portfolio credit risk - risklab toronto credit risk - part 1... · portfolio credit risk prof. luis...

Review of Basic ConceptsGoodrich-Morgan-Robabank Swap: A Fixed Rate Loan

Credit Loss

Portfolio Credit Risk

Prof. Luis Seco

Prof. Luis SecoUniversity of Toronto

Department of MathematicsDepartment of Mathematical Finance

July 27, 2011

Prof. Luis Seco Portfolio Credit Risk


Credit Loss

Table of Contents

1 Review of Basic ConceptsTime Value of MoneyCredit: Premium and SpreadA Two-State Markov Model

2 Goodrich-Morgan-Robabank Swap: A Fixed Rate LoanSetupValuation: CreditmetricsNon-Constant Spreads

Markov ProcessesCredit Rating AgenciesGeneral Framework and Multi-Step Markov Process

3 Credit LossCredit Concepts and Terminology

ExamplesExpected Losses



Credit Loss

The Goodrich-Rabobank Swap 1983

Rabobank: AAA Rated5.5 MillionB.F. Goodri h: BBB- Rated(11% Fixed)On e a Year On e a Year5.5 Million

Morgan Guarantee TrustSwap Swap(Libor-x)%Semiannual Semiannual(Libor-y )%

Eurodollar MarketU.S. Savings Banks

Makes (y − x)% Semiannual

11% AnnualLIBOR+0.5%Semiannual



Credit Loss

Time Value of MoneyCredit: Premium and SpreadA Two-State Markov Model

Review of Basic Concepts



Credit Loss


Cash Flow Valuation

Fundamental Principle:

TIME IS MONEY

The present value of cash flows is given by the value equation:

Value =n∑

i=1

pie−ri ti (1)

Where:

n is the number of payments

pi is theamount paid at time ti

ri is the continuously compounded interest rate at time ti

Equation (1) assumes payments will occur with probability 1 (nodefault risk)



Credit Loss


Credit Premium

The discounted value of cash flows, when there is probability ofdefault, is given by:

Value =

n∑

i=1

pie−rti qi (2)

In the equation above qi denotes the probability that thecounterparty is solvent at time ti . A large default risk (i.e. a smallq) implies that:

1 For a fixed set of pis the discounted present value will alwaysbe less than or equal to the value equation (equation (1))

2 To preserve the same present value of cashflows as in equation(1) the cashflows ({pi}

ni=1) need to be increased. The amount

by which each payment is increased is q−1i . This is the credit

premium at time ti .



Credit Loss


The Credit Spread

The credit spread.

Since qi ≤ 1 we can write qi as:

qi = e−hi ti (3)

which implies:

hi =−ln(qi )

ti,

where hi is the credit spread at time ti .

The value of a loan with cashflows {pi}ni=1 at times {ti}

ni=1

and credit spread {hi}ni=1 is:

Value =n∑

i=1

pie−(ri+hi )ti (4)



Credit Loss


Example: Default Yield Curve

Example (Default Yield Curve)

A senior unsecured BB rated bond matures exactly in 5 years, andis paying an annual coupon of 6%

One-year forward zero-curves for each credit rating (%)Category Year 1 Year 2 Year 3 Year 4AAA 3.60 4.17 4.73 5.12AA 3.65 4.22 4.78 5.17A 3.72 4.32 4.93 5.32

BBB 4.10 4.67 5.25 5.63BB 5.55 6.02 6.78 7.27B 6.05 7.02 8.03 8.52

CCC 15.05 15.02 14.03 13.52

Table: One-year forward zero-curves for each credit rating (%)*

*Source: Creditmetrics, JP Morgan



Credit Loss


Solution: Default Yield Curve

Using the on the previous slide find the 1-year forward price of thebond, if the obligor stays BB.

Solution (Default Yield Curve)

Solution: 102.0063

VBB = 6 +6

1.0555+

6

1.06022+

6

1.06783+

106

1.07274= 102.0063



Credit Loss


First Model: Two Credit States

A simple two credit state model, some considerations andassumptions:

What is the credit spread?

