Transcript Financial Institutions - George Mason University

CREDIT RISK
MEASUREMENT
Classes #14; Chap 11
Lecture Outline
2
Purpose: Gain a basic understanding of credit risk. Specifically,
how it is measured

Measuring Credit Risk


Qualitative Factors
Quantitative Models




Credit Score Models
Value-at-Risk (VaR)
RAROC
Other models (if time permits)
3
Measuring Loan Credit Risk
How Did we Adjust for Credit Risk?
4
What is the
probability of default E ( R)  P(1  k )  (1  P)(R)

We basically looked at two questions:
1. How much do we lose if the borrower defaults
2. How likely is it that the borrower defaults

What is Recovery (R)
Answering these two
questions gets us the
expected loss i.e., how
much do we expect to
lose on this loan?
We adjust for credit risk by considering (adding) the expected loss


If there is no expected loss then we earn the contractually promised return.
To adjust for credit risk we need to know 2 things
1.
What is the probability that the borrower defaults
2.
How much can we recover if the borrower defaults
Adjusting for Credit Risk
5
In the credit risk game we need good estimates of:
1. The loan’s probability of default
2. The recovery in default
3. The expected loss

Instead of estimating the probability of default and recovery separately
we can take them together and estimate the expected loss directly
Credit Risk Estimation - Methods
6
1. Qualitative Factors
2. Quantitative Models




Credit Score Models
Value at Risk (VaR)
RAROC Model
Other Models
7
Qualitative Factors
Qualitative Credit Risk Factors
8
Loan Interest Rate
The higher the interest rate on the loan the more difficult it is to make
payments and the more likely the borrower is to default.
Borrower Reputation
From prior borrowing experiences at the bank (high/low quality)
From prior borrowing in general – timely bill, rent … payments
Collateral
Physical assets that can be seized an sold to recover value in default
Capital
The insolvency buffer capital-to-asset or leverage ratio
Economic Conditions
How is the borrowers ability to repay affected by the business cycle –
type of business (industry), type of project, type of collateral …
Qualitative Credit Risk Factors
9

Capacity


The capacity of the borrower to repay depends on future income
These effects are usually quantified for use in CR models






Loan interest rate → I
Borrower reputation → FICO, credit report …
Collateral → Loan-to-value ratio
Capital → Leverage, Tier I, and Total capital ratios
Exposure to economic conditions → Industry’s Market Beta
Capacity → projected interest coverage ratio (earnings divided by
interest expense).
10
Quantitative Models of Credit Risk




Credit Score Models
Value at Risk (VaR)
RAROC Model
Other Models
11
Credit Score Model



Linear Probability Model
Logit Analysis
Linear Discriminate Model
Credit Score Models – Introduction
12
Credit Score
New
Credit
Credit Score
Model
Credit Score Models are models designed to analytically aggregate many
dimensions of credit worthiness into a single credit score that represents a
borrowers likelihood of default
Credit Score Models – Example
13
FICO Score
(Fair Isaac Company)
Payment
Types of
Amount
Length of
New
FICO = 0.35 History + 0.30
+
0.10
+
0.15
credit used
Owed
credit history + 0.10 Credit
FICO = 720
580
Sub-prime
Credit Score Models – Construction
14
How to build a credit score model
Estimation window – track loans
Default = 1
Survive = 0
Collect loan/borrower
characteristics
These characteristics should be related to the borrowers
likelihood of default – for example leverage
Credit Score Models – Construction
(Loan\Borrower Characteristics)
15
What do you want to know about the loan/borrower?
Loan/borrower characteristics:

Reputation: Years at the bank, borrowing history, # of loans repaid …

Leverage: Leverage ratio, Tier I and Total capital ratios

Future income: Earnings volatility (repayment capacity)

Collateral: Market value of physical assets backing the loan

Loan characteristics: Term, interest rate, type …


Business cycle effects: market beta, earnings sensitivity to GDP or other
economic indicators
Interest rates: earnings, profitability, investment … sensitivity to interest
rates .
Credit Score Models – Construction
(Basic Estimation)
16
Estimation:
Estimation Window
Default = 1
Survive = 0
Object is to build a model that we can use to predict
default in the next period
Credit Score Models – Construction
(Basic Estimation)
17
Estimation:
Get loan/borrower
information at the
beginning of the year
Default =a

