CHAPTER 3
1
THE THREE COMMON
APPROACHES FOR
CALCULATING VALUE AT
RISK
INTRODUCTION
2
 VaR is a good measure of risk
 To estimate the value's probability distribution, we
use two sets of information
the current position, or holdings, in the bank's trading
portfolio
 an estimate of the probability distribution of the price
changes over the next day.

INTRODUCTION
3
 The estimate of the probability distribution of the
price changes is based on the distribution of price
changes over the last few weeks or months.
 The goal of this chapter is to explain how to
calculate VaR using the three methods that are in
common use:
Parametric VaR
 Historical Simulation
 Monte Carlo Simulation.

LIMITATIONS SHARED BY ALL
THREE METHODS
4
 It is important to note that while the three calculation
methods differ, they do share common attributes and
limitations.
 Each approach uses market-risk factors

Risk factors are fundamental market rates that can be derived
from the prices of securities being traded

Typically, the main risk factors used are interest rates, foreign
exchange rates, equity indices, commodity prices, forward
prices, and implied volatilities

By observing this small number of risk factors, we are able to
calculate the price of all the thousands of different securities
held by the bank
LIMITATIONS SHARED BY ALL
THREE METHODS
5
 Each approach uses the distribution of historical
price changes to estimate the probability
distributions.

This requires a choice of historical horizon for the market
data

how far back should we go in using historical data to
calculate standard deviations?

This is a trade-off between having large amounts of
information or fresh information
LIMITATIONS SHARED BY ALL
THREE METHODS
6
Because VaR attempts to predict the future probability
distribution, it should use the latest market data with the
latest market structure and sentiment
 However, with a limited amount of data, the estimates
become less accurate
 There is less chance of having data that contains those
extreme, rare market movements which are the ones that
cause the greatest losses

LIMITATIONS SHARED BY ALL
THREE METHODS
7
 Each approach has the disadvantage of assuming
that past relationships between the risk factors will
be repeated

it assumes that factors that have tended to move together
in the past will move together in the future
LIMITATIONS SHARED BY ALL
THREE METHODS
8
 Each approach has strengths and weaknesses
when compared to the others, as summarized in
Figure 6-1
The degree to which the circles are shaded corresponds
to the strength of the approach
 The factors evaluated in the table are

the speed of computation
 the ability to capture nonlinearity



the ability to capture non-Normality


Nonlinearity refers, to the price change not being at linear function
of the change in the risk factors. This is especially important for
options
non-Normality refers to the ability to calculate the potential changes
in risk factors without assuming that they have a Normal
distribution
the independence from historical data
LIMITATIONS SHARED BY ALL
THREE METHODS
9
PARAMETRIC VAR
10
 Parametric VaR is also known as Linear VaR,
Variance-Covariance VaR
 The approach is parametric in that it assumes that
the probability distribution is Normal and then
requires calculation of the variance and covariance
parameters.
 The approach is linear in that changes in
instrument values are assumed to be linear with
respect to changes in risk factors.
 For example, for bonds the sensitivity is described
by duration, and for options it is described by the
Greeks
PARAMETRIC VAR
11
 The overall Parametric VaR approach is as follows:





Define the set of risk factors that will be sufficient to
calculate the value of the bank's portfolio
Find the sensitivity of each instrument in the portfolio to
each risk factor
Get historical data on the risk factors to calculate the
standard deviation of the changes and the correlations
between them
Estimate the standard deviation of the value of the portfolio
by multiplying the sensitivities by the standard deviations,
taking into account all correlations
Finally, assume that the loss distribution is Normally
distributed, and therefore approximate the 99% VaR as 2.32
times the standard deviation of the value of the portfolio
PARAMETRIC VAR
12
 Parametric VaR has two advantages:
It is typically 100 to 1000 times faster to calculate
Parametric VaR compared with Monte Carlo or Historical
Simulation.
 Parametric VaR allows the calculation of VaR
contribution, as explained in the next chapter.

PARAMETRIC VAR
13
 Parametric VaR also has significant limitations:
 It gives a poor description of nonlinear risks
 It gives a poor description of extreme tail events, such as
crises, because it assumes that the risk factors have a
Normal distribution. In reality, as we found in the
statistics chapter, the risk-factor distributions have a high
kurtosis with more extreme events than would be
predicted by the Normal distribution.
 Parametric VaR uses a covariance matrix, and this
implicitly assumes that the correlations between risk
factors is stable and constant over time
How to calculate volatility of each asset?
 JP Morgan's RiskMetrics system
 Equally-Weighted Moving Average (that is Simple Moving
Average (SMA); standard deviation)
2
(
R

