Dinosaurs in a portfolio Marc Freydefont Moody’s Investors Service Ltd. London, 13 March 2002 ISDA - PRMIA Seminar

Dinosaurs in a portfolioDinosaurs in a portfolio

Marc FreydefontMarc Freydefont

Moody’s Investors Service Ltd.Moody’s Investors Service Ltd.London, 13 March 2002London, 13 March 2002ISDA - PRMIA SeminarISDA - PRMIA Seminar

ContentsContents

Section 1 Introduction

Section 2 Rating CDOs

Section 3 The BET approach

Section 4 The Monte Carlo approach and the log-normal method

Section 5 Portfolio analysis

Section 6 Characteristic functions / Fourier transforms

Section 7 Conclusions

IntroductionIntroduction

Section 1

CDOs: Another Record Year in EuropeCDOs: Another Record Year in Europe

Transfer Of Credit Risk

0

20

40

60

80

100

120

1996 1997 1998 1999 2000 2001

$ b

illi

on

0

20

40

60

80

100

120

nu

mb

er

Rated Volume Number

Collateralised Risk Obligations ?Collateralised Risk Obligations ?

CLO: “Collateralised Loan Obligations” - Securitisation of a portfolio of corporate loans

CBO:“Collateralised Bond Obligations” - Securitisation of a portfolio of corporate bonds

CDO:“Collateralised Debt Obligations” - Can include CLOs, CBOs, or a combination thereof

CSO:“Collateralised Synthetic Obligations” - Securitisation of a portfolio of synthetic exposures via Credit Default Swaps

CFO: “Collateralised Fund Obligations” - Securitisation of a portfolio of exposures to hedge funds

CEO:“Collateralised Equity Obligations” - Securitisation of a portfolio of equity or private equity exposures

COO: “Collateralised Options Obligations” - Securitisation of a portfolio of Options

CRO: “Collateralised Risk Obligations” - Securitisation of a portfolio of Risks

Section 4 Portfolio Analysis

Section 5 Characteristic Functions applied to Portfolio Analysis

Conclusions

Rating CDOsRating CDOs

Section 2

Rating CDOsRating CDOs

IssuingSPV

...AssetPool

B2

Ba3

B1

Principal& Interest

SeniorInvestors

JuniorInvestors

Swap(Aa2)

Rating CDOs: Expected LossRating CDOs: Expected Loss

Expected Loss is the probability-weighted mean of losses arising from credit events

Expected Loss “=” Probability x Severity

Expected Loss = ,

Where L : Severity of credit loss p(L) : Probability density of credit loss

1

0)(. dLLpL

Rating CDOs: Various InputsRating CDOs: Various Inputs

Scheduled cash flow from assets

Cash flow allocation within the transaction

Probability of default of the assets

Diversification within the portfolio

Recovery rate in case of default

Rating CDOs: EL CalculationRating CDOs: EL Calculation...

Default Scenarios

Scenario 1

Scenario 2

Scenario 3

Senior Loss 1

Sub Loss 1

Senior Loss 2

Sub Loss 2

Senior Loss 3

Sub Loss 3

PV Losses to Notes

...

CashFlowsfrom

Assets

CashFlows

to Notes

Senior Expected

Loss

Junior Expected

Loss

Mean of Losses(Weighted byScenario Probability)Cash

Flow

Model

•Priorities

•Reserves

•OC & IC Triggers

•Swaps

•Leverage

Analysing CROsAnalysing CROs

Measuring the credit performance of a CRO means the following: determining the likelihood of occurrences of defaults and losses on the

underlying portfolio of assets determining the likelihood of occurrences of defaults and losses on

each class of notes issued by the CRO vehicle

Three classical approaches: Binomial / Multinomial Expansion Trees (“BET”) - Homogeneous pools Monte Carlo simulations - Heterogeneous pools Log-normal method

Innovative approach: Characteristic functions and Fourier transformsCharacteristic functions and Fourier transforms

The BET approachThe BET approach

Section 3

The BET approachThe BET approach

Description

The total expected loss of a pool of N assets having the same default probability p and the same recovery rate RR is calculated using the binomial formula

Key variables and assumptions

Homogeneous pool of identical independent assets

Main disadvantages

Does not account for heterogeneity in size, in risk (default probability), in recovery rate, in correlations

