Lecture 4

LECTURE FOUR

a. Introduction to market risk
b. Modelling volatility
c. VaR Models

1

Part 1

INTRODUCTION TO
MARKET RISK
a. Overview
b. Risk measurement
c. Classification of risks
d. Sources of market risks

2

1. Overview
Types of financial risk
•Market risk: movements in prices or volatility
•Liquidity risk: losses when a position is liquidated
•Credit risk: counterparty cannot fulfil contractual obligations Can interact
•Operational risk: related to an inadequate internal process,
or caused by an external event
Part 1. Introduction to market risk

Settlement risk Operational risk
Example
The time difference of two parties In the settlement date there is a
delayed the payment one day Currency swap blackout that lasts a couple of hours

Liquidity risk
Credit risk
One of the currencies is Iraqi dinar
Market risk One of the counterparties
goes to bankruptcy
During any day the exchange
rate can change
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The Value at Risk is key to measure market risk :
It includes probability and scenario analysis

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.
Fixed income: linear VaR This is the reason why it is so important Back
Options: non linear VaR Testing and Stress testing (scenario analysis)

Additional consideration
Position based (Risk of positions) VS Returns based (VaR based on returns):
• Offer data for new instruments, market and managers
• Capture style drift
Lecture 4

• More realistic
BUT
• More expensive (technologically)

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 VaR should be viewed as a
measure of dispersion that
SPX yearly return 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
• Management ignores important risk factors (i.g. political stress)
• Inaccurate distribution for a specific factor

SPX volatility from 2004-2007
is extremely low to forecast
2007’s.

Wrong distribution !!

Mapping process could be incorrect
UNKnown UNKnowns
Lecture 4

• Events totally outside the scenario: i.g. sudden restriction to short sales

4. Sources of market risks Bond
Where risk comes from?
1. Exposure to the factor
2. Movement to the factor itself Interest rate
3. Risk of the system
Volatility
𝑅 𝑖 = 𝛼 𝑖 + 𝛽𝑖 𝑹 𝑴
or …

Market loss = Exposure x Adverse movement
Currency Risk Fixed Income Risk
1. Volatility 1. Inflation: WHY???
2. Correlations 2. Correlations among bonds
3. Cross-rate volatility: 3. Short term bonds have little price risk (durat.)
Two currencies tied by a base currency 4. Reference rate (driven by expected inflation)
5. Credit spread: Bonds VS risk free
Lecture 4

Note: TERM SPREAD.
Long term-Short term

4. Sources of market risks

Equity Risk Commodity Risk
1. Volatility 1. Volatility
2. Future risks
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Part 2

MODELLING
VOLATILITY

a. Intro. models
b. Volatility standard approach
c. Garch (1,1)
d. EWMA
e. Risk Metrics ™
f. Details

9

1.Volatility Models
 Standard Approach to Estimating Volatility
 ARCH(m) Model
 EWMA Model
 GARCH model
 Maximum Likelihood Methods

2. Standard Approach to Estimating Volatility
Part 2. Volatility models

• 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}
Lecture 4

1 m
s 
2
n 
m  1 i 1
( Rn i  R ) 2 1 m
R   Rn i
m i 1
A measure of divergence from average
m-1 because there are m-1 returns

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).
Lecture 4

3. Garch (1,1) approximation
In GARCH (1,1) we assign some weight to the long-run average
variance rate

s  gVL + aR
2
n
2
n 1 + bs 2
n 1

Since weights must sum to 1
g + a + b 1

Setting w  gV, the GARCH (1,1) model

s n  w + aRn 1 + bs n 1
2 2 2
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w
And: VL 
1 a  b

p q
s  w + ai R
2
n
2
n i +  b js 2
n j
i 1 j 1

substitute for s n -1 in GARCH(1,1)
2

s n 1  w + aRn21 + b (w + aRn2 2 + bs n  2 )
2 2

 w + bw + aRn 1 + ab Rn  2 + b 2s n  2
2 2 2

substitute for s n -2
2

s n  2  w + bw + b 2w + aRn21 + ab Rn2 2 + ab 2 Rn23 + b 3s n 3
2 2

The weights decline exponentially at rate β
Lecture 4

Tomorrow’s variance is a weighted average of the long run variance!!


