Modeling Related Failures in Finance

Arkady ShemyakinMFM Orientation, 2010

Outline

• Relationships and Related Events• Related Failures: Insurance, Survival, Reliability• Failures in Finance• Probability Structure• Default Correlation (w/example)• Copula Models• Applications of Copulas• References• Conclusion

Relationships and Related Events

• Old, old story…• Relationships that do not matter (hypothesis of

independence)• Relationships that do matter• How to model relationships?• Random variables – height or weight, personal

income, stock prices• Random variables –length of life or age at

death

Related Failures

• Insurance (mortality structure on associated human lives)

• Survival (biological species)• Reliability (connected components in complex

engineering systems)• Finance (?)

Insurance

• Associated human lives (e.g., husbands and wives)

• Common lifestyles• Common disasters (accidents)• Broken-heart syndrome• Exclusions!

Survival

• Biological species within certain environment (e.g., life on an island)

• Common environmental concerns• Predator/prey interactions• Symbiosis

Reliability

• Interaction of components of a complex engineering system (e.g., power grid)

• Links in a chain (series or parallel)• High-load periods• Climate and natural disasters• Overloads• Sayano-Shushenskaya HPS

Finance

• Bank failures, credit events, defaults on mortgages

• Market situation• Macroeconomic indicators• Deficit of trust• Chain reaction of failures

Probability Distributions• Distribution function (d.f.; c.d.f)

• Survival function

• Distribution density function (d.d.f.)

( )F t P X t

( ) 1 ( )S t P X t F t

( ) ( ) t zd

f z F tdt

Joint Distributions

• Joint distribution function

• Joint survival function

• Joint density

( , ) ,H s t P X s Y t

( , ) , )K s t P X s Y t

2

,( , ) ( , ) s z t wd

h z w H s tdsdt

Independence

• For any

• For any

• For any

• Joint functions are built from marginals

, ( , ) ( ) ( ) ( ) ( )s t H s t F s G t P X s P Y t

, ( , ) ( ) ( )s t K s t P X s P Y t

, ( , ) ( ) ( )s t h s t f s g t

Pearson’s Moment Correlation

• Pearson’s moment correlation (correlation coefficient) is defined as

• It is a good measure of linear dependence, strongly connected with the first two moments, and is known not to capture non-linear dependence

,,

( )X Y

Cov X Y EXY EX EYX Y

Var X Var Y

Sample Pearson’s Correlation

• Given a paired (matched) sample

the sample correlation coefficient is defined as

1 1, ,..., , ,n nx y x yx,y

1 1 1

2 2

2 2

1 1 1 1

1

ˆ1 1

n n n

i i i ii i i

n n n n

i i i ii i i i

x y x yn

x x y yn n

x,y

Default Correlation

• Time-to-default random variables• CDS (Credit Default Swaps)• CDO (Collateralized Debt Obligations)• Recent crisis• Problem: mathematical models failed to

accurately predict the risks• Problems with default correlation• Example: three-mortgage portfolio

Example (Absolutely Unrealistic)

• We underwrite three identical mortgages, each with $100K principal

• Term: 1 year• Probability of default: 0.1 for each• Annual payment is made in the beginning of the

year• Interest rate of 11%• Expected gain: $1,000 per mortgage per year• Problem: relatively high risk of a big loss

Losses

• We can lose as much as over $250K while making on the average $3K!

• Expected gain = $11,000 x 0.9 - $89,000 x 0.1 = $1,000

• Potential loss = $89,000• We collect (three mortgages) the interest of

$33,000 = $ 30,000 + $3,000• We bear the risk of losing the principal 3

x $89,000 = $267,000

Selling the Risk

• Is it possible to hedge the risks (sell the risks)?• CDO structure: how many defaults?• Senior tranche (safe)• Mezzanine tranche (middle-of-the-road)• Equity tranche (risky)• Find the buyers (investors): those who will

receive our cash flows and accept responsibility for possible defaults

Default Probabilities - Independence

• P(all three defaults) = P(ABC) = 0.1 x 0.1 x 0.1 = 0.001

• P(at least two defaults) = 0.027 + 0.001 = =0.028

• P(at least one default) = 0.243 + 0.027 + 0.001 = 0.271

Investors’ Side - Independence

• Assume independence of failures• Senior tranche: expected loss of $100• Mezzanine tranche: expected loss of $2,800• Equity tranche: expected loss of $27,100• Expected losses of all tranches add up to