The BET approach : Binomial Expansion TreeThe BET approach : Binomial Expansion Tree

Real portfolio represented as a lesser number D of independent and identical assets

Each scenario, with defaults ranging from 0 to D, is considered

The loss for each tranche is recorded and then multiplied by the probability of the scenario

Probability of n defaults = nDnDn ppC 1

The BET approach : Binomial or multinomialThe BET approach : Binomial or multinomial

Senior % Fee 0.210% Issue date 28-Jan-02

Senior Expenses €250,000 Periodicity 2 per annum

Periods to Maturity 30

Random Recovery TRUE Re-investment to 5 yr

Mean Recovery 36.00% Delay to recovery 1 yr

50% defaults in year 2 50% defaults in year 1

Pool 1 representing 30.00% Pool 2 representing 70.00% Total

Proportion Portfolio 150.00 Proportion Portfolio 350.00 500.0

17% Collateral Coupon 6.90% 86% Collateral Coupon 5.60% 5.7000%

83% Collateral Spread 2.75% 14% Collateral Spread 1.20% 2.3071%

Rating Level Ba3 Rating Level Baa1

Length 9.00 yr Length 9.00 yr

Default Prob 16.710% Default Prob 2.270% 6.602%

Diversity 12 Diversity 33

# of defaults 0 # of defaults 0

Recovery Rate 30.0% Recovery Rate 50.0%

Class A OC test 108.50% IR Scenario Std Dev assumed 18%

Class A IC test 110.00% # of Std Dev 1

Class B OC test 103.00% Hedges Swap Float (rec) 0.00%

Class B IC test 108.00% Termination Payment ? FALSE

Class C OC test 101.25% Cap rate 9.00%

Class C IC test 105.00% Intra period Re-inv: Euribor -0.20%

Class C Reinvestment OC 102.45%

The BET approach : Cash flow allocationThe BET approach : Cash flow allocation

Remaining interest

Remaining principal

Class A OC ratio

Class A IC ratio

Targetted Redemption

from interestTo Class A Class A

Remaining interest

Remaining principal

108.50% 110.00%

5,043,408 0 112.5% 160.7% 0 0 440,000,000 5,043,408 0

4,489,642 0 111.3% 151.5% 0 0 440,000,000 4,489,642 0

3,184,932 0 109.4% 132.3% 0 0 440,000,000 3,184,932 0

2,590,716 0 107.1% 124.5% 5,607,544 2,590,716 435,481,214 0 0

2,553,945 0 106.7% 123.4% 7,118,795 2,553,945 432,927,269 0 0

2,526,063 0 106.3% 122.3% 8,666,816 2,526,063 430,401,206 0 0

2,468,525 0 105.9% 121.6% 10,242,719 2,468,525 427,932,680 0 0

1,545,340 0 105.5% 113.1% 11,876,160 1,545,340 426,387,340 0 0

1,489,276 0 104.8% 112.5% 14,432,786 1,489,276 424,898,065 0 0

1,007,629 0 104.1% 108.2% 17,045,477 1,007,629 423,890,435 0 0

1,677,587 27,678,038 103.3% 113.6% 20,139,814 20,139,814 403,750,621 0 9,215,810

1,336,547 27,458,371 102.9% 111.4% 20,193,410 20,193,410 374,341,401 0 8,601,507

1,639,987 35,009,780 103.5% 115.1% 16,705,747 16,705,747 349,034,147 0 19,944,019

1,544,028 35,009,780 104.4% 115.5% 12,323,058 12,323,058 316,767,069 0 24,230,749

