Financial Engineering 6

Lecture 25

Risk Assessment Conclusions

References: Villalobos, J.P. Morgan

Lecture Topics • The J.P. Morgan Guide to Credit Derivatives • Default risk • CreditMetrics Model • Discrete Markov Chain • Exposure • Time to Default • Examples • Conclusions

Credit Risk • Credit risk is the risk of loss that will be incurred with the

default of a counterparty. – The counterparty defaults when it fails to meet its

contractual payments.

• To assess potential credit losses we need to assess: – The counterparty credit exposure at the time of the default. – The probability of default. – The amount that can be recovered after the default.

• Assumption: Default occurs when the value of a firm’s assets drops below the market value of its liabilities.

• The potential exposure of contract c at time t it is defined as:

where T is the time to expiration of the contract and PV is the present value at time t.

• The actual exposure of contract c at time t is defined as:

where V(c,t) is the value of the contract at time t.

Credit Exposure • Credit exposure: The cost of replacing or hedging the contract at

the time of default. – It is the maximum value that will be lost if the counterparty

defaults.

( ) ( )[ ]{ }{ }tcVcVPVtcPE tTt ,,max,0max),( −= ≤< ττ

( )( , ) max 0, ,AE c t V c t=   

• Total exposure is the summation of actual and potential exposure.

Example • Suppose that you have a forward contract that you signed with

a company (your counterparty) two years ago for the delivery of 1000 gallons of unleaded regular gasoline a year from today for $2 per gallon.

• What is the current exposure today if the price of the gasoline is expected to be at $4 per gallon a year from now?

• Potential exposure depends on the behavior of the underlying asset from the current time to the end of the contract.

– Suppose that the future price remains the same, then the potential exposure is 0.

( )( , ) max 0, , 1000(4 2)AE c t V c t= = −  

( ) ( )[ ]{ }{ }tcVcVPVtcPE tTt ,,max,0max),( −= ≤< ττ

Portfolio Exposure • When you have several parties with several transactions in your

portfolio, you will have different ways to determine the individual credit exposures:

– Netting allowed: Positive and negative exposures of a party can cancel each other.

– Netting not allowed: Only the positive exposures are included and added together; pessimistic view.

Value of a Defaulted Security • Let’s look at an example of a zero coupon bond with one year to

maturity.

( )1 1 1 d d

PV P V P RV r

= − +  +

V

RV

1-Pd

Pd

PV

• With a certain probability (Pd), the bond defaults and recovers only a certain amount (RV) instead of the original payment (V).

• The problem with this approach is our assessment of this probability or risk.

– One way is to find a “risk neutral” (or arbitrage fee) estimate.

Example • Suppose that the par value of the previous bond is $1,000 and it is

currently trading at $930. – The current risk free rate is of 5% – This corresponds to a rate of return on the investment of

7.52%, which is significantly over the risk free rate. – We can considered the “excess” of return as the probability of

default. • Let’s assume that if the bonds defaults, the recovery value is zero.

– We therefore have:

• Solving we get Pd = 0.0235 • In some regards, this is similar to the Implied Volatility that we

have used in the past. • This approach also has some problems which we might explore in

a future lecture.

( ) ( )1 11 930 1 1000 0 1 1 0.05d d d d

PV p V p RV p p r

= − + = = − +      + +

Portfolio Risk Models • The goal for the models are to assess the risk of a portfolio.

• The models allow market participants to recognize sources of risk, while credit derivatives provide flexibility to manage these sources.

• Some models include: – CreditMetrics, published by J.P.Morgan, and further developed by

RiskMetrics. – CreditRisk+ published by CreditSuisse. – Credit Portfolio View, published by Thomas Wilson of McKinsey and

Company.

• The models share two features.

– They treat default as a significant downward jump in exposure value, which uses default probabilities and loss severities as their inputs.

– They develop some structure to describe the dependency between defaults of individual names.

Default Probability • The default probability for a given issuer or counterparty is analogous

to the volatility of an asset when considering market risk.

• Methods commonly used to assess default probability include: – Credit ratings (scores):

• Score or rank individual names. • Categorize the names historically according to their credit

score. • Measure the proportion of similar names that have defaulted

over time. • To get probabilities we need to extrapolate from this

information and assign A-rated names for the historical default probability for that rating.

– Utilize the equity (stock) markets: • Extract a firm’s default probability from its equity price, the

structure of its liabilities, and the observation that equity is essentially a call option on the assets of the firm.

