Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss...

116
Review of Basic Concepts Credit Loss Credit Models Portfolio Credit Risk Prof. Luis Seco Prof. Luis Seco University of Toronto Mathematical Finance Program February 28, 2017 Prof. Luis Seco Portfolio Credit Risk

Transcript of Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss...

Page 1: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

Portfolio Credit Risk

Prof. Luis Seco

Prof. Luis SecoUniversity of Toronto

Mathematical Finance Program

February 28, 2017

Prof. Luis Seco Portfolio Credit Risk

Page 2: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

Table of Contents

1 Review of Basic ConceptsTime Value of Money

Credit Rating AgenciesGeneral Framework and Multi-Step Markov Process

2 Credit LossExamples

Expected Loss

3 Credit ModelsCreditMetrics

Correlations in CreditMetricsSimulation approach for credit portfolios

KMV and Merton ModelMerton ModelKMVThe distribution of defaults

Exercises and Examples

Prof. Luis Seco Portfolio Credit Risk

Page 3: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

The Goodrich-Rabobank Swap 1983

Prof. Luis Seco Portfolio Credit Risk

Page 4: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsTime Value of Money

Review of Basic Concepts

Prof. Luis Seco Portfolio Credit Risk

Page 5: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsTime Value of Money

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)

Prof. Luis Seco Portfolio Credit Risk

Page 6: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsTime Value of Money

Credit Premium

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

Value =n∑

i=1

pie−ri tiqi (2)

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

1 For a fixed set of pi s the discounted present value will alwaysbe less than or equal to the value equation (equation (2))

2 To preserve the same present value of cashflows as in equation(2) the cashflows ({pi}ni=1) need to be increased. The amountby which each payment is increased is q−1i . This is the creditpremium at time ti .

Prof. Luis Seco Portfolio Credit Risk

Page 7: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsTime Value of Money

The Credit Spread

Let’s be mathematicians:

Since qi ≤ 1 we can write qi as:

qi = e−hi ti (3)

by simply defining

hi =−ln(qi )

ti,

This invented number, hi , the credit spread at time ti , and ishugely important.

Now we rewrite (2) in a familiar form:

Value =n∑

i=1

pie−(ri+hi )ti (4)

Prof. Luis Seco Portfolio Credit Risk

Page 8: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsTime Value of Money

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 4

AAA 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 13.52Table: One-year forward zero-curves for each credit rating (%)*

*Source: Creditmetrics, JP Morgan

Prof. Luis Seco Portfolio Credit Risk

Page 9: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsTime Value of Money

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

Prof. Luis Seco Portfolio Credit Risk

Page 10: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsTime Value of Money

First Model: Two Credit States

Two assumptions:

Assume only 2 possible credit states: solvency and default.

Assume the probability of solvency in a period (one year, forexample), conditional on solvency at the beginning of theperiod, is given by a fixed amount q.

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

A consequence of these assumptions is that:

qi = qti

which gives rise to a constant credit spread:

hi = h = −ln(q)

Prof. Luis Seco Portfolio Credit Risk

Page 11: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsTime Value of Money

The same model: formal version

Assume the default process follows a two-state Markov Chain withtransition probabilities given by the matrix:

Solvency Default

Solvency q 1− q

Default 0 1

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

Prof. Luis Seco Portfolio Credit Risk

Page 12: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsTime Value of Money

ApplicationCountry Risk

Prof. Luis Seco Portfolio Credit Risk

Page 13: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsTime Value of Money

Simple setup

Imagine a one-year zero coupon Greek bond providing a spread hover a (German) risk-free bond, with two assumptions

1 Notional N

2 (Risk Neutral) Probability of solvency after one year equal to q

When a country defaults, not all is lost: there is a recovery rate R,which tell us, in percentages, the value we recover in the case ofdefault.Assume R = 50%.

Prof. Luis Seco Portfolio Credit Risk

Page 14: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsTime Value of Money

Sovereign default probability

We adjust (2) to take into account recovery:

V = N e−rq + R · Ne−r (1− q) (5)

Using the spread:V = N e−(r+h) (6)

Therefore, solving for V :

q =e−h − R

1− R(7)

Prof. Luis Seco Portfolio Credit Risk

Page 15: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsTime Value of Money

Cases

We can calculate the risk-neutral probability of default of countriesin the Euro-zone:

Date Issuer Spread p

5/2012 Spain 419 bps 8.2%

5/2012 Portugal 909 bps 17%

5/2012 France 135 bps 2.7%

Table: Issuer risk

Prof. Luis Seco Portfolio Credit Risk

Page 16: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsTime Value of Money

Markov Chain Default models

Generalize the previous setting to include more than one solvencystate.Transition probabilities between the different states are constantover time:

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

p31 · · · · · · · · ·...

