Comparison of Estimation Methods of Structural Models of Credit Risk Jeff Blokker, Shafigh Mehraeen, Won Chase Kim, Bobak Javid, and John Weng MS&E 347.

Comparison of Estimation Methods of Structural Models of Credit Risk

Jeff Blokker, Shafigh Mehraeen, Won Chase Kim, Bobak Javid, and John Weng

MS&E 347 Term ProjectStanford University

June 2009

Structural Models

• Structural models refer to models that look at the evolution of the capital structure of the firm to evaluate their credit risk.

• Merton’s model (1974) was the first modern credit risk model that was considered a structural model.– It assumes the capital structure of the firm is composed of equity St and

a zero coupon bond of value Dt with face value F. – Then the asset value of the firm is the sum of the equity and debt.

– Assumptions• No transaction costs, no bankruptcy costs, no taxes, • infinite divisibility of assets, unrestricted borrowing and lending, • constant interest rate• GBM of firm’s asset value.

t t tV S D

Merton’s Model• If the value of the firm at the maturity date T is less than K then the firm will

be unable to repay the debt.

• The payoff structure at T is:

0T

T

DefaultS

V F Otherwise

TT

V DefaultD

F Otherwise

T

F

t

Vt

Default

Merton’s Model

• The firm’s equity St represents a European call option on the firm’s assets with maturity T.

• The Bond represents a risk free loan F with maturity T plus selling a European put option with strike F and maturity T

• Merton’s model assumes that the firm can only default at time T.• The value of the firm is assumed to follow the SDE

• With the volatility of the firm’s asset value, a constant interest rate r, and risk neutral Brownian motion

( )T TS V F

tV t V t

t

dVdt dW rdt dW

V

( )T TK F F V

VtW

Merton’s Model

• Applying the Black Scholes equation to the equity value of the firm yields

• To implement Merton’s model we need an estimate of :– Volatility of the asset value - – Drift of the asset value -

( )( ) ( )r T tt tS V d e F d

( )

21ln ( )

2

r T tt

V

V

e VT t

Fd

T t

Vd d T t

V

First Passage Model

• The first passage model is an extension of the Merton model

• Default at any time T1 < T if the asset value Vt crosses the barrier K.

T

F

t

Vt

DefaultK

T1

Default

First Passage Model

• At T the value of the equity is

• This is a Down and Out call option with formula

when F>=K

when F<K

[0 ]{ min ( ) }( ) 1

tt T

T T V KS V F

2 2

2 21 1

( )( , , , , ) ( ) ( )V V

r r

r T tt BS t V t

t t

K KS C V r F T t Fe h V h

V V

22

2

1ln ( )( )

2 Vt

V

Kr T t

FVh

T t

2

2

1ln ( )( )

2 Vt

V

Kr T t

Vh

T t

Model Calibration

• To implement the first passage model we need an estimate of

– Asset volatility -

– Default barrier - K

– Drift -

• We compare three methods for calibration:

– Inversion Method

– MLE

– Iterative Method - KMV

V

Inversion Method• for Merton’s model• for First Passage model

• From Ito’s formula we get

• Comparing coefficients of the two SDE equations we conclude that

where f is a simple call option (Merton) or down-and-out call option (First Passage model)

( , , )t t VS f V t

2 2 2

22V t

t t V t tt t t

Vf f f fdS V r dt V dW

t V V V

( , , )t VE t t V

t

f V tS V

V

( , , , )t t VS f V K t

Maximum Likelihood Estimate (MLE)

• Proposed by Duan (1994)• Given a time sequence of equity values , we can estimate a time

sequence of asset values , volatility , drift , and the barrier K.• We denote the probability density function for the equity value

at ti given the equity value at ti-1 and the parameter vector .• Then the log-likelihood is given by

• Using the previously defined function and assuming it is differentiable and invertible we can write

where is the P-density of Vt given Vt-1.

1,...,

nt tS S

1,...,

nt tV V V1

( | , )i it tg S S

12

( ) ln ( | , )i i

n

t ti

L h S S

( , )t tS f V t

1 11

1 1

( ; ) | ( ; ),( | , )

( ( ; ); )t t

t tt

g F S F Sh S S

F F S

1( | ; )t tg V V

Maximum Likelihood Estimate (MLE)

• MLE for the Merton’s Model– Letting be the time between observations

where

2

20 2

1 1 ( 1)

ˆ1 1ˆ( , ; , ,..., ) ln(2 ) ln( ) ln ( ) lnˆ2 2 2 2

n nE kh

h nh V khk kV k h

Vn nL S S S d

V

2ˆln

2ˆ

t V

t

V

Vt

Fd

t

h

Maximum Likelihood Estimate (MLE)

• MLE for the First Passage Model

22

0 21

2

1

ˆ( ( ) )1 2( , ; , ,..., ) ln exp22

ˆ ( )ln( ) ( )( )

2ln

n kE

h nhk

khn

k

R hL S S S

hh

Vr T kh

FT kh

Iteration Method - KVM

• Estimation of and • Asset values Vt are implied from equity value

– Returns and

– Volatility

– Drift

• Repeat until convergence.

• Equivalent to EM algorithm and asymptotically converges to ML• For the Merton’s model, much faster than ML• For the First Passage model, no analytical formula.