CreditMetrics Model • CreditMetrics models the changes in portfolio value that result

from significant credit quality moves, such as defaults or rating changes.

• The model takes information on the individual obligors (bond issuers) in the portfolio as inputs, and produces as output the distribution of portfolio values at some fixed horizon in the future.

• From this distribution, it is possible to produce statistics which quantify the portfolio’s absolute risk level.

– Such as the standard deviation of value changes. – Or the worst case loss at a given level of confidence.

CreditMetrics Model The three steps of the CreditMetrics model are:

1. The definition of the possible “states” for each obligor’s credit quality, and a description of how likely obligors are to be in any of these states at the horizon date.

2. The revaluation of exposures in all possible credit states.

3. The interaction and correlation between credit migrations of different obligors.

Step 1 – Defining the States • Commonly, the rating scores of the rating agencies are used. • For example:

• Which can be converted to a Markov transition probability matrix Aaa Aa A Baa Ba B Caa Default

Aaa 0.9338 0.0594 0.0064 0.0000 0.0002 0.0000 0.0000 0.0002 Aa 0.0161 0.9053 0.0746 0.0026 0.0009 0.0001 0.0000 0.0004 A 0.0007 0.0228 0.9235 0.0463 0.0045 0.0012 0.0001 0.0009 Baa 0.0005 0.0026 0.0551 0.8848 0.0476 0.0071 0.0008 0.0015 Ba 0.0002 0.0005 0.0042 0.0516 0.8691 0.0591 0.0024 0.0129 B 0.0000 0.0004 0.0013 0.0054 0.0635 0.8422 0.0191 0.0681 Caa 0.0000 0.0000 0.0000 0.0062 0.0205 0.0408 0.6919 0.2406 Default 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000

Start

Finish

Markov Chains • From Markov Chains we need to remember:

– One, two and nth step transition probabilities. – Probabilities for passage to a close state. – Mean recurrence and expected passage times.

• Define:

• Then the mean recurrence times are given by:

nn Ppp 0=

  





  





  





  





== −−

−−

−

mn MM

mn M

mn M

mn

m MM

m M

m M

m

mnmn

pp

pp

pp

pp













0

000

0

000

PPP

} MatrixIdentity I

statest K transien 0 =

  

 =

I RQ

P

( )-1E = I - Q

Step 2 – Revaluation • Assume a particular instrument’s value today is known and we

wish to estimate its value – At our risk horizon. – Conditional on any of the possible credit migrations that the

instrument’s issuer might undergo.

• Consider a Baa-rated, three year, fixed 6% coupon bond, currently valued at par.

– With a one year horizon, the revaluation step consists of estimating the bond’s value in one year under each possible transition.

– For the transition to default, we value the bond through an estimate of the likely recovery value.

Step 2 – Revaluation • One step transition probability matrix:

=   





  





= m MM

m M

m M

m

pp

pp







0

000 1P

== 112 PPP

Aaa Aa A Baa Ba B Caa Default Aaa 0.9338 0.0594 0.0064 0.0000 0.0002 0.0000 0.0000 0.0002 Aa 0.0161 0.9053 0.0746 0.0026 0.0009 0.0001 0.0000 0.0004 A 0.0007 0.0228 0.9235 0.0463 0.0045 0.0012 0.0001 0.0009 Baa 0.0005 0.0026 0.0551 0.8848 0.0476 0.0071 0.0008 0.0015 Ba 0.0002 0.0005 0.0042 0.0516 0.8691 0.0591 0.0024 0.0129 B 0.0000 0.0004 0.0013 0.0054 0.0635 0.8422 0.0191 0.0681 Caa 0.0000 0.0000 0.0000 0.0062 0.0205 0.0408 0.6919 0.2406 Default 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000

Aaa Aa A Baa Ba B Caa Default Aaa 0.8729 0.1094 0.0163 0.0005 0.0004 0.0000 0.0000 0.0004 Aa 0.0297 0.8222 0.1367 0.0082 0.0021 0.0003 0.0000 0.0009 A 0.0017 0.0419 0.8571 0.0840 0.0104 0.0027 0.0002 0.0020 Baa 0.0010 0.0060 0.1000 0.7879 0.0842 0.0152 0.0015 0.0042 Ba 0.0004 0.0012 0.0105 0.0910 0.7616 0.1016 0.0049 0.0288 B 0.0000 0.0008 0.0029 0.0128 0.1093 0.7139 0.0295 0.1309 Caa 0.0000 0.0000 0.0005 0.0111 0.0349 0.0638 0.4796 0.4101 Default 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000