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 VL

dV  a(VL  V )dt + Vdz

where time measured in days,a  1 - a - b ,   a 2
Lecture 4

14

s n  gVL + aRn1 + bs n 1 Volatility of past periods (lagged variance)
2 2 2

Long Run Variance Returns of past periods
Long Term Variance (Av. Variance of all) 0,050163%
Gamma 0,2
Alpha 0,3
Beta 0,5
Weights of each factor Garch 0,0275%
𝑅𝑖 From here we 1st: (1-lambda)
= 𝑖 ca n ca lculate 2nd: last*lamb = 𝑅 𝑖
𝑖 s i mple va riance

Yesterday
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%
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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
To weighted returns : recent past 0,014942%
will affect more!

4. Exponentially-Weighted Moving Average
4. EWMA approximation

Garch (1,1) s  w + aR
2
n
2
n 1 + bs 2
n 1
= 0 and ( 𝛼 + 𝛽 ) =1:

The equation simplifies to

s  (1  b ) R
2
n
2
n 1 + bs 2
n 1

This is now equivalent to the formula for exponentially weighted
moving average (EWMA):

EWMA  s  (1   ) R
2 2
n 1+ s
2
n 1
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n
In EWMA, the lambda parameter now determines the ―decay:‖ a
lambda that is close to one (high lambda) exhibits slow decay.

3. EWMA approach
EWMA  s n  s n 1 + (1   ) Rn 1
2 2 2

replace s n 1
2

s  [s
2
n
2
n2 + (1   ) R 2
n2 ] + (1   ) R 2
n 1

 (1   )( R 2
n 1 + R
2
n2 )+ s
2 2
n2

substitute s 2

n2

s  (1   )( R
2
n
2
n 1 + R 2
n2 + R
2 2
n 3 )+s3 2
n 3

continuing the way gives
Lecture 4

m
s n  (1   ) i 1 Rn2i + ms n  m
2 2

i 1

3. EWMA approach
For the large m, ms n  m is sufficiently small
2

the equation is same as the quation of weight scheme
where a i (1   )i 1 or a i +1  a i

Advantages
• 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
Lecture 4

Example EWMA Calculation
𝑅𝑖 From here we 1st: (1-lambda)
= 𝑖 ca n ca lculate 2nd: last*lamb = 𝑅 𝑖
𝑖 s i mple va riance

Today Yesterday
Weights of sq.
Date Close Daily Return Return to 2 Weights Weights
Returns
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
To weighted returns : recent past Volatility 0,016037% Volatility 0,014942%
will affect more! in a exponentially Is the summatory of weighted
declining fashion squared returns
--Proportional decay
-- = We do not have to calculate the complete series

EWMA  s n  (1   ) Rn1 + s n1
2 2 2
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0,016037%

Lambda
0,94

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Example

3. Risk Metrics 5. 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)
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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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Lecture 4
Part 2. Volatility models 6. Some important details

Part 3

VALUE AT RISK
MODELS

a. Overview
b. Initial considerations
c. Var Models (intro)
d. VaR Historical
e. Parametric Approach
f. Monte Carlo Approach
g. Basel 2
24

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.
Part 3. VaR Methodologies

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.
Lecture 4

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
Lecture 4

cannot be captured by the VaR methods.

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.

VaR
Conditional VaR.
Conditional VaR
The potential loss when the
portfolio is hit beyond VaR
Lecture 4

In JPM case it is $116.000

VaR: additional considerations
Standard Deviation approach

𝑉𝑎𝑅 = 𝑊 𝜎 𝛼

Portfolio: $1.000.000
= $48.404
SD: $23.300
Prob: Normal distribution. 1.65

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
Lecture 4

𝑥𝑖
𝑀𝑎𝑥𝐷𝑟𝑎 𝐵𝑎𝑐𝑘 = −
𝑀𝑎𝑥 𝑥 𝑟𝑜𝑙𝑙𝑖 𝑔 𝑖 𝑑𝑜

Maximum drawback

70% 45%

Main limitation: not comparable among portfolios

There is no one VaR number for a single portfolio, because different
methodologies used for calculating VaR produce different results.
Lecture 4

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)

Short time Long time
To check a specific portfolio To avoid bankruptcy
Lecture 4

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

3.VaR Models

Forecasts n
paths and find
the VaR

Order numbers

and obtain
quintiles and
using history,
losses could be…
Lecture 4

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.
Lecture 4


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.

