LECTURE FOUR
1
a. Introduction to market risk
b. Modelling volatility
c. VaR Models
INTRODUCTION TO MARKET RISK
Part 1
2
a. Overview
b. Risk measurement
c. Classification of risks
d. Sources of market risks
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1. Overview
•Market risk: movements in prices or volatility
•Liquidity risk: losses when a position is liquidated
•Credit risk: counterparty cannot fulfil contractual obligations
•Operational risk: related to an inadequate internal process,
or caused by an external event
Types of financial risk
Can interact
Currency swap
Operational risk
Credit risk
Market risk
Liquidity risk
Settlement risk
The time difference of two parties
delayed the payment one day
The Value at Risk is key to measure market risk :
It includes probability and scenario analysis
During any day the exchange
rate can change
One of the counterparties
goes to bankruptcy
In the settlement date there is a
blackout that lasts a couple of hours
One of the currencies is Iraqi dinar
Example
2. Risk measurement systems
• From market data, construct
the distribution of risk factors
• Collect portfolio positions
and collect then onto risk
factors
• Use the risk engine to
construct the distribution of
portfolio profits and losses
over the period.
Position based (Risk of positions) VS Returns based (VaR based on returns):
• Offer data for new instruments, market and managers
• Capture style drift
• More realistic
BUT
• More expensive (technologically)
Fixed income: linear VaR
Options: non linear VaR
This is the reason why it is so important Back
Testing and Stress testing (scenario analysis)
Additional consideration
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3. Classification of risks
Known Knowns
• All factors are identified
• All factors are measured correctly
• Appropriate description of distribution of risk factors
Losses explained by:
• Bad luck
• Too much exposure
SPX yearly return
VaR should be viewed as a
measure of dispersion that
should be exceeded with
some regularity
Conditional VaR here is really
important!
• Losses once the VaR is
broken
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3. Classification of risks Known UNKnowns (Some problems about known risks factors)
• Management ignores important risk factors (i.g. political stress)
• Inaccurate distribution for a specific factor
Mapping process could be incorrect
SPX volatility from 2004-2007
is extremely low to
forecast 2007’s.
Wrong distribution !!
UNKnown UNKnowns
• Events totally outside the scenario: i.g. sudden restriction to short sales
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4. Sources of market risks 1. Exposure to the factor
2. Movement to the factor itself
3. Risk of the system
or …
Market loss = Exposure x Adverse movement
Bond
Interest rate
Volatility
Currency Risk 1. Volatility
2. Correlations
3. Cross-rate volatility:
Two currencies tied by a base currency
Fixed Income Risk
1. Inflation: WHY???
2. Correlations among bonds
3. Short term bonds have little price risk (durat.)
4. Reference rate (driven by expected inflation)
5. Credit spread: Bonds VS risk free
Note: TERM SPREAD.
Long term-Short term
Where risk comes from?
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4. Sources of market risks
Equity Risk 1. Volatility
Commodity Risk 1. Volatility
2. Future risks
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MODELLING VOLATILITY
Part 2
9
a. Intro. models
b. Volatility standard approach
c. Garch (1,1)
d. EWMA
e. Risk Metrics ™
f. Details
1. Volatility Models
Standard Approach to Estimating Volatility
ARCH(m) Model
EWMA Model
GARCH model
Maximum Likelihood Methods
2. Standard Approach to Estimating Volatility
• Define sn as the volatility per day between day nt-1 and day nt,
as estimated at end of day nt-1
• Define Si as the value of market variable at end of day i
• Define Ri= ln (Si/Si-1) {KNOWN from previous lecture}
m
i
inn RRm 1
22 )(1
1s
m
i
inRm
R1
1
A measure of divergence from average
m-1 because there are m-1 returns
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3. Generalized AutoRegressive Conditional Heteroskedasticity
3. Garch (p,q) approximation
• GARCH (p, q) and in particular GARCH (1, 1)
• Autoregressive: tomorrow’s variance (or volatility) is a regressed
function of today’s variance — it regresses on itself
• Conditional: tomorrow’s variance depends—is conditional on —
the most recent variance. An unconditional variance would not
depend on today’s variance
• Heteroskedastic: variances are not constant, they flux over time
• GARCH (1, 1) “lags” or regresses on last period’s squared
return (i.e., just 1 return) and last period’s variance (i.e., just 1
variance).