Estimation Window
Default = 1
Survive = 0
Tier I
b1 Ratio
of loans
 b2 #w/
 b3
bank
Earnings
Volatility
 b4
Collateral
Value
 b5
Loan
Covenants
We use this information collected at the beginning of the
year to estimate the model parameters (weights)
Credit Score Models – Construction
(Prediction)
18
Prediction:
Estimation Window
Default = 1
Survive = 0
Tier I
Loan
of loans
Earnings
Collateral  b
Default
=0
.020.8
b2 #w/

b

b

.09
Default =a  b1 Ratio  0.1

1.2

0.01
3
5
4
Covenants
bank
Volatility
Value
After estimating the model, we can fill in the parameters and
the model can be used to forecast loan/borrower defaults
Credit Score Models – Construction
(Prediction)
19
Prediction:
Estimation Window
Default = 1
Survive = 0
Default =0.020.8
Tier I
Loan
# of loans
Earnings
Collateral
 0.1

.09

1.2

0.01
Ratio
Covenants
w/ bank
Volatility
Value
Example: Suppose we collect this information for Netflix at the beginning of 2012
Default =0.020.8 0.10  0.1 3  1.2 0.05  0.01 2  .09 0.45 = 0.04
We would expect Netflix to default with a
probability of 4% over the next 1 year
Credit Score Models – Estimation
20
Estimation Method
Estimation Window
Default = 1
Survive = 0
How can you estimate the parameters (weights)?
Default =a

Tier I
b1 Ratio
of loans
 b2 #w/
 b3
bank
Earnings
Volatility
 b4
Collateral
Value
 b5
Loan
Covenants
1
0.04
0
2.23
2M
4
0
0.07
3
0.45
2.3M
12
⁞
⁞
⁞
⁞
There are many⁞ different ways
to estimate the
parameters ⁞
1
7
1.23
0.32M
8
0.10
• 0 Ordinary Least
Squares (OLS) regression is the most straight forward
2
2.8
0.8M
6
0.02
• 0 Logit regression
0
1.2
5.2M
3
0.05
• Discriminate analysis
Its impossible to exactly
explain default with a model
so we allow for an error
Credit Score Models – Linear Probability
21
Default =a
 b1
Tier I
# of loans
Earnings
Collateral

b

b

b
Default
=
a

b
X

b
X

b
X
…

b
2
3
4
1
2 2 Volatility
3 3
nX
n e
Ratio
w/ 1bank
Value
 b5
Loan
Covenants
Linear Probability Model

Uses Ordinary Least Squares (OLS) regression to estimate parameters
Choose a, b1, b2, b3, b4, …bn, so that the sum of the squared residuals
is as small as we can get it (minimized) e
Find a, b1, b2, b3, b4, …bn to minimize 
#loans
e
i 1

2
i
PROBLEMS:


Can predict default probabilities outside of the range from 0-1
OLS should only be used with continuous dependent variables
Credit Score Models – Logit
22
Default =a
 b1 X1  b2 X2  b3 X3 …  bn Xn  e
Logit Model
Adjustment to correct for problems with the Linear Probability Model
1.
Estimate the probability of default using the Linear Probability Model
2.
Transform the probability using the Logit transformation
PD Logit 
1
1  e  PDLP
In practice: (only if you are interested)



In practice, a more sophisticated estimation is used.
We say that e follows a Logit distribution, then we use an estimation technique called maximum
likelihood to find a, b1, b2, b3, b4, …bn,
it is more precise
Credit Score Models – Discriminate Analysis
23
Default =a
 b1 X1  b2 X2  b3 X3 …  bn Xn  e
Linear Discriminate Analysis




Just another way of coming up with a, b1, b2, b3, b4, …bn
The estimation technique is more complicated than either the Linear
Probability model or the Logit model
The end result is still just a set of parameters
The reason that we talk about it is because there is a famous application
called the Altman Z – Score that estimates a firm’s probability of
default
Credit Score Models – Discriminate Analysis
(Altman Z–Score)
24
Altman Z-score

Used Linear Discriminate Analysis to estimate the model below
Calculation:
Z  1.2 X1  1.4 X 2  3.3X 3  .6 X 4  1X 5
X1 = Working Capital/Total Assets
X2 = Retained Earnings/Total Assets
X3 = EBIT/Total Assets
X4 = MV Equity/BV Long-Term Debt
X5 = Sales/Total Assets
Evaluation:



Z > 2.99 -“Safe” Zone
1.81  Z  2. 99 -“Grey” Zone
Z  1.81 -“Distress” Zone
Kaplan Associates has estimated the following linear probability model using loan defaults over the past 4 years.
PD  0.001 0.003( Leverage)  0.0085( FICO)  0.032(lengthof credit history)
Suppose that North Star restaurant applies for a loan. They have a leverage ratio of 0.25, a FICO score of 720 and a 10 year credit history.
a) Calculate the probability that North Star defaults over the next year using the linear probability model
b) Calculate the probability that North Star defaults over the next four years using the linear probability model
c) Calculate the probability that North Star defaults over the next four years using the Logit model
25
Credit Score Models