Assume only 2 possible credit states: solvency and default.

Assume the probability of solvency in a fixed period (one year,for example), conditional on solvency at the beginning of theperiod, is given by a fixed amount q. For period ti+1 we have:

Pr(Solvent at time ti+1|Solvent at time ti ) = qi

According to this model, we have:

qi = qti

which gives rise to a constant credit spread:

hi = h = −ln(q)



Credit Loss


The General Markov Model

In other words, when the default process follows a Markov Chainthe probabilities of default/solvency for period (ti , ti+1] are givenby the matrix:

Solvency Default

Solvency qi 1− qiDefault 0 1

Table: Markov Chain for the Constant Credit Spread hi = h = −ln(qi )



Credit Loss

SetupValuation: CreditmetricsNon-Constant Spreads

Goodrich-Morgan-Robabank SwapA Fixed Rate Loan



Credit Loss


Setup

Conisder:

An 8 year fixed rate loan of 50$M at 11%

With cashflows (in millions of USD):1 0.125 upfront2 5.5 per year during 8 years

Assume

Constant spread of h = 180bpi over all periods

2 state transition probability matrix

Calculate

Expectd Cashflows

Expected Loss



Credit Loss


Expected Cashflows

The expected cashflows:

E [cashflows] = 0.125 +

8∑

i=1

5.5e−(ri+1.8)ti



Credit Loss


Probability of Default I

Under our assumptions, the conditional probability of non-defaultfor a given year is constant. This implies:

Pr(Non-Default ∈ (ti , ti+1]|Non-Default ∈ (0, ti ])

= e−h = e−0.018 = 0.98216 ≈ 0.9822

Similarly:

Pr(Default ∈ (ti , ti+1]|Non-Default ∈ (0, ti ])

= 1− e−h = 1− 0.98216 = 0.01784 ≈ 0.018



Credit Loss


Probability of Default II

So the 2 state matrix is written as:

BBB D

BBB 0.9822 0.178

D 0 1Table: Two-State Transition Matrix

Graphically: BBB D0.98216 10.01784Prof. Luis Seco Portfolio Credit Risk


Credit Loss


Computing Cashflows I

Inputs:

Pr(Non-Default ∈ (ti , ti+1]|Non-Default ∈ (0, ti ]) = 0.9822

The government zero curve for August 1983 was:

Year r

1 0.08852 0.092973 0.096564 0.09878555 0.105506 0.1043557 0.117708 0.118676

1 2 3 4 5 6 7 8

0.09

00.

095

0.10

00.

105

0.11

00.

115

Time(years)

r



Credit Loss


Computing Cashflows II

Now we can compute the expected cashflows:

E[Cashflows] = 0.125 + 5.5

8∑

i=1

e−ri ·ti (0.9822)ti

= 0.125 + 5.5

8∑

i=1

e−ri ·i (0.9822)i Since ti = i

= 23.0527 ≈ 23.05

Similarly we can compute the riskless cashflows:

Risk-Less Cashflows = 0.125 + 5.58∑

i=1

e−ri ·i

= 24.67581 ≈ 24.68



Credit Loss


Expected Losses

Therefore the Expected Percentage Loss is:

E [%Loss] = 1−

(E [Cashflows]

E [Non-Risk Cashflow]

)

= 6.5776% ≈ 6.58%

Similarly the Expected Loss is ≈ 6.58% of 24.67581$(USDmillion), that is

E [Loss] = 0.065776 × 24.67581 ≈ 1.623$(USD million)



Credit Loss


How Do We Deal With Non-Constant Spreads?

A default/no-default model (suchas CreditRisk+) leads to constantsspreads unless probabilities varywith time.

