Credit Risk Modelling
Matthias -Watt University, Edinburgh
January–March 2018
MAF based on AJM (HWU)
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Credit Risk Modelling 2018
QRM textbook
[McNeil et al., 2015]
MAF based on AJM (HWU) Credit Risk Modelling 2018
1 Introduction to Credit Risk
2 Measuring Credit Quality
3 Structural Models of Default
4 Bond and CDS Pricing in Hazard Rate Models
5 Threshold Models
6 Mixture Models
7 Asymptotics for Large Portfolios Based亚 大型投资组合的渐近性
8 Monte Carlo Methods
9 Statistical Inference in Portfolio Credit Risk Models
MAF based on AJM (HWU) Credit Risk Modelling 2018
1 Introduction to Credit Risk Introductory Material
Background Material: Risk Measures Credit Risky Instruments
PD, LGD and EAD
MAF based on AJM (HWU) Credit Risk Modelling 2018
Defining Credit Risk
“Credit risk is the risk of a loss arising from the failure of a coun-
terparty to honour its contractual obligations. This subsumes both default risk (the risk of losses due to the default of a borrower or a
deterioration in the credit quality of a counterparty that translates into a downgrading in some rating system). ”
Obligor = a counterparty who has a financial obligation to us; for example, a debtor who owes us money, a bond issuer who promises interest, or a counterparty in a derivatives transaction.
Default = failure to fulfil that obligation, for example, failure to repay loan or pay interest/coupon on a loan/bond; generally due to lack of liquidity or insolvency; may entail bankruptcy.
trading partner) and downgrade risk (the risk of losses caused by a
MAF based on AJM (HWU) Credit Risk Modelling 2018
Where Does Credit Risk Arise?
Credit risk is omnipresent in the portfolio of a typical financial institution. A portfolio of loans is obviously affected by credit risk.
Bond portfolios are subject to credit risk. 衍生品交易
Credit risk accompanies any OTC (over-the-counter,
ransaction such as a swap, because the default of one of the parties involved may substantially affect
the actual pay-off of the transaction.
There is a specialized market for credit derivatives, such as credit default swaps, which are primarily designed to mitigate credit risk, but are also used to speculate on changes in credit risk.
Credit risk relates to the core activities of most banks but is also highly relevant to insurance companies.
i.e. non-exchange-guaranteed) derivativ
Insurers are exposed to substantial credit risk in their investment
portfolios and counterparty default risk in their reinsurance treaties.
MAF based on AJM (HWU) Credit Risk Modelling 2018
Credit Risk Management
Managing credit risk at financial institutions involves a range of tasks:
An enterprise needs to determine the capital it requires to absorb losses
due to credit risk, both for regulatory and economic capital purposes. It also needs to manage the credit risk on its balance sheet.
Portfolios of credit-risky instruments should be well diversified and optimized according to risk-return considerations.
The risk profile of a portfolio can be improved by hedging risk concentrations with credit derivatives or by transferring risk to investors through securitization证. 券化
Institutions need to manage their portfolio of traded credit derivatives,
which involves pricing, hedging and managing collateral for such trades. 交易对手
Financial institutions need to control the counterparty credit risk in their trades and contracts with other institutions. This has particularly been the case since the 2007–2009 financial crisis.
MAF based on AJM (HWU) Credit Risk Modelling 2018
1 Introduction to Credit Risk Introductory Material
Background Material: Risk Measures
Credit Risky Instruments PD, LGD and EAD
MAF based on AJM (HWU) Credit Risk Modelling 2018
Risk Measures
dBasicideaofbankingregulation
Capital requirements are determined by applying risk measures to the
distribution of potential balance sheet losses. 分布函数
Let 0 < ↵ < 1. Consider a loss L with distribution function (df) FL.
management
Value at Risk is defined as ⻛险价值
.Bhanr where we use the notation q↵(FL) or q↵(L) for a quantile of the df FL.