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3. Garch (1,1) approximation
In GARCH (1,1) we assign some weight to the long-run average
variance rate
Since weights must sum to 1
g + a + b 1
2
1
2
1
2
++ nnLn RV bsags
Setting w gV, the GARCH (1,1) model
2
1
2
1
2
++ nnn R bsaws
ba
w
1LVAnd:
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3. Garch (1,1) approximation
The weights decline exponentially at rate β
Tomorrow’s variance is a weighted average of the long run variance!!
2
1 1
22
jn
p
i
q
j
jinin R
++ sbaws
2
3
32
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for substitute
)(
GARCH(1,1)in for substitute
++++++
++++
++++
nnnnn
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RRR
RR
RR
sbababawbbwws
s
sbababww
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3. Garch (1,1) approximation
Note the Mean reversion!
The GARCH(1,1) model recognizes that over time the variance tends
to get pulled back to a long-run average level
The GARCH(1,1) is equivalent to a model where the variance V
follows the stochastic process
14
LV
2,--1days,in measured timewhere
)(
aba
+
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VdzdtVVadV L
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3. Garch (1,1) approximation
2
1
2
1
2
++ nnLn RV bsagsLong Run Variance Returns of past periods
Volatility of past periods (lagged variance)
Weights of each factor
Long Term Variance (Av. Variance of all) 0,050163%
Gamma 0,2
Alpha 0,3
Beta 0,5
Garch 0,0275%
Date Close Daily Return Return to 2 Weights
23/10/2012 41,33 -1,822% 0,0332%
22/10/2012 42,09 -0,545% 0,0030% 6,00% 0,0002%
19/10/2012 42,32 -1,617% 0,0262% 5,64% 0,0015%
18/10/2012 43,01 -0,718% 0,0052% 5,30% 0,0003%
17/10/2012 43,32 1,138% 0,0129% 4,98% 0,0006%
16/10/2012 42,83 1,056% 0,0112% 4,68% 0,0005%
15/10/2012 42,38 1,810% 0,0327% 4,40% 0,0014%
31/10/2011 34,76 -5,404% 0,2920% 0,00% 0,0000%
28/10/2011 36,69 -0,895% 0,0080% 0,00% 0,0000%
27/10/2011 37,02 7,982% 0,6371% 0,00% 0,0000%
26/10/2011 34,18 2,039% 0,0416% 0,00% 0,0000%
25/10/2011 33,49 -3,174% 0,1007% 0,00% 0,0000%
24/10/2011 34,57 3,383% 0,1145% 0,00% 0,0000%
21/10/2011 33,42
0,014942%
Yesterday
From here we can ca lculate
s imple variance
To weighted returns : recent past will affect more!
1st: (1-lambda)2nd: last*lamb
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4. Exponentially-Weighted Moving Average
4. EWMA approximation
The equation simplifies to
This is now equivalent to the formula for exponentially weighted
moving average (EWMA):
In EWMA, the lambda parameter now determines the
“decay:” a lambda that is close to one (high lambda) exhibits slow
decay.
Garch (1,1)
= 0 and ( + ) =1:
2
1
2
1
2
++ nnn R bsaws
2
1
2
1
2 )1( + nnn R bsbs
2
1
2
1
2 )1( + nnn REWMA ss
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3. EWMA approach
2m
1
212
2
3
32
3
22
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3. EWMA approach
i
i
i
mn
aaa
s
+
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or )1( where
scheme weight ofquation theas same isequation the
smallly sufficient is m, large For the
• Relatively little data needs to be stored
• We need only remember the current estimate of the variance rate and the most recent observation on the market variable
• Tracks volatility changes
Advantages
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3. EWMA approximation
Example
Date Close Daily Return Return to 2 WeightsWeights of sq.