In order to fit non-constantspreads and be able to fit themodel to market observations, oneneeds to assume either:

1 Time-varying default probabilites2 Multi-rating systems (such as

CreditMetrics)

1 2 3 4 5 6 7

24

68

10

Constant Credit Spread

Time

%

Interest RateInterest Rate + Credit Spread

1 2 3 4 5 6 7

24

68

10

Non−Constant Credit Spread

Time

%

Interest RateInterest Rate + Credit Spread



Credit Loss


Markov Processes

123456

123456

123456p65

p11p12

p34

t = 0 t = 1 t = 2

p11

Transition ProbabilitiesConstant in Time



Credit Loss


Transition Probabilities

Conditional probabilities, which give rise to a matrix with n creditstates:

p11 p12 · · · · · · p1np21 p22 · · · · · · p2n

p31 · · · · · · · · ·...

... · · · · · · · · ·...

pn1 · · · · · · · · · pnn

pij is the conditional probability of changing from state i to state j .More formally:

pij = Pr(State j |State i)



Credit Loss


Credit Rating Agencies I

Credit rating agencies:

There are corporations whose business is to rate the creditquality of corporations, governments and also specific debtissues.

The main ones are:1 Moody’s Investors Service2 Standard & Poor’s3 Fitch IBCA4 Duff and Phelps Credit Rating Co



Credit Loss


Credit Rating Agencies II

S & P’s Rating System

AAA - Highest Quality; Capacity to pay interest and repay principalis extremely strong

AA - High Quality

A - Strong payment capacity

BBB - Adequate payment capacity

BB - Likely to fulfill obligations; ongoing uncertainty

B - High risk obligations

CCC - Current vulnerability to default

D - In bankruptcy or default, or other marked shortcoming



Credit Loss


Standard and Poor’s Markov Model

Multi-state transition matrix for Standard and Poor’s MarkovModel:

AAA AA A BBB BB B CCC DAAA 0.9081 0.0833 0.0068 0.0006 0.0012 0.0000 0.0000 0.0000AA 0.0070 0.9065 0.0779 0.0064 0.0006 0.0014 0.0002 0.0000A 0.0009 0.0227 0.9105 0.0552 0.0074 0.0026 0.0001 0.0006

BBB 0.0002 0.0033 0.0595 0.8693 0.0530 0.0117 0.0012 0.0018BB 0.0003 0.0014 0.0067 0.0773 0.8053 0.0884 0.0100 0.0106B 0.0000 0.0011 0.0024 0.0043 0.0648 0.8346 0.0407 0.0520

CCC 0.0022 0.0000 0.0022 0.0130 0.0238 0.1124 0.6486 0.1979D 0 0 0 0 0 0 0 1



Credit Loss


Long Term Transition Probabilites

Transition probabilities:

The transition probability between state i and state j , in twotime steps, is given by:

p(2)ij =

∑

k

pik · pkj (5)

In other words, if we donte by A the one-step conditionalprobability matrix, the two-step transition probability matrix isgiven by:

A2



Credit Loss


Transition Probabilities in General

If A denotes the transition probability matrix at one step (onyear, for example), the transition probability after n steps (30is especially meaningful for credit risk) is given by:

An

For the same reason, the quarterly transition probabilitymatrix should be given by:

A1/4

This gives rise to some important practical issues.



Credit Loss


Matrix Fractional Powers

Matrix Expansion

We can expand a matrix as folows:

Aa =

a∑

k=0

(α

k

)

(A− 1)k

Where (α

k

)

=α(α− 1) · · · (α− k + 1)

k!



Credit Loss


Example: PRM Exam Question

Example:Transition Matrix Problem

The following is a simplified transition matrix for four states:

Starting StateEnding State

Total ProbabilityA B C Default

A 0.97 0.03 0.00 0.00 1.00

B 0.02 0.93 0.03 0.03 1.00

C 0.01 0.12 0.23 0.23 1.00

Default 0.00 0.00 0.00 1.00 1.00

Calculate the cumulative probability of default in year 2 if the intialrating in year 0 is B.