1,697,682 35,009,780 105.5% 119.4% 8,036,329 8,036,329 284,499,991 0 28,671,132

1,655,458 35,009,780 107.0% 121.3% 3,595,945 3,595,945 252,232,913 0 33,069,292

1,733,041 35,009,780 108.9% 126.2% 0 0 219,163,621 1,733,041 35,009,780

1,688,474 35,009,780 110.7% 130.1% 0 0 183,933,295 1,688,474 35,009,780

1,696,479 35,009,780 113.4% 138.2% 0 0 148,762,136 1,696,479 35,009,780

1,661,710 35,009,780 117.7% 148.4% 0 0 113,557,793 1,661,710 35,009,780

890,632 7,551,409 125.8% 137.2% 0 0 78,401,760 890,632 7,551,409

811,838 7,551,409 128.5% 137.1% 0 0 70,850,351 811,838 7,551,409

The BET approach : Loss calculationThe BET approach : Loss calculation

Class A Class B Class C

Size in million 440.00 27.50 15.00

OC (%) 113.6% 107.0% 103.6%

Target Aaa A1 Baa3

Stress 1 1.30 1.14 1.07

Stress 2 1.22 1.07 1.00

Coupon Floating Floating Floating

Spread 0.48% 1.60% 2.50%

Loss 0.0000% 44.7% 97.2%

Exp Loss 0.0008% 0.352% 3.764%

Scenario Life 7.84 13.25 15.00

WAL 7.84 10.94 11.67

Rating Aaa+ Aa3 Baa3+

Mid Point 0.0035% 0.3455% 4.5230%

The Monte Carlo approach and the log-normal methodThe Monte Carlo approach and the log-normal method

Section 4

The Monte Carlo approachThe Monte Carlo approach

Description

The total expected loss of a portfolio can be calculated as the average of the losses generated by running a high number of default simulations on the pool of assets and applying to each defaulted asset the relevant recovery rate

Key variables and assumptions

Default probability, recovery rates, default correlations

Main disadvantages

Difficult to implement, convergence problems, calculation time

The log-normal methodThe log-normal method

Lognormal Default DistributionMean 5% - Std Dev/Mean =40%

0.0%

0.2%

0.4%

0.6%

0.8%

1.0%

1.2%

1.4%

0% 5% 10% 15%

Portfolio analysisPortfolio analysis

Section 5


In the following, consider an ever increasing number of independent assets in the pool:

N = 1 bond A 2 cases : no default or A defaults

N = 2 bonds A and B 4 cases : no default, A or B defaults, A and B default

N = 3 bonds A, B and C 8 cases : no default, A, B or C defaults, (A and B)

(B and C) or (A and C) default and (A, B and C) default

...

N bonds 2N cases


N=1N=1

N=4N=4N=3N=3

N=2N=2

Losses in %Losses in %

Prob.Prob.

(%)(%)


Prob.Prob.

(%)(%)


Prob.Prob.

(%)(%)


Prob.Prob.

(%)(%)


N=5N=5

N=30N=30N=20N=20

N=10N=10


Prob.Prob.

(%)(%)


Prob.Prob.

(%)(%)


Prob.Prob.

(%)(%)


Prob.Prob.

(%)(%)


N=40N=40

N=200N=200N=100N=100

N=50N=50


Prob.Prob.

(%)(%)


Prob.Prob.

(%)(%)


Prob.Prob.

(%)(%)


Prob.Prob.

(%)(%)


N=300N=300

N=600N=600N=500N=500

N=400N=400


Prob.Prob.

(%)(%)


Prob.Prob.

(%)(%)


Prob.Prob.

(%)(%)


Prob.Prob.

(%)(%)


N=700N=700

N=1000N=1000N=900N=900

N=800N=800


Prob.Prob.

(%)(%)


Prob.Prob.

(%)(%)


Prob.Prob.

(%)(%)


Prob.Prob.

(%)(%)

The log-normal methodThe log-normal method

Lognormal Default DistributionMean 5% - Std Dev/Mean =40%

0.0%

0.2%

0.4%

0.6%

0.8%

1.0%

1.2%

1.4%

0% 5% 10% 15%

Characteristic functionsCharacteristic functions

Section 6


In probability theory, a characteristic function is defined by:

XtiX eEt ..)(


The characteristic function of simple random variables can be computed easily

For instance, when X is a Bernouilli variable :

X = 1 with probability p

X = 0 with probability q = 1 - p

Therefore for a bond with a default probability p and a size S:

tititiXtiX epqepeqeEt .1..0.... ...)(

StiX epqt ...)(


Consider now a portfolio of N independent bonds / assets :

Sizes: S1, S2, … SN

Default probabilities: p1, p2, …, pN

Define X (random variable) as being the defaulted amount of the portfolio:

X = S1. X1 + S2. X2 + S3. X3 + … SN. XN

NNNN XStiXStiXStiXSXSXStiX eEeEeEeEt .........).......(. ......)( 22112211

).).....(.).(.()( ....22

..11

21 NStiNN

StiStiX epqepqepqt

In a Nutshell:

Fourier transform theoryFourier transform theory

Functions of a space variable (x)

Real Space

Functions of a time/ frequency variable (t)

Fourier Dual SpaceFourier Tranform

Inverse Fourier Tranform

dxexftf ixt).()(ˆ

dtetgxg ixt).()(2

1

f(x)

g(t)

ff ˆ :FormulaInversion

If you know the Fourier Transform of a function , it is easy (at least theoretically) to get the original function by applying the inverse Fourier Transform (hence the name of the inverse Fourier Transform)

Portfolio analysis : why did dinosaurs come into the Portfolio analysis : why did dinosaurs come into the picture ?picture ?