The quantile of a df is given by its generalized inverse FL : q↵(FL) = FL (↵) = inf{x : FL(x) ↵}.下限
creditrisk VaR↵ = q↵(FL) ,marketrisk
Provided E(|L|) < 1 expected shortfZall is defined as ⻰会 du
naveragevalue ES↵ = 1 1 qu(FL)du. (2)
at 㸠 1 ↵ ↵ formarketriskin based亚
For a continuous loss distribution
ES↵ =E(L|L VaR↵).
MAF based on AJM (HWU) Credit Risk Modelling 2018
Losses and Profits
Loss Distribution
Mean loss = -2.4
95% VaR = 1.6
95% ES = 3.3
5% probability
Pitof241Va2l95 195 quan
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probability density
0.0 0.05 0.10 0.15 0.20 0.25
1 Introduction to Credit Risk Introductory Material
Background Material: Risk Measures
Credit Risky Instruments
PD, LGD and EAD
MAF based on AJM (HWU) Credit Risk Modelling 2018
May be categorized into: retaiil loans (to individuals and small or
medium-sized companies), corporate loans (to larger companies),
interbank loans and sovereign loans (to governments). -
In each of these categories there may be a number of different products.
For example, retail customers may borrow money using mortgages 抵押物 against property, credit cards and overdrafts.
A sum of money, known as the principal, is advanced to the borrower for
a particular term in exchange for a series of defined interest payments,
which may be at fixed or floating interest rates. At the end of the term the
borrower is required to pay back the principal.
A useful distinction to make is between secured and unsecured lending. If a loan is secured the borrower has pledged an asset as collateral for the loan. In a mortgage the collateral is a property. In the event of default, the lender may take possession of the asset to mitigate the loss.
In an unsecured loan the lender has no such claim on a collateral asset.
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Bonds are publicly traded securities issued by companies and governments which allow the issuer to raise funding on financial markets.
Bonds issued by companies are corporate bonds and bonds issued by governments are known as treasuries, sovereign bonds or, particularly in the UK, gilts (gilt-edged securities).
The security commits the bond issuer (borrower) to make a series of interest payments to the bond buyer (lender) and pay back the principal at a fixed maturity.
The interest payments, or coupons, may be fixed at the issuance of the bond (so-called fixed-coupon bonds). Alternatively, there are also bonds where the interest payments vary with market rates (so-called floating-rate notes).
The reference rate for the floating rates is often a LIBOR rate (London Interbank Offered Rate).
There are also convertible bonds which allow the purchaser to convert them into shares of the issuing company at predetermined time points.
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Risks Faced by Bondholders
A bond holder is subject to a number of risks, particularly interest-rate risk, default risk, downgrade risk and spread risk.
Changes in the term structure of interest rates affect the value of bonds. As for loans, default risk is the risk that promised coupon and principal
payments are not made.
Downgrade risk is the risk that the bond loses value because the issuer’s credit rating is lowered.
Historically government bonds issued by developed countries have been considered default-free; for obvious reasons, after the European debt crisis of 2010–2012, this notion was called into question.
Spread risk is a form of market risk that refers to changes in credit spreads. The credit spread of a defaultable bond measures the difference in the yield of the bond and the yield of an equivalent default-free bond.
An increase in the spread of a bond means that the market value of the bond falls, which is generally interpreted as indicating that the financial markets perceive an increased default risk for the bond.
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Derivative Contracts Subject to Counterparty Risk
A substantial part of all derivative transactions is carried out over the counter and there is no central clearing counterparty such as an organized exchange to guarantee the fulfilment of the contractual obligations.
These trades are subject to the risk that one of the contracting parties defaults during the transaction, thus affecting the cash flows that are actually received by the other party. This risk, known as counterparty credit risk, received a lot of attention during the financial crisis of 2007-2009.
Some of the institutions heavily involved in derivative transactions experienced worsening credit quality or—in the case of —even a default event.
