Based on Risk Management, Crouhy, Galai, Mark, McGraw-Hill, 2000 Credit Risk Modeling Economic...

Based on Based on Risk Management, Risk Management, Crouhy, Galai, Mark,Crouhy, Galai, Mark,McGraw-Hill, 2000McGraw-Hill, 2000

Credit Risk ModelingCredit Risk Modeling

Economic Models of Credit RiskLectures 10 & 11

The Contingent Claim ApproachThe Contingent Claim Approach - - Structural Approach:Structural Approach: KMV KMV

(Kealhofer / McQuown / Vasicek)

3

KMV challenges CreditMetrics on several KMV challenges CreditMetrics on several fronts:fronts:

1.1. Firms within the same rating class have the Firms within the same rating class have the same default ratesame default rate

2.2. The actual default rate (migration The actual default rate (migration probabilities) are equal to the historical probabilities) are equal to the historical default rate (migration frequencies) default rate (migration frequencies)

• Default rates change continuously while ratings Default rates change continuously while ratings are adjusted in a discrete fashion.are adjusted in a discrete fashion.

• Default rates vary with current economic and Default rates vary with current economic and financial conditions of the firm.financial conditions of the firm.

The Option Pricing Approach: KMVThe Option Pricing Approach: KMV

4

KMV challenges CreditMetrics on several KMV challenges CreditMetrics on several fronts:fronts:

3.3. Default is only defined in a statistical sense Default is only defined in a statistical sense without explicit reference to the process without explicit reference to the process which leads to default.which leads to default.

• KMV proposes a structural model which relates KMV proposes a structural model which relates default to balance sheet dynamicsdefault to balance sheet dynamics

• Microeconomic approach to default: a firm is in Microeconomic approach to default: a firm is in default when it cannot meet its financial default when it cannot meet its financial obligationsobligations

• This happens when the value of the firm’s This happens when the value of the firm’s assets falls below some critical levelassets falls below some critical level


5

KMV’s model is based on the option pricing approach KMV’s model is based on the option pricing approach to credit risk as originated by Merton (1974)to credit risk as originated by Merton (1974)

1. 1. The firm’s asset value follows a standardThe firm’s asset value follows a standard geometric Brownian motion, i.e.: geometric Brownian motion, i.e.:

tt ttVV )

2(exp

2

0

tt

t dZdtVdV


6


Equity: St

Assets Liabilities / Equity

Risky Assets: VtDebt: Bt

(F)

Total: Vt

Vt

2.2. Balance sheet of Merton’s firm Balance sheet of Merton’s firm

7

Equity value at maturity of debt obligation:Equity value at maturity of debt obligation:

0,max FVS TT Firm defaults ifFirm defaults if

FVT with probability of default (“real world” probability with probability of default (“real world” probability measure) measure)

2

2

0

0

2dN

T

TFV

Ln

PFVP TT


8

Assets Value

VT

V0

Probability of default

TimeT

F

E V V eT OT( )

V V T TT O T

exp 2

2

3.3. Probability of default Probability of default (“real world” probability measure) (“real world” probability measure)


• Distribution of asset values at maturity of the debt obligation

9

Time 0 T

Value of Assets V0 VT F VT > F

Bank’s Position:

· make a loan -B0 VT F

buy a put -P0 F - VT O

Total -B0 -P0 F F

B0 + P0 = Fe-rT

Bank’s pay-off matrix at times 0 and T for making a loan Bank’s pay-off matrix at times 0 and T for making a loan to Firm ABC and buying a put on the value of ABCto Firm ABC and buying a put on the value of ABC

Corporate loan = Treasury bond + short a put


·

10

Firm ABC is structured as follows:Vt = Value of Assets (at time t)

St = Value of Equity

Bt = Value of Debt (zero-coupon)

F = Face Value of Debt

1 Po = f ( Vo, F, v, r, T ) (Black-Scholes option price)

2 Bo = Fe-rT - Po

3 So = Vo - Bo (assuming markets are frictionless)

4 Bo = Fe-YTT where YT is yield to maturity

5 Probability of Default = g (Vo, F, v, r, T) = N ( - d2 )

(“risk neutral” probability measure)

