Summer 08, MFIN7011, Tang

1
MFIN 7011 Credit RiskSummer, 2008Dragon Tang

• Lecture 16
• Credit Value-at-Risk I
• Tuesday, August 12, 2007
• Readings RiskMetrics Technical Documents
• http//www.riskmetrics.com/techdoc.html

2
(No Transcript)
3
How Bad Can Things Get?

• Amaranth (6.5 billion in one week in September
  2006)
• Credit Lyonnais (5.0 billion in 1990)
• LTCM (4.6 billion in 1998)
• Sumitomo (2.6 billion in 1996)
• Orange County (2 billion in 1994)
• Barings (1.4 billion in 1995)
• Daiwa Bank (1.1 billion in 1995)
• Enrons Counterparties
• Allied Irish Bank (0.7 billion in 2002)
• China Aviation Oil (0.6 billion in 2004)
• Kidder Peabody (0.4 billion in 1994)
• China State Reserve Bureau (0.2 billion in 2006)
• Procter and Gamble (0.2 billion in 1994)

4
Risk Limits

• Risk must be quantified and risk limits set
• Exceeding risk limits not acceptable even when
  profits result
• Do not assume that you can outguess the market
• Be diversified
• Scenario analysis and stress testing is important
• Do not give too much independence to star traders

5
Credit Value-at-Risk I

• Objectives
• Measuring Value-at-Risk (VaR)
• Credit VaR

6
The Question Being Asked in VaR

• What loss level is such that we are X
  confident it will not be exceeded in N business
  days?

7
VaR and Regulatory Capital

• Regulators base the capital they require banks
  to keep on VaR
• The market-risk capital is k times the 10-day 99
  VaR where k is at least 3.0
• Under Basel II capital for credit risk and
  operational risk is based on a one-year 99.9 VaR

8
Advantages of VaR

• It captures an important aspect of risk in a
  single number
• It is easy to understand best know market risk
  measure since 1993
• It asks the simple question How bad can things
  get?
• Particularly useful for senior management, which
  does not want to know the delta, gamma, vega for
  each individual equity, FX, Interest Rates, and
  commodity
• Known as the 415 report (first developed by J.P.
  Morgan RiskMetrics released in 1994)

9
VaR vs. Expected Shortfall

• VaR is the loss level that will not be exceeded
  with a specified probability
• Expected shortfall is the expected loss given
  that the loss is greater than the VaR level (also
  called C-VaR and Tail Loss)
• Two portfolios with the same VaR can have very
  different expected shortfalls

10
Distributions with the Same VaR but Different
Expected Shortfalls
VaR
VaR
11
Coherent Risk Measures

• Define a coherent risk measure as the amount of
  cash that has to be added to a portfolio to make
  its risk acceptable
• Properties of coherent risk measure
• If one portfolio always produces a worse outcome
  than another its risk measure should be greater
• If we add an amount of cash K to a portfolio its
  risk measure should go down by K
• Changing the size of a portfolio by l should
  result in the risk measure being multiplied by l
• The risk measures for two portfolios after they
  have been merged should be no greater than the
  sum of their risk measures before they were merged

12
VaR vs Expected Shortfall

• VaR satisfies the first three conditions but not
  the fourth one
• Expected shortfall satisfies all four conditions.
• Example Two 10 million one-year loans each of
  which has a 1.25 chance of defaulting. All
  recoveries between 0 and 100 are equally likely.
  If there is no default the loan leads to a profit
  of 0.2 million. If one loan defaults it is
  certain that the other one will not default.

13
Normal Distribution Assumption

• The simplest assumption is that daily
  gains/losses are normally distributed and
  independent
• It is then easy to calculate VaR from the
  standard deviation (1-day VaR2.33s)
• The N-day VaR equals times the one-day VaR
• Regulators allow banks to calculate the 10 day
  VaR as times the one-day VaR

14
Independence Assumption in VaR Calculations

• When daily changes in a portfolio are identically
  distributed and independent the variance over N
  days is N times the variance over one day
• When there is autocorrelation equal to r the
  multiplier is increased from N to

15
Impact of Autocorrelation Ratio of N-day VaR to
1-day VaR
16
Choice of VaR Parameters

• Time horizon should depend on how quickly
  portfolio can be unwound. Regulators in effect
  use 1-day for bank market risk and 1-year for
  credit/operational risk. Fund managers often use
  one month
• Confidence level depends on objectives.
  Regulators use 99 for market risk and 99.9 for
  credit/operational risk. A bank wanting to
  maintain a AA credit rating will often use 99.97
  for internal calculations. (VaR for high
  confidence level cannot be observed directly from
  data and must be inferred in some way.)

