Risk Management completo

EXAMPLE:

EAD = drawn part + undrawn part*CCF

Drawn part is already in balance. With other two parameters we estimate what

part we will have on balance in the next 1 year. For a credit 100 , 80 drawn, on

balance 20 off balance, this latter part when we have to evaluate the riskiness

of our portfolio we have to consider it, how much I expect my client will use

(drawn) in 1 year? I use the CCF to estimate the amount I will drawn in one

year.

For each asset class we have several weights we can apply, two dimension

system, depends on the portfolio characteristics and the rating.

IRB approach:

Internal models evaluated and accepted by regulators. Internal rating is a

number that synthetize the riskiness of an asset. In order to develop a rating

model the Default definition is the important part to be understood. Without

going in depth the default 3 main default status, passed due, unlikely to pay,

non performing, with respected the time horizon of delay of payments.

Parameters used are PD, intended as a PD in 1 year. Higher PD doesn’t mean

higher expected loss, because the result depends also by LGD and EAD.

ECL = PD*LGD*EAD. LGD is the loss I expect to have in one year in case of

default. It can be measured transaction based, observing the recovery process

of the transaction, observe the default event and the end of the process,

formula on slides. EAD is equal to notional amount you observe on balance,

plus the ratio of undrawn part.

A bank can use the internal rating also for a single or some asset class, eg only

for corporate receivebles. Two different approach of IRB

- Banks estimate only the PD and uses supervisory data for other parameters

- Own estimates on all parameters

Calculation of capital requirement with IRB approach

The result is a constrain, it must result a value equal or higher than 8%.

Regulatory capital depends on the credit risk parameters internally estimated.

It represent the % of….. with that function we calculate also the unexpected

loss, that depends by EAD and LGD.

The standardized approach is the line with stairs. For the internal rating

approach we have a continuous function that depends from the single value

assumed for PD. When PD began 100% unexpected loss began 0 and expected

loss began 100%. Lines calculated for a given level of LGD.

2)Managerial use

Use of credit risk parameters in business activities. First use is for accounting

purposes, the banks have to calculate provisions to cover expected losses,

given by the product of EAD and LGD. Banks have to be compliant with

accounting principles, IAS 39 and IFSR 9, calculate expected credit loss with a

forward looking approach, using multiperiod risk parameters, calculated using a

time horizon of lifetime perspective, until maturity. The provisions is a cost item

of profit/loss accounts (IS), higher provisions causes a reduction of profitability.

Expanding horizon we have higher provisions.

- Other use is calculate the rating of their clients, by that depends the

price of loan for a customer.

- Other use is calculate the risk adjusted price of loans, adjust the price

using the credit risk parameters

- Parameters used for the definition of RAF, plans of the bank.

Regressione logistica

Partendo dal modello lineare classico:

y = + , where = + x + … + x ; ~ N(0,

        )

i i i i 0 1 1,i k k,i i

hence y ~ N( ,

 )

i i

y è una misura, ma se volessimo interpretarla come una probabilità?

i

La variabile di risposta y potrebbe assumere valori 0 o 1, corrispondenti a

i

successo o insuccesso.

Media del vettore delle osservazioni:

p = P(y =1) probabilità che la variabile di risposta assuma valore 1



i i

L’estensione p = + x + … + x (valore atteso y ) è problematica:

   

i 0 1 1,i k k i

- I valori previsti possono cadere fuori dall’intervallo [0,1]

]

- Passare ai log non risolve, l’intervallo diventa [-inf, 0

E allora come facciamo ad applicare le tecniche della regressione lineare per

trattare probabilità?

Dobbiamo adottare una trasformazione. Innanzitutto definiamo l’ODDS:

p i

ODDS odds = ovvero trasformazione monotona, che mappa

 i 1−p i

[0,1] su [0,+inf)

Esempio:

p = 0,75 odds = 0,75/0,25 = 3 il successo è 3 volte più probabile

 

i i

dell’insuccesso.

- odds > 1 significa che il successo è più probabile dell’insuccesso

i

Noto l’odds, si può sempre trovare la probabilità corrispondente:

p = odds /(1+odds )

i i i

Esempio:

odds = 4 successo 4 volte più probabile dell’insuccesso p = 4/5 = 0,80

 

i i

(1 – p ) = P(y =0) = 0,2 la nostra attesa è di osservare 4 successi ogni



i i

insuccesso.