6 Conditional recovery when default = VT

KMV: Merton’s ModelKMV: Merton’s Model

11

Problem:Problem:

Vo ( say =100 ), F ( say = 77 ), v ( say = 40% ), r ( say =10% ) and T ( say = 1 year)

Solve for Bo,So,YT and Probability of Default


12

`P0( = 3.37) Bo( = 66.63) So( = 33.37) YT ( =15.6%) T ( = 5.6%)

Note: In solving for P0 we get Probability of Default ( = 24.4% )

BB FeFe PPoorTrT

oo SS VV BBoo oo oo YY LL

FF

BBYY rrTT NN

ooTT TT

Solution:Solution:

PP ff VV TToo oo (( ,, ))


13

Default spread ( ) for corporate debtDefault spread ( ) for corporate debt

( For V( For V00 = 100, T = 1, and r = 10% ) = 100, T = 1, and r = 10% )

rYTT

0V

FeLR

rT

Leverage ratio:Leverage ratio:


LR 0.05 0.10 0.20 0.40

0.5 0 0 0 1.00.6 0 0 0.1% 2.5%0.7 0 0 0.4% 5.6%0.8 0 0.1% 1.5% 8.4%0.9 0.1% 0.8% 4.1% 12.5%1.0 2.1% 3.1% 8.3% 17.3%

14

KMV: EDFs KMV: EDFs (Expected Default Frequencies)(Expected Default Frequencies)

4. Default point and distance to default

Observation:

Firms more likely to default when their asset values reach a certain level of total liabilities and value of short-term debt.

Default point is defined as

DPT=STD+0.5LTD

STD - short-term debt

LTD - long-term debt

15

Asset Value

Time1 year

DPT = STD + ½ LTD

Expected growth ofassets, net

E(V)1

DD

Probability distribution of V

0

V0


Default point Default point (DPT)(DPT)

16


DistanceDistance--toto--default default (DD)(DD)

DDDD -- is the distance between the expected is the distance between the expected asset value in asset value in T T yearsyears, E(V, E(VTT) ) , and the , and the

default point, DPT, expressed in standard default point, DPT, expressed in standard deviation of future asset returns:deviation of future asset returns:

TA

T DPTVEDD

,

17

5.5. Derivation of the probabilities of default from Derivation of the probabilities of default from the distance to defaultthe distance to default

5 643 DD21

EDF

40 bp

2dNEDF

KMV also uses historical data to compute EDFs


18


Example:Example: V0 = 1,000

20%V1 = V0(1.20) = 1,200

Current market value of assets: Net expected growth of assets per annum: Expected asset value in one year: Annualized asset volatility, Default point

:

100800

DD

1 200 800100

4,

Assume that among the population of all the firms with DD of 4 at one Assume that among the population of all the firms with DD of 4 at one point in time, e.g. 5,000, 20 defaulted one year later, then:point in time, e.g. 5,000, 20 defaulted one year later, then:

EDF year1

205 000

0 004 0 4% ,

. . or 40 bp

The implied rating for this probability of default is BBThe implied rating for this probability of default is BB++

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Example:Example:Federal Express ($ figures are in billions of US$)Federal Express ($ figures are in billions of US$)

November 1997November 1997 February 1998February 1998

Market capitalization (SMarket capitalization (S00 ) )

(price* shares outstanding)(price* shares outstanding)

Book liabilitiesBook liabilities

Market value of assets (VMarket value of assets (V0 0 ))

Asset volatilityAsset volatility

Default pointDefault point

Distance to default (DD)Distance to default (DD)

EDFEDF

$ 7.8$ 7.8

$ 4.8$ 4.8

$ 12.6$ 12.6

15%15%

$ 3.4$ 3.4

12.6-3.412.6-3.4

0.15·12.60.15·12.6

0.06%(6bp)0.06%(6bp)