17
Volatility Estimation

• The most important parameter for VaR is
  volatility
• Volatility is not observable and have to be
  estimated.
• There exist many different methods
• The non-weighted moving average (Standard)
• Exponential weighted average (EWMA)
• ARCH and GARCH (Generalized Autoregressive
  Conditional Heteroskedasticity)

18
Standard Approach

• Assuming a lognormal process for the underlying
  market variable
• Si is the value of the variable at the end of
  day i
• si is the daily volatility estimated at the end
  of day i

19
Simplified Approach

• For risk management purposes, the following
  simplification is often applied, which has little
  effect on accuracy
• Si is the value of the variable at the end of
  day i
• si is the daily volatility estimated at the end
  of day i

20
Non-Equal Weighting

• Some risk managers would put more weights on
  recent observations, instead of equal-weighting
  (1/m)

21
EWMA

• Exponentially weighted moving average model
  requires the weights to decline exponentially
  with time, which gives
• The above equation can be verified by recursive
  substitution
• RiskMetrics advocates the use of exponentially
  weighted moving average, with ? 0.94
• Advantages of EWMA
• Few data needs to be stored Only need to
  remember the current estimate of the variance
  rate and the most recently observed value of the
  market variable
• Tracks volatility changes

22
GARCH(1,1)

• Volatility of asset returns appears to be
  serially correlated (i.e. volatility clustering).

23
GARCH(1,1)

• GARCH(1,1) puts weights on the long-run average
  variance V (i.e. the unconditional variance). The
  model is

24
GARCH(1,1)

• The innovation, un1, is assumed to have a normal
  distribution conditional on time n information

25
GARCH(1,1)

• Setting w gV, the model becomes

26
Updating

• Parameters have to be estimated
• We can update the volatility estimate on a daily
  basis, with the newly observed underlying variable

New market price info
27
Calibrating GARCH

• GARCH parameters can be estimated using Maximum
  Likelihood
• In the maximum likelihood method, we are seeking
  parameter values that maximize the likelihood of
  the observations occurring

28
Maximum Likelihood Estimation

• Example We are given a false die. We rolled the
  die 100 times, and observed that the side with
  six dots comes up 5 times in total. What should
  be our estimate of the probability p that six
  will come up in the next roll?
• Common Sense Solution p 5/100 0.05
• Statistical Solution
• If p is known, the probability of the outcome (in
  the order in which it is observed) is
• If we consider p to be a variable, above equation
  is called a likelihood function.
• This likelihood function is maximized for p
  0.05
• We say the maximum likelihood estimate (MLE)
  for p is 0.05

29
GARCH Maximum Likelihood

• For GARCH(1,1), the variance is not constant
• Let vi(?) be the conditional variance implied by
  the parameters and the history of returns for day
  i, where ?T (? a ß)
• We assume the distribution of ui1 conditional on
  vi is normal
• Now, we have to numerically maximize
• v0 is needed, the choice of which does not
  affect the consistency of the estimator.

30
Market Risk VaR Historical Simulation Approach

• Collect data on the daily movements in all market
  variables.
• The first simulation trial assumes that the
  percentage changes in all market variables are as
  on the first day
• The second simulation trial assumes that the
  percentage changes in all market variables are as
  on the second day
• and so on
• Suppose we use n days of historical data with
  today being day n
• Let vi be the value of a variable on day i
• There are n-1 simulation trials
• The ith trial assumes that the value of the
  market variable tomorrow (i.e., on day n1) is

31
Market Risk VaR The Model-Building Approach

• The main alternative to historical simulation is
  to make assumptions about the probability
  distributions of the returns on the market
  variables and calculate the probability
  distribution of the change in the value of the
  portfolio analytically
• This is known as the model building approach or
  the variance-covariance approach

32
Model Building vs Historical Simulation

• Model building approach is used for investment
  portfolios, but it does not usually work well for
  portfolios involving options that are close to
  delta neutral

33
Back-Testing

• Backtesting a VaR calculation methodology
  involves looking at how often exceptions
  (lossVaR) occur
• Back-testing is a way to test the performance of
  the VaR system
• It is asking the question Does 1 percentile of
  all daily losses exceed the 99 VaR?
• Alternatives a) compare VaR with actual change
  in portfolio value and b) compare VaR with change
  in portfolio value assuming no change in
  portfolio composition
• Suppose that the theoretical probability of an
  exception is p (1-X). The probability of m or
  more exceptions in n days is