Credit LossCredit Concepts and Terminology

Credit Loss




Credit Exposure

Definition (Credit Exposure)

Credit exposure is the maximum loss that a portfolio canexperience at any time in the future, taken with a certain level ofconfidence.




Example: Credit Exposure

Example (Credit Exposure)

Evolution ofthe Mark-to-Market of a20-MonthSwap, where:-99% Exposure-95% Exposure




Recovery Rate

Recovery Rate:

When default occurs, a portion of the value of the portfoliocan usually be recovered.

Because of this, a recovery rate is always considered whenevaluating credit losses, more specifically:

Definition (Recovery Rate)

The recovery rate (R) represents the percentage value which weexpect to recover, given default.




Loss Given Default

Similarly:

We can define a related term called loss given default:

Definition (Loss-Given Default)

Loss-given default (LGD) is the percentage we expect to lose whendefault occurs. Mathematically this is equivalent to:

R = 1− LGD

In both cases R and LGD may be modelled as randomvariables. However in simple exercises one may assume theyare constant.




Recovery Rate for Bond Tranches

For corporate bonds, there are two primary studies of recoveryratse which arrive at similar estimates (Carty & Liebermanand Altman & Kishore).

This study has the largest sample of defaulted bonds that weknow of:

Seniority Carty and Liberman Study Altman and Kishore StudyClass Number Average Std. Dev Number Average Std. Dev

Senior Secured 115 $53.80 $26.86 85 $57.89 $22.99Senior Unsecured 278 $51.13 $25.45 221 $47.65 $26.71

Senior subordinated 198 $38.52 $23.81 177 $34.38 $25.08Subordinated 226 $32.74 $20.18 214 $31.34 $22.42

Junior Subordinated 9 $17.09 $10.90 - - -

Table: Recovery Statistics by Seniority Class

Par (face value) is $100.00




Comments Regarding Studies

The table in the previous slide shows:

The subordinated classes are appreciably different from oneanother in their recovery realizations

In contrast, the difference between secured versus unsecureddebt is not statistically significant. It is likely that there is aself- selection affect here.

There is a greater chance for security to be requested in thecases where an underlying firm has questionable hard assetsfrom which to choose.




Default Probability (Frequency)

Let’s develop the notion of default probability (frequency):

Each counterparty has a certain probability of defaulting ontheir obligations.

Some models include a random variable which indicateswhether the counterparty is solvent or not.

Other models use a random variable which measures thecredit quality of the counterparty.

For the moment, we will denote by I{Counterparty Defaults?}the random variable which is 1 when the counterpartydefaults, and 0 when it does not, i.e.:

I{Counterparty Defaults?} =

{

1 Counterparty Defaults

0 Counterparty Does Not Default(6)




Default Probability (Frequency) Continued...

Continuing ...

The modelling of how I{Counterparty Defaults?} changesfrom 0 to 1 will be dealt with later.




Measuring the Distribution of Credit Losses I

For an instrument or portfolio with only one counterparty wedefine:

Credit Loss = I{Counterparty Defaults?} ×Credit Exposure× LGD

Note that:

Credit-Loss(Random Variable): Depends on the credit qualityof the counterparty

Credit Exposure(Number): Depends on the market risk of theinstrumnet or portfolio

LGD(number): Usually this number is a universal constant(55%) but more refined models relate it to the market and thecounterparty




Measuring the Distribution of Credit Losses II

For a portfolio with several counterparties we define:

Credit Loss =∑

i

I{Counterparty i Defaults?}×Credit Exposurei×LGD

Note that:

Credit-Loss(Random Variable): Normally different for differentcounterparties

Credit Exposure(Number): Normally different for differentportfolios, same for the same portfolios

LGD(number): Usually this number is a universal constant(55%) but more refined models relate it to the market and thecounterparty




Net Replacement Value

Net Replacement Value:

The traditional approach to measuring credit risk is toconsider only the net replacement value(NRV )

NRV =∑

i

(Credit Exposures)i .