... · · · · · · · · ·...

pn1 · · · · · · · · · pnn

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

Prof. Luis Seco Portfolio Credit Risk

Page 17: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsTime Value of Money

Credit Rating Agencies I

Credit rating agencies:

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

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

Prof. Luis Seco Portfolio Credit Risk

Page 18: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsTime Value of Money

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

Prof. Luis Seco Portfolio Credit Risk

Page 19: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsTime Value of Money

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

Prof. Luis Seco Portfolio Credit Risk

Page 20: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsTime Value of Money

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 (8)

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

A2

Prof. Luis Seco Portfolio Credit Risk

Page 21: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsTime Value of Money

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.

Prof. Luis Seco Portfolio Credit Risk

Page 22: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsTime Value of Money

Matrix Fractional Powers

Matrix Expansion

We can expand a matrix as folows:

Aα =∞∑k=0

(αk

)(A− 1)k

Where (αk

)=α(α− 1) · · · (α− k + 1)

k!

Prof. Luis Seco Portfolio Credit Risk

Page 23: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsTime Value of Money

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 0.02 1.00

C 0.01 0.12 0.23 0.64 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.

Prof. Luis Seco Portfolio Credit Risk

Page 24: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Credit Loss

Prof. Luis Seco Portfolio Credit Risk

Page 25: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

A more general framework

To better understand the loss distribution due to credit events, wewill break it down to three basic building blocks

1 Credit exposure

2 Recovery

3 Default

When the credit loss distribution is understood, we will focus onthe basic regulatory aspects of credit risk:

Expected loss

Unexpected loss and CreditVaR.

Prof. Luis Seco Portfolio Credit Risk

Page 26: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected 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.

Prof. Luis Seco Portfolio Credit Risk

Page 27: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Example: Credit Exposure

Example (Credit Exposure)

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

Prof. Luis Seco Portfolio Credit Risk

Page 28: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

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.

Prof. Luis Seco Portfolio Credit Risk

Page 29: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

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.

Prof. Luis Seco Portfolio Credit Risk

Page 30: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

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

Prof. Luis Seco Portfolio Credit Risk

Page 31: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

The Default Process

Consider It the random process which is

It =

{1 Counter-party is at default at time t

0 Counter-party is solvent at time t

The probability of default at time t is merely the expectation of Iat time t:

1− qt = E(It).

Prof. Luis Seco Portfolio Credit Risk

Page 32: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Distribution of Credit Losses I

For a portfolio with only one counter-party we define:

Credit Loss = I× Credit Exposure× LGD

Note that:

I: Depends on the credit quality of the counter-party

Credit Exposure: Depends on the market risk of theinstrument or portfolio

LGD(number): Related it to situation of the portfolio withinthe capital structure of the counter-party

Prof. Luis Seco Portfolio Credit Risk

Page 33: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Distribution of Credit Losses II

For a portfolio with several counter-parties we define:

Credit Loss =∑i

I(i)× Credit Exposurei × LGD

Prof. Luis Seco Portfolio Credit Risk

Page 34: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Credit Loss Distribution

The credit loss distribution is often very complexAs with Markowitz theory, we try to summarize its statisticswith two numbers: its expected value (µ), and its standarddeviation (σ).

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

Prof. Luis Seco Portfolio Credit Risk

Page 35: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Credit VaR/ Worst Credit Loss

In words:

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

CreditVaR (CVaR) represents the credit loss which will not beexceeded in excess of the expected credit loss, with some levelof confidence over a certain time horizon:

Prof. Luis Seco Portfolio Credit Risk

Page 36: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Credit VaR/ Worst Credit Loss

In numbers:

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

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 5%.

Prof. Luis Seco Portfolio Credit Risk

Page 37: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

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

Prof. Luis Seco Portfolio Credit Risk

Page 38: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Netting I

What is netting:

When two counter-parties 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.

Prof. Luis Seco Portfolio Credit Risk

Page 39: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Expected Credit Loss: General Framework

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

ECL = E [I× 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

Prof. Luis Seco Portfolio Credit Risk

Page 40: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Expected Credit Loss: Special Case

Because calculating the joint probability distribution of allrelevant variables is hard, 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

(9)

Prof. Luis Seco Portfolio Credit Risk

Page 41: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

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.

Prof. Luis Seco Portfolio Credit Risk

Page 42: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Solution 1

Solution (Commercial Mortgage)

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.

