Model for Base Correlations

Credit Derivatives Strategy London 6 May 2004 2004

A Model for Base Correlation Calculation

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JPMorgan’s implementation of Base Correlations through the Large Pool Model

Credit Derivatives Strategy Lee McGinty*

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Open, transparent approach to analysing standardised tranche correlations § §

Observable calculations No add-ins required

(44-20) 7325-5482 [email protected]

CDO/Credit Derivatives Strategy Rishad Ahluwalia (44-20) 7777-1045 [email protected]

The certifying analyst(s) analyst(s) is indicated by an asterisk (*). (*). See last page of the report for analyst certification and important legal and regulatory disclosures.

http://mm.jpmorgan.com

Lee McGinty (44-20) 7325-5482 [email protected]

Credit Derivatives Strategy London 6 May 2004

Introduction In previous research, we have described the Base Correlation framework ( Credit Correlation: A Guide and Introducing Base Correlations , Lee McGinty and Rishad Ahluwalia, April 2004) and At-the-money Correlation (A Relative Value Framework for Credit Correlation, Lee McGinty and Rishad Ahluwalia, April 2004). In this document, we describe the specific implementation of the Large Pool Model that we provide to market participants to illustrate the process. The model is provided as a description in the appendix and as a spreadsheet. The variables in the appendix have been given long names to correspond to the relevant range names in the spreadsheet. Note that the spreadsheet and model description are provided as an educational and reference tool only. Many thanks to Siobhan Cooper for her assistance in building and documenting this model.

The Large Pool Model The model that we use for calculating tranche values and spreads from market inputs is called the Homogeneous Large Pool Gaussian Copula Model (HLPGC, or the Large Pool Model). This model is not new; it is a simple methodology that is almost identical to the original Credit Metrics Model (Gupton et al, 1997). We use it as a standard mechanism for translating market prices to an implied correlation. The Large Pool Model works on the assumption that the portfolio can be modeled by a very large number of credits of uniform size, and that the portfolio is homogeneous. The Appendix gives calculation details of our implementation of the Large Pool Model.

From Large Pool to Base Correlations The use of the Large Pool Model isn’t essential to the use of Base Correlations – they are two separate stages to the process. It is possible to think of compound correlations with the Large Pool Model, or more likely, to use a different model in a Base Correlation Framework. In either case, there are some more steps to calculating Base Correlations. Firstly, the market level of tranche spreads are used to calculate the expected loss on the relevant tranche. These are then combined to form expected losses on the relevant first loss tranche (for instance expected losses on 0-6 tranche are calculated as expected loss on 0-3 plus expected loss on 3-6). The model then iterates to find the single correlation that when entered as an input for this (0-6 in this example) tranche provides the given expected loss.

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Using the Model On opening the model, you will be presented with a disclaimer. To proceed to the main valuation page, choose the “Large Pool Model Calculations” workbook tab at the bottom. The model allows users to type in market prices for tranches and calculates the Base Correlation curve consistent with these prices. It can also be used in reverse to calculate tranche prices from a given input of the Base Correlation Curve. The spreadsheet looks like Figure 1, and is split into three or four main parts. At the top, you define market-wide factors or defaults. The middle section is where the current market tranches are input – typically the liquid tradable tranches. The section below that is for valuing off-market tranches. All of the calculations are far off to the right and below, and are not needed for dayto-day use, but are provided in as transparent way as possible to ensure the model is open and simple to understand. Almost all of the calculations are provided as worksheet functions – there are macros in the spreadsheet, but they deal mostly with interpolation and solving etc. There is a consistent colour scheme on the model to help users find their way around. Cells with a yellow background are labels and grey cells have formulae in and should not be changed. Users should only enter values in cells with a blue background. The darker blue cells are user inputs to define a deal and relevant conventions. For instance in our standard definition of the Large Pool Model, we define discount rates to be zero, and recovery to be 40%. Light blue cells are where users can type in values (or have the model calculate values for them). Once you have defined the market in terms of constants, attachment points etc., the next stage will usually be to enter to market quoted spreads (cells I:20 to J:24), and then click on the “SolveForCorrs” button to calculate the Base Correlation from these spreads. The Base Correlation is the single correlation that gives the correct tranche values for the tranches with lower attachment point at zero. At-the-money correlation and correlation skew are given in cells L13 and L14. From here, it is possible to value off-market tranches in the lower section. The upper and lower attachment points are entered, and the interpolated Base Correlations for each point are given (you can overwrite with alternative correlations). If these are new tranches, then the fair (par) spread is given in column N, if they are existing tranches, then enter the deal spreads and the mark to market value will be calculated in column L. The worksheet “Correlations” contains a chart of the Base Correlation structure.

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Figure 1 Screenshot of Large Pool Model in Excel

Deal input and conventions

On-the-run liquid tranche

Off-the-run tranches

Source: JPMorgan

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Appendix: Large Pool Model Details Introduction In this Appendix we give a detailed background to the calculations performed in the Large Pool Model. The emphasis is on making the model as transparent as possible, so almost all of the operations performed in the spreadsheet are described in detail. We will first look at the algorithm for determining expected loss, that is the Large Pool Model, and then the Fair Spread and MTM calculations.

Expected Loss Calculations We consider a portfolio of total notional value Ntnl, an average credit spread of spd in basis points and an average recovery of recov. We also have a value date and maturity date for the portfolio. We calculate the horizon of interest

horizon =

( MaturityDate − ValueDate ) 360

,

the clean spread of the portfolio

cleanSprea d =

spd

1 − recov

,

and the average cumulative default probability of the portfolio

PD(horizon ) = 1 − e

−

cleanSpread 10000

⋅horizon

.

