Risk and Value in Banking and Insurance

Risk and Value in Banking and Insurance

Credit Portfolio Models

Risk and Value in Banking and

Insurance

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Agenda

Portfolio models– the main concepts behind credit VaR– Non normality of credit risk– Creditmetrics

• A multinomial approach and “mark-to-model”• The risk of a portfolio

– CreditRisk+• The analogy with insurance companies and the

mathematical framework• The issue of correlations


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Portfolio models

Main portfolio models• CreditMetricsTM (J.P. Morgan)• Portfolio ManagerTM (KMV)• CreditPortfolioViewTM (McKinsey)

• We’ll focus on the first two


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Credit VaR models: common goals

• To estimate the probability distribution of all possible future losses

• To use that distribution to isolate a measure of Value at Risk associated with a given confidence level

• To estimate a measure of– Expected loss– Unexpected loss


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For a portfolio of loans, the normal distribution cannot be reasonable:

• Returns are not normal, and significantly skewed to the left

• They reflect:– Limited earnings that

are highly likely– A limited probability of

huge losses0

0.5

1

1.5

2

-150% -100% -50% 0% 50%

Vera

NormaleTrueNormal

Same and


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In the case of loans, moreover:

• Since the default is a rare event, no historical databases exist to measure the correlation between a given pair of borrowers.

0

0.5

1

1.5

2

-150% -100% -50% 0% 50%

Vera

NormaleTrueNormal


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Credit VaR models: time horizon• Time horizon

– Can be set to just one common value for all loans (e.g., 1 year):

• This makes it easier to collect, file and use the parameters (e.g., PD, EAD, correlations, transitions) needed to feed credit VaR models

• This is the time period needed to raise new capital when the bank experiences high losses

– Alternatively, it can be set – for each asset – to its true liquidation horizon:

• This means that a 5-year loan for which no secondary market is available has a risk horizon of 5 years, not of 12 months


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Credit VaR models: the time horizon

Objective Key factors Ideal horizon Risk measurement and control

Effective liquidity of positions and holding period of the bank

Residual life of exposures

Simplification Consistency with the time horizon used in estimating PD and other risk parameters

1 year

Measurement of the risk-adjusted performance (RAP) of the various bank units

Frequency of the budgeting process Frequency of the reporting process Capital allocation

1 year 1 year 1 year

Consistency between risk and capital

Time necessary to collect new capital

1 year

Implementing corrective actions on the portfolio

Average portfolio turnover period 1 year

Pricing Maturity of exposures Frequency of rate revisions

Residual life of exposures 1 year


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Credit VaR models: definition of loss

• Definition of loss:– Only default (“default mode paradigm”)

• Credit risk can then be modeled by means of binomial models

• Loans can be kept at “book value”• BUT: an important source of losses can be overlooked, above

all if the model adopts a short risk horizon (e.g., one year) – Any change in value (“mark-to-market”)

• Rather, it is a “mark-to-model” paradigm• Two future states are not enough• Loans must be evaluated based on their market value


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Credit VaR models: correlations

• Correlations among loans– Can be modeled explicitly

• The values of firm A and firm B change together• This means that their defaults tend to occurr together and

their PDs tend to change together– Can be modeled implicitly/indirectly

• For every state of the economy, two loans can be thought of as independent

• However, their PDs depend on the state of the economy– For example, a recession increases the PDs of both loans– The uncertainty on the future state of the economy, therefore,

makes two loans correlated


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Creditmetrics

• The multinomial approach– the role of ratings and transition matrices– Wilson’s critique and CreditPortfolioView– spread curves and mark-to-model

• The “present value in future”– Expected value, percentiles, VaR– The normal distribution is a false friend

• The case of 2 or more loans– Joint transitions, VaR, diversification effects– Estimating joint transitions: multinomial Merton

• Montecarlo simulations


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Creditmetrics: default is just one among a set of credit events

