Measuring Credit Risk

1
Measuring Credit Risk

• FIN 653 Lecture Notes
• From
• Saunders and Cornett
• Ch. 11

2
I. Credit Scoring Models

• Credit Scoring Models use data on observed
  borrower characteristics either to calculate the
  probability of default or to sort borrowers into
  different default risk classes.

3
I. Credit Scoring Models

• By selecting and combining different economic and
  financial borrower characteristics, a financial
  institution manager may be able to
  
• 1. Numerically establish which factors are
• important in explaining default risk
• 2. Evaluate the relative degree or importance
  of
  
• these factors intelligently
• 3. Improve the pricing of default risk
• 4. Be better able to screen out bad loan
  applicants
  
• 5. Be in a better position to calculate any
  reserves
  
• needed to meet expected future loan
  losses.
  

4
I. Credit Scoring Models

• 1. Linear Probability Model
• Uses past data as input into a model to explain
  repayment experience on old loans
  
• Zi ??j Xi,j ?i
• The relatively important factors used in
  explaining past repayment performance are then
  used to forecast repayment probability on new
  loans.
• Zi ??j Xi,j ?i
• The expected Z value can be interpreted as the
  probability of default.
  
• E(Zi) 1 - Pi.

5
I. Credit Scoring Models

• E.g., there were two factors influencing the past
  default the leverage or debt-equity ratio (D/E)
  and the sales-asset ratio (S/A). Based on the
  past default experience, the linear probability
  model is estimated as
• Zi .5(D/Ei) .1(S/Ai).
•  
• Assume a prospective borrower has a D/E .3 and
  an S/A 2.0. Its expected probability of
  default is then
  
• Zi .5(.3) .1(2.0) .35.

6
I. Credit Scoring Models

• The major weakness of Credit Scoring Models is
  that the estimated probabilities of default can
  often lie outside the interval (0 to 1).
  
• The logit and probit models overcome this
  weakness by restricting the estimated range of
  default probability to lie between 0 to 1.

7
I. Credit Scoring Models

• 2. The Logit Model
• The logit model constrains the cumulative
  probability of default on a loan to lie between 0
  and 1 and assumes the probability of default to
  be logistically distributed according to the
  functional form
• 1
• F (Zi) ------------------
• 1 e -Zi
• where F (Zi) the cumulative probability of
  default on the loan.
  
•   Zi estimated by regression in a
  similar fashion to the linear
  
• probability model.

8
I. Credit Scoring Models

• It's major weakness is the assumption that the
  cumulative probability of default takes on a
  particular functional form that reflects a
  logistic function. 

9
I. Credit Scoring Models

• 3. The Probit Model
• The probit model also constrains the projected
  probability of default to lie between 0 and 1,
  but differs from the logic model in assuming that
  the probability of default has a (cumulative)
  normal distribution rather than the logistic
  function.
• However, when multiplied by a fixed factor, logit
  estimates may produce appropriately correct
  probit values.

10
I. Credit Scoring Models

• 4. Linear Discriminant Models
• Discriminant models divide borrowers into high or
  low default risk classes contingent on their
  observed characteristics (Xj).
  
• The discriminant model by E.I. Altman
• Z 1.2X1 1.4 X2 3.3 X3 0.6 X4 1.0
  X5
  
• Where Z an overall measure of the default
  risk
  
• X1 Working Capital/Total Assets
• X2 Retained Earnings/Total Assets
• X3 Earnings before Interest
  Taxes/Total Assets
  
• X4 Market Value of Equity/Book Value
  of Long-term Debt
  
• X5 Sales/Total Assets

11
I. Credit Scoring Models

• According to Altman's credit scoring model
• Z 1.81 ? Low default risk class
•   Z
• Ex Suppose that the financial ratios for a
  potential borrowers
  
• X1 .20 X2 .0
• X3 -.20 X4 .10 X5 -2.0
• Then Z score 1.64
• The FI should not make a loan to this borrower
  until it improves its earnings.
  