This is a rough statistic, which measures the amount thatwould be lost if all counter-parties default at the same time,and at the time when all portfolios are worth most, and withno recovery rate.




Credit Loss Distribution

The credit loss distribution is often very complex:

As with Markowitz theory, we try to summarize its statisticswith two numbers: its expected value (µ), and its standarddeviation (σ).In this context, this gives us two values:

1 The expected loss (µ) 2 The unexpected loss (σ)

Loss ValueLikelihood

Expe ted Loss

Unexpe ted Loss Credit Loss Distribution




Credit VaR/ Worst Credit Loss

Worst credit loss:

Worst Credit Loss represents the credit loss which will not beexceeded with some level of confidence, over a certain timehorizon.

A 95%-WCL of $5M on a certain portfolio means that theprobability of losing more than $5M in that particular portfoliois exactly 5%.

Credit-VaR:

CVaR represents the credit loss which will not be exceeded inexcess of the expected credit loss, with some level ofconfidence over a certain time horizon:

A daily CVaR of $5M on a certain portfolio, with 95% meansthat the probability of losing more than the expected loss plus$5M in one day in that particular portfolio is exactly 5%.




Economic Credit Capital

Capital:

Capital is traditionallydesigned to absorb unexpectedlosses

Credit Var, is therefore, themeasure of capital.

It is usually calculated witna one-year time horizon

Losses can come from eitherdefaults or migrations

Credit Reserves:

Credit reserves are set aside toabsorb expected losses

Worst Credit Loss measuresthe sum of the capital and thecredit reserves

Losses can come from eitherexpected losses




Netting I

What is netting:

When two counterparties enter into multiple contracts, thecashflows over all the contracts can be, by agreement, mergedinto one cashflow.

This practice, called netting, is equivalent to assuming thatwhen a party defaults on one contract it defaults in all thecontracts simultaneously.




Netting II

Properties of netting:

Netting may affect the credit-risk premium of particularcontracts.

Assuming that the default probability of a party isindependent from the size of exposures it accumulates with aparticular counter-party, the expected loss over severalcontracts is always less or equal than the sum of the expectedlosses of each contract.

The same result holds for the variance of the losses (i.e. thevariance of losses in the cumulative portfolio of contracts isless or equal to the sum of the variances of the individualcontracts).

Equality is achieved when contracts are either identical or theunderlying processes are independent.




Expected Credit Loss: General Framework

In the general framework, the expected credit loss (ECL) is givenby:

ECL = E [I{Counterparty i Defaults?} × CE× LGD]

=

∫ ∫ ∫

[(I× CE× LGD)× f (I,CE, LGD)] dI · dCE · dLGD

Note that:

f (I,CE, LGD) is the joint probability density function of the:

Default status (I)Credit Expousre (CE)Loss Given Default(LGD)

The ECL is the expectation using the jpdf of I,CE and LGD




Expected Credit Loss: Special Case

Because calculating the joint probability distribution of allrelevant variables is hard, most often one assumes that theirdistributions are independent.

In that case, the ECL formula simplifies to:

ECL = E [I]︸︷︷︸

Probability of Default

× E [CE]︸︷︷︸

Expected Credit Exposure

× E [LDG]︸︷︷︸

Expected Severity

(7)




Example 1

Example (Commercial Mortgage)

Consider a commercial mortgage, with a shopping mall ascollateral. Assume the exposure of the deal is $100M, an expectedprobability of default of 20% (std of 10%), and an expectedrecovery of 50% (std of 10%).Calculate the expected loss in two ways:

1 Assuming independence of recovery and default (call it x)

2 Assuming a −50% correlation between the default probabilityand the recovery rate (call it y).

What is your best guess as to the numbers x and y?

a) x = $10M, y = $10M.

b) x = $10M, y = $20M.

c) x = $10M, y = $5M.

d) x = $10M, y = $10.5M.




