III MODELLING WEEK UCM Master in Mathematical

Engineering -

UCM Madrid, June 22-30 , 2009PDa

PDb

VaR

Contents1.

Introduction

2

.

Values

of

constants

3.

VaR

for single units

4

.

Combining

the

units

5.

Macroeconomics

6.

Conclusions

Complubank has two businesses: credit cards and mortgages. Therefore, it has two RU’, each composed of a number of contracts.

IT staff at CompluBank keep in their database each contract of the bank with its LGD, LossGiven Default, and EAD, Exposure at default.

Along the problem LGD and EAD will be considered deterministic variables. That’s to say, along the horizon of risk measurement they will not change.

However, they did not keep track of the PD of each contract. All we know is that, at themoment of measure, all contracts in a given RU have same PD, which corresponds to theprobability of default within 1 year horizon.

This PD should be an unconditional default probability. That means this probabilityshould not be conditionned to any particular state of the economy realization.

Introduction

Moreover, for each RU we have a time series of conditional PDz for a whole economic cycle(from 1990 to 2007). In each of these series, for every three month period, we have the PD of the RU conditionned to the macroeconomic situation in that moment.

We assume that the relation between quarterly known PDz and the unconditionnal PD (unknown) is given by the Vasicek function:

where:

Z is assumed to be a N(0,1), common to all contratcs in a RU. It is called systemicfactor, and represents the state of the economy in given time horizon. It is thefundamental risk driver of the loss.

ρ is an unknown parameter between 0 and 1.

Correlation Ө between the Z factors of risk units can be derived from the correlation betweensystemic factors of each RU.

Introduction

( )⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

=−

ρρ

1

1 zpdNNpd Z

Tha aims of the week have been:

-

Be able

to

stablish

a causal relation

between

the

systemic

factor os each

RU

and some

macroeconomic

factors.

-

Be able

to

compute the

loss

distribution

of

both

A (credit

cards)

and

B (mortgage

loans)

risk units

at CompluBank.

-

Be able

to

compute aggregated

loss

distribution

of

CompluBank, taking

into

account

the

diversification

/ correlation

of

both

risk units.

Once the above is reached:

-

Analyze

the

sensibility

of

percentile

of

loss

distribution

of

both

RUs

A

and

B

and

global, to

changes

in PD, ρ

and Ө.

-

Estimate

losses

in macroeconomic

stress scenarios.

Introduction

To reach our aims, we base our study in Vasicek model, which aim is to measure the loss associated with the risk of an amount of loans, and other definitions.

Assuming EAD and LGD are deterministic, in a given horizon, each RU has a lossdistribution given by:

where:

N is the number of contratcs in the risk unit.

PDiz is the probability of default along the horizon, conditionned to a state of the

economy. Such state of the economy is the risk factor that codifies the evolution ofthe losses.

( )∑=

⋅⋅=N

iii

zi LGDEADPDBerL

1

Introduction

Loss distribution of M risk units of a bank will be given by:

In order to properly measure the risk, it is essential to take into account the correlationbetween contracts within the same risk unit and the correlation between different risk units.

Introduction

MLLL ++= ...1

Contents1.

Introduction

2

.

Values

of

constants

3.

VaR

for single units

4

.

Combining

the

units

5.

Macroeconomics

6.

Conclusions

PD values are the probability of default independentof the economic situation (i.e. unconditionalprobability of default).

Using

the

data we

have

obtained

PD’s values

as the

average of

the

PD at each

year

for each

risk

unit. 6,67%

Year Conditional PD A Conditional PD B1990 1.40% 7.84%1991 2.16% 5.44%1992 3.08% 6.38%1993 4.56% 9.57%1994 5.03% 5.56%1995 1.88% 4.86%1996 2.41% 3.21%1997 1.88% 1.80%1998 1.74% 0.99%1999 1.52% 0.97%2000 1.63% 0.87%2001 1.38% 0.54%2002 1.18% 0.44%2003 1.04% 0.40%2004 0.63% 0.47%2005 0.52% 0.43%2006 0.44% 0.44%2007 0.89% 0.62%