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.

Prof. Luis Seco Portfolio Credit Risk

Page 43: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

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%

Prof. Luis Seco Portfolio Credit Risk

Page 44: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

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−+

Solving for p++, p+−, p−+, p−−:

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

Prof. Luis Seco Portfolio Credit Risk

Page 45: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

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

Prof. Luis Seco Portfolio Credit Risk

Page 46: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Goodrich-Rabobank example

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).

What is your best guess as to the numbers x and 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.

Prof. Luis Seco Portfolio Credit Risk

Page 47: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Solution 2

Solution (Goodrich-Rabobank)

What is your best guess as to the numbers x and 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.

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

Prof. Luis Seco Portfolio Credit Risk

Page 48: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

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

Prof. Luis Seco Portfolio Credit Risk

Page 49: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

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

Prof. Luis Seco Portfolio Credit Risk

Page 50: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

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

Prof. Luis Seco Portfolio Credit Risk

Page 51: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

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

3 50% (they are in the same sector)

Prof. Luis Seco Portfolio Credit Risk

Page 52: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

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

Prof. Luis Seco Portfolio Credit Risk

Page 53: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

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.

Prof. Luis Seco Portfolio Credit Risk

Page 54: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

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

Prof. Luis Seco Portfolio Credit Risk

Page 55: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

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.”

Prof. Luis Seco Portfolio Credit Risk

Page 56: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Solution 23-3: FRM exam 1999

Solution (Example 23-3)

The expected exposures times expected LGD are:

a) $500,000

b) $1M

c) $2.4M. Riskiest.

d) $1.6M

Prof. Luis Seco Portfolio Credit Risk

Page 57: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

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 counter-party 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.”

Prof. Luis Seco Portfolio Credit Risk

Page 58: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Solution 23-4: FRM Exam 1999

Solution (Example 23-4)

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 (*)

Prof. Luis Seco Portfolio Credit Risk

Page 59: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Expected and Unexpected loss over time

Prof. Luis Seco Portfolio Credit Risk

Page 60: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

An example

Consider a bond issued from a default-prone party, paying two$5 coupons after the end of the second and fourth years. Weassume throughout the duration of the bond the interest ratesare 0% (this assumption simplifies discounting).

The default-prone party has a yearly default probability of 7%and when it defaults no money can be recovered (recoveryrate= 1−severity= 0).

We assume that the default-free party maintains a risk-capitalto cover the standard deviation of losses that is is adjustedannually and that it demands a certain return on thisrisk-capital.

Prof. Luis Seco Portfolio Credit Risk

Page 61: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Survival and Default Probabilities

Survival and Default Probabilities.∗.

where

D =Default.

ND =Not Default.

∗Nodes are one year apart.

Prof. Luis Seco Portfolio Credit Risk

Page 62: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Expected Loss Calculation

Expected loss calculation:

There are two, equivalent in this case, ways to compute theexpected loss.Since the value of the contract is always non-negative to thedefault-free party we do not need to discard any future events(as already explained this not a limitation, as every contractcan be decomposed into contracts that have alwaysnon-negative or non-positive value).One way to compute the expecetd loss is to compute theexpected cashflows.Recall that there are two such cashflows:

1 $5 at t = 2,2 $5 at t = 4.

but we also need to factor in the probabilities of default withinthese periods.

Prof. Luis Seco Portfolio Credit Risk

Page 63: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Expected Loss Calculation Continued...

Continuing...

There are two cashflows of $5 each, and the expectedcashflow is:

EC = 5 · pND∈(0,2] + 5 · pND∈(0,4] = $8.065

where pND∈(i ,j] is the probability that the default-prone partydoes not default in the time interval between years i and j(i < j).

The expected loss is:

EL = 10− EC = $1.935

Prof. Luis Seco Portfolio Credit Risk

Page 64: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

An Equivalent Way to Calculate Expected Loss

The second way is to calculate loss:

Is based on the yearly exposure:

Exposure(year 1−) = $10

Exposure(year 2−) = $10

Exposure(year 3−) = $5

Exposure(year 4−) = $5

where no correction is due to discounting was included, sinceinterst rates are flat at %0 and Exposure(year 1−), the valueof the contract just before year 1.

Prof. Luis Seco Portfolio Credit Risk

Page 65: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

An Equivalent Way to Calculate Expected LossContinued...

Continuing...