Step 2 – Revaluation • The resulting probabilities under one, two and three years

planning horizons for our Baa bond are:

Year 1 Year 2 Year 3 Year 6 Aaa 0.0005 0.0010 0.0015 0.0032 Aa 0.0026 0.0060 0.0098 0.0227 A 0.0551 0.1000 0.1366 0.2098 Baa 0.8848 0.7879 0.7062 0.5287 Ba 0.0476 0.0842 0.1121 0.1604 B 0.0071 0.0152 0.0235 0.0465 Caa 0.0008 0.0015 0.0022 0.0039 Default 0.0015 0.0042 0.0079 0.0248

Step 2 – Revaluation • An example for the valuation of bond after one year:

Step 2 – Revaluation

• For our example, Q is highlighted in yellow.

} MatrixIdentity I

statest K transien 0 =

  

 =

I RQ

P

Aaa Aa A Baa Ba B Caa D Aaa 0.9338 0.0594 0.0064 0.0000 0.0002 0.0000 0.0000 0.0002 Aa 0.0161 0.9053 0.0746 0.0026 0.0009 0.0001 0.0000 0.0004 A 0.0007 0.0228 0.9235 0.0463 0.0045 0.0012 0.0001 0.0009 Baa 0.0005 0.0026 0.0551 0.8848 0.0476 0.0071 0.0008 0.0015 Ba 0.0002 0.0005 0.0042 0.0516 0.8691 0.0591 0.0024 0.0129 B 0.0000 0.0004 0.0013 0.0054 0.0635 0.8422 0.0191 0.0681 Caa 0.0000 0.0000 0.0000 0.0062 0.0205 0.0408 0.6919 0.2406 D 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000

Aaa Aa A Baa Ba B Caa Aaa 22.3672 26.6742 48.8709 27.7477 15.8582 7.7553 0.6922 Aa 7.5842 27.9929 48.9901 27.8417 15.8926 7.7746 0.6940 A 4.6563 15.8934 50.1935 28.0800 15.9504 7.8094 0.6976 Baa 3.3966 11.3311 34.2532 30.3949 16.2602 7.9078 0.7069 Ba 2.0221 6.6877 20.0321 16.9201 18.5389 8.0404 0.6933 B 1.0286 3.4170 10.1789 8.4973 8.5137 10.1744 0.7224 Caa 0.3391 1.1255 3.3701 2.8627 2.6882 2.0415 3.4017

Rating Years to Default Aaa 149.9657 Aa 136.7701 A 123.2806 Baa 104.2507 Ba 72.9346 B 42.5323 Caa 15.8287

( ) =-1E = I - Q

• E and the average number of years to default are:

Step 3 – Building Correlations • We assume an unseen "driver" of credit migrations, which we

think of as changes in asset value. – Default occurs when the value of a firm’s assets drops

below the market value of its liabilities.

• The stand alone information for each name is used as input through the specification of the transition matrix.

– Assets are used only to build the interaction between obligors.

• We assume that asset value changes are normally distributed.

Step 3 – Building Correlations • We then partition the asset change distribution for each name

according to the name's transition probabilities. – For the Baa-rated obligor, with default probability equal to

0.15%, the default partition is chosen as the point beyond which lies 0.15% probability.

– The Caa partition is then chosen to match the obligor's probability of migrating to Caa, and so on.

Step 3 – Building Correlations • Once we define the partitions for each obligor, we must

describe the correlation between asset value changes. • We take correlations in equity returns as a proxy for the asset

value correlations. • We calculate the probabilities of all joint rating transitions.

– Such as obligor 1 defaults, obligor 2 downgrades, obligor 3 stays the same rating, etc.

• In practice a Monte Carlo approach is used due to the 100s or even 1,000s of bonds in a portfolio.

• Thus, for a single scenario: – We draw from a multivariate normal distribution to produce

asset value changes. – Read from the partitions to identify the changes with new

rating states and exposure values. – Aggregate the individual exposures to arrive at a portfolio

value for the scenario.

Conclusion

• We just scratched the surface of risk assessment and risk management for credit and credit derivatives.

• The risk assessment of individual assets and portfolios is a very hot area of research.

– Recent economic issue has put a lot of pressure on the agencies who assigned the corresponding credit ratings.

Assignments • Finish reading The J.P. Morgan Guide to Credit Derivatives.