12
I. Credit Scoring Models

• Problems with the discriminant analysis
• 1. This model usually discriminates only between
  two extreme cases of borrowers behavior, default
  and no default.
  
• 2. No obvious economic reason to expect the
  weights in the discriminant function to be
  constant over short periods. The same concern
  also applies to the variables (Xj).
• 3. These models ignore important
  hard-to-quantify factors that may play a crucial
  role in the default or no default decision.

13
II. Term Structure Derivation of Credit Risk

• One market-based method of assessing credit risk
  exposure and default probabilities is to analyze
  the risk premiums inherent in the current
  structure of yields on corporate debt or loans to
  similar risk-rated borrowers.
• The spreads between risk free deep-discount bonds
  issued by the Treasury deep-discount bonds
  issued by corporate borrowers of different
  quality may reflect perceived credit risk
  exposure of corporate borrowers for single
  payments at different times in the future.

14
II. Term Structure Derivation of Credit Risk

• 1. Probability of Default on a One-Period Debt
  Instrument
  
• Assume that the Financial Institution requires an
  expected return on a one year corporate debt at
  least equal to the risk free return on T-bonds of
  one year's maturity.

15
II. Term Structure Derivation of Credit Risk

• 1. Probability of Default on a One-Period Debt
  Instrument
  
• Let P be the probability that the debt will be
  repaid, then (1-P) is the probability of default.
  The FI would be indifferent when
  
• P (1 k) (1-P) 0 1 i
• Then P (1i)/(1k) and
• 1-P (k-i)/(1k)

16
II. Term Structure Derivation of Credit Risk

• Suppose, i 10, k 15.8
• Then the probability of repayment as perceived by
  the market is
  
• 1 i 110
• P --------- --------------- 0.95
• 1 k 115.8
• Probability of default (l-P) 5.
• And a probability of default of 5 on the
  corporate bond requires the FI to set a risk
  premium of 5.8
  
• ? k - i 5.8

17
II. Term Structure Derivation of Credit Risk

• Let ? be the proportion of the loan's principal
  and interest that is collectable on default. The
  FI manager would set the expected return on the
  loan to equal the risk free rate
• ? (1k) (1-P) P(lk) 1 i
• Then the probability of default is
• (k-i)
• 1- P -------------------
• (1-?) (1k)

18
II. Term Structure Derivation of Credit Risk

• Collateral requests are a method of controlling
  default risk they act as direct substitute for
  risk premiums in setting required loan
  rates  (1 i)
• k - i ? -------------------- - (1
  i)
  
• (? p - p? )
•  As ? goes up, (k-i) goes down.

19
II. Term Structure Derivation of Credit Risk

• If i 10, p 0.95 as before, but the FI can
  expect to collect 90 of the promised proceeds if
  default occurs, then the required risk premium
  
•   ? 0.6
• and the required rate of return on the corporate
  bond would be
  
• k 10.6

20
II. Term Structure Derivation of Credit Risk

• 2. Probability of Default on a Multiperiod Debt
  Instrument
  
• What is the probability of default on a two-year
  bond? We must estimate the probability that the
  bond could default in the second year conditional
  on the probability that it does not default in
  the first year.
• The probability that a bond would default in any
  one year is the marginal default probability for
  that year.

21
II. Term Structure Derivation of Credit Risk

• Suppose the marginal default probability
• 1-P1 0.05 prob. of default for yr. 1
• 1-P2 0.07 prob. of default for yr. 2
• The probability of the borrower surviving - not
  defaulting at any time between now (t0) and the
  end of period 2 is
  
•   P1 P2 (.95)(.93) .8835
•  The cumulative default probability (CP) is
  therefore
  
•   CP 1 - P1P2 1 - (.95)(.93) .1165

22
II. Term Structure Derivation of Credit Risk

• Given the presence of both one- and two-year
  discount bonds for Treasury issues and corporate
  issues of a particular risk classes, we can
  derive P2 from the term structure of interest
  rates.
•   Maturity
• 1
  yr. 2 yrs
  