The expected losses are:

EL = Exposure(year 1−) · pD∈(0,1] + Exposure(year 1−) · pD∈(1,2]+ Exposure(year 3−) · pD∈(2,3] + Exposure(year 4−) · pD∈(3,4]= 10× 0.07 + 10× 0.0651 + 5× 0.0605 + 5× 0.0563

= $1.935

where pD∈(2,3] is the probability that the default-prione partydefaults in the time interval between years 2 and 3.

Prof. Luis Seco Portfolio Credit Risk

Page 66: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

The Unexpected Loss

Recall that the unexpected loss is the variance of the losses, so:

V(L[0,1)) = Exposure(year 1−)2 · pD∈(0,1] −

(Exposure(year 1−) · pD∈(0,1]

)2= (EL(1))2

(1

pD∈[0,1)− 1

)= 6.51

V(L[1,2)) = Exposure(year 2−)2 · pD∈(1,2] −

(Exposure(year 2−) · pD∈(1,2]

)2= (EL(2))2

(1

pD∈[1,2)− 1

)= 6.08

V(L[2,3)) = (EL(3))2(

1

pD∈[2,3)− 1

)= 1.42

V(L[3,4)) = (EL(4))2(

1

pD∈[3,4)− 1

)= 1.33

whereV(X ) = Var(X ) = E

[X 2]− (E [X ])2

Prof. Luis Seco Portfolio Credit Risk

Page 67: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Credit Reserve

Credit Reserve:

If, for any example, a risk-capital of two standard deviations isrequired, the default-free party anticipates to use risk-capitalequal to:

1 $5.10 at year 0,2 $4.93 at year 1,3 $2.38 at year 2 and4 $2.31 at year 3.

A yearly return of 10% on such capital leads to an additionalsurcharge of $1.47.

Remark

Notice that a high enough return rate would lead to the possibilityof arbitrage (in this case arbitrage corresponds to an intialcredit-risk premium of more than $10).

Prof. Luis Seco Portfolio Credit Risk

Page 68: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Credit VaR

Prof. Luis Seco Portfolio Credit Risk

Page 69: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Credit VaR

Credit VaR:

Credit VaR is the unexpected credit loss, at some confidencelevel, over a certain time horizon.

If we denote by f (x) the distribution of credit losses over theprescribed time horizon (typically one year), and we denote byc the confidence level (i.e. 95%), then the Worst-Credit-Loss(WCL) is defined to be:∫ ∞

WCLf (x)dx = 1− c

and

Credit VaR = (Worst-Credit Loss)− (Expected Credit Loss)︸ ︷︷ ︸Leads to Reserve Capital

Prof. Luis Seco Portfolio Credit Risk

Page 70: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Example 23-5: FRM Exam 1998

Example (Example 23-5: FRM Exam 1998)

A risk analyst is trying to estimate the Credit VaR for a riskybond.

The Credit VaR is defined as the maximum unexpected loss ata confidence level of 99.9% over a one month horizon.

Assume that the bond is valued at $1M one month forward,and the one year cumulative default probability is 2% for thisbond

→ What is your estimate of the Credit VaR for this bondassuming no recovery?

a) $20,000

b) $1,682

c) $998,318

d) $0

Prof. Luis Seco Portfolio Credit Risk

Page 71: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Solution 23-5: FRM Exam 1998

Solution (23-5: FRM Exam 1998)

→ What is your estimate of the Credit VaR for this bond assumingno recovery?

a) $20,000

b) $1,682

c) $998, 318

d) $0

Why?

If d is the monthly probability of default then:

(1− d)12 = (0.98), so d = 0.00168,ECL =$ 1,682,WCL(0.999)=WCL(1-0.00168)=$1,000,000,

∴ CVaR=$1,000,000−1, 682 =$998,318.

Prof. Luis Seco Portfolio Credit Risk

Page 72: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Example 23-6: FRM exam 1998

Example (Example 23-6: FRM exam 1998)

A risk analyst is trying to estimate the Credit VaR for aportfolios of two risky bonds.

The Credit VaR is defined as the maximum unexpected loss ata confidence level of 99.9% over a one month horizon.

Assume that both bonds are valued at $500,000 one monthforward, and the one year cumulative default probability is 2%for each of these bonds.

→ What is your best estimate of the Credit VaR for this portfolioassuming no default correlation and no recovery?

a) $841

b) $1,682

c) $10,000

d) $249,159

Prof. Luis Seco Portfolio Credit Risk

Page 73: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Solution: Example 23-6: FRM exam 1998

Solution (Solution: Example 23-6: FRM exam 1998)

→ What is your best estimate of the Credit VaR for this portfolioassuming no default correlation and no recovery?

a) $841

b) $1,682

c) $10,000

d) $249, 159

Why?