ReturnsWeights
23/10/2012 41,33 -1,822% 0,0332% 6,00% 0,0020%
22/10/2012 42,09 -0,545% 0,0030% 5,64% 0,0002% 6,00% 0,0002%
19/10/2012 42,32 -1,617% 0,0262% 5,30% 0,0014% 5,64% 0,0015%
18/10/2012 43,01 -0,718% 0,0052% 4,98% 0,0003% 5,30% 0,0003%
17/10/2012 43,32 1,138% 0,0129% 4,68% 0,0006% 4,98% 0,0006%
16/10/2012 42,83 1,056% 0,0112% 4,40% 0,0005% 4,68% 0,0005%
15/10/2012 42,38 1,810% 0,0327% 4,14% 0,0014% 4,40% 0,0014%
31/10/2011 34,76 -5,404% 0,2920% 0,00% 0,0000% 0,00% 0,0000%
28/10/2011 36,69 -0,895% 0,0080% 0,00% 0,0000% 0,00% 0,0000%
27/10/2011 37,02 7,982% 0,6371% 0,00% 0,0000% 0,00% 0,0000%
26/10/2011 34,18 2,039% 0,0416% 0,00% 0,0000% 0,00% 0,0000%
25/10/2011 33,49 -3,174% 0,1007% 0,00% 0,0000% 0,00% 0,0000%
24/10/2011 34,57 3,383% 0,1145% 0,00% 0,0000% 0,00% 0,0000%
21/10/2011 33,42
Volatility 0,016037% Volatility 0,014942%
We do not have to calculate the complete series
0,016037%
Lambda
0,94
Today Yesterday
Is the summatory of weighted
squared returns
EWMA Calculation
2
1
2
1
2 )1( + nnn REWMA ss
From here we
can ca lculate
s imple variance
To weighted returns : recent past will affect more! in a exponentially declining fashion--Proportional decay--
1st: (1-lambda)2nd: last*lamb
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3. EWMA approximation
Example
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3. Risk Metrics
RiskMetrics is a branded form of the exponentially weighted moving
average (EWMA) approach:
The optimal (theoretical) lambda varies by asset class, but the overall
optimal parameter used by RiskMetrics has been 0.94. In practice,
RiskMetrics only uses one decay factor for all series:
• · 0.94 for daily data
• · 0.97 for monthly data (month defined as 25 trading days)
5. Risk Metrics L
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6. Some important details
EWMA is (technically) an infinite series but the infinite series elegantly
reduces to a recursive form:
R
R
R
R
R
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6. Some important details L
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VALUE AT RISK MODELS
Part 3
24
a. Overview
b. Initial considerations
c. Var Models (intro)
d. VaR Historical
e. Parametric Approach
f. Monte Carlo Approach
g. Basel 2
1. Overview VaR
There are many models that measure risk. However the Value at Risk
is the most popular and also answers all requirements in a financial
institution
Definition: . VaR is a measure of the
1. worst expected loss that a firm may suffer
2. over a period of time that has been specified by the user,
3. under normal market conditions and
4. a specified level of confidence.
Specifically, it is the maximum loss which can occur with X% confidence
over a holding period of n days.
It has several advantages, but the most important ones are:
• It gives a clear number (only one) and
• It is easy to implement and interpret.
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1. Overview VaR
Limitations:
1. These methods use past historical data to provide an estimate
for the future. What happened in the past does not mean that will
happen again in the future
2. VaR number can be calculated by using several methods. These
methods try to capture volatility's behavior. However, there is an
argument on which is the method that performs best.
3. Methods for computing it are based on different assumptions. These
assumptions help us with the calculation of VaR but they are not
always true (like distributional assumptions).
4. There are many risk variables (political risk, liquidity risk, etc ) that
cannot be captured by the VaR methods.
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2. Additional Considerations • Does NOT describe the worst loss
• Only describes the probability that a value occurs
• VaR number indicates that 1% of days of a period of time, the losses could
be higher
• The previous VaR depends on history!. So it will be very important that data
have at least one crisis.
Conditional VaR.
The potential loss when the
portfolio is hit beyond VaR
In JPM case it is $116.000
VaR
Conditional VaR
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VaR: additional considerations
Portfolio: $1.000.000
SD: $23.300
Prob: Normal distribution. 1.65
= $48.404
2. Additional Considerations
Maximum drawback
Main limitation: not comparable among portfolios
70% 45%
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There is no one VaR number for a single portfolio, because different
methodologies used for calculating VaR produce different results.
The VaR number captures only those risks that can be measured in quantitative
terms; it does not capture risk exposures such as operational risk, liquidity risk,
regulatory risk or sovereign risk.
2. Additional Considerations
VaR parameters
Short time
To check a specific portfolio
Long time
To avoid bankruptcy
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2. Additional Considerations
3. VaR Models L
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Order numbers
and obtain
quintiles and
using history,
losses could be…
Forecasts n
paths and find
the VaR
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4. Historical simulation
Definition Tries to find an empirical distribution of the rates of
return assuming that past history carries out into the future.
• Uses the historical distribution of returns of a portfolio to
simulate the portfolio's VaR.
