Financial Risk Management and Governance

Credit Risk Portfolio Management

Prof. Hugues Pirotte

Beyond simple estimations Credit risk includes counterparty risk and therefore there is

always a residual credit risk

Identification of credit risk exposure » Not very easy since can be intimately linked to the market value

» In particular, for derivatives

» Contracts can be

1. Assets to the firm

2. Liabilities to the firm

3. Maybe both...

Examples of vulnerable assets » (Defaultable) Options (insurances)

» (Defaultable) Swaps

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Some algebra about this... Assume that default probability is independent of the value of

the derivative

Define t1, t2,…tn: times when default can occur

qi: default probability at time ti.

fi: The value of the contract at time ti

R: Recovery rate

The expected loss from defaults at time ti is

qi (1-R) E[max(fi ,0)]

Defining ui=qi(1-R) and vi as the value of a derivative that provides a payoff of max(fi,0) at time ti, the PV of the cost of defaults is

n

i

iivu1

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IRS vs. CS Expected exposure on pair of offsetting interest rate swaps and a

pair of offsetting currency swaps (Hull, RMFI, Figure 12.2, page 281)

Exposure

Maturity

Currency

swaps

Interest Rate

Swaps

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Two-sided default risk In a swap operation

» There are two reciprocal credit exposures

» The market value of the swap will evolve through time depending on the underlying market conditions

» The creditworthiness of both counterparts may also evolve during the life of the swap

If both exposures (market & credit) propose scenarios that are equally likely for both counterparts » Then the credit risk spread on the transaction should be 0.

Otherwise, the value of the swap to a counterpart (let’s take A) should be...

But swap credit risk remains a complex subject...

A B

B Af Exp Exp

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Other issues Netting

» We replace fi by in the definition of ui to calculate the expected cost of defaults by a counterparty where j counts the contracts outstanding with the counterparty

» The incremental effect of a new deal on the exposure to a counterparty can be negative!

Collateralization » Contracts are marked to markets periodically (e.g. every day) » If total value of contracts Party A has with party B is above a specified

threshold level it can ask Party B to post collateral equal to the excess of the value over the threshold level

» After that collateral can be withdrawn or must be increased by Party B depending on whether value of contracts to Party A decreases or increases

Downgrade triggers » A downgrade trigger is a clause stating that a contract can be closed out by

Party A when the credit rating of the other side, Party B, falls below a certain level

» In practice Party A will only close out contracts that have a negative value to Party B

» When there are a large number of downgrade triggers they are counterproductive

j

ijf

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CreditVaR: Methods Definition

» Can be defined analogously to Market Risk VaR » A one year credit VaR with a 99.9% confidence is the loss level that we are

99.9% confident will not be exceeded over one year

Vasicek’s model » For a large portfolio of loans, each of which has a probability of Q(T) of

defaulting by time T the default rate that will not be exceeded at the X% confidence level is

where r is the Gaussian copula correlation

Basle II » One-factor Gaussian copula model (Vasicek’s model) with special formulations

for the correlation parameter (that depends on PD in many cases)

Portfolio models » How do we aggregate individual credit risks?

1 1( ) ( )

1

N Q T N XN

r

r

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Portfolio Models: CreditRisk+ Simplified

» A financial institution has N counterparts with a PD each (p).

» Assuming independent defaults and that p is small, the probability of n defaults is given by a Poisson process of the form

where

» Combining this to a probability distribution of default losses on a single counterpart, this can produce a distribution of total losses for our current portfolio

Also » Estimation per category of counterparts

» Varying default rates

Use a probability distribution based on historical data and link each category’s PD to this distribution.

CSFP provides an analytical form under some assumptions

!

ne

n

Np

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Portfolio Models: CreditRisk+ (2) Otherwise, use a Monte Carlo simulation where the steps are:

1. Sample overall default rate

2. Calculate a PD for each category

3. Sample number of defaults for each category

4. Sample size of loss for each default

5. Calculate total loss

6. Repeat the simulation procedure many times

Assuming categories linked to an overall PD distribution implied “default correlations”. The distribution of total losses will be thus positively skewed.

Total losses

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Portfolio Models: CreditMetrics Idea

» Calculates credit VaR by considering possible rating transitions

» A Gaussian copula model is used to define the correlation between the ratings transitions of different companies

Framework

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Portfolio Models: CreditMetrics

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Portfolio Models: CreditMetrics

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Results for the stand-alone case

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Source: CreditMetrics technical document

Add-in Incorporating the additional uncertainty around default:

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Source: CreditMetrics technical document

Portfolio Models: CreditMetrics

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Portfolio Models: CreditMetrics

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Portfolio Models: CreditMetrics

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Portfolio Models: CreditMetrics

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(See the accompanying Excel file for a more precise matrix and calculations…there is a property mismatch in this original application…find it!)

Portfolio Models: CreditMetrics

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Results from the portfolio case

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Portfolio Models: CreditPortfolioView Uses a factor models that takes into account macro variables

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References Some papers

» Baz, Jamil (1995), “Three Essays on Contingent Claims”, Harvard PhD Thesis, August 1995.

» Cossin & Pirotte (1999), “Swap Credit Risk: An Empirical Investigation on Transaction Data”, Journal of Banking and Finance, Vol. 21, No.10, October 1997, pp. 1351-1373.

» Cossin & Pirotte (1998), “How well do classical credit risk pricing models …t swap transaction data?”, European Financial Management, Vol. 4, No.1, March 1998, pp. 65-77.

» Duffe, Darrel and Ming Huang (1996), “Swap Rates and Credit Quality”, Journal of Finance, 51(3), July 1996, 921-949.

» Duffee, G.R., (1995a), “On Measuring Credit Risks of Derivative Instruments”, Working paper, Federal Reserve Board, February 1995.

Other documents » Documents by Credit Suisse Financial Products, RiskMetrics and McKinsey

» Hull RMFI’s slides

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