We consider a specific tranching of the portfolio. In order to calculate the expected tranche loss, we need to use a model giving the loss distribution and using as input the average default probability. The use of Base Correlations described above requires us to price the tranche as the difference between two first loss pieces: the (0% to tranche upper bound) and (0% to tranche lower bound) tranches.

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Base Correlations are defined as the correlation inputs required for a series of equity tranches that give the tranche values consistent with quoted spreads using the standardised Large Pool Model. Where the standardised Large Pool Model uses a recovery rate of 40%, a market spread equal to the mid level of an equivalent quoted DJ TRAC-X unfunded swap and a discount rate of 0. We then use the Credit Metrics model with uniform pair-wise correlations corr given by the Base Correlations on these 2 first loss tranches. (Please refer to the Credit Metrics Technical documentation.). The uniform correlation assumption allows us to reduce the dimensionality of the Credit Metrics problem by introducing a single normal variable m, such that the name Credit Metrics gaussian factor can be decomposed into

X n

=

corr ⋅ m + 1 − corr ⋅ Z n

Where Z n is an idiosyncratic factor, and m is the common factor. We then calculate the following results conditional on m using the fact that the names defaults are independent conditional on m.

PCum(m): This is the expected percentage default of portfolio P given m.

æ Φ 1 (PD(horizon ) ) − corr ⋅ m ö ÷ PCum (m) = Φç ç ÷ 1 − corr è ø −

Where Φ is the standard normal cumulative distribution function.

Portfolio Loss: This is the percentage expected loss of the portfolio given m.

PL= PCum( m) ⋅ (1 − recov )

Tranche Loss: We use the large pool assumption here to justify that the portfolio loss will be concentrated at the expected portfolio loss, so that the expected tranche loss is

MIN ( MAX (PL ⋅ Ntnl ,0 ), TrancheUpp er ⋅ Ntnl ) Ntnl

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Note that here we do not consider the subordination or the tranche size as all the tranches that we are interested in have a lower attachment point of 0. To get the unconditional expected tranche loss, we do the weighted sum of conditional tranche loss. We calculate the expectation conditional on all m ∈ M ={-5, -4.9, -4.8,…, 4.8, 4.9 ,5} and then use

TotalExpec tedTranche Loss

=

å w(m) ⋅ TrancheLos s(m ) m∈ M

where the weight w(m) is

w( m) = 0.1 ⋅

d Φ( m) dm

Once we have carried out the above calculations for the first loss tranches, we can easily calculate the tranche loss for the tranche of interest. Let LowerLoss be the expected loss of the 0 to tranche lower bound portion of the portfolio priced using the base correlation for the 0 to tranche lower tranche. Let UpperLoss be the expected loss of the 0 to tranche upper bound portion of the portfolio priced using the base correlation for that tranche. Then tranche loss is

TrancheLos s

= UpperLoss − LowerLoss

and the expected survival of the tranche at horizon is

ExpectedSu rvival (horizon ) =

(TrancheSiz e − TrancheLos s ) TrancheSiz e

where

TrancheSize = (TrancheUpper − TrancheLower ) ⋅ Ntnl

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Fair Spread Determination & MTM Calculations We want to use the information that we have from the Large Pool Model to evaluate the price of protection against expected loss in the tranche of interest. We assume that the protection buyer pays the protection seller a quarterly fee of a fixed spread on the remaining notional at the payment date. Let T = {t 0 , t 1 ,…, t n } be the set of payment dates where t 0 is the value date, define for each t∈T

DayCount (t i ) =

t i

− t i −1

360

For the remainder of this discussion we assume that the discount factors and spreads are quoted on an ACTUAL/360 quarterly basis. For each t ∈T the discount factor for a given input quarterly rate can be calculated

æ discountRa te ö DF (t ) = ç1 + ÷ 4 è ø

−4 t

We treat the ExpectedSurvival as a discount factor for the period from value date to horizon. We can then convert the ExpectedSurvival to a quarterly survival rate and use the rate to get expected survival factors (or risky discount factors) for each payment date as follows

æ SurvivalRa te ö ExpectedSu rvival (t ) = ç1 + ÷ 4 è ø

−4 t

where the SurvivalRate is chosen such that this equation holds for t = horizon. We now have all the ingredients to calculate risk free and risky duration

n

RiskFreeDu ration = å DF (t i ) ⋅ DayCount (t i ) i =1 n

RiskyDurat ion = å DF (t i ) ⋅ ExpectedSu rvival (t i ) ⋅ DayCount (t i ) i =1

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We can now calculate expected incremental tranche loss occuring in the period from t i-1 to t i where ExpectedSurvival(0)= 1.

PD(t i , t i −1 ) = ExpectedSu rvival (t i −1 ) − ExpectedSu rvival (t i ) Given these default probabilities we can evaluate the contingent leg n

PV (Contingent Leg ) = å PD(t i , t i −1 ) ⋅ DF (t i ) ⋅ TrancheSiz e i =1

We are now also in a position to calculate the “Fair Spread” i.e. the spread at which the PV of the contingent leg is equal to the PV of the fee leg

FairSpread =

PV (Contingent Leg ) RiskyDurat ion ⋅ TrancheSiz e

Then the MTM of the protection bought by the client given the actual spread paid is

MTM = ( FairSpread − Spread ) ⋅ RiskyDurat ion ⋅ TrancheSiz e Note that both FairSpread and Spread have been quoted in Actual/360 quarterly notation.

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Model for Base Correlations

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