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
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4. We can construct a frequency distribution of daily returns
5. We can calculate our value at risk

VaR

Deviation from the average
return

With 99% prob, With 99% prob,
the loss won’t the loss won’t
be higher than be higher than
$75.000 per M $75.000 per M
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I have N observations, and it is easy to find what observation is 1%

Advantages
• 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
Lecture 4

significantly.

5. Parametric approach

Variance – Covariance 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
Lecture 4

Using them it could be possible to plot a normal distribution curve.

We use the familiar curve instead of actual data

5.Variance – Covariance approach
The problem

• Var-Cov VaR methodology is based on

𝑉𝑎𝑅 𝑝 = 𝛼 𝜎 𝑃 𝑊
Wealth
Volatility (sd) of the portfolio

Confidence level (normal equivalent)

𝑉𝑎𝑅 𝑝 = 𝛼 𝑋′ 𝑋
Variance-Covariance matrix showing wealth
Lecture 4

$ Positions x VaR-CoV x $ Positions

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

Variance Covariance VaR
Goog Nok
Portfolio Value $ 100,00
Weights 1/3 2/3
Stock worths $ 33,33 $ 66,67
Volatility 1,703% 3,879%
Correlation 0,26109

Var-Cov Matrix
Goog Nok X' Var-Cov Matrix X

Goog 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
1X 2 2X 2 2x 1
Variance = Covariance:
Volatility * Volatility
0,02116696 7,778
σxy=ρxy σx σy 0,10608465

Variance $ 7,78
Volatility $ 2,79
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Confidence 95%
Critical value 1,645

VaR $ 4,59


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.
Lecture 4

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
Lecture 4

• Allows the risk manager to use actual historical distributions for risk
factor returns rather than having to assume normal returns.

6. Monte-Carlo Simulation approach
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
Lecture 4

Theory
Lecture 4

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
Lecture 4

The price of a stock is lognormally distributed


S Very short period of time
 t + s t
S
Certain component Uncertain component

The return is uncertain or random
Stochastic component
Deterministic component (drift)

• ε is the ranom component of the
• μ is the expected rate of return
standard normal distribution (mean
• If the price of the stock today is S0,
0 and sd of 1).
then its price ST at time T in the
• σ is the volatility
future would be:
• The longer the time interval, the
ST=S0 e(μT)
more variable the return
Lecture 4

S
 N ( t , s t )
S

S
 t + s t
S
• S=$10
• μ=12% per year
• σ=40% per year
• t= 1 day, that is 0.004 of a year
What this
S number
 0.12 * 0.004 + 0.4 * 0.8 * 0.0632  2.07% means ?

BUT: S
Draws for e will be sometimes negative, the proportional return can be
positive and negative
Lecture 4

Prices Log-normal distributed
The natural log of S are normally distributed

 S   s2  
ln   N   
 T , s T 
 S   2  
The price path will be
 S t + t   s2 
ln    
  t + s t
 S 2 

 t   
 s2  
St + t  St exp  
 t + s t 
 2  
Lecture 4

Prices Log-normal distributed
Apply my formula
Expected return (yearly) 20%
Daily return 0,08%
Volatility (yearly) 40% Price
Daily volatility 2,52% Day Random Uniform Normal
Count 10
Time t (in days) 1 1 0,28878 -0,557 9,65851
Stock price $ 10 N(0,1). Exp value of 0 and sd of 2 0,53095 0,07766 9,71073
1 3 0,70087 0,52692 10,0446
s 2
(  ) (yearly) 12,000% Standard normal cumulative 4 0,44846 -0,1296 9,96742
2 distribution. Value btw -3 and 3. 5 0,15337 -1,0221 9,34798
6 0,16773 -0,9632 8,79975

To randomize my volatility
7 0,47247 -0,0691 8,7656
8 0,2168 -0,783 8,34607
9 0,39029 -0,2786 8,20426
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Other models (interest rates)
Lecture 4

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
Lecture 4

• 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
Lecture 4

Independent review of risk measurement systems by internal audit

Board and senior management should be actively involved

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)

MRCt = Max (Avg VaR over 60 days, VaR t-1) + SRC
SRC – Specific Risk Charge
60
Lecture 4

𝑀𝑅𝐶 𝐼𝑀𝐴 = 𝑀𝑎𝑥 𝑘 𝑉𝑎𝑅 𝑡 𝑖 , 𝑉𝑎𝑅 𝑡 + 𝑅𝐶
60
𝑖=

K>3 is a multiplier

Lecture 4

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