• ________________________________________
• T-Bond 10 11
• Corporate Bond 15.8 18
• ________________________________________
•  

23
II. Term Structure Derivation of Credit Risk

• From the T-Bond Yield Cure
• (1i2)2 (1i1)(1f1)
• when f1 the expected one-year forward rate
• (1f1) (1i2)2/(1i1) 1.12
• Current One-Year Rate
  Expected one-Year Rate
  
• __________________________________________________
  ______
  
• Treasury 10.0 (i1) 12.0
  (f1)
  
• Corporate 15.8 (k1) 20.2 (c1)
• Spread 5.8 8.2
• __________________________________________________
  ______

24
II. Term Structure Derivation of Credit Risk

• The expected rates on one -year bonds can
  generate an estimate of the expected probability
  of repayment on one-year corporate bonds in
  one-year's time, or
•   P2 (1f1)/(1c1) .9318
• where f1 and c1 are expected one-year rate on
• Treasury bill and corporate bond, respectively.
• Thus the expected probability of default in year
  2
  
•   (1 - P2) 6.82.  

25
II. Term Structure Derivation of Credit Risk

• In a similar fashion, the one-year rates expected
  in two-year's time can be derived from the
  Treasury corporate term structures. The
  probability of repayment on one-year loans
  originates in two-year's time is
•  
• P3 (1f2)/(1c2)
•  
• Thus we can derive a whole term structure of
  expected future one-year default probabilities
  for grade B bonds.

26
II. Term Structure Derivation of Credit Risk

• The cumulative probability of default would tell
  the FI the probability of a loan or bond
  investment defaulting over a particular time
  period. In the example, the corporate bond would
  default over the next two year is
•  
• CP 1 - (P1)(P2) 1 - (.95) (.9318)
• 11.479

27
II. Term Structure Derivation of Credit Risk

• Advantages
• 1. It is forward looking and based on market
  expectations
  
• 2. If there are liquid markets for Treasury
  corporate discount bonds - such as T-strips
  corporate zeros-, then we can easily estimate
  expected future default rates.
• Disadvantage
• 1. While the market for T-strips is now quite
  deep, the market for corporate discount bonds is
  still small. It might not be able to extract
  default risk premium for corporate discount
  bonds.

28
III. Mortality Rate Derivation of Credit Risk

• The FI manager may analyze the historic or past
  default risk experience, the mortality rates, of
  bonds or loans of a similar quality. Let 
  
• P1 the prob. of a grade B bond or loan
  surviving the first year of its issue
  
• 1 - P1 the marginal mortality rate or the
  prob. of the bond or loan dying or
  defaulting in the first year of issue.
  
• For each grade of corporate borrower quality, a
  marginal mortality rates (MMR) curve can show the
  historical default rate experience of bonds in
  any specific quality class in each year after
  issue on the bond or loan.

29
III. Mortality Rate Derivation of Credit Risk

• Total value of grade B defaulting in year 1 of
  issue
  
• MMR1 -----------------------------------------
  ------------------------------
  
• Total value of grade B bonds
  outstanding in year 1 of issue
  
•  
•   Total value of grade B bonds defaulting in
  year 2 of issue
  
• MMR2 -----------------------------------------
  ------------------------------
  
• Total value of grade B bonds
  outstanding in year 2 of
  
• issue adjusted for defaults, calls, sinking
  fund and redemption, and maturities in the
  prior year.
  