Counterparty risk management is now a key issue for all financial institutions and the focus of many new regulatory developments.
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A管B Two parties A and B agree to exchange a series of interest payments on
Example of Interest-Rate Swap
a given nominal amount of money for a given period.
A receives payments at a fixed interest rate and makes floating payments at a rate equal to the three-month LIBOR rate.
Suppose that A defaults at time ⌧A before the maturity of the contract.
If interest rates have risen relative to their value at inception of contract:
I The fixed interest payments have decreased in value and the value of the contract has increased for B.
I The default of A constitutes a loss for B.
I The exact size will depend on the term structure of interest rates at ⌧A.
If interest rates have fallen relative to their value at t = 0:
I The fixed payments have increased in value so that the swap has a negative
value for B.
I B will still has to pay the value of the contract into the bankruptcy pool, I There is no upside for B in A’s default.
If B defaults first the situation is reversed: falling rates lead to a counterparty-risk-related loss for A.
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Management of Counterparty Credit Risk
Counterparty risk has to be taken into account in pricing and valuation. This has led to the notion of credit value adjustments (CVA).
Counterparty risk needs to be controlled using risk-mitigation techniques
such as netting and collateralization.
Under a netting agreement the value of all derivatives transactions between A and B is computed and only the aggregated value is subject to counterparty risk; since offsetting transactions cancel each other out, this has the potential to reduce counterparty risk substantially.
Under a collateralization agreement the parties exchange collateral (cash and securities) that serves as a pledge for the receiver. The value of the collateral is adjusted dynamically6to reflect changes in the value of the underlying transactions.
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Credit Derivatives
Credit derivatives are securities which are primarily used for the hedging and trading of credit risk.
The promised pay-off of a credit derivative is related to credit events affecting one or more firms.
Major participants in the market for credit derivatives are banks,
insurance companies and investment funds.
Retail banks are typically net buyers of protection against credit events; other investors such as hedge funds and investment banks often act as both sellers and buyers of credit protection.
Credit default swaps (CDSs) are the workhorses of the credit derivatives market and the market fIor CDSs written on larger corporations is fairly liquid.
信用违约掉期 quotedinFTWSJ
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Structure of CDS
Consider contract with maturity T and ignore counterparty credit risk. Three
parties are involved (only two directly):
min(⌧C , T ). 周期性的
B (protection seller); makes default payment to A if ⌧C < T .
referenceentity
C (reference entity); default at time ⌧C < T triggers default payment. A (protection buyer); makes periodic premium payments to B until
premium payment at fixed dates until default or maturity
Protectionseller
Protectionbuyer
default of C occurs ?
yes: default payment
no: no payment
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CDS: Payment Flows
If the reference entity experiences a default event before the maturity date T of the contract, the protection seller makes a default payment to the protection buyer, which mimics the loss due to the default of a bond issued by the reference entity (the reference asset); this part of a CDS is called the default payment leg.
In this way the protection buyer has acquired financial protection against the loss on the reference asset he would incur in case of a default.
As compensation the protection buyer makes periodic premium payments (typically quarterly or semiannually) to the protection seller (the premium payment leg); after the default of the reference entity, premium payments stop. There is no initial payment.
The premium payments are quoted in the form of an annualized percentage x⇤ of the notional value of the reference asset; x⇤ is termed the (fair or market quoted) CDS spread. 传播
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Use of CDS
hedgingvs speculation
Investors enter into CDS contracts for various reasons. Bond investors with a large credit exposure to the reference entity may buy CDS protection to insure themselves against losses due to the default of a bond.
This may be easier than reducing the original bond position because CDS markets are often more liquid than bond markets. Moreover, CDS positions are quickly settled.
CDS contracts are also held for speculative reasons. In particular, so-called naked CDS positions, where the protection buyer does not own the bond are often assumed by investors who are speculating on the widening of the credit spread of the reference entity. These positions are similar to short-selling bonds issued by the reference entity.
