Lecture 4

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

Le

ctu

re 4

Pa

rt 1

. In

tro

du

cti

on

to

ma

rke

t ri

sk

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

Le

ctu

re 4

Pa

rt 1

. In

tro

du

cti

on

to

ma

rke

t ri

sk

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

Le

ctu

re 4

Pa

rt 1

. In

tro

du

cti

on

to

ma

rke

t ri

sk

3. Classification of risks Known UNKnowns

• 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

Le

ctu

re 4

Pa

rt 1

. In

tro

du

cti

on

to

ma

rke

t ri

sk

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 𝑅𝑖 = 𝛼𝑖 + 𝛽𝑖𝑹𝑴

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?

Le

ctu

re 4

Pa

rt 1

. In

tro

du

cti

on

to

ma

rke

t ri

sk

4. Sources of market risks

Equity Risk 1. Volatility

Commodity Risk 1. Volatility

2. Future risks

Le

ctu

re 4

Pa

rt 1

. In

tro

du

cti

on

to

ma

rke

t ri

sk

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

Le

ctu

re 4

Pa

rt 2

. V

ola

tility

mo

de

ls

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).

Le

ctu

re 4

Pa

rt 2

. V

ola

tility

mo

de

ls

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:

Le

ctu

re 4

Pa

rt 2

. V

ola

tility

mo

de

ls


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

3

22

2

2

1

22

2

2

2-n

2

2

22

2

2

1

2

2

2

2

2

1

2

1

2

1-n

for substitute

)(

GARCH(1,1)in for substitute

++++++

++++

++++

nnnnn

nnn

nnnn

RRR

RR

RR

sbababawbbwws

s

sbababww

bsawbaws

s

Le

ctu

re 4

Pa

rt 2

. V

ola

tility

mo

de

ls


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

+

a

VdzdtVVadV L

Le

ctu

re 4

Pa

rt 2

. V

ola

tility

mo

de

ls


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

𝑅𝑖

= 𝑖

𝑖

From here we can ca lculate

s imple variance

To weighted returns : recent past will affect more!

1st: (1-lambda)2nd: last*lamb = 𝑅 𝑖

Le

ctu

re 4

Pa

rt 2

. V

ola

tility

mo

de

ls

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 bsaws𝛼 𝛽

2

1

2

1

2 )1( + nnn R bsbs

2

1

2

1

2 )1( + nnn REWMA ss

Le

ctu

re 4

Pa

rt 2

. V

ola

tility

mo

de

ls

3. EWMA approach

2m

1

212

2

3

32

3

22

2

2

1

2

2

2

2

2

22

2

2

1

2

1

2

2

2

2

2

2

1

)1(

gives way thecontinuing

))(1(

substitute

))(1(

)1(])1([

replace

mn

m

i

in

i

n

nnnnn

n

nnn

nnnn

n

R

RRR

RR

RR

+

+++

++

++

ss

ss

s

s

ss

s

2

1

2

1


Le

ctu

re 4

Pa

rt 2

. V

ola

tility

mo

de

ls

3. EWMA approach

i

i

i

mn

aaa

s

+

1i

1

2m

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

Le

ctu

re 4

Pa

rt 2

. V

ola

tility

mo

de

ls


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

𝑅𝑖

= 𝑖

𝑖

2

1

2

1


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 = 𝑅 𝑖

Le

ctu

re 4

Pa

rt 2

. V

ola

tility

mo

de

ls


Example

Le

ctu

re 4

Pa

rt 2

. V

ola

tility

mo

de

ls

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

ec

ture

4

Pa

rt 2

. V

ola

tility

mo

de

ls

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

Le

ctu

re 4

Pa

rt 2

. V

ola

tility

mo

de

ls

6. Some important details L

ec

ture

4

Pa

rt 2

. V

ola

tility

mo

de

ls

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 worst expected loss that a firm

may suffer over a period of time that has been specified by the user,

under normal market conditions and 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.

Le

ctu

re 4

Pa

rt 3

. V

aR

Me

tho

do

log

ies

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.

Le

ctu

re 4

Pa

rt 3

. V

aR

Me

tho

do

log

ies

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

Le

ctu

re 4

Pa

rt 3

. V

aR

Me

tho

do

log

ies

Le

ctu

re 4

Pa

rt 3

. V

aR

Me

tho

do

log

ies

VaR: additional considerations

Standard Deviation approach

𝑉𝑎𝑅 = 𝑊 𝜎 𝛼

But this measure is symmetrical and cannot distinguish between large gains and

small looses

Maximum Drawback

Is defined as the largest value that can be lost in a rolling window of time (it

could be the complete serie).