30
III. Mortality Rate Derivation of Credit Risk

• TABLE 8-9 Adjusted Mortality Rates by Original
  Standard Poor's Bond
  
• Rating (Defaults and Issues,
  1971-88)
  
•  
•  
• Years after Issuance (percentage)
•  _________________________________________________
  ____________________________________________
  
• Original Rating 1 2 3 4
  5 6 7 8 9 10
  
•  
• AAA Yearly 0.00 0.00
  0.00 0.00 0.00 0.15 0.05 0.00
  0.00 0.00
  
• Cumulative 0.00 0.00
  0.00 0.00 0.00 0.15 0.21 0.21
  0.21 0.21
  
•  
• AA Yearly 0.00 0.00
  1.39 0.33 0.20 0.00 0.27 0.00
  0.11 0.13
  
• Cumulative 0.00 0.00
  1.39 1.72 1.92 1.92
  2.18 2.18 2.29 2.42
  
• A Yearly 0.00 0.39 0.32
  0.00 0.00 0.11 0.11 0.07
  0.13 0.00
  
• Cumulative 0.00 0.39
  0.71 0.71 0.71 0.82 0.93
  1.00 1.13 1.13
  

31
III. Mortality Rate Derivation of Credit Risk

• Years after Issuance (percentage)
•  _________________________________________________
  ____________________________________________
  
• Original Rating 1 2 3 4
  5 6 7 8 9 10
  
•  
• BBB Yearly 0.03 0.20
  0.12 0.26 0.39 0.00 0.14 0.00
  0.21 0.80
  
• Cumulative 0.03 0.23
  0.35 0.61 1.00 1.00 1.14 1.14
  1.34 2.13
  
•  
• BB Yearly 0.00 0.5 0
  0.57 0.26 0.53 2.79 3.03
  0.00 0.00 3.48
  
• Cumulative 0.00 0.50
  1.07 1.34 1.86 4.59
  7.48 7.48 7.48 10.70
  
• B Yearly 1.40 0.65 2.73
  3.70 3.59 3.86 6.30 3.31
  6.84 3.70
  
• Cumulative 1.40 2.04
  4.72 8.24 11.54 14.95 20.31
  22.95 28.22 30.88
  
• CCC Yearly 1.97 1.88 4.37 16.35
  2.06 0.00 0.00 0.00 0.00
  0.00
  
• Cumulative 1.97 3.81
  8.01 23.05 24.64 24.64 24.64 24.64
  24.64 24.64
  
• __________________________________________________
  ___________________________________________
  

32
III. Mortality Rate Derivation of Credit Risk

• Problems with the Mortality Rate Approach
• 1.   Like the credit scoring model, it produces
  historic or backward looking measures.
  
• 2.   The estimates of default rates and,
  therefore implied future default probability tend
  to be highly sensitive to the period over which
  the FI manager calculates the MMRs.
• 3.   The estimates tend to be sensitive to the
  number of issues and the relative size of issues
  in each investment grade.

33
IV. Option Models of Default Risk

• 1. Theoretical Framework
• When a firm raises funds either by issuing bonds
  or increasing its bank loans, it holds a very
  valuable default or repayment option. That is, if
  a borrower cannot repay the bond holder or the
  bank, it has the option of defaulting on its debt
  repayment and turning any remaining assets over
  to the debt holders.
• On the other hand, if things go well, the
  borrower can keep most of the upside returns on
  asset investments after the promised principal
  and interest on the debt have been paid.

34
IV. Option Models of Default Risk

• 2. The Borrower's Payoff from Loans (Loan as a
  call option)
  
• If the investments turn out badly, the
  stockholder - owners of the firm would default on
  the firms debt, turn its assets (Al) over to the
  debt holders, and lose only their initial stake
  in the firm(s). By contrast, if the firm does
  well and the assets of the firm are valued highly
  (A2), the firm's stockholders would payoff the
  firm's debt (OB) and keep the difference (A2 - B).