Solution 1

Solution (Commercial Mortgage)


a) x = $10M, y = $10M.

b) x = $10M, y = $20M.

c) x = $10M, y = $5M.

d) x = $10M, y = $10.5M.

First notice that theanswer cannot be a) orc) since x has to besmaller than y

Now we can take one of two approaches to find the solution.One is the tree-based approach whereas the other uses thecovariance structure of the random variables.

Only the tree based approach is considered.




Tree Based Model

Note: Tree Based Model

Under the tree-based model we assume:

1 Two equally likely future credit states, given by defaultprobablites of 30% and 10%

2 Two equally likely future recovery rates states, given by 60%and 40%




Tree Based Model Continued...

Solution (Continued: Tree Based Model)

−0.5 = p++ − p+−

− p−+ + p−−

0.5 = p++ − p+−

0.5 = p−+ + p−−

0.5 = p++ + p−+

1-10

1-10

Credit State Re overy StateSolving for p++

, p+−, p−+

, p−−:

p++ = 0.125, p+− = 0.375, p−+ = 0.125, p−− = 0.375




Tree Based Approach Continued...

Solution (Continued-Correlating Default and Exposure)

Given a −50% correlation between the recovery rates and creditstates, along with the probabilities p++

, p+−, p−+

, p−−, theexpected loss (EL) is:

EL = $100M × (0.375 × 0.6 × 0.3 + 0.375 × 0.4 × 0.1

+ 0.125 × 0.4× 0.3 + 0.125 × 0.6 × 0.1)

= $100M × (0.0825 + 0.0225)

= $10.5M




Example 2

Example (Goodrich-Rabobank)

Consider the swap between Goodrich and MGT. Assume a totalexposure averaging $10M (50% std), a default rate averaging 10%(3% std), fixed recovery (50%).Calculate the expected loss in two ways:

1 Assuming independence of exposure and default (call it x)

2 Assuming a 50% correlation between the default probabilityand the exposure (call it y).


a) x = $500, 000 y = $460, 000.

b) x = $500, 000 y = $1M.

c) x = $500, 000 y = $500, 000.

d) x = $500, 000 y = $250, 000.




Solution 2

Solution (Goodrich-Rabobank)


a) x = $500, 000, y = $460, 000.

b) x = $500, 000, y = $1M.

c) x = $500, 000, y = $500, 000.

d) x = $500, 000, y = $250, 000.

Notice that the answer cannot be a) or c) since x has to be largerthan y




Solution 2 Continued...

Solution

Correlating Default and Exposure Using the tree-based model weassume:

1 Two equally likely future credit states, given by defaultprobabilities of 13% and 7%.

2 Two equally likely exposures, given by $15M and $5M.

With a 50% correlation between them, the expected loss (EL) is:

EL = 0.5× (0.125 × $15M × 0.13 + 0.125 × $5M × 0.7

+ 0.375 × $15M × 0.07 + 0.375 × $5M × 0.13)

= 0.5× ($0.24M + $0.40M + $0.24M)

= $460, 000




Example 23-2: FRM Exam 1998, Question 39

Example (23-2: FRM exam 1998,Question 39)

“Calculate the 1 yr expected loss of a $100M portfolio comprising10 B-rated issuers. Assume that the 1-year probability of default ofeach issuer is 6% and the recovery rate for each issuer in the eventof default is 40%.”




Example 23-2: FRM Exam 1998, Question 39

Example (23-2: FRM exam 1998,Question 39)

“Calculate the 1 yr expected loss of a $100M portfolio comprising10 B-rated issuers. Assume that the 1-year probability of default ofeach issuer is 6% and the recovery rate for each issuer in the eventof default is 40%.”