• It is generally used in a trend

Portfolio: $1.000.000

SD: $23.300

Prob: Normal distribution. 1.65

= $48.404

𝑀𝑎𝑥𝐷𝑟𝑎 𝐵𝑎𝑐𝑘 =𝑥𝑖

𝑀𝑎𝑥 𝑥 𝑟𝑜𝑙𝑙𝑖 𝑔 𝑖 𝑑𝑜 −

2. Additional Considerations

Maximum drawback

Main limitation: not comparable among portfolios

70% 45%

Le

ctu

re 4

Pa

rt 3

. V

aR

Me

tho

do

log

ies

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.


VaR parameters

1. Confidence level: depends on the number of observations: the more

observations, the more quintiles .

• Confidence level should be tested: that give a real perception of

money at risk. 80% and 99,9% are non-real scenarios

2. Horizon

𝑉𝑎𝑟 𝑇 𝑑𝑎𝑦𝑠 = 𝑉𝑎𝑅 𝑑𝑎𝑦 𝑇

i. Distribution of returns should be unchanged in long horizons

ii. Characteristics of the portfolio: if the positions have a high (low)

rotation, the horizon should be short(long)

iii. Not long VaR periods: The more data points, the more accurate could

be the model

iv. Main recommendation: daily VaR according to MtM policies

Short time

To check a specific portfolio

Long time

To avoid bankruptcy

Le

ctu

re 4

Pa

rt 3

. V

aR

Me

tho

do

log

ies


3. VaR Models L

ec

ture

4

Pa

rt 3

. V

aR

Me

tho

do

log

ies

Order numbers

and obtain

quintiles and

using history,

losses could be…

Forecasts n

paths and find

the VaR

Le

ctu

re 4

Pa

rt 3

. V

aR

Me

tho

do

log

ies

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

Le

ctu

re 4

Pa

rt 3

. V

aR

Me

tho

do

log

ies


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

pe riod).

• The third step assumes that the historical distribution that the

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

ctu

re 4

Pa

rt 3

. V

aR

Me

tho

do

log

ies


Advantages

Le

ctu

re 4

Pa

rt 3

. V

aR

Me

tho

do

log

ies

• It does not depend on assumptions about the distribution of

returns. Therefore, the mistakes of assuming parametric

distributions with thin tails where in reality the distributions of

returns have fat tails are avoided.

• 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.


Le

ctu

re 4

Pa

rt 3

. V

aR

Me

tho

do

log

ies

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

Le

ctu

re 4

Pa

rt 3

. V

aR

Me

tho

do

log

ies

The problem

• Var-Cov VaR methodology is based on

𝑉𝑎𝑅𝑝 = 𝛼 𝜎𝑃 𝑊

𝑉𝑎𝑅𝑝= 𝛼 𝑋′ 𝑋

The problem to solve is the solution of the var-con matrix

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

Le

ctu

re 4

Pa

rt 3

. V

aR

Me

tho

do

log

ies


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

Le

ctu

re 4

Pa

rt 3

. V

aR

Me

tho

do

log

ies


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.

Le

ctu

re 4

Pa

rt 3

. V

aR

Me

tho

do

log

ies

6. Monte Carlo methods

Overview

• Monte Carlo simulation involves trying 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.

Le

ctu

re 4

Pa

rt 3

. V

aR

Me

tho

do

log

ies

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

Le

ctu

re 4

Pa

rt 3

. V

aR

Me

tho

do

log

ies

Theory


Le

ctu

re 4

Pa

rt 3

. V

aR

Me

tho

do

log

ies

Geometric Brownian Motion


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

Le

ctu

re 4

Pa

rt 3

. V

aR

Me

tho

do

log

ies



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

s

Le

ctu

re 4

Pa

rt 3

. V

aR

Me

tho

do

log

ies



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

Le

ctu

re 4

Pa

rt 3

. V

aR

Me

tho

do

log

ies

Prices Log-normal distributed


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

Le

ctu

re 4

Pa

rt 3

. V

aR

Me

tho

do

log

ies

Prices Log-normal distributed


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

Le

ctu

re 4

Pa

rt 3

. V

aR

Me

tho

do

log

ies

Other models (interest rates)


Le

ctu

re 4

Pa

rt 3

. V

aR

Me

tho

do

log

ies

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

Le

ctu

re 4

Pa

rt 3

. V

aR

Me

tho

do

log

ies

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

𝑀𝑅𝐶𝐼𝑀𝐴 = 𝑀𝑎𝑥 𝑘

60 𝑉𝑎𝑅𝑡 𝑖 , 𝑉𝑎𝑅𝑡

60

𝑖=

+ 𝑅𝐶

K>3 is a multiplier

Le

ctu

re 4

Pa

rt 3

. V

aR

Me

tho

do

log

ies

Documents

Lecture 4