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
Portfolio analysisPortfolio analysis
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
Portfolio analysisPortfolio analysis
N=1N=1
N=4N=4N=3N=3
N=2N=2
Losses in %Losses in %
Prob.Prob.
(%)(%)
Losses in %Losses in %
Prob.Prob.
(%)(%)
Losses in %Losses in %
Prob.Prob.
(%)(%)
Losses in %Losses in %
Prob.Prob.
(%)(%)
Portfolio analysisPortfolio analysis
N=5N=5
N=30N=30N=20N=20
N=10N=10
Losses in %Losses in %
Prob.Prob.
(%)(%)
Losses in %Losses in %
Prob.Prob.
(%)(%)
Losses in %Losses in %
Prob.Prob.
(%)(%)
Losses in %Losses in %
Prob.Prob.
(%)(%)
Portfolio analysisPortfolio analysis
N=40N=40
N=200N=200N=100N=100
N=50N=50
Losses in %Losses in %
Prob.Prob.
(%)(%)
Losses in %Losses in %
Prob.Prob.
(%)(%)
Losses in %Losses in %
Prob.Prob.
(%)(%)
Losses in %Losses in %
Prob.Prob.
(%)(%)
Portfolio analysisPortfolio analysis
N=300N=300
N=600N=600N=500N=500
N=400N=400
Losses in %Losses in %
Prob.Prob.
(%)(%)
Losses in %Losses in %
Prob.Prob.
(%)(%)
Losses in %Losses in %
Prob.Prob.
(%)(%)
Losses in %Losses in %
Prob.Prob.
(%)(%)
Portfolio analysisPortfolio analysis
N=700N=700
N=1000N=1000N=900N=900
N=800N=800
Losses in %Losses in %
Prob.Prob.
(%)(%)
Losses in %Losses in %
Prob.Prob.
(%)(%)
Losses in %Losses in %
Prob.Prob.
(%)(%)
Losses in %Losses in %
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
Characteristic functionsCharacteristic functions
In probability theory, a characteristic function is defined by:
XtiX eEt ..)(
Characteristic functionsCharacteristic functions
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 ...)(
Characteristic functionsCharacteristic functions
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 ?
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
Tackling dependencies between assets ?Tackling dependencies between assets ?
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
Tackling dependencies between assets ?Tackling dependencies between assets ?
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)
Tackling dependencies between assets ?Tackling dependencies between assets ?
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