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
11-26
27
Value at Risk (VaR)
Thinking About Credit Risk
28
What have we done so far:



Followed a group of firms – some defaulted and some did not.
Used the actual defaults vs. non-default to try to understand, in
general, what causes a firm to default on its loans.
Problem - we have to wait for firms to default to understand what
causes default
Another way of thinking:




If no firms defaulted would that mean that there is no credit risk?
The value of a loan can change simply because the probability of
default or what we expect to recover in default changes.
This is credit risk! It exists even if no firms default
Value-at-Risk is one method used to measure this
Value-at-Risk (VaR) – Concept
29

VaR asks:



Based on what has happened in the past
On a really bad day, how much will I lose on my loan position?
How to answer this question:
1.
2.
3.
4.
5.
Collect past returns – for example, one year of daily returns
Calculate the mean and standard deviations of daily returns
Assume a normal distribution
Declare a significance level – for example 99%
Find the Value-at-risk (VaR) – Value that the company’s losses
will exceed only 1% of the time – over the return horizon (next
day)
Value-at-Risk (VaR) – Assumption
30


Returns are considered normally distributed, but this assumption
can cause problems
What do we need to define a normal distribution

Mean & Standard deviation – (that’s it!)
Is this normal?
Mean = 0.0090
Stdev = 0.0034
Value-at-Risk (VaR) – Assumption
31


Returns are considered normally distributed, but this assumption
can cause problems
What do we need to define a normal distribution

Mean & Standard deviation – (that’s it!)
Is this normal?
Mean = 0.0053
Stdev = 0.046
Value-at-Risk (VaR) – Example
32
Find the one-day 95% value at risk for a bond with 1,000 face value if the price
is currently $723.98.

First of all, what are we looking for?


We are looking for a threshold value for daily losses that will only be exceeded 5% of
the time. That is, we have a 5% chance of losing more than this value tomorrow.
Lets start by collecting a year of historical daily returns

We work with returns because they are usually normally distributed – prices are not!
Based on the data that we have collected, there is a 5% probability
of experiencing a return below this threshold tomorrow
Once we find this
return, we can back
out the VaR(95%)
Value-at-Risk (VaR) – Example
33
Find the one-day 95% value at risk for a bond with 1,000 face value if the price
is currently $723.98.
Step #1: Calculate the mean and standard deviation
Mean = 0.005473
Stdev = 0.009128
Step #2: Find the 95% VaR
Mean = 0.005473
Stdev = 0.009128
We want to find the return
that gives us 5% of the area
under the curve in the tail
How do you do it?
5%
Value-at-Risk (VaR) – Example
34
Find the one-day 95% value at risk for a bond with 1,000 face value if the price
is currently $723.98.
Step #2: Find the 95% VaR (continued )
These areas and their corresponding z-values
are all tabulated for the standard normal
distribution. So, we can go to the normal
tables and find the z-value for which 5% of
the area under the curve is in the left tail.
What does that tell us?
Standard Normal
Mean = 0.0
Stdev = 1
5%
-1.64
For any normal distribution, this value occurs
1.64 standard deviations below the mean
Value-at-Risk (VaR) – Example
35
Find the one-day 95% value at risk for a bond with 1,000 face value if the price
is currently $723.98.
Step #2: Find the 95% VaR (continued )
These areas and their corresponding z-values
are all tabulated for the standard normal
distribution. So, we can go to the normal
tables and find the z-value for which 5% of
the area under the curve is in the left tail.
Standard Normal
Mean = 0.0
Stdev = 1
5%
-1.64
We know that “X” is 1.64 standard deviations
below the mean
Our Distribution
Mean = 0.005473
Stdev = 0.009128
X = 0.005473 – 1.64(0.009128) = -0.0095
Start at the mean
subtract 1.64 Standard deviations
in this case 0.009128
5%
X
Value-at-Risk (VaR) – Example
36
Finds the one-day 95% value at risk for a bond with 1,000 face value if the price
is currently $723.98.
Step #2: Find the 95% VaR (continued )
Our Distribution
Mean = 0.005473
Stdev = 0.009128
5%
-0.0095
X