Average (PD) 1,85% 2,82%

Values of constants

6,67%

Values of constants

( 1 ) ( )P z c N cρ ρ ε+ − < =

( 1) ( )P Default P X c= = <

1( )c N PD=

RU A RU B

C -2.5850 -1.97020

Values of constants

As we now the unconditional probability of default, PD, we can calculate ρ:

( )

( ) ( )( )ρ

ρρρ

ρρρ

ρ

−=−+

−=

=−−

=⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

=

−

−−

−

1)(

11

)()1(

11

)())((

1

12

11

zVarianzapdNVarianza

zpdNVarianzazpdN

VarianzapdNVarianza z

( )

( )))((1))((

)))((1())((1))((

1

1

11

1

z

z

zz

z

pdNVarianzapdNVarianza

pdNVarianzapdNVarianzapdNVarianza

−

−

−−

−

+=

+⋅=

=−⋅

ρ

ρ

ρρ

ρ values are a measure of the independence of the debtors. ρ = 1 implies alldebtors are 100% correlated (if one defaults, everybody will) but if ρ = 0 all debtorsdefault independently.

We

use the

following

relation

to

obtain

ρ

values

for both

RU

giving

( )1

1

( ( ))1 ( ( ))

z

z

Varianza N pdVarianza N pd

ρ−

−=

+6,67%

RU A RU B

ρ 6,67% 19,38%

6,67%6,67%6,67%

Values of constants

Values of constants

One of the last variables we have to know is Ө, that we estimate:

( ) ( )( ) ( )

( ) ( )( )ZB

ZA

BBAA

BABA

pdNpdNnCorrelatio

zpdNzpdNnCorrelatio

ZZnCorrelatioZZianceCo

11

11

,

1,

1

,,var

−−

−−

=

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

=

===Θ

ρρ

ρρ

Θ value

We

have

estimated

θ, that

in our

model

represents

correlation

between

the

probabilities

of

default

of

the

two

risk

units, through

the

following

formula

where and represents

the

contitioned

probability, obtaining

the

value

6,67%( ) ( )( )1 1,Z ZA BCorrelation N pd N pd− −

ZApd Z

Bpd

Θ=79,7%

Values of constants

Contents1.

Introduction

2

.

Values

of

constants

3.

VaR

for single units

4

.

Combining

the

units

5.

Macroeconomics

6.

Conclusions

VaR

for single units

Procedure

Generate a random (N(0,1)) variable z to describe a macroeconomic situation

Get the PD for this z

Generate a Bernoulli distributed random variable epsilon (N(0,1)) based on PD for each client like a individual feature of the client

•

If PD is higher than epsilon then the default event occurs, not

in the other case

1 1

( 1/ ) ( 1 )

( ) ( )( ) ( )1 1

t t

zt t

P D z z P z c

N PD z N PD zP PD N

ρ ρ ε

ρ ρε

ρ ρ

− −

= = = + − <

− −< = =

− −

Continue Procedure

The loss generated for this client is

The total loss of the portfolio is the sum of the loss of each client

where

n denotes the

number

of

clients

for

each

units

( )zClientLoss Ber PD EAD LGD= ⋅ ⋅

( )1

nzi i i

iL Ber PD EAD LGD

=

= ⋅ ⋅∑

VaR

for single units

VaR

for single units

We simulate the process 10000 times and get the distribution show below

VaR

for single units

PORTFOLIO A PORTFOLIO B

Results

for

100.000 Simulations.

99.90

99.97

799.21891.1975

VaRVaR

=

=

99.90

99.97

1684.51890.2

VaRVaR

=

=

VaR

for single units

Varying the rho we get how VaR varies in the graph below.

99.9%€8990AVaR

AA ρρ∂

=∂

99.97%€9810AVaR

AA ρρ∂

=∂

This also gives us the numerical derivative with respect to var around rhoA = 0.067

VaR

for single units

99.9%€9140BVaR

BB ρρ∂

=∂

99.97%€11040BVaR

BB ρρ∂

=∂

This also gives us the numerical derivative with respect to var around rhoA = 0.067

Contents1.

Introduction

2

.

Values

of

constants

3.

VaR

for single units

4

.

Combining

the

units

5.

Macroeconimics

6.

Conclusions

Combining the units

What if portfolios are correlated ?