$30,000 • For us: margin of $3,000 and no risk!• We might have to split the margin

Diagram 1 (Independence)

Correlation

• Assume that there is no independence and we expect pair-wise correlations (Pearson’s moment correlations) between the individual defaults at 0.5

• That corresponds to joint probability of two defaults being 0.055

• Sadly, it says next to nothing about the joint probability of three defaults

• Different scenarios are possible

Calculation of the Multiple Default Probabilities

( ) ( ) ( ),

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

( ) 0.1 0.10.5 ( ) 0.055

0.1 0.9

( ) ?

EXY EX EY P AB P A P BX Y

P A P A P B P BVar X Var Y

P ABP AB

P ABC

Diagram X - Correlation

Diagram 2 (Extreme Scenario 2)

Default Probabilities – Scenario 2

• P(all three defaults) = 0.01• P(at least two defaults) = 0.145• P(at least one default) = 0.145

Investors’ Side – Scenario 2

• Assume default correlations of 0.5• Senior tranche: expected loss of $1,000• Mezzanine tranche: expected loss of $14,500• Equity tranche: expected loss of $14,500• Expected losses of all tranches add up to

$30,000

Diagram 3 (Extreme scenario 3)

Default Probabilities – Scenario 3

• P(all three defaults) = 0.055• P(at least two defaults) = 0.055• P(at least one default) = 0.19

Investors’ Side – Scenario 3

• Assume default correlations equal to 0.5• Senior tranche: expected loss of $5,500• Mezzanine tranche: expected loss of $5,500• Equity tranche: expected loss of $19,000• Expected losses of all tranches add up to

$30,000

What do we conclude?

• Correlation between the default variables is important in order to estimate expected losses (i.e., to price) the tranches

• Results are sensitive to the value of the correlation coefficient

• Knowing pair-wise correlation coefficients is not enough to price the tranches in case of more than 2 assets

• It would be enough under assumption of normality

Definition of Copula Function

• A function

is called a copula (copula function) if:

1. For any

2. It is 2-monotone (quasi-monotone).

3. For any

2: 0,1 [0,1] 0,1C I I

, ( ,1) ; (1, )u v I C u u C v v

, ( ,0) (0, ) 0u v I C u C v

Frechet Bounds

• For any copula the following inequalities (Frechet bounds) hold:

( , )C u v

( , ) ( , ) ( , ),

( , ) max 1,0 ,

( , ) min ,

W u v C u v M u v

W u v u v

M u v u v

Maximum Copula ( , ) min{ , }M u v u v

Maximum Copula ( , ) min{ , }M u v u v

Minimum Copula ( , ) max 1,0W u v u v

Minimum Copula ( , ) max 1,0W u v u v

Product Copula ( , )P u v uv

Product Copula ( , )P u v uv

Sklar’s Theorem

• Theorem: 1. For any correctly defined joint distribution function with marginals

, there exists such a copula function that

2. If the marginals are absolutely continuous, then this representation is unique.

( , )H x y( ), ( )F x G y

( , ) ( ), ( )H x y C F x G y

Applications of Copulas

• Going beyond correlation• Extreme co-movements of currency exchange

rates• Mutual dependence of international markets• Evaluation of portfolio risks• Pricing CDOs

References

• Joe Nelsen; An Introduction to Copulas, Springer

• Umberto Cherubini, Elisa Luciano, Walter Vecchiato; Copula Methods in Finance, Wiley

• Attilio Meucci; Computational Methods in Decision-making, Kluwer

• Robert Engle et al.• Paul Embrechts et al.

Conclusions

• Work in progress – the world is in search for better models (?)