• Credit events = all events that can alter the value of a loan

• The default is only the most significant one:– The new value of the loan is equal to the value that can be

reasonably recovered (e.g. il 30%)• But all changes in the borrower’s rating are credit events,

in that they change the fair value of the loan– The future cash flows from downgraded loans have to be

discounted using higher rates (higher spreads) and their present value decreases


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CreditMetrics™

6 steps1. Market value of exposures2. Estimate migration probabilities3. Estimate recovery rate4. Market values associated to different

rating grades5. Distribution of market values at the end of

the year6. Portfolio risk


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CreditMetrics™

Model inputs Time horizon Rating system (S&P, Moodys, internal) Transition matrix Recovery rates Forward spreads associated to different

rating grades


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CreditMetrics™

Second step: estimate migration probabilities

One-year transition matrix

RATING AT YEAR-END (%) INITIAL RATING AAA AA A BBB BB B CCC Default

AAA 90.81 8.33 0.68 0.06 0.12 0 0 0.00

AA 0.70 90.65 7.79 0.64 0.06 0.14 0.02 0.00

A 0.09 2.27 91.05 5.52 0.74 0.26 0.01 0.06

BBB 0.02 0.33 5.95 86.93 5.30 1.17 0.12 0.18

BB 0.03 0.14 0.67 7.73 80.53 8.84 1.00 1.06

B 0.00 0.11 0.24 0.43 6.48 83.46 4.07 5.20

CCC 0.22 0.00 0.22 1.30 2.38 11.24 64.86 19.79 Source: S&P CreditWeek (15 April 1996)


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CreditMetrics™

Third step: recovery rates

Category Senior Secured

Senior Unsecured

Senior Subordinated

Subordinated Junior Subordinated

Mean 53.80 51.13 38.52 32.74 17.09

Std. dev. (%) 26.86 25.45 23.81 20.18 10.90 Source: Gupton, Finger and Bhatia (1997).


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CreditMetrics™

Fourth step: market values associated to different rating grades

One-year forward zero coupon rate curve (%)

Maturity 1 year 2 years 3 years 4 years Rating class AAA 3.60 4.17 4.73 5.12

AA 3.65 4.22 4.78 5.17

A 3.72 4.32 4.93 5.32

BBB 4.10 4.67 5.25 5.63

BB 5.55 6.02 6.78 7.27

B 6.05 7.02 8.03 8.52

CCC 15.05 15.02 14.03 13.52


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CreditMetrics™ Example: BBB bond, 5 years, 6% fixed coupon

Remains BBB (probability = 86.93%)

If downgraded to BB

Loss 5.52 = 107.53-102.01

53.107%)63.51(

106%)25.51(

6%)67.41(

6%)10.41(

66 432,1

BBBFV

01.102%)27.71(

106%)78.61(

6%)02.61(

6%)55.51(

66 432,1

BBFV


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CreditMetrics™ Fifth step: distribution of market value changes

Distribution of one-year market values of a BBB bond

State at year-end (j) Present value in a year's time

(FVj)

Probability, pj (%)

Vj = FVj – E(FV)

AAA 109.35 0.02 2.28

AA 109.17 0.33 2.10

A 108.64 5.95 1.57

BBB 107.53 86.93 0.46

BB 102.01 5.3 -5.07

B 98.09 1.17 -8.99

CCC 83.63 0.12 -23.45

Default 53.80 0.18 -53.27

Mean, E(FV)= pjFVj 107.07


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CreditMetrics™ Vj Probability,

pj (%) Cumulative

probability (%)

ij VV

ji pc

-53.27 0.18 0.18 -23.45 0.12 0.3

-8.99 1.17 1.47 -5.07 5.3 6.77 0.46 86.93 93.7 1.57 5.95 99.65

2.1 0.33 99.98 2.28 0.02 100

We can compute the value at risk associated with a certain confidence level, by “cutting” the distribution of value changes at the desired percentile VaR 99% = 8.99VaR 95% = 5.07

75.69.232.2%99 VaR77.49.264.1%95 VaR

If we had used a parametric approach based on the normal distribution, we would have found quite different values