If d is the monthly probability of default then:

(1− d)12 = (0.98), so d = 0.00168,ECL =$ 839.70,WCL(0.999)=WCL(1-0.00168)=$250,000,

∴ CVaR= $250, 000− $840 =$249,159.

Prof. Luis Seco Portfolio Credit Risk

Page 74: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit ModelsExpected Loss

Solution: Example 23-6 Continued...Credit Loss Distribution

As before, the monthlydiscount is d = 0.00168

The 99.9% loss quantile isabout $500,000

Also we have that:

EL =$ 839.70,

WCL(0.999)=WCL(1-0.00168)=$250,000,

∴ CVaR=$250, 000 − $840 =$249,159.

Default Probability Loss P×L2 Bonds d2 = 0.00000282 $500,000 $1.41 Bond 2 · d · (1− d) = 0.00336 $250,000 $839.700 Bonds (1− d)2 = 0.9966 $0 $0

Prof. Luis Seco Portfolio Credit Risk

Page 75: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Credit Models

Prof. Luis Seco Portfolio Credit Risk

Page 76: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Portfolio: Credit Risk Models

CreditMetrics

Introduced by JPMorganBased on bond pricesConsiders several credit statesModelled with a Markov Chain

KMV

Based on the Merton credit modelBased on equity pricesOnly two credit states: solvency and defaultModelled with the Black-Scholes theory

Prof. Luis Seco Portfolio Credit Risk

Page 77: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

CreditMetrics

CreditMetrics:

Credit risk is driven by movements in bond ratings.

Credit events are rating downgrades obtained through amatrix of migration probabilities.

Each instrument is valued using the credit spread for eachrating class.

Recovery rates are obtained from historical similarities.

Correlations between defaults are inferred from equity prices,assigning each obligor to a combination of 152 indices (factordecomposition)

It does not integrate market and credit risk.

Prof. Luis Seco Portfolio Credit Risk

Page 78: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Simulation of one Asset: a Bond

Bond Rating Probability(%) Value∑

i piVi∑

i pi (Vi −m)2x AAA 0.02 $109.37 0.02 0.00AA 0.33 $109.19 0.36 0.01A 5.95 $108.66 6.47 0.15

BBB · −→ BBB 86.93 $107.55 93.49 0.19yBB 5.30 $102.02 5.41 1.36B 1.17 $98.10 1.15 0.95

CCC 0.12 $83.64 0.10 0.66Default 0.18 $51.13 0.09 5.64

Table: Figure 23-3 Building the Distribution of Bond Values∗

Note:

The boldface values are the statistics for the MtM.

We also have that:99% CVaR [$11,$24]

2σ $6∗ Source: CreditMetrics

Prof. Luis Seco Portfolio Credit Risk

Page 79: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Correlations in CreditMetrics

Two counter-parties (For example, Dupont and Allianz):

Models were generated using 152 country indices, 28 countryindices, 19 worldwide indices including:

1 US Chemical Industries2 German Insurance Index3 German Banking Index

Linear Regression

r1 = 0.9× rUS,Ch︸︷︷︸US Chemical Industries

+k1ξ1

r2 = 0.74× rGE,In︸︷︷︸German Insurance Index

+0.1× rGE,Ba︸︷︷︸German Banking Index

+k2ξ2

In the linear regression models above we further assume thatthe residuals ξ1 and ξ2 are uncorrelated.

Prof. Luis Seco Portfolio Credit Risk

Page 80: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Correlations in CreditMetrics Continued...

The correlation on returns are however modelled as:

ρdef(r1, r2) = 0.90×0.74ρ(rUS,ChrGE,In)+0.90×0.15ρ((rUS,ChrGE,Ba) = 0.11

Note:

Correlations on returns drive the correlations on credit ratings

Prof. Luis Seco Portfolio Credit Risk

Page 81: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Simulation of more than one Asset

Simulating more than one asset:

Consider a portfolio consisting of m counter-parties, and atotal of n possible credit states.

We need to simulate a total of mn states; their multivariatedistribution is given by their marginal distributions (as before)and the correlations given by the regression model.

To obtain accurate results, since many of these states havelow probabilities, large simulations are often needed.

It does not integrate market and credit risk: losses areassumed to be due to credit events alone.

Prof. Luis Seco Portfolio Credit Risk

Page 82: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

KMV and Merton Model

Prof. Luis Seco Portfolio Credit Risk

Page 83: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

The Merton Model

Merton Model:

Merton (1974) introduced the view that equity value is a calloption on the value of the assets of the firm, with a strikeprice equal to the firm’s debt.