$ 7.3$ 7.3

$4.9$4.9

$ 12.2$ 12.2

17%17%

$ 3.5$ 3.5

12.2-3.512.2-3.5

0.17·12.20.17·12.2

0.11%(11bp)0.11%(11bp)

= 4.9= 4.9 = 4.2= 4.2

20

KMV: EDFs (Expected Default KMV: EDFs (Expected Default Frequencies)Frequencies)

4.4. EDF as a predictor of EDF as a predictor of defaultdefaultEDF of a firm which EDF of a firm which

actually defaulted actually defaulted versus EDFs of firms in versus EDFs of firms in various quartiles and various quartiles and the lower decile. the lower decile.

The quartiles and The quartiles and decile represent a decile represent a range of EDFs for a range of EDFs for a specific credit class.specific credit class.

21


4.4. EDF as a predictor of EDF as a predictor of defaultdefault

S& P

E D F

EDF of a firm which EDF of a firm which actually defaulted actually defaulted versus Standard & versus Standard &

Poor’s rating.Poor’s rating.

22


4.4. EDF as a predictor of EDF as a predictor of defaultdefault

Assets value, equity Assets value, equity value, short term debt value, short term debt and long term debt of and long term debt of a firm which actually a firm which actually defaulted.defaulted.

IVIV The Actuarial Approach: The Actuarial Approach: CreditRisk+ CreditRisk+

Credit Suisse Financial Products

24

In CreditRisk+ no assumption is made about the causes In CreditRisk+ no assumption is made about the causes of default: an obligor A is either in default with of default: an obligor A is either in default with probability Pprobability PAA, or it is not in default with probability 1-P, or it is not in default with probability 1-PAA. .

It is assumed that:It is assumed that:

• for a loan, the probability of default in a given for a loan, the probability of default in a given period, say one month, is the same for any other period, say one month, is the same for any other monthmonth

• for a large number of obligors, the probability of for a large number of obligors, the probability of default by any particular obligor is small and the default by any particular obligor is small and the number of defaults that occur in any given period number of defaults that occur in any given period is independent of the number, of defaults that is independent of the number, of defaults that occur in any other periodoccur in any other period

The Actuarial Approach: CreditRisk+The Actuarial Approach: CreditRisk+

25

Under those circumstances, the probability distribution Under those circumstances, the probability distribution for the number of defaults, during a given period of time for the number of defaults, during a given period of time (say one year) is well represented by a Poisson (say one year) is well represented by a Poisson distribution:distribution:

= average number of defaults per year

A

AP

,...2,1for !

defaults

nn

enP

n

where

It is shown that can be approximated as

The Actuarial Approach: CreditRisk+The Actuarial Approach: CreditRisk+

26

CreditRisk+: Frequency of default CreditRisk+: Frequency of default eventsevents

Credit Rating

AaaAaA

BaaBaB

Average (%)

0.000.030.010.131.427.62

Standard deviation (%)

0.00.10.00.31.35.1

Note, that standard deviation of a Poisson distribution is . For instance, for rating B: .

CreditRisk+ assumes that default rate is random and has Gammadistribution with given mean and standard deviation.

One year default rate

Source: Carty and Lieberman (1996)

1.5 76.262.7 versus

27

Distribution of default eventsDistribution of default events

CreditRisk+: Frequency of default CreditRisk+: Frequency of default eventsevents

ProbabilityExcluding default rate volatility

Including default rate volatility

Number of defaultsSource: CreditRisk+

28

CreditRisk+: Loss distributionCreditRisk+: Loss distribution

• In CreditRisk+, the exposure for each obligor is adjusted by the anticipated recovery rate in order to produce a loss given default (exogenous to the model)

29

1.1. Losses (exposures, net of recovery) are Losses (exposures, net of recovery) are divided into bands, with the level of divided into bands, with the level of exposure in each band being exposure in each band being approximated by a single number.approximated by a single number.