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CreditMetrics™

The distribution of forward values


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CreditMetrics™

Sixth step: portfolio VaR Example: 2 independent bonds with rating A and

BB Probability they both remain in their initial grating

grade 80.53% x 91.05% = 73.32% Probability they both default 0.06% x 1.06% =

0.00% Proceeding this way, one can construct a joint

transition matrix Problem: in reality migrations are not independent


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CreditMetrics™ Probability of joint migration of two issuers with ratings A and BB, assuming

independence of the relative migration rates

Issuer A

AAA AA A BBB BB B CCC Default

Issuer BB 0.09 2.27 91.05 5.52 0.74 0.26 0.01 0.06

AAA 0.03 0.00 0.00 0.03 0.00 0.00 0.00 0.00 0.00

AA 0.14 0.00 0.00 0.13 0.01 0.00 0.00 0.00 0.00

A 0.67 0.00 0.02 0.61 0.40 0.00 0.00 0.00 0.00

BBB 7.73 0.01 0.18 7.04 0.43 0.06 0.02 0.00 0.00

BB 80.53 0.07 1.83 73.32 4.45 0.60 0.20 0.01 0.05

B 8.84 0.01 0.20 8.05 0.49 0.07 0.02 0.00 0.00

CCC 1.00 0.00 0.02 0.91 0.06 0.01 0.00 0.00 0.00

Default 1.06 0.00 0.02 0.97 0.06 0.01 0.00 0.00 0.00


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CreditMetrics™

• Assumption of independence not realistic rating changes and defaults are partly the result of common factors (e.g. economic cycle, interest rates, changes in commodity prices, etc.)

• CreditMetrics™– uses a modified version of the Merton model, where

not only defaults but also migrations depend on changes in the value of corporate assets (asset value returns, AVR)

– estimates the correlation between the asset value returns of the two obligors

– based on that correlation, derives a distribution of joint probabilities


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CreditMetrics™ Example BB

A “multinomial” Merton model with default and migrations

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

-4 -3 -2 -1 0 1 2 3 4

f(x)

Z-def

Z-CCC

Z-B Z-BB

Z-BBB

Z-A

Z-AA

80,53%

7,73%

0,67%

0,14%

1,06%

1,00%

8,84%

0,03%

-2,30 -1,23 1,37 2,39 2,93 3,43-2,04


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CreditMetrics™

In Merton’s model, the estimate of Zdef involves an analysis of corporate debt, the current value of corporate assets and its volatility

In Creditmetrics (reduced-form model), all AVRTs are derived from the probabilities of the transition matrix graphically, each transition probability is equivalent to the area below the relevant section of the asset distribution

For example, Zdef is selected such that the default probability (area under the curve left of the value Zdef) is the PD of the transition matrix (1.06%) in the case of a BB bond

defZ

defBBBB PDZFdrrf %06.1)()(


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CreditMetrics™

Since the probability density function of the AVRs is normal, the condition becomes

Similarly, ZCCC will be selected such that the area included between Zdef and ZCCC is equal to the probability of migration from BB to CCC

defZ

defBBBB PDZFdrrf %06.1)()(

%06.1)( defZN Zdef = N-1(1.06%) -2.3

CCC

def

Z

Z CCCBBdefCCCBBBB pZNZNdrrf %1)()()(


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CreditMetrics™ Probability of migration and associated AVR thresholds

for a BB company

State at year end (j)

Transition probability

(pBB j)

Cumulative probability

AVRT (Zj)

Default 1.06% 1.06% -2.3 CCC 1.00% 2.06% -2.04 B 8.84% 10.90% -1.23 BB 80.53% 91.43% 1.37 BBB 7.73% 99.16% 2.39 A 0.67% 99.83% 2.93 AA 0.14% 99.97% 3.43 AAA 0.03% 100.00%


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CreditMetrics™ The same logic can be followed for an A rated company

Migration probabilities and relative thresholds for a BB company

State at end-year (j)

Transition probability

(pA j)

Cumulative probability

AVRT (Zj)

Default 0.06% 0.06% -3.24 CCC 0.01% 0.07% -3.19 B 0.26% 0.33% -2.72 BB 0.74% 1.07% -2.30 BBB 5.52% 6.59% -1.51 A 91.05% 97.64% 1.98 AA 2.27% 99.91% 3.12 AAA 0.09% 100.00%


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CreditMetrics™

If the stand.zed AVRs of the two companies are described by a std normal the joint AVR distribution is described by a standardized bivariate normal

Its cumulative density function (probability that x is less than X and, at the same time, y is less than Y) is given by the following double integral

both functions depend on the parameter correlation between the asset value returns

)1(22

2

2

22

12

1);;(

yxyx A

eyxf

);;(12

1;Pr )1(22

2

2

22

YXNdxdyeYyXxY X yxyx


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Joint probabilities of two obligors (with A and BB ratings) assuming correlation between asset value returns of 20% - Values %