In particular, the stock price embodies a forecast of the firm’sdefault probabilities, in the same way that an option embodiesan implied forecast of the option being exercised.

Prof. Luis Seco Portfolio Credit Risk

Page 84: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

A Simple Setting

In its simplest situation, assume:

The firm’s value equal to V .

The firm issued a zero-couponbond due in one time unit equalto K .

Solvency: The firm’s value is higherthan K :

1 The bondholders get their bondpayment and,

2 The remainder value of spread isdistributed amongst theshareholders.

Default The value of the firm is lessthan K :

1 The bondholders get the value of thefirm V and

2 Equity value is 0.

Prof. Luis Seco Portfolio Credit Risk

Page 85: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Equity Values

When debt is due:

The value of a firm can be determined at the time debt is due:

ST = max(VT − K , 0)

Since the firm’s value equals equity plus bonds:

BT = Vt −max(Vt − K , 0)

= min(VT , k)

Before debt is due

Stocks and bonds are priced as a call option and a short put.

Prof. Luis Seco Portfolio Credit Risk

Page 86: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Corollaries

A long position in a risky bond is equivalent to a long positionin a risk-free bond plus a short put option.

The short put option is really a credit derivative, like the riskybond.

This explains the left skewness in credit losses for risky debt(similar to a short option position.)

It also shows that equity is equivalent to an option on thevalue’s assets; due to the limited liability of the firm, investorscan lose no more than their original investment.

Prof. Luis Seco Portfolio Credit Risk

Page 87: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Pricing Equity

We assume the firm’s assets are geometric Brownian:

dV = µVdt + σVdz

We also assume no transaction costs (including bankruptcy costs)

V = B + S

We can price S with the Black-Scholes methodology, obtaining:

S = VN(d1)− Ke−rtN(d2) (10)

where

d1 =−ln(Ke−eτ/V )

σ√τ

+σ√τ

2, d2 = d1 − σ

√τ

Prof. Luis Seco Portfolio Credit Risk

Page 88: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Asset Volatility

Asset Volatility:

In practice, only equity volatility is observed, not assetvolatility, which we must derive as follows:

The hedge ratio

dS =∂S

∂VdV

yields a relationship between the stochastic differentialequations for S and V, from where we get

σSS = σVV∂S

∂V

and

σV = σSS∂V

V ∂S

Prof. Luis Seco Portfolio Credit Risk

Page 89: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Pricing Debt

The value of the bond is given by B = V − S , therefore

B = Ke−rtN(d2) + N[1− N(d1)]

orB

Ke−Rτ= N(d2) +

(V

Ke−rτ

)N(−d1)

Black-Scholes tells us that N(d2) is the probability of exercisingthe call, or probability that the bond will not default

Prof. Luis Seco Portfolio Credit Risk

Page 90: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Credit Loss

The expected credit loss is the value of the risk-free bond minusthe risky bond:

ECL = Ke−rτ − Ke−rτN(d2)− V [1− N(d1)]

= Ke−rτN(−d2)− VN(= d1)

= N(−d2)

[Ke−rτ − V

N(−d1)

N(−d2)

](”morally”)

=Probability × Exposure × Loss-Given-Default

1 Probability of not exercising the call, or probability that the bond willdefault.

2 PV of face value of the bond.

3 PV of expected value of the firm in case of default

Prof. Luis Seco Portfolio Credit Risk

Page 91: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Advantages

Advantages:

Relies on equity prices, not bond prices: more companies havestock prices than bond issues.

Correlations among equity prices can generate correlationsamong default probabilities.

It generates movements in EDP that can lead to credit ratings.

Prof. Luis Seco Portfolio Credit Risk

Page 92: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Disadvantages

Disadvantages:

Cannot be used for counter-parties without traded stock(governments, for example).

Relies on a static model for the firm’s capital and riskstructure:

The firm could take on operations that will increase stockprice but also its volatility, which may lead to increased creditspread; this is in apparent contradiction with its basic premise,which is that higher equity prices should be reflected in lowercredit spreads.

Prof. Luis Seco Portfolio Credit Risk

Page 93: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

KMV

KMV:

KMV was a firm founded by Kealhofer, McQuown andVasicek, (and then sold to Moody’s), which was a vendor ofdefault frequencies for 29,000 companies in 40 differentcountries.