35
IV. Option Models of Default Risk
Payoff to shareholders

Assets
A1
A2
B(debt)
36
IV. Option Models of Default Risk

• 3. The Debtholder's Payoff from Loans
• The maximum amount the bank or bondholder can get
  back is B, the promised payment. However, the
  borrower who possesses the default or repayment
  option would only rationally repay the loan if A
  B. A borrower whose asset value falls below B
  would default and turn over any remaining assets
  to the debtholders.
• Thus the value of the loan from the perspective
  of the lender is always the minimum of B or A, or
  min B,A. That is, the payoff function to the
  debtholder is similar to writing a put option on
  the value of the borrowers' assets with B, the
  face-value of debt, as the exercise price.

37
IV. Option Models of Default Risk

• Payoff to
• Debt Holders

Assets
A1
B(debt)
A2
38
IV. Option Models of Default Risk

• 4. Applying the Option Valuation Model to the
  Calculation of Default Risk Premiums
  
• The market value of a risky loan made by a lender
  to a borrower can be expressed as
  
• F(?) Be-? i (1/ d)N(h1 ) N(h2)
• Where
• B amount of debt
• i risk-free interest rate
• ? the length of time remaining to loan
  maturity
  
• d the borrowers leverage ratio measured
  as Be-? i /A
  
• N(h) the probability that a deviation
  exceeding the calculated value of h will occur.
  
• h1 -1/2 ?2 ? - ln(d)/ ? ??
• h2 -1/2 ? 2 ? ln(d)/ ? ?? .

39
IV. Option Models of Default Risk

• 4. Applying the Option Valuation Model to the
  Calculation of Default Risk Premiums
  
• Written in terms of a yield spread, this equation
  reflects an equilibrium default risk premium that
  the borrower should be charged
  
• k(? )- i (-1/? ) ln N(h2) (1/2)N (h1)
• where k(? ) Required yield on risky debt
• i Risk-free rate on debt of
  equivalent maturity

40
IV. Option Models of Default Risk

• Thus Merton has show that the lender should
  adjust the required risk premium as d and ?2
  change , i.e., as leverage and asset risk change.
  Specifically,
•   ?k(?)-i
• ---------- 0
• ? d
•  
• ?k(?)-i
• ---------- 0
• ? ?

41
IV. Option Models of Default Risk

• Problems with the Option Pricing Models
• 1.  The assumption of continuously traded claims
  on the assets of the borrower. Since many loans
  are never, or at best, infrequently traded, this
  assumption is difficult to accept in many
  real-world applications.
• 2. The value of ?2 - the volatility of the
  underlying assets of the borrower - plays a
  crucial role in setting the equilibrium risk
  premium. The value of option-based premium is
  extremely sensitive to errors made in measuring
  ?2. Moreover, volatility itself is variable over
  time.

42
IV. Option Models of Default Risk

• An Option Model Application 
• B 100,000 ? 1 year
• i 5 d 90 or .9 ?2 12
• Substituting these values into the equations for
  hl and h2 and solving for the areas under the
  standardized normal distributions, we find
  N(hl) .174120, and N(h2) .793323

43
IV. Option Models of Default Risk

• An Option Model Application 
• where
• -1/2(.12) 2 - ln (.9)
• h1 ------------------------ -.938
• .12
• and
• -1/2(.12) 2 ln (.9)
• h2 ------------------------- .818
• .12

44
IV. Option Models of Default Risk

• The current market value of the loan is
• L (t) Be?i (1/ d)N(h1) N(h2)
• 100,000
• ----------- .793323(1.1111)(.17412)
  
  
• 1.05127
• 100,000
• ------------ .986788
  93,866.18
  
• 1.05127

45
IV. Option Models of Default Risk

• and the required risk spread or premium is
•  
• k(? )- i (-1/? ) ln N(h2) (1/2)N (h1)
  (-1)ln .986788 1.33
  
• Thus, the risky loan rate k(? ) should be set at
  6.33 percent when the risk-free rate (i) is 5
  percent.

46
V. RAROC Models

• RAROC (Risk Adjusted Return On Capital) was
  pioneered by Bankers Trust and has now been
  adopted by virtually all the large banks,
  although with some proprietary differences
  between them.

