Chapter 19 The Analysis of Credit Risk. The Analysis of Credit Risk.
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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
Review of Basic ConceptsGoodrich-Morgan-Robabank Swap: A Fixed Rate Loan
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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
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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
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Time Value of MoneyCredit: Premium and SpreadA Two-State Markov Model
Review of Basic Concepts
Prof. Luis Seco Portfolio Credit Risk
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Time Value of MoneyCredit: Premium and SpreadA Two-State Markov Model
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)
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Time Value of MoneyCredit: Premium and SpreadA Two-State Markov Model
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 .
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Time Value of MoneyCredit: Premium and SpreadA Two-State Markov Model
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)
Prof. Luis Seco Portfolio Credit Risk
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Time Value of MoneyCredit: Premium and SpreadA Two-State Markov Model
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
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Time Value of MoneyCredit: Premium and SpreadA Two-State Markov Model
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
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Time Value of MoneyCredit: Premium and SpreadA Two-State Markov Model
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)
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Time Value of MoneyCredit: Premium and SpreadA Two-State Markov Model
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 )
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SetupValuation: CreditmetricsNon-Constant Spreads
Goodrich-Morgan-Robabank SwapA Fixed Rate Loan
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SetupValuation: CreditmetricsNon-Constant Spreads
Setup
Conisder:
An 8 year fixed rate loan of 50$M at 11%
With cashflows (in millions of USD):1 0.125 upfront2 5.5 per year during 8 years
Assume
Constant spread of h = 180bpi over all periods
2 state transition probability matrix
Calculate
Expectd Cashflows
Expected Loss
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Expected Cashflows
The expected cashflows:
E [cashflows] = 0.125 +
8∑
i=1
5.5e−(ri+1.8)ti
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Probability of Default I
Under our assumptions, the conditional probability of non-defaultfor a given year is constant. This implies:
Pr(Non-Default ∈ (ti , ti+1]|Non-Default ∈ (0, ti ])
= e−h = e−0.018 = 0.98216 ≈ 0.9822
Similarly:
Pr(Default ∈ (ti , ti+1]|Non-Default ∈ (0, ti ])
= 1− e−h = 1− 0.98216 = 0.01784 ≈ 0.018
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Probability of Default II
So the 2 state matrix is written as:
BBB D
BBB 0.9822 0.178
D 0 1Table: Two-State Transition Matrix
Graphically: BBB D0.98216 10.01784Prof. Luis Seco Portfolio Credit Risk
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Computing Cashflows I
Inputs:
Pr(Non-Default ∈ (ti , ti+1]|Non-Default ∈ (0, ti ]) = 0.9822
The government zero curve for August 1983 was:
Year r
1 0.08852 0.092973 0.096564 0.09878555 0.105506 0.1043557 0.117708 0.118676
1 2 3 4 5 6 7 8
0.09
00.
095
0.10
00.
105
0.11
00.
115
Time(years)
r
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Computing Cashflows II
Now we can compute the expected cashflows:
E[Cashflows] = 0.125 + 5.5
8∑
i=1
e−ri ·ti (0.9822)ti
= 0.125 + 5.5
8∑
i=1
e−ri ·i (0.9822)i Since ti = i
= 23.0527 ≈ 23.05
Similarly we can compute the riskless cashflows:
Risk-Less Cashflows = 0.125 + 5.58∑
i=1
e−ri ·i
= 24.67581 ≈ 24.68
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Expected Losses
Therefore the Expected Percentage Loss is:
E [%Loss] = 1−
(E [Cashflows]
E [Non-Risk Cashflow]
)
= 6.5776% ≈ 6.58%
Similarly the Expected Loss is ≈ 6.58% of 24.67581$(USDmillion), that is
E [Loss] = 0.065776 × 24.67581 ≈ 1.623$(USD million)
Prof. Luis Seco Portfolio Credit Risk
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SetupValuation: CreditmetricsNon-Constant Spreads
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
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Markov Processes
123456
123456
123456p65
p11p12
p34
t = 0 t = 1 t = 2
p11
Transition ProbabilitiesConstant in Time
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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)
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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
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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
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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
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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
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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.
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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!
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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.
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Credit Loss
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Review of Basic ConceptsGoodrich-Morgan-Robabank Swap: A Fixed Rate Loan
Credit LossCredit Concepts and Terminology
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.
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Example: Credit Exposure
Example (Credit Exposure)
Evolution ofthe Mark-to-Market of a20-MonthSwap, where:-99% Exposure-95% Exposure
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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.
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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.
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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
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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.
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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)
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Default Probability (Frequency) Continued...
Continuing ...
The modelling of how I{Counterparty Defaults?} changesfrom 0 to 1 will be dealt with later.
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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
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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
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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.
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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
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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%.
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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
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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.
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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.
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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
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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)
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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.
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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.
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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%
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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
Prof. Luis Seco Portfolio Credit Risk
Review of Basic ConceptsGoodrich-Morgan-Robabank Swap: A Fixed Rate Loan
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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
Review of Basic ConceptsGoodrich-Morgan-Robabank Swap: A Fixed Rate Loan
Credit LossCredit Concepts and Terminology
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).
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
Review of Basic ConceptsGoodrich-Morgan-Robabank Swap: A Fixed Rate Loan
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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 a) or c) since x has to be largerthan y
Prof. Luis Seco Portfolio Credit Risk
Review of Basic ConceptsGoodrich-Morgan-Robabank Swap: A Fixed Rate Loan
Credit LossCredit Concepts and Terminology
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
Review of Basic ConceptsGoodrich-Morgan-Robabank Swap: A Fixed Rate Loan
Credit LossCredit Concepts and Terminology
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%.”
Prof. Luis Seco Portfolio Credit Risk
Review of Basic ConceptsGoodrich-Morgan-Robabank Swap: A Fixed Rate Loan
Credit LossCredit Concepts and Terminology
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
Review of Basic ConceptsGoodrich-Morgan-Robabank Swap: A Fixed Rate Loan
Credit LossCredit Concepts and Terminology
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)”
Prof. Luis Seco Portfolio Credit Risk
Review of Basic ConceptsGoodrich-Morgan-Robabank Swap: A Fixed Rate Loan
Credit LossCredit Concepts and Terminology
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
Review of Basic ConceptsGoodrich-Morgan-Robabank Swap: A Fixed Rate Loan
Credit LossCredit Concepts and Terminology
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
Review of Basic ConceptsGoodrich-Morgan-Robabank Swap: A Fixed Rate Loan
Credit LossCredit Concepts and Terminology
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
Review of Basic ConceptsGoodrich-Morgan-Robabank Swap: A Fixed Rate Loan
Credit LossCredit Concepts and Terminology
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
Review of Basic ConceptsGoodrich-Morgan-Robabank Swap: A Fixed Rate Loan
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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
Review of Basic ConceptsGoodrich-Morgan-Robabank Swap: A Fixed Rate Loan
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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.”
Prof. Luis Seco Portfolio Credit Risk
Review of Basic ConceptsGoodrich-Morgan-Robabank Swap: A Fixed Rate Loan
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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
Review of Basic ConceptsGoodrich-Morgan-Robabank Swap: A Fixed Rate Loan
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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)
Prof. Luis Seco Portfolio Credit Risk
Review of Basic ConceptsGoodrich-Morgan-Robabank Swap: A Fixed Rate Loan
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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.
Prof. Luis Seco Portfolio Credit Risk