• Often historical simulation is called non-parametric approach,
because parameters like variances and covariances do not have
to be estimated, as they are implicit in the data.
• The choice of sample period influences the accuracy of VaR
estimates.
• Longer periods provide better VaR estimates than short ones.
Illustration:
We have 1 M pounds in JPM Stocks, and we want to figure out what could be
the value at risk of this position
1. Obtain the data. In this case from 2000 Jan to 2012 Oct
2. Calculate daily return (or weekly), depend on the VaR
3. Estimate daily (weekly) gain/loss
4. We can construct a frequency distribution of daily returns
5. We can calculate our value at risk
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4. Historical simulation
The methodology:
• Identifying the instruments in a portfolio and collecting a sample of
their historical returns.
• Calculate the simulated price of every instrument using the weights
of the current portfolio (in order to simulate the returns in the next
period).
• Assumption: returns follow is a good proxy for the returns in the
next period.
VaR
Deviation from the average
return
With 99% prob,
the loss won’t
be higher than
$75.000 per M
With 99% prob,
the loss won’t
be higher than
$75.000 per M
I have N observations, and it is easy to find what observation is 1% Le
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4. Historical simulation
Advantages
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• It does not depend on assumptions about the distribution of
returns. I would avoid fat tails issues
• There is no need for any parameter estimation.
• There are not different models for equities, bonds and
derivatives
Disadvantages
• Results are dependent on the data set from the past, which may
be too volatile or not, to predict the future.
• Assumes that returns are independent and identically
distributed.
• It uses the same weights on all past observations. If an
observation from the distant past is excluded the VaR estimates
may change significantly.
4. Historical simulation
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5. Parametric approach
Definition This approach for calculating the value at risk is also known
as the delta-normal method.
• This is the most straightforward method of calculating Value at
Risk.
• It is the method used by the RiskMetrics methodology, the VaR
system originally developed by JP Morgan.
• Assumes that returns are normally distributed. It ONLY requires
that we estimate two factors
• expected (or average) return and
• a standard deviation
Using them it could be possible to plot a normal distribution curve.
Variance – Covariance approach
We use the familiar curve instead of actual data
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5. Variance – Covariance approach
Volatility (sd) of the portfolio
Wealth
Confidence level (normal equivalent)
Variance-Covariance matrix showing wealth
$ Positions $ Positions VaR-CoV x x
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5. Variance – Covariance approach
Goog NokPortfolio Value
Weights 1/3 2/3
Stock worths 33,33$ 66,67$
Volatility 1,703% 3,879%
Correlation
Goog Nok XGoog 0,000290 0,000172 33,33 66,67 0,000290 0,000172 33,33
NoK 0,000172 0,001505 0,000172 0,001505 66,67
1 X 2 2 X 2 2 x 1
0,02116696 7,778
0,10608465
Variance 7,78$
Volatility 2,79$
Confidence 95%
Critical value 1,645
VaR 4,59$
Variance Covariance VaR
100,00$
0,26109
Var-Cov Matrix
Var-Cov MatrixX'
Variance =
Volatility * VolatilityCovariance:
σxy=ρxy σx σy
Pearson Correlation
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5. Variance – Covariance approach
Advantages
• Easy to capture relations among data
Disadvantages
• assuming normal distribution of returns for assets and portfolios
with non-normal skewness or excess kurtosis. Using unrealistic
return distributions as inputs can lead to underestimating the real
risk with VAR.
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6. Monte Carlo methods
Overview
• Monte Carlo simulation try to simulate the conditions, which
apply to a specific problem, by generating a large number of
random samples
• Each simulation will be different but in total the simulations will
aggregate to the chosen statistical parameters
• After generating the data, quantities such as the mean and variance of
the generated numbers can be used as estimates of the unknown
parameters of the population
• It is more flexible
• Allows the risk manager to use actual historical distributions for risk
factor returns rather than having to assume normal returns.
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Theory
• Consider a stock S, with a price of $20
• The price can only rise (drop) $1 each day for successive days
• Forecast instrument: a coin: this is a RANDOM VARIABLE
• The more days, the more simulation paths
What can we assure?
• The EXPECTED mean of the price will be $20 (no matter how many
periods ahead!!!)
• It is possible to calculate standard deviation and probabilistic
statements
We cannot determine what could be the price at the end of a period
6. Monte-Carlo Simulation approach
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Theory
6. Monte-Carlo Simulation approach
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Geometric Brownian Motion
6. Monte-Carlo Simulation approach
Continuity: The paths are continuous in time and value. (stock prices
can be observed at all times and they are changing).