Their method is a simplified version of Merton’s model

Basic model inputs are:1 Value of the liabilities (calculated as liabilities (¡1 year) plus

one half of long term debt)2 Stock value3 Asset volatility4 Assets

Prof. Luis Seco Portfolio Credit Risk

Page 94: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Basic Terms

Prof. Luis Seco Portfolio Credit Risk

Page 95: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Basic Formula: Distance to Default

Distance to Default

[Distance

to Default

]=

[Market Value

of Assets

]−[

DefaultPoint

][

Market Valueof Assets

] [Asset

Volatility

]

Prof. Luis Seco Portfolio Credit Risk

Page 96: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Example

Example

Consider a firm with:

$100M Assets,

$80M liabilities,

Volatility of $10M (annualized)

Then the distance from default is calculated as:

A− K

σ= 2

The default probability is then 0.023 (using a Gaussian).

Prof. Luis Seco Portfolio Credit Risk

Page 97: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Captial Requirement under BIS

Corollary

The captial requirement under BIS is:

K = LGD×

[N

(N−1(PD)−

√RN−1(0.999)√

1− R

)− PD

]×MF

where:

N: Cumulative Normal,√R:1-Factor asset correlation,

MF:Maturity function:

→ Emprirically adjusts for the maturity of the portfolio.

Prof. Luis Seco Portfolio Credit Risk

Page 98: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Exercises and Examples

Exercises and Examples

Prof. Luis Seco Portfolio Credit Risk

Page 99: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Exercise 1: The Merton Model

Example (The Merton Model)

Consider a firm with total asset worth $100, and assetvolatility equal to 20%.

The risk free rate is 10% with continuous compounding.

Time horizon is 1 year.

Leverage is 90% (i.e., debt-to-equity ratio 900%)

Find:1 The value of the credit spread.2 The risk neutral probability of default.3 Calculate the PV of the expected loss.

Prof. Luis Seco Portfolio Credit Risk

Page 100: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Solution Part 1: Finding the Credit Spread

Solution (The Merton Model-Finding the Credit Spread)

A leverage of 0.9 implies that

Ke−0.1/V = 0.9

which says that K = 99.46.

Using Black-Scholes, we get that the call option is worth S =$13.59.

The bond price is then

B = V − S = $100− $13.59 = $86.41

for a yield of

ln

(K

B

)= ln

(99.46

86.41

)= 14.07%

or a credit spread of 4.07%.

Prof. Luis Seco Portfolio Credit Risk

Page 101: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Solution Part 2: Option Calculation

Solution (The Merton Model-Option Calculation)

Prof. Luis Seco Portfolio Credit Risk

Page 102: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Solution Part 2: The Risk Neutral Probability of Default

Solution (The Merton Model-Risk Neutral Probability of Default)

The risk neutral probability of default is given by:

N(d2) = 0.6653, EDF = 1− N(d2) = 33.47%

Prof. Luis Seco Portfolio Credit Risk

Page 103: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Solution Part 3: The Expected Loss

Solution (The Merton Model-The Expected Loss)

The expected loss is given by:

ECL = N(−d2)

[Ke−rτ − V

N(−d1)

N(−d2)

]= 0.3347×

[$90− $100× 0.2653

0.3347

]= $3.96

Prof. Luis Seco Portfolio Credit Risk

Page 104: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Additional Considerations

Variations of the same problem:

If debt-to-equity ratio is 233%, the spread is 0.36%.

If debt-to-equity ratio is 100%, the spread is about 0.

In other words, the model fails to reproduce realistic, observedcredit spreads.

Prof. Luis Seco Portfolio Credit Risk

Page 105: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

The Goodrich Corporation

Example (The Goodrich Corporation)

The following information about the Goodrich corporation isavailable to us:

From the Company’s financials:

Debt/equity ratio: 2.27Shares out: 117,540,000.Expected dividend: $0.20/share.

From NYSE, ticker symbol GR:

Stock volatility: 49.59%Real rate of return (3 years): 0.06%Share price: $17.76 (May 2003)

From Interest rate market:

Annual risk free rate: 3.17%

Prof. Luis Seco Portfolio Credit Risk

Page 106: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

The Goodrich Corporation Continued...

Example (The Goodrich Corporation Continued...)

Stock and Company Value:

S = 117, 540, 000× $17.76

= $2, 087B

V = S + B = 3.27S = 6.826B

Current Debt: $4.759B

Future debt (strike price):

K = $4.759e0.0317 = $4.912

Dividend:

Dividend = $0.20× 117, 540, 000

= $23, 508, 000.