Solution (Example 23-2)

Note that the recory rate is 1− 0.6 = 40%, this implies:

0.06 × $100M × 0.6 = $3.6M




Variation of Example 23-2 I

Example (Variant of Example 23-2)

“Calculate the 1 yr unexpected loss of a $100M portfoliocomprising 10 B-rated issuers. Assume that the 1-year probabilityof default of each issuer is 6% and the recovery rate for each issuerin the event of default is 40%. Assume, also, that the correlationbetween the issuers is

1 100% (i.e., they are all the same issuer)

2 50% (they are in the same sector)

3 0% (they are independent, perhaps because they are indifferent sectors)”




Variation of Example 23-2 II

Solution (Variation of Example 23-2)

1. The loss distribution is a random variable with two states:default (loss of $60M, after recovery), and no default (loss of 0).The expectation is $3.6M. The variance is

0.06 × ($60M − $3.6M)2 + 0.94 × (0− $3.6M)2 = 200($M)2

The unexpected loss is therefore

√

(200) = $14M




Variation of Example 23-2 III

Solution (Variation of Example 23-2 Continued...)

2. The loss distribution is a sum of 10 random variable, each withtwo states: default (loss of $6M, after recovery), and no default(loss of 0). The expectation of each of them is $0.36M. Thestandard deviation of each is (as before) $1.4M.The standard deviation of their sum is

√

(10)× $1.4M = $5M

Note: the number of defaults is given by a Poisson distribution. This will be of

relevance later when we study the CreditRisk+ methodology.




Variation of Example 23-2 IV

Solution (Variation of Example 23-2 Continued...)

3. The loss distribution is a sum of 10 random variables Xi , eachwith two states: default (loss of $6M, after recovery), and nodefault (loss of 0). The expectation of each of them is $0.36M.The variance of each is (as before) 2. The variance of their sum is:

Var

(∑

i

Xi

)

=∑

i ,j

E [XiXj ]− µiµj

=∑

i

σ2i +

∑

i 6=j

σi ,j

=∑

i

σ2i +

∑

i 6=j

0.5× σiσj

= 110




Example 23-3: FRM exam 1999

Example (23-3: FRM exam 1999)

“Which loan is more risky? Assume that the obligors are rated thesame, are from the same industry, and have more or less the samesized idiosyncratic risk: A loan of

a) $1M with 50% recovery rate.

b) $1M with no collateral.

c) $4M with a 40% recovery rate.

d) $4M with a 60% recovery rate.”




Solution 23-3: FRM exam 1999


The expected exposures times expected LGD are:

a) $500,000

b) $1M

c) $2.4M. Riskiest.

d) $1.6M




Example 23-4: FRM Exam 1999

Example (23-4: FRM Exam 1999)

“Which of the following conditions results in a higher probability ofdefault?

a) The maturity of the transaction is longer

b) The counterparty is more creditworthy

d) The price of the bond, or underlying security in the case of aderivative, is less volatile.

d) Both 1 and 2.”




Solution 23-4: FRM Exam 1999


a) True

b) False, it should be “less”, nor “more”

c) The volatility affects (perharps) the value of the portfolio, andhence exposure, but not the probability of default (*)




Expected Loss Over the Life of the Asset

Expected and unexpected losses must take into account, notjust a static picture of the exposure to one cash flow, but thevariation over time of the exposures, default probabilities, andexpress all that in today’s currency.

This is done as follows: the PV ECL is given by:

PV(ECL) =∑

t

E [CLt ]× PVt (8)




Expected Loss: An Approximation

We can re-write formula (8) as:

PV(ECL) =∑

t

E [CLt ]× PVt

=∑

t

pt × E [CEt ]× (1− f )× PVt︸︷︷︸

Note that each of these numbers changes with time

≈ Avet{pt} × Avet{E [CEt ]} × (1− f )∑

t

PVt

︸︷︷︸

Each term is replaced by an amount indepdent of time: their average

Remark

In the book, the term (1− f ) is assumed to be independent oftime. In some situations, such as commercial mortgages, this willunderestimate the credit risk.


portfolio credit risk - risklab toronto credit risk - part 1... · portfolio credit risk prof. luis...

Documents