La trasformazione adottata è quindi la variabile “logit”:

( )

p i =¿

- logit = ln trasformata logistica

 

i i

1− p i

Mappa [0,1] in (-inf,+inf):

( )

p

i =¿

Il modello ln + x + … + x si chiama modello di

   

0 1 1,i k k,i

1− p

i

regressione logistica, che è lineare su scala logit, ma non è lineare sulla

scala delle probabilità. il parametro rappresenta il tasso di

  1

crescita o decrescita della curva, più è grande in valore assoluto più

velocemente la curva cresce.

Esempio:

P(y = 1) = 0,5 odds = p /(1-p ) = 1 = ln(p /1-p ) = 0

  

i i i i i i i

Considerando il nostro modello lineare: = b + b x = 0 x = - b /b

 

i o 1 1 i o 1

Quando x assume questo valore allora p è 0.5!

i i

WEEK 4 – RM

EXCEL FILE how a bank calculates credit risk for their clients or SME.

If you are not selective enough you will lose part of loans, they will become

NPL, you are going to lose money and in addition it costs a lot in terms of

regulatory capital and don’t allow to give other loans. Also important for

corporates, important to know what banks looks on. Basically, we use a logistic

regression model, based on historical data, to build a credit score (very

simplified model, with very simplified ass.):

- Get historical data, needs to contain the events that you want to model

which will be for your history a series of default and non-default for

historical clients loans.

- For each position a series of explicative variables, variables which the

bank thinks that are important to check to understand whether a client is

risky or not, the credit risk of a client. Financial statement info for

corporates. For SMA in addition important to know how this firm is

behaving in its relationship with banks (existed client), see the CFs on his

accounts.

Assume we already extracted those data. At the end we keep two explicative

variables for peoples. Retail credit risk model:

- Performance column, how those clients behave in the past, 1=default,

0=non default. There are two kinds of rating models, for new clients, to

decide if accept it or not, and for existing client, called behavioral score

because as a bank you actually know how the client behave. My historical

data can be monthly for a certain number of years, 12 months

observations at least to have a reliable model. In that column the default

we can observe can refer to different dates on the past 12 months.

- Constant column, design metrics, this is “x”

- Age, important for retail credits, for different reasons, in a certain age

better work contract, or pension. Someone older better client difficult to

fire. Higher the age lower the risk.

- Salary/Installment, the fraction between how much is your income and

how much you want to borrow. If I see a number to low or to high I have

to check, maybe can be a particular situation. There could be a

guarantee for example.

These are 20 observations. The reality is that for a retail model you have

million of observations and number of variables we have to take in

consideration are 700. It is very simplified but with same steps.

ODDS in this case is 1, because there is 50% default rate and so on, not

realistic. ODDS = number of Default/number of non default. On 20 observations

10 defaults. PD = 50%, impossible to stay in business. In reality at the

beginning I calculate a PD that is not realistic, select random a sample of 50%

of bad clients and 50% of good clients to calculate the initial PD, to understand

which one of variables better explain the risk of default. At the end we

recalibrate the model for the actual PD.

In order to understand what are the statistics that explain better credit risk we

use small samples.

Then we calculate:

- Logit function: is a linear predictor, looks like to a linear regression,

something linked to my PD if we interpret this expected value of PD given

the risk factors we use.

( )

p i =¿

Logit = ln + x + … + x

  

i 0 1 1,i k k,i

1− p i

- Logistic function (predicted probability): we can find it from logit

function, formula in the previous notes.

- Likelihood: calculation related to past defaults and logistic. I do that for

every single observatation. Once I have the likelihood for single

observation which is the likelihood to observe these data given the PD 

make the sum of them and maximize it.

Higher the Likelihood better is the model. Choose parameters in order to

make this likelihood as likely as possible. Utilizzare la funzione “solver”in excel.

(Tools Solver Use GRG non lineare). It is an optimizer. The optimization

 

depends on the initial parameters . Excel is not considered professional solver.

 i

Once we have our parameters we can estimate the ODDS.

Form a methodological point of view credit score is not complicated. I rewrite

the linear predictor and predicted probability. Just to remind the logistic

function is the transformation of logit data form linear regression to log

function. I obtain a logistic function in the graph so I achieve that there aren’t

probabilities of default lower than 0 or higher than 1.

Inference

Once build this model we have to make inference. We have to understand the

reliability parameter of the model estimated, if it is a good model. We have to

understand if we are in good case or case that in reality is not really liked to

risk variables chosen.

The PD must be very linked to our variables, otherwise the model will not work,

useless regression.

The standard way for the regulator for assess the quality of credit risk model.

Variance of a binomial variable looks like ad PD times probability of

survive. It is something I need in order to estimate the covariance matrix of

the coefficients. Matrix approach, perform calculations in different steps 

(X VX) you get a 3x3 matrix with at dia