There is a 5% chance that the daily return will be less than –0.0095
-0.0095
Value-at-Risk (VaR) – Example
37
Finds the one-day 95% value at risk for a bond with 1,000 face value if the price
is currently $723.98.
Step #2: Find the 95% VaR (continued )
Our Distribution
Mean = 0.005473
Stdev = 0.009128
5%
-0.0095
X


There is a 5% chance that the daily return will be less than –0.0095
So your Value-at-Risk is:
VaR = ($723.98)(-0.0095) = –$6.88
There is a 95% chance that the maximum daily loss (tomorrow) will be less than $6.88
OR
There is 5% chance that we will lose $6.88 or more tomorrow
Lorden Investments has a loan portfolio with a current value of $172M . The mean an variance
of the value weighted daily return on their portfolio is 0.0181 and 0.0004 respectively. Find the
99% value at risk for the loan portfolio
38
2.33
39
RAROC Model
RAROC Models
40
Risk Adjusted Return on Capital
RAROC 
1 year net income on loan
 Funding Costs
Loan Risk
Accept loan
1-year net income on the loan:
1 year net
income on loan =
Interest
Earned
Funding
+ Fees – cost
RAROC Models
41
Risk Adjusted Return on Capital
RAROC 
1 year net income on loan
 Funding Costs
Loan Risk
Accept loan
Loan Risk:
Extreme change in the
credit risk premium

Option #1

Option #2
Use value at risk to estimate the expected loss in loan value for the extreme
case
LN
ΔR
 -DLN
LN
(1  R)
The Lucre Island Community Bank (LICB) is planning to make a loan of $5,000,000 to the Dunder-Mifflin
Paper Company. It will charge a servicing fee of 50 bps, the loan will have a maturity of 8 years, and a
duration of 7.5 years. The cost of funds (RAROC benchmark) for the bank is 10%. Assume that LICB has
estimated the maximum change in the risk premium on the paper processing sector to be approximately
4.2%, The current market interest rate for loans in this sector is 12
42
Lecture Summary
43

We looked at three different ways to measure credit risk:

Measuring Credit Risk

Credit Score Models






Linear Probability
Logit Model
Linear Discriminant
Value-at-Risk (VaR)
Risk Adjusted Return on Capital - RAROC
Other Models
44
Appendix
Other Models
Term Structure Based Methods

We can use the credit spread in the market to determine the
level of risk probability of default using zero coupons and
strips

Suppose the contractual promised return on a corporate bond
is k –the expected return is then
p (1+ k)+(1-p)(0)
Assuming zero recovery

Suppose the FI require a return equal to the risk free rate i
p (1+ k)+(1-p)(0)= 1+i
11-45
Term Structure Based Methods
46

Then the probability of survival is:
P

1 i
1 k
Suppose we are looking at a
corporate bond that has secondary
market prices. How would this change?
If we allow for recovery as a percent of repayment
p(1 k)  (1- p) (1 k)  1  i
Over what horizon?
(1  i)   (1  k )
P
(1  k )(1   )
Term Structure Based Methods
 May
be generalized to loans with any maturity or to
adjust for varying default recovery rates
 The loan can be assessed using the inferred probabilities
from comparable quality bonds
11-47
Mortality Rate Models
Similar to the process employed by insurance companies to
price policies; the probability of default is estimated from
past data on defaults
 Marginal Mortality Rates:

MMR1 ( B grade bonds) 
Value of B grade bonds defaultedin year1 of issue
Value of grade B outstanding in year1
MMR2 ( B grade bonds) 

Value of B grade bonds defaultedin year 2 of issue
Value of grade B outstanding in year 2
Has many of the problems associated with credit scoring
models, such as sensitivity to the period chosen to calculate
the MMRs
11-48
Option Models
Equity holders:

Equity holders view the bond as the purchase of a call option on the
value of the firm
Bond Principal
(B)
Assets (A)
If the project fails the
managers (equity holders)
default on the bond
If the project succeeds the
managers (equity holders)
pay off the bond and keep
A-B proceeds
11-49
Option Models
Debt holders:

Debt holders view the bond as the sale of a put option
Bond
Principal (B)
Assets (A)
If the project fails the Debt
holders receive the
remaining collateral A
If the project succeeds the
debt holders receive full
repayment (B)
11-50
Applying Option Valuation Model

Merton showed value of a risky loan:


F (t )  Beit Beit N (h1 )  N (h2 )
 2t  ln(Beit / A)
h1 
 t
 2t  ln(Beit / A)
h2 
 t
where
F(t) = value of risky debt
ln = Natural logarithm
i = Risk-free rate on debt of equivalent maturity
t  remaining time to maturity
B = principal amount on the bond
11-51