We assume that traders and systems are working weekends
and nights
Markov process: GBM follows a Markov process, meaning that only
the current stock’s price history is relevant for predicting future
prices (stock price history is irrelevant).
Weak form of the efficient market hypothesis.
No momentum when occurs a trend
No technical analysis
Normality: the proportional return over infinite increments of time
for a stock is normally distributed
The price of a stock is lognormally distributed
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Geometric Brownian Motion
6. Monte-Carlo Simulation approach
ttS
S+
s
Very short period of time
Certain component Uncertain component
The return is uncertain or random
Deterministic component (drift)
• μ is the expected rate of return • If the price of the stock today is S0,
then its price ST at time T in the future would be:
ST=S0 e(μT)
Stochastic component
• ε is the ranom component of the standard normal distribution (mean 0 and sd of 1).
• σ is the volatility
• The longer the time interval, the more variable the return
),( ttNS
S
sWe already know how to
prove this!!
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Geometric Brownian Motion
6. Monte-Carlo Simulation approach
ttS
S+
s
• S=$10
• μ=12% per year • σ=40% per year • t= 1 day, that is 0.004 of a year
%07.20632.0*8.0*4.0004.0*12.0 +
S
SWhat this
number
means ?
BUT:
Draws for e will be sometimes negative, the proportional return can be
positive and negative
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Prices Log-normal distributed
6. Monte-Carlo Simulation approach
TTN
S
Ss
s ,
2ln
2
The natural log of S are normally distributed
The price path will be
+
+
+
+
ttSS
ttS
S
ttt
t
tt
ss
ss
2exp
2ln
2
2
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Prices Log-normal distributed
6. Monte-Carlo Simulation approach
Expected return (yearly) 20%
Daily return 0,08%
Volatility (yearly) 40% Price
Daily volatility 2,52% Day CountRandom Uniform Normal 10
Time t (in days) 1 1 0,28878 -0,557 9,65851
Stock price $ 10 2 0,53095 0,07766 9,71073
3 0,70087 0,52692 10,0446
(yearly) 12,000% 4 0,44846 -0,1296 9,96742
5 0,15337 -1,0221 9,34798
6 0,16773 -0,9632 8,79975
7 0,47247 -0,0691 8,7656
8 0,2168 -0,783 8,34607
9 0,39029 -0,2786 8,20426
)2
(2s
N(0,1). Exp value of 0 and sd of
1
Standard normal cumulative
distribution. Value btw -3 and 3.
To randomize my volatility
Apply my formula
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Other models (interest rates)
6. Monte-Carlo Simulation approach
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6. Monte Carlo simulation approach Advantages
• Able to model instruments with non-linear and path-
dependent payoff functions (complex derivatives).
• Moreover, is not affected as much as Historical Simulations VaR by
extreme events
• We may use any statistical distribution to simulate the returns
as far as we feel comfortable with the underlying assumptions that
justify the use of a particular distribution.
Disadvantages
• The main disadvantage of Monte Carlo Simulations VaR is the
computer power that is required to perform all the simulations
• Cost associated with developing a VaR
7. Basel 2 (a quick view) Qualitative Criteria
VaR is a robust Risk Measurement and Management Practice
Banks can use their own VaR models as basis for capital requirement for Market Risk
Regular Back-Testing
Initial and on-going Validation of Internal Model
Bank’s Internal Risk Measurement Model must be integrated into Management decisions
Risk measurement system should be used in conjunction with Trading and Exposure Limits.
Stress Testing
Risk measurement systems should be well documented
Independent review of risk measurement systems by internal audit
Board and senior management should be actively involved
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3. Basel 2 (a quick view) Quantitative Parameters :
VaR computation be based on following inputs :
• Horizon of 10 Trading days
• 99% confidence level
• Observation period – at least 1 year historical data
Correlations : recognise correlation within Categories as well as across
categories (FI and Fx, etc)
Market Risk charge : General Market Risk charge shall be – Higher of
previous day’s VaR or Avg VaR over last 60 business days X Multiplier factor K
(absolute floor of 3)
SRC – Specific Risk Charge
MRCt = Max (Avg VaR over 60 days, VaR t-1) + SRC
K>3 is a multiplier
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LECTURE FOUR
52
End Of The Lecture
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