1. Topic 3. Measuring Credit Risk (Individual Loan)
3.1 Components of credit risk
3.2 Usefulness of credit risk measurement
3.3 The return of a loan
3.4 Default risk models

2. 3.1 Components of credit risk
Credit risk consists of
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.
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.

3. 3.2 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. 3

4. 3.2 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. 4

5. 3.2 Usefulness of credit risk measurement
(John Hull (2007), “Risk Management and Financial Institutions”, Pearson International Edition) 5

6. 3.3 The return of a loan Loan rate (per annum) = BR + m (3.1)
In the coupon bond case, no margin is discussed…
The annual loan interest rate (in short, loan rate) is made up of
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).
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 (3.1) 6

7. 3.3 The return of a loan 7

8. 3.3 The return of a loan
A compensating balance is the portion of a loan that a borrower must keep on deposit with the credit-granting depository FI. Thus, the funds are not available for use by the borrower. Without the compensating balance, a FI would lend the full amount, the gross return reduces.
A reserve requirement is the amount of capital that central banks charges the FI from the compensating balance the FI withholds from loan borrower. It is proportional to the amount of compensating balance. Eliminating the reserve requirement, less capital is taken out of the FI, would increase the gross return. 8

9. 3.3 The return of a loan 9

10. 3.3 The return of a loan 10

11. 3.3 The return of a loan Example 3.1
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. (3.4), 11

12. 3.3 The return of a loan Expected return
Taking the default risk into account, the annual gross return, r, per actual dollar of the loan is less than k.
The effective amount received by FI is (random):
D*×(1 + r), where,
D* = D(1 – b(1  RR)) is default-risk-free amount
If no default, the FI gets back D* (1+k) ;
If default occurs, we assume the lender recover
R (0  R  1) of D* when default occurs.
r is thus a random variable, depending on default probability. 12

13. 3.3 The return of a loan
Denote q as the probability that no default in one year. More precisely, q is expressed as
q = Pr( > 1)
where  is defined as the default time of a borrower.
It is a random variable.
q is known as the 1-year survival probability
(1-q) is known as 1-year default probability 13

14. 3.3 The return of a loan q
1  q
survive
default
D*·(1+k)
D*R D* 14

15. 3.3 The return of a loan Ways of increasing E(r):
Increase default-risk-free return k by :
1) increasing margin charge m and/or
2) withholding more compensating balance b and/or
3) charge more origination fees of.
2. Increase survival probability q by screening and/or credit rationing (do not make the loan or limiting the size of the loan to the borrowers with unsatisfactory credit quality). 15

16. 3.3 The return of a loan
In general, you cannot increase k and q simultaneously since the high fees and base rates may reduce the probability of repayment. In other words, when interests burden is too high, borrower might not be able to make all the payments. 16

17. 3.4 Default risk models Qualitative models (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 the judgment is based on the FI manager experience, this model is also called expert system. 17

18. 3.4 Default risk models Borrower-specific factors:
Reputation
Leverage (Debt/Equity)
Earnings stability/volatility
Collateral
Market-specific factors:
Business cycle ( expansion or recession )
Level of interest rates (high or low) 18

19. 3.4 Default risk models Quantitative Model I: Credit scoring model
Credit scoring model is a quantitative model.
It uses data on observed borrower’s 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. 19

20. 3.4 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. 20

21. 3.4 Default risk models 21

22. 3.4 Default risk models
The most famous discriminant score is Altman’s Z-score, which is developed by Edward Altman in It uses accounting ratios to predict default occurrence.
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 22

23. 3.4 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. 23

24. 3.4 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 certain market factors such as business cycle effects. 24

25. 3.4 Default risk models Quantitative Model II: Logit 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. 25

26. 3.4 Default risk models Pc DS 26

27. 3.4 Default risk models Example 4.3
Assume c = –3. From Example 4.2, the estimated probability of default under logit model is 27

28. 3.4 Default risk models Marginal and cumulative default probabilities
Marginal default probability p(s,t) over the period (s, t]
The probability a borrower defaults over the period (s, t], given that there was no default strictly before s.
p(s,t) = Pr (s <   t |  > s) (4.9)
Cumulative default probability cp(t) at the time t:
The probability a borrower defaults 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) 28

29. 3.4 Default risk models
For n  2, we let 0 = t0 < t1 < t2 < … < tn. The marginal and cumulative default probabilities are related by
where
Cumulative default probability equals one minus cumulative survival probability. 29

30. 3.4 Default risk models Proof of (4.11)
We prove (4.11) by using Mathematical Induction (MI).
When n = 2, 30

31. 3.4 Default risk models Assume (4.11) is true when n = k,
Let n = k + 1, 31

32. 3.4 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 32

33. 3.4 Default risk models Quantitative Model III: Mortality rate model
The probability of defaults is estimated from the past financial market data on defaults. It is a type of market-based model, via historical data.
The marginal mortality rate in year i (MMRi) for a particular rating class, say B, is estimated as 33

34. 3.4 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. 34

35. 3.4 Default risk models 35

36. 3.4 Default risk models
Quantitative Model IV: Current 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). 36

37. 3.4 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). 37

38. 3.4 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
cp(N) 38

39. 3.4 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, 39

40. 3.4 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 2
81.16
B-rated US corporate zero coupon bond
86.36
71.82 40

41. 3.4 Default risk models
From Eq. (4.14), 41

42. 3.4 Default risk models Quantitative Model V: RAROC model
The RAROC (risk-adjusted return on capital) model evaluates and prices credit risk based on the modeling of loss from 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. 42

43. 3.4 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. 43

44. 3.4 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. 44

45. 3.4 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%. 45

46. 3.4 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.) 46

47. 3.4 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 47

48. 3.4 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:
A credit rating (AAA, AA, and so on) from certain credit rating agency (such as S&P, Moody) is assigned to borrower. 48

49. 3.4 Default risk models
Procedures in estimating the distribution of m (cont.):
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.
The m in Eq. (4.17) equals the -th percentile of the distribution of m in step 2. 49

50. 3.4 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. 50

51. 3.4 Default risk models
Hypothetical Frequency Distribution of Yield Spread Changes for All AAA Bonds in 2012 51

52. 3.4 Default risk models Taking  = 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 52

53. 3.4 Default risk models If 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. 53