), the loan will be approved.

If it is less, the loan will either be rejected outright or the borrower will be asked to pay higher fees and/or a higher spread to increase the RAROC to acceptable levels.

%1.11000,27$

000,3$RAROC