Risk and Value in Banking and Insurance

Risk and Value in Banking and Insurance
Credit Portfolio Models

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 Risk and Value in Banking and Insurance

Portfolio models Main portfolio models CreditMetricsTM (J.P. Morgan)
Portfolio ManagerTM (KMV) CreditPortfolioViewTM (McKinsey) We’ll focus on the first two Risk and Value in Banking and Insurance

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 Risk and Value in Banking and Insurance

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 losses Same m and s True Normal Risk and Value in Banking and Insurance

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. True Normal Risk and Value in Banking and Insurance

Credit VaR models: 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 Risk and Value in Banking and Insurance

Credit VaR models: the time horizon
Risk and Value in Banking and Insurance

Credit VaR models: 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 Risk and Value in Banking and Insurance

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 Risk and Value in Banking and Insurance

Creditmetrics The multinomial approach The “present value in future”
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 Risk and Value in Banking and Insurance

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 Risk and Value in Banking and Insurance

CreditMetrics™ 6 steps Market value of exposures
Estimate migration probabilities Estimate recovery rate Market values associated to different rating grades Distribution of market values at the end of the year Portfolio risk Risk and Value in Banking and Insurance

CreditMetrics™ Model inputs Time horizon
Rating system (S&P, Moodys, internal) Transition matrix Recovery rates Forward spreads associated to different rating grades Risk and Value in Banking and Insurance

Second step: estimate migration probabilities
CreditMetrics™ Second step: estimate migration probabilities Risk and Value in Banking and Insurance

CreditMetrics™ Third step: recovery rates

CreditMetrics™ Fourth step: market values associated to different rating grades Risk and Value in Banking and Insurance

CreditMetrics™ Example: BBB bond, 5 years, 6% fixed coupon
Remains BBB (probability = 86.93%) If downgraded to BB Loss  5.52 = Risk and Value in Banking and Insurance

Fifth step: distribution of market value changes
CreditMetrics™ Fifth step: distribution of market value changes Risk and Value in Banking and Insurance

CreditMetrics™ 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% = 8.99 VaR 95% = 5.07 Risk and Value in Banking and Insurance If we had used a parametric approach based on the normal distribution, we would have found quite different values

CreditMetrics™ The distribution of forward values

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 Risk and Value in Banking and Insurance

CreditMetrics™ Risk and Value in Banking and Insurance

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 Risk and Value in Banking and Insurance

CreditMetrics™ Example BB Risk and Value in Banking and Insurance

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 Risk and Value in Banking and Insurance

CreditMetrics™ Zdef = N-1(1.06%)  -2.3
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 Zdef = N-1(1.06%)  -2.3 Risk and Value in Banking and Insurance

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

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 Risk and Value in Banking and Insurance

The estimation of the r 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 Risk and Value in Banking and Insurance

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 Risk and Value in Banking and Insurance

E.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 Company A B Industry / Country USA - - Banking 1 50% - Insurance 2 40% - Italy - Automotive 3 - 80% Idiosyncratic risk 10% 20% Total 100% 100% Risk and Value in Banking and Insurance

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 Risk and Value in Banking and Insurance

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 Risk and Value in Banking and Insurance

MC simulation: a simple (but general) example with 2 loans:
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) Risk and Value in Banking and Insurance Scenario # 1 2 3 4 … 20.000 Firm 1 +34% -8% +1% -20% … +43% Firm 2 +20% -10% +0% -22% … +52%

Scenario # 1 2 Firm 1 Moves from B to A … Firm 2 Stays in A …
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) Firm 1 Firm 2 Scenario # 1 2 Firm 1 Moves from B to A … Firm 2 Stays in A … 3. For every credit event the value of the loan is computed (based on the forward rates or on the recovery rate) Firm 1 Loan value: 108.7 Firm 2 Loan value: 106.3 Risk and Value in Banking and Insurance Total Portfolio value: 215 Note steps 2 e 3 are repeated for each of the 20,000 generated at step 1 Total

Due to the high number of observations, the sample distribution
Total 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 Risk and Value in Banking and Insurance Due to the high number of observations, the sample distribution Is very close to the theoretical one

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 r between 2 firms, a correlation matrix is needed, reporting correlations between each firm and the other 9 Steps 2, 3 and 4 remain unchanged Risk and Value in Banking and Insurance

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 Risk and Value in Banking and Insurance

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 Risk and Value in Banking and Insurance

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 Risk and Value in Banking and Insurance

The insurance approach
23 April 2013 Today the customers pay a premium against future risks…. Bank JJJJJJJJJ JJJJJJJJJJ JJJJJJ Risk and Value in Banking and Insurance Risk-free rate Costs Premium Lending rate

23 April 2016 Tomorrow the bank pays the cost of their defaults Bank JLJJJJJJJ JJLLJJLJJJ JJJJJL Risk and Value in Banking and Insurance Guarantees Losses Lent principal

Bank LL LLL JJJJJJJJJ JJJJJJJJJJ JJJJJJ Build up the reserves Risk and Value in Banking and Insurance Face losses Premi Premi Premi Premi Premi Premia Reward the shareholder’s capital

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) Risk and Value in Banking and Insurance

The Poisson variable Given an expected number of defaults m (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 Risk and Value in Banking and Insurance

An example Example with 3 loans i Borrower Default probability (p ) 1
Rossi 1% 2 Bianchi 2% Risk and Value in Banking and Insurance 3 Verdi 0.5% # of expected defaults ( m ): 0.035

(continued) An example

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 Risk and Value in Banking and Insurance