47
V. RAROC Models

• The essential idea behind RAROC is to balance
  expected loan income against the loan's risk.
  Thus, rather than dividing loan income by assets,
  it is divided by some measure of asset (loan)
  risk
• One year income on a loan
• RAROC ------------------------------------
• Loan (asset) risk or Risk Capital

48
V. RAROC Models

• A loan is approved only if RAROC is sufficiently
  high relative to a benchmark cost of capital for
  the bank. Alternatively, if the RAROC on an
  existing loan falls below a bank's RAROC
  benchmark, the lending officer should seek to
  adjust the loan's terms to make it "profitable"
  again.

49
V. RAROC Models

• One problem in estimating RAROC is the
  measurement of loan risk. Duration showed that
  the percentage change in the market value of an
  asset such as a loan (?L/L) is related to the
  duration of the loan and the size of the interest
  rate shock (?R/1R)
• (?L/L) -DL (?R/1R)

50
V. RAROC Models

• The same concept is applied here, except that
  interest late shocks are replaced by credit
  quality shocks
  
•   ? L - DL L (?R/1R)
• (dollar capital (duration of (risk
  amount (expected risk change of risk
  exposure the loan) or size
  amount) in credit factor on
• or loss amount) loan)

51
V. RAROC Models

• While the loan's duration (2.7 years) and the
  loan amount (1 million) are easily estimated, it
  is more difficult to estimate the maximum change
  in the credit risk premium on the loan. Since
  publicly available data on loan risk premium are
  scarce, we turn to publicly available corporate
  bond market data to estimate premium.
• First, an SP credit rating (AAA,AA, A, and so
  on) is assigned to borrower.
  
• Thereafter, the risk premium changes of all the
  bonds traded in that particular rating class over
  the last year are analyzed.

52
V. RAROC Models

• The ?R in the RAROC equation equals
•  
• ?R Max. ?(RI - RG) 0
•  
• Where ?(RI - RG ) is the change in the yield
  spread between corporate bonds of credit rating
  class I (RI) and matched duration treasury bonds
  (RG) over the last year. In order to consider
  only the worst-case scenario, the maximum change
  in yield spread is chosen, as opposed to the
  average change.

53
V. RAROC Models

• Example To evaluate the credit risk of a loan to
  a AAA borrower. Assume there are currently 400
  publicly traded bonds in that class.
  
• The first step is to evaluate the actual changes
  in the credit risk premium (RI - RG) on each if
  these bonds for the past year. The range from a
  fall in the risk premiums of negative 2 percent
  to an increase of 3.5 percent. Since the largest
  increase maybe a very extreme (unrepresentative)
  number, the 99 percent worst-case scenario is
  chosen (i.e., only 4 bonds out 400 have risk
  premium increases exceeding the 99 percent worst
  case).

54
V. RAROC Models

• The estimate of loan(or capital) risk, assuming
  that the current average level of rates ( R )on
  AAA bond is 10 percent, is
  
•  
• ? L DL L ( ?R/ 1R)
• -(2.7) (1 million) (.011/1.1)
• -27,000
• Thus, while the face value of the loan amount is
  1 million, the risk amount or change in the
  loan's market value due to a decline in its
  credit quality is 27,000.

55
V. RAROC Models

• To determine whether the loan is worth making,
  the estimated loan risk is compared to the loan
  income (spread over the FI's cost of funds plus
  fees on the loan). Suppose the projected spread
  plus fees is as follows
•   Spread 0.2 1 million 2,000
• Fees 0.1 1 million 1,000
• Total Loan Income 3,000

56
V. RAROC Models

• The loan's RAROC is
• One-year income on loan
• RAROC -----------------------------------------
  
  
• Loan risk (or Capital risk) (?L)
• 3,000/ 27,000 11.1
• If the 11.1 percent exceeds the bank's internal
  RAROC benchmark (based on its cost of funds), the
  loan will be approved. If it 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.