Issuer A

Issuer BB AAA AA A BBB BB B CCC Default Total

AAA 0.00 0.00 0.03 0.00 0.00 0.00 0.00 0.00 0.03

AA 0.00 0.01 0.13 0.00 0.00 0.00 0.00 0.00 0.14

A 0.00 0.04 0.61 0.01 0.00 0.00 0.00 0.00 0.67

BBB 0.02 0.35 7.10 0.20 0.02 0.01 0.00 0.00 7.69

BB 0.07 1.79 73.65 4.24 0.56 0.18 0.01 0.04 80.53

B 0.00 0.08 7.80 0.79 0.13 0.05 0.00 0.01 8.87

CCC 0.00 0.01 0.85 0.11 0.02 0.01 0.00 0.00 1.00

Default 0.00 0.01 0.90 0.13 0.02 0.01 0.00 0.00 1.07

Total 0.09 2.29 91.06 5.48 0.75 0.26 0.01 0.06 100.00 Source: Gupton, Finger and Bhatia (1997).

CreditMetrics™


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The estimation of the between the asset values of two borrowers

• Creditmetrics uses an approach by large building blocks:– Correlations are first estimated among a large set

of industries and countries (“risk factors”)– For each borrower, a set of weights must be

specified, expressing his sensitivity to different risk factors and to idiosyncratic risk

– Combining those weights and the risk factor correlations, an estimate of the pairwise correlation of two firms can be obtained


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The estimation of the between the asset values of two borrowers• In practice: the asset returns are proxied by the return on

stock indices• The AVR of a firm is decomposed into one or more

systematic components (connected with the dynamics of country- or industry-specific stock indices, e.g., chemical, banking, automotive, etc.), plus an idiosyncratic term which is typical of each individual company

I1,I2,… In = common factors (country/industry indices)j = specific component for company j

jjnjnjjj IIIr ,2,21,1 ...


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The estimation of the between the asset values of two borrowersE.g., Companies A and B

Since the idiosyncratic component is not correlated to any country/industry index, the correlation between company A and company B boils down to

AAAAA IIr 2,21,1

BBBB Ir 3,3

3,2,31,23,1,3,1, BABABA

21

3

Company A BIndustry / CountryUSA -- Banking 50%- Insurance 40% -Italy- Automotive - 80%Idiosyncratic risk 10% 20%Total 100% 100%


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Montecarlo simulations

• When the number of obligors increases, the analytical computation of joint probabilities becomes more and more complex– 2 borrowers = 16 cases – 3 borrowers = 64 cases

• Then, a different approach can be more effective to estimate the distribution of the future portfolio values


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Montecarlo simulations

• A large number of scenarios is generated. The number is so high (e.g. 20,000) that the empirical distribution obtained in the end is a good approximation of the “true” (theoretical) one

• For each scenario, a change in the asset values of n obligors is generated. This is made by means of random draws, but the random numbers generator takes into account the correlation among different borrowers

• For each firm, the asset value change is compared to its thresholds. After checking for rating changes (or default) the value of the loan is computed, and so is the total value of the portfolio


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MC simulation: a simple (but general) example with 2 loans:

-50% -25% 0% 25% 50%-50%

-15%

20%

Variazioni valore impresa BBB

Variazioni valore impresa

A

1. 20.000 random asset value changes are generated for firms 1 e 2(the values shown below suggest that the two firms’ asset values are strongly and positively correlated)

Scenario # 1 2 3 4 … 20.000Firm 1 +34% -8% +1% -20% … +43%Firm 2 +20% -10% +0% -22% … +52%


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N . s c e n a r i o 1 2 3 4 … 2 0 . 0 0 0I m p r e s a 1 + 3 4 % - 8 % + 1 % - 2 0 % … + 4 3 %I m p r e s a 2 + 2 0 % - 1 0 % + 0 % - 2 2 % … + 5 2 %

Scenario # 1 2Firm 1 Moves from B to A

…Firm 2 Stays in A

N. scenario 1 2Im presa 1 Il credito vale 108.7 -8%Im presa 2 Il credito vale 106.3 -10%Totale Il portafoglio vale 215

N . sc e n a r io 1 2 3 4 … 2 0 .0 0 0T o ta le 2 1 5 2 1 2 .3 2 1 5 .2 2 1 4 .7 … 2 0 3 .2