ConclusionConclusion

In most cases, it is impossible to derive tractable formulas for default distributions in the space domain.

However, under certain sets of modelling assumptions, the formulas simplify if we translate ourselves in the Fourier domain.

In order to get back to the “real” space, apply the Inverse Fourier Transform.

Computing an Inverse Fourier Transform basically costs nothing in terms of computation time. Fast Fourier Transform algorithms were discovered some 50 years, (it is a powerful technique that made possible technical revolutions in many industries - electronics, CD, DVD, radio, telecommunications, medical systems,…

This is precisely because computing a Fourier Transform costs practically nothing in terms of computation times that this new numerical method is of interest for getting default distributions.

ConclusionConclusion

Theoretical framework that permits to justify the log-normal approach

Framework that permits much more: analysis of tricky portfolio risk profile that results from aggregation of heterogeneous assets and analysis of dependence between recovery rates and default rates (not really addressed so far)

Avoids Monte Carlo simulations and calibration/convergence issues

Could potentially be applied to any kind of portfolio indicators (default, losses, but also PV, etc,)

Surprisingly, almost nothing was written by academics on portfolio loss/default distributions. However, very recently, Vasicek (1997- KMV) and Finger (1999 - Credit Metrics) obtained very promising results

Using a simple factor model (i.e. a model that assumes that default correlation between the loans is created by exposure of all the loans to a common market index), they find that the portfolio defaults have a normal inverse probability distribution

Tackling dependencies between assets ?Tackling dependencies between assets ?


Default correlation is primarily the result of individual companies being linked to one another through the general economy

Beyond that induced by the general economy, default correlation exists between firms in the same industry because of industry-specific economic conditions

Default correlations also exist between companies in different industries that rely on the same production inputs and among companies that rely on the same geographical market


Given the probability of default for each asset, we calculate a threshold such that given a variable Z normally distributed (i.e. having a density of probability function N(0,1)):

is the threshold against which we will need to compare the random gaussian variable to determine if a default on that specific obligor has occurred

)(.2

1 2

2

dxeZp

x


The model’s assumptions:

Let’s define Zj as a (normalized) credit risk measure of the jth debtor at the end of the

horizon. The lower Zj, the higher the credit risk of the jth debtor.

Debtor j defaults during the period if Zj <j, i.e. j is determined by Pr(Zj <j) = pj.

j is the default threshold. If Zj is normalized, j = -1(pj), where (x) is the cumulative

standard normal distribution function.

The credit risk indicator of the jth debtor is split between a systemic risk (exposure to

a common normalised market index Z - for instance, economic growth, … ) and an idiosyncratic risk (normalised risk that can be only attributed to the jth debtor):

jjjj ZZ .1. 2

Systemic riskSystemic risk IdiosyncraticIdiosyncratic riskriskExposure to Common IndexExposure to Common Index

(correlation parameter)(correlation parameter)


The jth debtor will default if Zj < ji.e. if

Using the standard normality of j, the default probability of the jth

debtor (conditional to a fixed Z) will simply be

Conditional Fourier transform of asset j (given Z) is:

Conditional Fourier transform of the portfolio (given Z) is:

Fourier transform of the portfolio is:

21

.

j

jjj

Z

)1

.()(

2j

jjj

ZZp

jj

j

Stijj

XtiZX eZpZqZeEt ....

/ ).()()/()(

)).()(()/()(1

....

/

1

1

N

j

Stijj

Xti

ZX

j

N

jj

N

jj

eZpZqZeEt

dzeZpZqetN

j

Stijj

z

X

jN

jj

.)).()((..2

1)(

1

..22

1

Documents

Dinosaurs in a portfolio Marc Freydefont Moody’s Investors Service Ltd. London, 13 March 2002 ISDA - PRMIA Seminar