Another example: Example of bad approximations
Default probabilities of the borrowers Rossi Bianchi Verdi 25.0% 50.0% 12.5% Probability of n defaults occurring 1 2 3 Approximated 41.7% 36.5% 16.0% 4.7% True 32.8% 48.4% 17.2% 1.6% Risk and Value in Banking and Insurance

(continued) Another example:
>> << > Extremes are overestimated Risk and Value in Banking and Insurance 98.7%

A more realistic example
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% Risk and Value in Banking and Insurance 9 Astuti 2.50% 10 Codardi 2% 11 Stupazzoni 0.50% 12 Molinari 2% 13 Vasari 1% # of expected defaults ( m ): 0.1850

Results for this example
Probability Risk and Value in Banking and Insurance # of defaults

From the number of defaults to 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)? Risk and Value in Banking and Insurance

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 ei=li/L with l i£piLi 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 Risk and Value in Banking and Insurance

A practical example: ) ) ) ) Rossi 1% 11,000 1 110 Name Probability
Exposure standardized Expected of default exposure loss (p ) (L ) ( v ) ( l ) i i i i Rossi 1% 11,000 1 110 11,000 / 1 11,000 x 1% (recovery rate=0) Risk and Value in Banking and Insurance Note: this is where recovery rates fit into the Creditrisk+ model, in a simple, deterministic way

A practical example (continued)
Risk and Value in Banking and Insurance All into band # 1

Practical example (continued)

For every band we can compute
# of expected defaults in band j While # of expected defaults in the whole portfolio Risk and Value in Banking and Insurance Total expected loss in band j

In our example we get: Summary data for the 3 bands Exposure
# of expected Expected v defaults m loss e j j j 1 0.060 0.06 2 0.052 0.10 Risk and Value in Banking and Insurance 3 0.082 0.25 Totale m =0.1934 0.41

Each band is a small portfolio, with losses proportional to defaults
Probability of n defaults occurring in the j-the band, that is, of losing an amount nvjL Alternatively, we can write: Risk and Value in Banking and Insurance Probability associated with nvj losses, each one of amount L, all coming from bank j

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 Risk and Value in Banking and Insurance

In our example, we get: Loss Probability (nL) - 82.41% 1 10,000 4.90%
- 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% Risk and Value in Banking and Insurance 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%

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 Risk and Value in Banking and Insurance

Default probabilities of Messrs. Rossi, Bianchi and Verdi
4.5% 4.0% 4.0% 3.5% 3.5% 3.0% 3.0% 2.5% Gdp chg. 2.5% Verdi 2.0% 2.0% Bianchi 1.5% Risk and Value in Banking and Insurance 1.5% Rossi 1.0% 1.0% 0.5% 0.5% 0.0% 0.0% 95 96 97 98 99

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 Risk and Value in Banking and Insurance Stochastic disturbance Average long-term value

The logic driving this enhanced model
Risk and Value in Banking and Insurance 1. For every possible draw of x, the individual PDs and the future loss distribution can be worked out accordingly

2. Step 1 is repeated looping over all possible values of x and generating n scenarios, each one with its own probability distribution: 1 2 Risk and Value in Banking and Insurance 3

3. The “weighted average” of those scenarios (each one conditional to a given value of x) provides the unconditional distribution of future losses 2 1 N Risk and Value in Banking and Insurance

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 Risk and Value in Banking and Insurance

A tiny example: Default probabilities of 2 borrowers in 2 possible states of the world (a) Boom Bianchi Defaults Survives Total 0.08% 1.92% 2% 3.92% 94.08% 98% Rossi 4% 96% 100% (b) Recession 0.60% 5.40% 6% 9.4 0% 84.60% 94% 10% 90% Risk and Value in Banking and Insurance

Unconditional distribution
Bianchi Fails Survives Total Fails 0.34% 3.66% 4% Rossi Survives 6.66% 89.34% 96% Total 7% 93% 100% Risk and Value in Banking and Insurance 0,34% > 7% x 4% (0,28%) r > 1%

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 Risk and Value in Banking and Insurance

Questions & exercises 1. Which of the following is used by the CreditMetrics model to estimate default correlations? CreditMetrics does not use correlations and implicitly assumes they are zero, i.e. independence The equity returns correlations The correlations between corporate bond spreads with respect to Treasury returns The correlations between historical default rates Risk and Value in Banking and Insurance

Questions & exercises 2. 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). Risk and Value in Banking and Insurance

Questions & exercises 3. Which of the following statements is true?
Creditrisk+ does not take into account migration risk and is based on the assumption of independence between the different bank’s borrowers CreditRisk+ takes into account migration risk and allows to indirectly model the correlations between the bank’s different borrowers While not taking into account migration risk, CreditRisk+ allows to indirectly model the correlations between the bank’s different borrowers CreditRisk+ does take into account migration risk and allows to indirectly model the correlations between the bank’s different borrowers Risk and Value in Banking and Insurance

Questions & exercises 4. The probability of obtaining a given number of defaults on a loan portfolio can be well proxied by a Poisson distribution only if: individual default probabilities are low and defaults are correlated; individual default probabilities are high and defaults are not correlated; individual default probabilities are low and defaults are not correlated; individual default probabilities are low, the number of loans is sufficiently large, and defaults are not correlated Risk and Value in Banking and Insurance

Questions & exercises 5. Default correlation is incorporated into the CreditRisk+ model… … by making the expected number of defaults a stochastic variable; … by making the expected number of customers in the portfolio a stochastic variable; … by estimating the correlation coefficient between the value changes of the assets of the different clients’ couples; …by “banding” loans into a number of subportfolios where all exposures are approximately equal. Risk and Value in Banking and Insurance

Questions & exercises 6. 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. Risk and Value in Banking and Insurance

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