34
Basel Committee Rules for Market Risk VaR

• If number of exceptions in previous 250 days is
  less than 5 the regulatory multiplier, k, is set
  at 3
• If number of exceptions is 5, 6, 7, 8 and 9
  supervisors may set k equal to 3.4, 3.5, 3.65,
  3.75, and 3.85, respectively
• If number of exceptions is 10 or more k is set
  equal to 4

35
Bunching

• Bunching occurs when exceptions are not evenly
  spread throughout the backtesting period
• Statistical tests for bunching have been developed

36
Stress Testing

• Considers how portfolio would perform under
  extreme market moves
• Scenarios can be taken from historical data (e.g.
  assume all market variable move by the same
  percentage as they did on some day in the past)
• 22.3 std dev drop in SP during Oct 19, 1987
• 7.7 std dev rise in 10 year gilt yield in April
  10, 1992
• Alternatively they can be generated by senior
  management

37
Credit Risk in Derivatives Transactions

• Three cases
• Contract always an asset
• Contract always a liability
• Contract can be an asset or a liability

38
General Result

• Assume that default probability is independent of
  the value of the derivative. Define
• t1, t2,tn times when default can occur
• qi default probability at time ti.
• fi The value of the contract at time ti
• R Recovery rate
• The expected loss from defaults at time ti is
• qi(1-R)Emax(fi,0).
• Defining uiqi(1-R) and vi as the value of a
  derivative that provides a payoff of max(fi,0) at
  time ti, the PV of the cost of defaults is

39
Applications

• If contract is always an asset so that fi0 then
  vi f0 and the cost of defaults is f0 times the
  total default probability, times 1-R
• If contract is always a liability then vi 0 and
  the cost of defaults is zero
• In other cases we must value the derivative
  max(fi,0) for each value of i

40
Expected Exposure on Pair of Offsetting Interest
Rate Swaps and a Pair of Offsetting Currency
Swaps
Exposure
Currency swaps
Interest Rate Swaps
Maturity
41
Interest Rate vs Currency Swaps

• The uis are the same for both
• The vis for an interest rate swap are on average
  much less than the vis for a currency swap
• The expected cost of defaults on a currency swap
  is therefore greater.

42
Two-Sided Default Risk

• In theory a company should increase the value of
  a deal to allow for the chance that it will
  itself default as well as reducing the value of
  the deal to allow for the chance that the
  counterparty will default

43
Credit Risk Mitigation

• Netting
• Collateralization
• Downgrade triggers

44
Netting

• We replace fi by in the definition of ui
• to calculate the expected cost of defaults by a
  counterparty where j counts the contracts
  outstanding with the counterparty
• The incremental effect of a new deal on the
  exposure to a counterparty can be negative!

45
Collateralization

• Contracts are marked to markets periodically
  (e.g. every day)
• If total value of contracts Party A has with
  party B is above a specified threshold level it
  can ask Party B to post collateral equal to the
  excess of the value over the threshold level
• After that collateral can be withdrawn or must be
  increased by Party B depending on whether value
  of contracts to Party A decreases or increases

46
Downgrade Triggers

• A downgrade trigger is a clause stating that a
  contract can be closed out by Party A when the
  credit rating of the other side, Party B, falls
  below a certain level
• In practice Party A will only close out contracts
  that have a negative value to Party B
• When there are a large number of downgrade
  triggers they are counterproductive

47
Credit VaR

• Can be defined analogously to Market Risk VaR
• A one year credit VaR with a 99.9 confidence is
  the loss level that we are 99.9 confident will
  not be exceeded over one year

48
Vasiceks Model

• For a large portfolio of loans, each of which has
  a probability of Q(T) of defaulting by time T the
  default rate that will not be exceeded at the X
  confidence level is
• Where r is the Gaussian copula correlation

49
CreditRisk

• This calculates a loss probability distribution
  using a Monte Carlo simulation where the steps
  are
• Sample overall default rate
• Sample number of defaults for portfolio under
  consideration
• Sample size of loss for each default

50
CreditMetrics

• Calculates credit VaR by considering possible
  rating transitions
• A Gaussian copula model is used to define the
  correlation between the ratings transitions of
  different companies

51
Summary

• Integration of market and credit risk
• Market risk VaR
• Credit VaR
• Next CreditMetrics