Notation

Obligor AExposure (net of recovery)Probability of defaultExpected loss A=LAxPA

LAPA


30

Obligor

A

Exposure ($)(loss given

default)LA

Exposure

(in $100,000)

j

Round-offexposure

(in $100,000)

j

Band

j

1 150,000 1.5 2 22 460,000 4.6 5 53 435,000 4.35 5 54 370,000 3.7 4 45 190,000 1.9 2 26 480,000 4.8 5 5

The unit of exposure is assumed to be L=$100,000. The unit of exposure is assumed to be L=$100,000. Each band j, j=1, …, m, with m=10, has an average common Each band j, j=1, …, m, with m=10, has an average common exposure: vexposure: vjj=$100,000j=$100,000j

Example: 500 obligors with exposures between $50,000 and $1M (6 obligors are shown in the table)


31

Notation

Common exposure in band j in units of L j

Expected loss in band j in units of L j

(for all obligors in band j)Expected number of defaults in band j j

In Credit Risk+ each band is viewed as an independent In Credit Risk+ each band is viewed as an independent portfolio of loans/bonds, for which we introduce the portfolio of loans/bonds, for which we introduce the following notation:following notation:

j = $100,000, $200,000, …, $1M

j can be expressed in terms of the individual loan characteristics

j = j x j


32

Band:j

Numberof

obligorsej mj

1 30 1.5 (1.5x1) 1.5

2 40 8 (4x2) 4

3 50 6 (2x3) 2

4 70 25.2 6.3

5 100 35 7

6 60 14.4 2.4

7 50 38.5 5.5

8 40 19.2 2.4

9 40 25.2 2.8

10 20 4 (0.4x10) 0.4


33

To derive the distribution of losses for the entire portfolio we proceed as follows:

Step 1:Step 1: Probability generating function for each band. Probability generating function for each band.

Each band is viewed as a portfolio of exposures by itself. The probability generating function for any band, say band j, is by definition:

jn

n

n

nj zdefaultsnPznLlossjPzG

00

)()()(

where the losses are expressed in the unit L of exposure.

Since we have assumed that the number of defaults follows a Poisson distribution (see expression 30) then:

jjjj

j

znnj

nj ez

ne

zG

!)(

0


34

Step 2:Step 2: Probability generating function for the entire portfolio. Probability generating function for the entire portfolio.

Since we have assumed that each band is a portfolio of exposures, independent from the other bands, the probability generating function for the entire portfolio is just the product of the probability generating functions for all bands.

m

jj

m

jj

jjj

jzz

m

j

eezG 11

1

)(

where

m

jj

1

denotes the expected number of defaults for the entire portfolio.


35


Given the probability generating function (33) it is straightforward to derive the loss distribution, since

these probabilities can be expressed in closed form, and depend only

on 2 sets of parameters: j and j . (See Credit Suisse 1997 p.26)

,...2,1|)(

!1

)(0

nfor

dzzGd

nnLoflossP

znn

Step 3:Step 3: loss distribution for the entire portfolio loss distribution for the entire portfolio

( ) LvnPn

nLP jnvj

j

j

=

of loss of loss:

( ) ( ) ==

j j

j

veeGP 0loss 0

V Reduced Form ApproachV Reduced Form Approach

Duffie-Singleton - Jarrow-Turnbull

37

Reduced Form ApproachReduced Form Approach

• Reduced form approach uses a Poisson process like environment to describe default.

• Contrary to the structural approach the timing of default takes the bond-holders by surprise. Default is treated as a stopping time with a hazard rate process.

• Reduced form approach is less intuitive than the structural model from an economic standpoint, but its calibration is based on credit spreads that are “observable”.

38

Example: a two-year defaultable zero-coupon bond that pays 100 if no default, probability of default , LGD=L=60%. The annual (risk-neutral) risk-free rate process is :


06.0

5.01p

08.8612.1

1004.006.010094.011

V

64.871.1

1004.006.010094.012 V

52.77

08.14.006.094.05.04.006.094.05.0 12121111

0 VVVVV

%8r

%10r5.02p

%12r

39


“Default-adjusted” interest at the tree nodes is:

%2.16108.86

10011 R %1.141

64.87100

12 R

%12152.77

64.875.008.865.00 R

)1(1 Ltt

tLtrtR

In all three cases R is solution of the equation ( ):

)1()1(1

1

1

1Ltt

trtR

1t

If , then , where is the risk-neutral expected loss rate, which can be interpreted as the spread over the risk-free rate to compensate the investor for the risk of default.