N . s c e n a r i o 1 2 3 4 … 2 0 . 0 0 0T o t a l e 2 1 5 2 1 2 . 3 2 1 5 . 2 2 1 4 . 7 … 2 0 3 . 2

2. For every scenario the change in the firm’s asset value is compared to the asset value thresholds (AVRTs), and translated into a rating (or default)

3. For every credit eventthe value of the loanis computed (based onthe forward rates oron the recovery rate)

Note steps 2 e 3 are repeatedfor each of the 20,000 generated at step 1

Firm 2

Total

Firm 2Firm 1

Loan value: 106.3Firm 1 Loan value: 108.7

Portfolio value: 215Total

…


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N . sc e n a r io 1 2 3 4 … 2 0 .0 0 0T o ta le 2 1 5 2 1 2 .3 2 1 5 .2 2 1 4 .7 … 2 0 3 .2

N . s c e n a r i o 1 2 3 4 … 2 0 . 0 0 0T o t a l e 2 1 5 2 1 2 . 3 2 1 5 . 2 2 1 4 . 7 … 2 0 3 . 2

4. The 20,000 values generated in the previous steps are now ranked in increasing order, and used to compute the expected value, the standard deviation, the percentile and the VaR

0%

2%

4%

6%

8%

10%

102 122 142 162 182 202

80%

81%

82%

83%

84%

85%

102 122 142 162 182 202

Due to the high number of observations, the sample distributionIs very close to the theoretical one

Total


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A MC simulation with more than 2 loans

• To generate all asset value changes, a bivariate normal distribution is not enough. A multivariate normal, e.g. with n=10, must be used

• Instead of one correlation coefficient between 2 firms, a correlation matrix is needed, reporting correlations between each firm and the other 9

• Steps 2, 3 and 4 remain unchanged


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Creditmetrics: advantages and disadvantages

– Uses objective and forward looking market data– Interest rate curves and stock indices correlations

– Evaluates the portfolio market value– Takes into account migration risk

• but:– Needs a lot of data: forward rates, transition

matrices– Assumes the bank is price-taker– Assumes stable transition matrices– Proxies correlations with stock indices– Maps counterparties to industries and countries in

an arbitrary and discretionary way


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Creditrisk+

• The insurance approach• Estimating the number of defaults

– The poisson distribution• Estimating losses

– Banding• Injecting correlation into the model

– Conditional PDs and their macroeconomic drivers


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Creditrisk+ and the “insurance”(actuarial) approach

• Banks and insurance companies are similar because they trade immediate payments for future ones

• Credit can be seen as an insurance contract:– the mark-up is a premium– The default is a “contractual right”

of the borrower


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The insurance approach

Bank

Risk-free rate Costs Premium

Lending rate

Today the customers pay a premium against future risks….

23April

2013


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Bank

Guarantees Losses

Lent principal

Tomorrow the bank pays thecost of their defaults

23April

2016


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Bank

Face losses

PremiPremiPremiPremiPremiPremia

Build up thereserves

Reward theshareholder’s

capital


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• If loans can be seen as insurance contracts, then actuarial methods that were originally developed for the pricing and provisioning of insurance policies can be borrowed by bank risk managers

• BUT the correlation among individuals must be treated with care– It can be ignored for simple insurance contracts (e.g., life

insurance)– It must be injected into the model for more sophisticated

contracts (e.g. fire insurance, loans)


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The Poisson variable

• Given an expected number of defaults (e.g. 4 over a portfolio of 100 counterparties with PD 4%) the probability of having n defaults can be approximated by

• The approximation requires the default events to be independent…– Not realistic, we’ll get back to it

• …and works well only if the PDs are low

!)(

nenp

n


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An example

Example with 3 loansi Borrower Default

probability (pi)1 Rossi 1%2 Bianchi 2%3 Verdi 0.5%

# of expected defaults (): 0.035


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(continued) An example

%56,96!0

035,0)0( 035,00035,0

eep

%38,3035,0!1

035,0)1( 035,01035,0

eep

%001,0)3( %059,0)2( pp


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Note: we traded precision for ease of computation