Prof. Luis Seco Portfolio Credit Risk

Page 107: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Boostrapping Asset Volatility

Solution (Boostrapping Asset Volatility)

Prof. Luis Seco Portfolio Credit Risk

Page 108: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Boostrapping Asset Volatility (Iterative Process)

Solution (Boostrapping Asset Volatility -Iterative Process)

Prof. Luis Seco Portfolio Credit Risk

Page 109: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Example 23-7: FRM Exam 1999

Example (23-7: FRM Exam 1999)

Which of the following is used to estimate the probability ofdefault for a firm in the KMV model?

I Historical probability of default based on the credit rating of the firm(KMV have a method to assign a rating to the firm if unrated).

II Stock price volatility.

III The book value of the firm’s equity

IV The market value of the firm’s equity

V The book value of the firm’s debt

VI The market value of the firm’s debt

a) I

b) II, IV and V

c) II, III, VI

d) VI only

Prof. Luis Seco Portfolio Credit Risk

Page 110: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Solution 23-7: FRM Exam 1999

Solution ( 23-7: FRM Exam 1999)

Which of the following is used to estimate the probability ofdefault for a firm in the KMV model?

I Historical probability of default based on the credit rating of the firm(KMV have a method to assign a rating to the firm if unrated).

II Stock price volatility.

III The book value of the firm’s equity

IV The market value of the firm’s equity

V The book value of the firm’s debt

VI The market value of the firm’s debt

a) I

b) II , IV and V

c) II, III, VI

d) VI only

Prof. Luis Seco Portfolio Credit Risk

Page 111: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Example 23-8: FRM Exam 1999

Example (23-8: FRM Exam 1999)

J.P. Morgan’s CreditMetrics uses which of the following toestimate default correlations?

I CreditMetrics does not estimate default correlations; itassumes zero correlations between defaults.

II Correlations of equity returns.

III Correlations between changes in corporate bond spreads totreasury.

IV Historical correlation of corporate bond defaults.

Prof. Luis Seco Portfolio Credit Risk

Page 112: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Solution 23-8: FRM Exam 1999

Solution (23-8: FRM Exam 1999)

J.P. Morgan’s CreditMetrics uses which of the following toestimate default correlations?

I CreditMetrics does not estimate default correlations; itassumes zero correlations between defaults.

II Correlations of equity returns

III Correlations between changes in corporate bond spreads totreasury

IV Historical correlation of corporate bond defaults

Prof. Luis Seco Portfolio Credit Risk

Page 113: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Example 23-9: FRM Exam 1998

Example (23-9: FRM Exam 1998)

J.P. Morgan’s CreditMetrics uses which of the following toestimate default correlations?

a) Bond spreads to treasury.

b) History of loan defaults.

c) Assumes zero correlations and simulates defaults.

d) None of the above.

Prof. Luis Seco Portfolio Credit Risk

Page 114: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Solution 23-9: FRM Exam 1998

Solution (23-9: FRM Exam 1998)

J.P. Morgan’s CreditMetrics uses which of the following toestimate default correlations?

a) Bond spreads to treasury.

b) History of loan defaults.

c) Assumes zero correlations and simulates defaults.

d) None of the above.

Prof. Luis Seco Portfolio Credit Risk

Page 115: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Example 23-10: FRM Exam 2000

Example (23-10: FRM Exam 2000)

The KMV credit risk model generates an estimated defaultfrequency (EDF) based on the distance between the currentvalue of the assets and the book value of the liabilities.

Suppose that the current value of a firm’s assets and the bookvalue of its liabilities are $500M and 300M, respectively.

Assume that the standard deviation of returns on the assets is$100M, and that the returns of the assets are normallydistributed.

Assuming a standard Merton Model, what is the approximatedefault frequency (EDF) for this firm?

1. 0.010

2. 0.015

3. 0.020

4. 0.030

Prof. Luis Seco Portfolio Credit Risk

Page 116: Portfolio Credit Risk - RiskLab Toronto Credit Risk.pdf · Review of Basic Concepts Credit Loss Credit Models Time Value of Money Cash Flow Valuation Fundamental Principle: TIME IS

Review of Basic ConceptsCredit Loss

Credit Models

CreditMetricsKMV and Merton ModelExercises and Examples

Solution 23-10: FRM Exam 2000

Solution (23-10: FRM Exam 2000)

→ Assuming a standard Merton Model, what is the approximatedefault frequency (EDF) for this firm?

1. 0.010

2. 0.015

3. 0.020

4. 0.030Why?

Distance from default is calculated as:

A− K

σ= 2.

The default probability is then 0.023 (using a Gaussianmodel).

Prof. Luis Seco Portfolio Credit Risk