0t LrR L

40


General case: is hazard rate, so that if denotes the time to default, the survival probability at horizon t is

t

tdssEtProb

0))(exp()(

E is expectation under risk-neutral measure. For the constant we have:

t

)exp()( tEtProb The probability of default over the interval provided no default has happened until time t is:

ttt ,

tttttProb )((similar to the example above).

41


Term structure of interest rates

t

tR

Treasury curve

Corporate curve

Yield spread = L

r

R

rR,

Maturity

42


By modelling the default adjusted rate we can incorporate other factors which affect spreads such as liquidity:

lLrR where l denotes the “liquidity” adjustment premium.

if there is a shortage of bonds and one can benefit from holding the bond in inventory,

if it becomes difficult to sell the bond.

0l

0l

Identification problem : how to separate and in . Usually is assumed to be given. Implementations differ with respect to assumptions made regarding default intensity .

L LL

43


How to compute default probabilities and

Example. Derive the term structure of implied default probabilities from the term structure of credit spreads (assume L=50%).

Maturity

t (years)

Treasurycurve

(%)

Company Xone-year

forward rates(%)

One-yearforward credit

spreads

FS t (%)

1 5.52 5.76 0.24

2 6.30 6.74 0.44

3 6.40 7.05 0.65

4 6.56 7.64 1.08

5 6.56 7.71 1.15

6 6.81 8.21 1.40

7 6.81 8,47 1.65

44


For example, for year 4: , then %08.144 LFS 16.24

Cumulative probability: 74.41 4334 PPP

Conditional probability: 10.21 434 Pp

Maturityt (years)

Forwardprobabilities

of defaultt (%)

Cumulativedefauilt

probabilitiestP (%)

Conditionaldefault

probabilitiestp (%)

1 0.48 0.48 0.48

2 0.88 1.36 0.88

3 1.30 2.64 1.28

4 2.16 4.74 2.10

5 2.30 6.93 2.19

6 2.80 9.54 2.61

7 3.30 12.52 2.99

45


Generalizations:

• Intensity of the default is modeled as a Cox process (CIR model), conditional on vector of state variables , such as default free interest rates, stock market indices, etc.

where is a standard Brownian motion, is the long-run mean of is mean rate of reversion to the long-run mean, is a volatility coefficient.

Properties:

• ,

• Conditional survival probability , where

and are known time-dependent functions of time,

• The volatility of is

t tX

, tdBtdttktd tB

k

0 t

tststestp ,

stp , ttsstp ,

46


Generalizations:Generalizations:

• Intensity of the default can be modeled as a jump process:

where , - cumulative jumps by at Poisson arrival times, is mean arrival rate, is mean jump size.

t

, tdZdttktd JttNtZ t

JtN

(b.p.)

Years0

Take jumps sizes to be, say, independent and exponentially distributed.

47


Generalizations:Generalizations:

• Risk free spot rate is modeled as one factor extended Vasicek process.

where , , are similar to parameters in CIR model,

is a function defined from current term structure of interest rates. is correlated with Brownian motion of the default intensity process.

Closed form solutions for the bond prices.

t1

tr

, 1111 tdBdttrtktdr

tB1 1k 1

tB1

48


Inputs :Inputs :

• the term structure of default-free rates

• the term structure of credit spreads for each credit category

• the loss rate for each credit category Model assumptions :Model assumptions :

• zero correlations between credit events and interest rates

• deterministic credit spreads as long as there are no credit events

• constant recovery rates

Based on Risk Management, Crouhy, Galai, Mark, McGraw-Hill, 2000 Credit Risk Modeling Economic...

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