• The p(n)s are greater than zero even when n>3

• The quality of the approximation declines when the pis are not small enough

• Let us have a look at an example, to see both of these drawbacks


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Another example:

Example of bad approximationsDefault probabilities of the borrowers

RossiBianchiVerdi

25.0%50.0%12.5%

Probability of n defaults occurring

0123

Approximated41.7%36.5%16.0%4.7%

True32.8%48.4%17.2%1.6%


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Page: 53 Example of bad approximations

Default probabilities of the borrowersRossiBianchiVerdi

25.0%50.0%12.5%


0123

Approximated41.7%36.5%16.0%4.7%

True32.8%48.4%17.2%1.6%

Example of bad approximationsDefault probabilities of the borrowers

RossiBianchiVerdi

25.0%50.0%12.5%


0123

Approximated41.7%36.5%16.0%4.7%

True32.8%48.4%17.2%1.6%

(continued) Another example:

98.7%

>><<

>

Extremesare

overestimated


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A more realistic examplei Borrower Default probability (p i)1 Rossi 1%2 Bianchi 2%3 Verdi 0.50%4 Gialli 2%5 Neri 1%6 Mori 1%7 Grossi 1%8 Piccoli 2%9 Astuti 2.50%10 Codardi 2%11 Stupazzoni 0.50%12 Molinari 2%13 Vasari 1%

# of expected defaults ( ): 0.1850


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Results for this example

70%

75%

80%

85%

90%

0 1 2 3 4 5 6 7 8 9 10 11 12 13

N. di insolvenze

0%

5%

10%

15%

20%

0 1 2 3 4 5 6 7 8 9 10 11 12 13

# ofdefaults

Probability


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From the number of defaultsto the amount of losses

• The model above foresees the number of defaults

• Yet, credit risk models (and the pricing, provisioning, capitalization schemes that rely on these models) require an estimate of the probability distribution of future losses

• Are the tools above – which refer to discrete random variables – still suitable for losses (a continuous random variable)?


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The answer is called banding

• Fix a measurement unit L (e. g., 10,000 €)• Divide all exposures Li by L and round them up,

getting standardized values vi

• Also re-write expected losses using the new measurement unit. They become i=i/L with ipiLi

– Do not round up, as it would lead to useless loss of precision

• Group into a single bucket (“band”) all loans of equal size vi


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A practical example:

11,000 / 10,000 1 11,000 x 1%(recovery rate=0)

011,010000

110

Li

i

Name Probability of default

(pi)

Exposure

(Li)

standardizedexposure

( vi)

Expectedloss (i)

Rossi 1% 11,000 1 110

Note: this is where recoveryrates fit into the Creditrisk+ model, in a simple, deterministic way


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A practical example (continued)

R o s s i 1 % 1 1 , 0 0 0 1B i a n c h i 2 % 1 2 , 0 0 0 1V e r d i 0 . 5 0 % 1 1 , 0 0 0 1G i a l l i 2 % 9 , 5 0 0 1

All into band # 1


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Practical example (continued)1 Rossi 1% 11,000 1 1102 Bianchi 2% 12,000 1 2403 Verdi 0.50% 11,000 1 554 Gialli 2% 9,500 1 190

5 Neri 1% 22,000 2 2206 Mori 1% 21,000 2 2107 Grossi 1% 19,500 2 1958 Piccoli 2% 20,800 2 416

9 Astuti 2.50% 33,000 3 82510 Codardi 2% 28,500 3 57011 Stupazzoni 0.50% 31,000 3 15512 Molinari 2% 30,800 3 61613 Vasari 1% 29,000 3 290


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For every band we can compute

ji vvi i

ij v

m

jj

1

jvi

iji

# of expected defaults in band j

Total expected loss in band j

While # of expected defaults in the whole portfolio


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In our example we get:

Summary data for the 3 bandsExposurevj

# of expecteddefaults j

Expectedloss j

1 0.060 0.062 0.052 0.103 0.082 0.25Totale =0.1934 0.41


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Each band is a small portfolio, with losses proportional to defaults

!)(

ne

npnj

j

Probability of n defaults occurring in the j-the band, that is, of losing an amount nvjL

!)(

ne

nvpnj

j

j

Probability associated with nvj losses, each one of amount L,all coming from bank j

Alternatively, we can write:


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In order to obtain the distribution of losses one needs to combine these p• Why? Think at the example of a 120,000 euro loss (12L).