V

( , )t tS f V

1

1 ˆn

kk

R Rn

( 1)ˆ ˆ ˆln( / )i i iR V V

2 2

1

1 ˆˆ ( )n

V kk

R Rn

21 1

ˆ ˆ2 VR

Monte Carlo Simulation Environment

• Asset value paths are generated by GBM with constant parameters– V0=1.5– F = 1.0– K/F = 0.8 or 1.2– T = 2– volatility = 0.3– Drift = 0.1– R = 5%

• 2500 samples generated and down-sampled to 250 per year– To reduce bias (In reality, we only observe daily equity values)– Only keep the value process which does not default

• Converted to equity value paths by BS formula (call or DOC)• Use equity paths in each model to recover parameters

Results – Merton Model

Merton Model Method

Mean ML 0.2984 0.1275

STD ML 0.0174 0.2615

Mean Inversion 0.3245 0.1366

STD Inversion 0.0347 0.2632

Mean Iterative 0.2992 0.1278

STD Iterative 0.0175 0.2617

Volatility Drift

Results –First Passage Model F>=K

Cox Model (F>K) Method

Mean ML 0.2957 0.1362 0.7426

STD ML 0.0224 0.2576 0.2257

Mean Inversion 0.3468 0.1607 0.4330

STD Inversion 0.0848 0.2579 0.1524

Volatility Drift Default Barrier

Results –First Passage Model F<K

Cox Model (F<K)

Method

Mean ML 0.3033 0.2729 1.1894

STD ML 0.0516 0.2313 0.1364

Mean Inversion 0.6641 0.6632 0.2211

STD Inversion 0.2243 0.2925 0.1982

Volatility Drift Default Barrier

DJIA 2003 - Merton Model ML Volatility Drift Equity to Debt ratio (S/F)

3M 0.1527 0.2818 5.6689

ALCOA 0.1698 0.3005 1.0928

PHILIP MORRIS 0.1689 0.2322 1.2633

AMERICAN EXPRESS 0.0672 0.0896 0.3730

AIG 0.0671 0.0358 0.2926

BOEING 0.1037 0.1091 0.5879

CATERPILLAR 0.1143 0.2688 0.7583

CITI 0.0398 0.0660 0.2073

DU PONT 0.1375 0.0704 1.5864

EXXON 0.1316 0.1435 3.0847

GE 0.0820 0.0902 0.5388

GM 0.0152 0.0364 0.0599

HP 0.2499 0.1958 1.7116

HONEYWELL INTERNATIONAL 0.1522 0.1959 1.1639

IBM 0.1510 0.1132 1.9118

INTEL 0.3384 0.6756 17.4645

CHASE 0.0224 0.0436 0.0865

JOHNSON & JOHNSON 0.1882 -0.0268 8.6067

MCDONALDS 0.2055 0.2989 1.8293

MERCK 0.1983 -0.0950 4.0999

MICROSOFT 0.2722 0.0637 17.6890

PFIZER 0.2007 0.1386 7.3363

SBC 0.1848 -0.0063 1.2807

COCA COLA 0.1810 0.1469 8.4725

HOME DEPOT 0.2829 0.3646 8.2828

PROCTER & GAMBLE 0.1076 0.1285 4.2341

UNITED TECHNOLOGIES 0.1426 0.2803 1.6363

Verizon 0.1240 -0.0261 0.7291

WAL MART 0.1825 0.0471 4.8501

DISNEY 0.1856 0.2096 1.4929

DJIA (2003) - Cox Model ML Volatility Drift Barrier Level Barrier to Debt ratio (K/F)

3M 0.1527 0.2818 0.5625 0.5878

ALCOA 0.1698 0.3005 1.4516 0.7102

PHILIP MORRIS 0.1689 0.2322 4.8873 0.7004

AMERICAN EXPRESS 0.0672 0.0896 6.4314 0.4375

AIG 0.0671 0.0358 43.0674 0.8367

BOEING 0.1037 0.1091 2.3736 0.5186

CATERPILLAR 0.1143 0.2688 1.5075 0.5371

CITI 0.0398 0.0660 95.0543 0.9163

DU PONT 0.1375 0.0704 1.4721 0.5567

EXXON 0.1316 0.1435 6.7947 0.8492

GE 0.0820 0.0901 26.6027 0.5073

GM 0.0152 0.0364 23.6416 0.6336

HP 0.2499 0.1956 2.6907 0.7619

HONEYWELL INTERNATIONAL 0.1522 0.1959 1.2275 0.6426

IBM 0.1510 0.1132 4.6293 0.6127

INTEL 0.3384 0.6756 1.1676 1.3008

CHASE 0.0224 0.0436 10.7869 0.1467

JOHNSON & JOHNSON 0.1882 -0.0268 1.6901 0.9232

MCDONALDS 0.2054 0.2990 1.1378 0.8107

MERCK 0.1983 -0.0950 2.7054 0.8988

MS 0.2722 0.0637 2.3658 1.4921

PFIZER 0.2007 0.1386 3.8797 1.4332

SBC 0.1848 -0.0062 4.7038 0.7418

COCA COLA 0.1810 0.1469 0.7987 0.6134

HOME DEPOT 0.2092 0.2534 4.7739 5.6025

PROCTER & GAMBLE 0.1076 0.1285 0.9861 0.3515

UNITED TECHNOLOGIES 0.1426 0.2803 1.2504 0.5882

Verizon 0.1240 -0.0261 9.0159 0.6522

WAL MART 0.1825 0.0471 5.2550 1.0602

DISNEY 0.1855 0.2096 2.0560 0.7540

Empirical analysis: an example

• From the model we can calculate corporate default probability

Conclusion

• Three estimation methods are compared for two structural credit models

• For Merton’s model, ML and KMV are equivalent and superior to inversion

• For the first passage model, ML is the only option but estimation of barrier is not an easy problem.

• Drift estimation is also difficult but it is out of our interest• When K/F is small, two models does not make much difference• Further research must be done for benefits of the first passage

model• Results from this projects can be extended for various applications

– Default probability estimation– Term structure of credit spread