If so we need two normal distributions with a given correlation

Where Θ is the correlation between portfolios

2

( (0,1))

1 ( (0,1))

PortfolioA

PortfolioB PortfolioA

Z random N

Z Z random N

=

= Θ + −Θ

Combining the units

Like we did for a single units we can do now. And we get the Loss distribusion

Var99.9 Var99.97RU A

810.5599 1050.1

RU B 1902.7 2297.5

RU AB 2532.9 2982.9

Combinig

the units

Rho a

Rho b

VaR

99.9

Rho a

Rho b

VaR

99.97

Variation of rho respect the whole portfolio

Combinig

the units

We now estimate the first derivate of VaR (99.97% and 99.9%) with respect of Θ.The plotobtained is shown below:

This also gives us the numerical derivative with respect to

99.97%€698A BVaR +∂

= Θ∂Θ

99.9%€501A BVaR +∂

= Θ∂Θ

0.8Θ ≈

Combining the units

We see the VAR obtained for the different PD

PDa

PDb

VaR

99.9

PDa

PDb

VaR

99.97

Contents1.

Introduction

2

.

Values

of

constants

3.

VaR

for single units

4

.

Combining

the

units

5.

Macroeconomics

6.

Conclusions

Macroeconomics

The aim is to establish a relationship between the macro economic factors and the systemicsfactors for each risk unit.

For this purpose we decided to use a linear regression model after a long study of theproblem.

Firstly, we calculated the systemic factors and then normalized both the systemics and themacro economic factors.

Macroeconomics

Using Excel, we calculate the regression coefficients and the errors that are involved since weneed them in our further work.

Risk

Unit A Risk

Unit Bb1 0,266197489 -0,077627772

b2 -0,79495506 -0,621544539

b3 0,08264054 -0,12181275

b4 0,048293754 -0,564335208

b5 0,193449818 -0,039757321

b6 -0,021242591 -0,073168947

b_epsilon 0,362743864 0,15134132

Macroeconomics

So we conclude that the systemic factors depend of the values of all our macro ecomonic factors.

where,

has a Normal(0,1) as distribution function.

If we fix one of the factors, we have:

but

the

macroeconomic

factors

don’t follow

a N(0,1) distribution

anymore

}{6

1, ,j j j j

i ii

z F j A Bεβ β ε=

= + ∈∑

}{6

1 12

, ,j j j j ji i

iz F F j A Bεβ β β ε

=

= + + ∈∑

}{, 1, 2, 3, 4, 5, 6 ,jiF i j A Bε ∀ = ∈

Macroeconomics

For simulating the systemic factor, we apply the Cholesky factorization using the correlationmatrix between the macro economic factors.

GDP Unemployme

nt

House_Price E3M IBEX Inflation

GDP 1 -0,37297 -0,30282 -0,40533 -0,11382 -0,02879

Unemployment -0,37297 1 -0,31707 0,57914 -0,06146 -0,39557

House_Price -0,30282 -0,31707 1 0,26396 -0,11747 0,58063

E3M -0,40533 0,57914 0,26396 1 -0,38748 -0,11351

IBEX -0,11382 -0,06146 -0,11747 -0,38748 1 -0,19195

Inflation -0,02879 -0,39557 0,58063 -0,11351 -0,19195 1

We now fix the values of the change of GDP and do a simulation to obtain a plot of VaR as a function of GDP. The plot obtained is shown below:

Macroeconomics

99.9% 99.97%€ €-16500 -18600A B A BVaR VaR

GDP GDPGDP GDP+ +∂ ∂

= =∂ ∂

This also gives us the numerical derivative around GDP growth around 2.5%:

99.9% 99.97%€ €89 1053 33 3

A B A BVaR VaRE M E ME M E M

+ +∂ ∂= =

∂ ∂

Macroeconomics

Contents1.

Introduction

2

.

Values

of

constants

3.

VaR

for single units

4

.

Combining

the

units

5.

Macroeconimics

6.

Conclusions

The variation of VAR is influenced by the combination of many factors.

To define the state of the economy the highest ifluence is given by the unemployment.

The most importan factor is the correlation of each portfolio to the economy, represented byRho.

This shows that diversification reduces risk

Conclusions