This can derive from– 12 defaults of band 1– 6 defaults of band 2– 4 defaults of band 3– 2 defaults of band 6…

• All these cases must be combined in order to obtain the probability of a 120,000 euro loss in the entire portfolio

• This can be done deriving, for each band, a pgf similar to the one for each band and combining these pgf– The pgf of the portfolio-sum is the product of each

individual pgf


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In our example, we get:

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Loss(nL)

Probability

0 - 82.41%1 10,000 4.90%2 20,000 4.44%3 30,000 7.01%4 40,000 0.52%5 50,000 0.37%6 60,000 0.30%7 70,000 0.03%8 80,000 0.02%9 90,000 0.01%

10 100,000 0.00%… … …30 300,000 0.00%


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How to inject correlations into the model

• The probability distribution for future losses has been derived in a (relatively) painless way because of the assumption that all loans are uncorrelated

• We now show how correlations can be added to the basic model


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Default probabilities of Messrs. Rossi, Bianchi and Verdi

0.0%0.5%1.0%1.5%2.0%2.5%3.0%3.5%4.0%4.5%

95 96 97 98 990.0%

0.5%

1.0%

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3.0%

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Gdp chg.VerdiBianchiRossi


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Default probabilities of Messrs. Rossi, Bianchi and Verdi• Individual PDs are not constant, but swing

around a long-term average because of macroeconomic factors

• They can be seen in the following way

xpp RossiRossi~

Average long-termvalue

Stochastic disturbance


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The logic driving this enhanced model

1 R o s s i 1 % 1 1 , 0 0 0 1 1 1 02 B i a n c h i 2 % 1 2 , 0 0 0 1 2 4 03 V e r d i 0 . 5 0 % 1 1 , 0 0 0 1 5 54 G i a l l i 2 % 9 , 5 0 0 1 1 9 0

5 N e r i 1 % 2 2 , 0 0 0 2 2 2 06 M o r i 1 % 2 1 , 0 0 0 2 2 1 07 G r o s s i 1 % 1 9 , 5 0 0 2 1 9 58 P i c c o l i 2 % 2 0 , 8 0 0 2 4 1 6

9 A s t u t i 2 . 5 0 % 3 3 , 0 0 0 3 8 2 51 0 C o d a r d i 2 % 2 8 , 5 0 0 3 5 7 01 1 S t u p a z z o n i 0 . 5 0 % 3 1 , 0 0 0 3 1 5 51 2 M o l i n a r i 2 % 3 0 , 8 0 0 3 6 1 61 3 V a s a r i 1 % 2 9 , 0 0 0 3 2 9 0

x~0%

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1. For every possible draw of x, the individual PDs and thefuture loss distribution can be worked out accordingly


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1~x




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2~x

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9 A s t u t i 2 . 5 0 % 3 3 , 0 0 0 3 8 2 51 0 C o d a r d i 2 % 2 8 , 5 0 0 3 5 7 01 1 S t u p a z z o n i 0 . 5 0 % 3 1 , 0 0 0 3 1 5 51 2 M o l i n a r i 2 % 3 0 , 8 0 0 3 6 1 61 3 V a s a r i 1 % 2 9 , 0 0 0 3 2 9 0 0%

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2. Step 1 is repeated looping over all possible values of x and generating n scenarios, each one with its own probability distribution:

2

1

3


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3. The “weighted average” of those scenarios (each one conditionalto a given value of x) provides the unconditional distributionof future losses

)( Nxp

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N )( 1xp ...)( 2xp

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In the final distribution risk has increased

• Compared to the “basic model”, where individual PDs are thought to be known without error, now we have two risk sources instead of just one::– Will Rossi & friends really default?– But then, what is the probability that they may default?