Note that, in contrast to insurance, there is no requirement for the protection buyer to have insurable interest, that is, to actually own a bond issued by the reference entity.
The speculative motive for holding CDS is at least as important as the insurance motive.
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1 Introduction to Credit Risk Introductory Material
Background Material: Risk Measures Credit Risky Instruments
PD, LGD and EAD
MAF based on AJM (HWU) Credit Risk Modelling 2018
Exposure LEAD_exposure atdefault
If we make a loan or buy a bond, our exposure is relatively easy to determine, since it is mainly the principal that is at stake. There is some additional uncertainty about the value of the interest payments that could be lost.
A further source of exposure uncertainty is due to the widespread use of credit lines, essentially a ceiling up to which a corporate client can borrow money at given terms.
For OTC derivatives the counterparty risk exposure is even more difficult to quantify, since it is a stochastic variable depending on the unknown time at which a counterparty defaults and the evolution of the value of the derivative up to that point.
In practice the concept that is used to describe exposure is exposure at default (EAD), which recognises that the exposure for many instruments will depend on the exact default time.
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Probability of Default (PD) twostandardnine 90days 1year
When measuring the risk of losses over a fixed time horizon, for example one year, we are particularly concerned with estimating the probability that obligors default by the time horizon, a quantity known to practitioners as probability of default or PD.
The probability of default is related to the credit quality of an obligor and we discuss models of credit quality next.
For instruments where the loss is dependent on the exact timing of default, for example OTC derivatives with counterparty risk, the risk of default is described by the whole distribution of possible default times and not just the probability of default by a fixed horizon.
In simple models of default time, the probability of default may be expressed in terms of a hazard function which measures the risk of default at any instant in time.
MAF based on AJM (HWU) Credit Risk Modelling 2018
Loss Given Default (LGD)
In the event of default, it is unlikely that the entire exposure is lost.
When a mortgage holder defaults on a residential mortgage, and there is no realistic possibility of restructuring the debt, the lender can sell the property (the collateral asset) and the proceeds from the sale will make good some of the lost principal.
When a bond issuer goes into administration, the bond holders join the group of creditors who will be partly recompensed for their losses by the sale of the firm’s assets.
Practitioners use the term loss given default or LGD to describe the proportion of the exposure that is actually lost in the event of default, or its converse, the recovery, to describe the amount of the exposure that can be recovered through debt restructuring and asset sales.
EAD, PD and LGD are dependent quantities. For example, in a period of financial distress, when PDs are high, asset values of firms are depressed and firms are defaulting, recoveries are likely to be correspondingly low, so that there is positive dependence between PDs and LGDs.
given as EAD
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2 Measuring Credit Quality
Measures of Credit Quality
Credit Rating Migration
Rating Transitions as a Markov Chain
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Scores, Ratings & Measures Inferred from Prices
Broadly speaking, there are two philosophies for quantifying the credit quality or default probability of an obligor.
1 Credit quality can be described by a credit rating or score that is based on empirical data describing the borrowing and repayment history and other characteristics of二the obligor, or similar obligors.
2 For obligors whose equity is traded on financial markets, prices can be used to infer the market’s view of the credit quality of the obligor.
Credit ratings and credit scores fulfill a similar function—they both allow us to order obligors according to their credit risk and map that risk to an estimate of default probability.
Credit ratings tend to be expressed on an ordered categorical scale whereas credit scores are often expressed in terms of points on a metric scale.
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Ratings and Scores
The task of rating obligors is often outsourced to a rating agency such as Moody’s or Standard & Poor’s (S&P) although internal systems can also be used.
In the S&P rating system there are seven pre-default rating categories labelled AAA, AA, A, BBB, BB, B, CCC, with AAA being the highest and CCC the lowest rating;
Moody’s uses nine pre-default rating categories labelled Aaa, Aa, A, Baa, Ba, B, Caa, Ca, C.
A finer alpha-numeric system is also used by both agencies.
Credit scores are traditionally used for retail customers and
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