• Extreme events are now more likely• In other words, the model now shows the effects

of the correlation linking Rossi, Bianchi, Verdi and friends– Portfolio diversification works less well


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A tiny example:Default probabilities of 2 borrowers in 2 possible states of the world

(a) Boom Bianchi Defaults Survives Total

Defaults 0.08% 1.92% 2%

Survives 3.92% 94.08% 98%

Rossi

Total 4% 96% 100%

(b) Recession Bianchi Defaults Survives Total

Defaults 0.60% 5.40% 6%

Survives 9.40% 84.60% 94%

Rossi

Total 10% 90% 100%


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Unconditional distribution

0,34% > 7% x 4% (0,28%)

> 1%

Bianchi

Fails Survives Total

Fails 0.34% 3.66% 4%

Survives 6.66% 89.34% 96%

Rossi

Total 7% 93% 100%


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CreditRisk+: advantages and disadvantages

– Simple inputs: PDs and exposures (book value) net of recovery are enough

• The “correlated” version requires also sensitivities to the economic cycle factors

• An analytical solution exists• Possibility to obtain the distrbution of losses without

recurring to simulation techniques– But:

• Only looks at default risk• Does not consider migration risk• Assumes constant exposures does not consider

recovery risk


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Questions & exercises

1. Which of the following is used by the CreditMetrics model to estimate default correlations?

A. CreditMetrics does not use correlations and implicitly assumes they are zero, i.e. independence

B. The equity returns correlationsC. The correlations between corporate bond

spreads with respect to Treasury returnsD. The correlations between historical default rates


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Questions & exercises2. A bank, using the Creditmetrics model, has issued a loan to a company

classified as “Rating 3” by its internal rating system. The loan will pay a coupon of 5 million euros after exactly one year, another coupon of 5 million euros after exactly two years and a final flow (coupon + principal) of 105 million euros after exactly three years. The one-year transition matrix of the bank is the following:

Assume that the zero-coupon curve is flat at 4% (yearly compounded), and that issuers falling into different rating grades pay the following premia, which are constant for all maturities: R1=0.26%, R2=0.51%, R3=0.76%, R4=1.26%, R5=2.52% Based on the above and assuming that the loan has a recovery value (in the event of default) of 70 million euro, compute:• the probability distribution of the future values of the loan after one year;• the expected value of the loan (after one year);• VaR, with a confidence level of 95%, of the loan (after one year).

Final status Rating 1 Rating 2 Rating 3 Rating 4 Rating 5 Default

Rating 1 90.0% 5.0% 3.0% 1.0% 0.5% 0.5% Rating 2 4.0% 88.0% 4.0% 2.0% 1.0% 1.0% Rating 3 2.0% 4.5% 85.0% 5.0% 2.0% 1.5% Rating 4 1.0% 4.0% 9.0% 80.0% 3.5% 2.5%

Initial status

Rating 5 0.5% 3.5% 6.0% 10.0% 75.0% 5.0%


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3. Which of the following statements is true?A. Creditrisk+ does not take into account migration risk and is

based on the assumption of independence between the different bank’s borrowers

B. CreditRisk+ takes into account migration risk and allows to indirectly model the correlations between the bank’s different borrowers

C. While not taking into account migration risk, CreditRisk+ allows to indirectly model the correlations between the bank’s different borrowers

D. CreditRisk+ does take into account migration risk and allows to indirectly model the correlations between the bank’s different borrowers


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4. The probability of obtaining a given number of defaults on a loan portfolio can be well proxied by a Poisson distribution only if:

A. individual default probabilities are low and defaults are correlated;

B. individual default probabilities are high and defaults are not correlated;

C. individual default probabilities are low and defaults are not correlated;

D. individual default probabilities are low, the number of loans is sufficiently large, and defaults are not correlated


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5. Default correlation is incorporated into the CreditRisk+ model…

A. … by making the expected number of defaults a stochastic variable;

B. … by making the expected number of customers in the portfolio a stochastic variable;

C. … by estimating the correlation coefficient between the value changes of the assets of the different clients’ couples;

D. …by “banding” loans into a number of subportfolios where all exposures are approximately equal.


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Questions & exercises6. In case of a recession, the default probability of company

Alfa is equal to 2% while the one of company Beta is 4%. In case of an economic expansion, both probabilities are halved. Based on a given economic scenario (recession or expansion), the defaults of the two companies can be considered independent. The analysts estimate that a recession has a 40% probability to occur while an expansion has a 60% probability to occur.

Calculate the non conditional default probability of Alfa and Beta;

Calculate the joint default probability of company Alfa and Beta conditional to the two scenarios;

Calculate the joint default probability not conditional to any economic scenario, and indicate whether it signals a positive correlation between the two defaults.

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Risk and Value in Banking and Insurance