R
)
̂ t2   t
T 1
i 1
 Exponentially-Weighted Moving Average (EWMA)
T
ˆ t2  ˆ t21  (1   ) Rt2
 λ: decay rate, 0<λ<1. The more the λ value, the less last
observation affects the current dispersion estimation.
 The formula of the EWMA model can be rearranged to the

following form:
2
t 1 2
ˆ t  (1   )  Rt i
i 0
14
How to calculate volatility of each asset?
15
 The EWMA model has an advantage in comparison with
SMA, because the EWMA has a memory.
 Using the EWMA allows one to capture the dynamic
features of volatility. This model uses the latest
observations with the highest weights in the volatility
estimate. However, SMA has the same weights for any
observation.
 JP Morgan suggests: the optimal value for current daily
dispersion (volatility) is =0.94; the optimal value for
current monthly dispersion (volatility) is =0.97
Example: One Asset
16
 The first example calculates the stand-alone VaR
for a bank holding a long position in an equity. The
stand-alone VaR is the VaR for the position on its
own without considering correlation and
diversification effects from other positions
 The present value of the position is simply the
number of shares (N) times the value per share,
(Vs)
 PV$ = N * Vs
Example: One Asset
17
 The change in the value of the position is simply the number of
shares multiplied by the change in the value of each share:
ΔPV$ = N * ΔVs

 The standard deviation of the value is the number of shares
multiplied by the standard deviation of the value of each share

σ v = N * σs
 we have assumed that the value changes are Normally
distributed, there will be a 1chance that the loss is more than 2.32
standard deviations; therefore, we can calculate the 99 VaR as
follows
VaR = 2.32 * N *σs
Example: Two Assets
 Portfolio: P=A1 (amount: w1)+A2 (amount: w2)
 Portfolio variance
 p2  (W11  W2 2 )2  (W1212  W22 22  2W1W2 121 2 )
Portfolio’s VaR
VaRp  Z * p
1
 Z * (W12 12  W22 22  2W1W2 12 1 2 ) 2
 (VaR12  VaR22  212VaR1VaR2 )
1
2
 VaR1 and VaR2 is the single asset’s VaR:
*
 VaR1= (-Z  1W1 )
and VaR2= (-Z* 2W2 )
18
Example: n Assets
 If the number of asset is n, the portfolio’s variance
 2p

  12

 W1 ,...,Wn  
 n1

 The portfolio VaR is
VaRp  Z * p 
  1n  W1 
 

      WW T
  n2  Wn 
1
T
Z * (W W ) 2
 ij   ij i i i, j  1,2,....., n
 Hence, the portfolio VaR depends on the each
asset’s variance and the correlation coefficient
between the returns of each asset.
19
Example
20
HISTORICAL-SIMULATION VAR
21
 Conceptually, historical simulation is the most simple VaR
technique, but it takes significantly more time to run than
parametric VaR.
 The historical-simulation approach takes the market data
for the last 250 days and calculates the percent change for
each risk factor on each day
 Each percentage change is then multiplied by today's
market values to present 250 scenarios for tomorrow's
values.
 For each of these scenarios, the portfolio is valued using full,
nonlinear pricing models. The third-worst day is the
selected as being the 99% VaR.
How to calculate historical simulation VaR
22
 Step 1: download the historical data of asset price




(last 250 days)
Step 2: calculate the change rate (returns) of asset
Step 3: sort the returns of asset from the lowest to
highest
Step 4: given the significant level to find the
historical simulation VaR
Ex: 1001 price observations (1000 returns
observations), significant level=1%, then
1000*1%=10. That is find the historical simulation
VaR is the tenth returns
HISTORICAL-SIMULATION VAR
23
 There are two main advantages of using historical
simulation:
It is easy to communicate the results throughout the
organization because the concepts are easily explained
 There is no need to assume that the changes in the risk
factors have a structured parametric probability
distribution
 no need to assume they are Normal with stable
correlation

HISTORICAL-SIMULATION VAR
24
 The disadvantages are due to using the historical
data in such a raw form:
The result is often dominated by a single, recent, specific
crisis, and it is very difficult to test other assumptions.
 The effect of this is that Historical VaR is strongly
backward-looking, meaning the bank is, in effect,
protecting itself from the last crisis, but not necessarily

preparing itself for the next
HISTORICAL-SIMULATION VAR
25
There can also be an unpleasant "window effect."
 When 250 days have passed since the crisis, the crisis
observation drops out of our window for historical data
and the reported VaR suddenly drops from one day to the
next.
 This often causes traders to mistrust the VaR because
they know there has been no significant change in the risk
of the trading operation, and yet the quantification of risk
has changed dramatically

MONTE CARLO SIMULATION VAR
26
 Monte Carlo simulation is also known as Monte
Carlo evaluation (MCE). It estimates VaR by
randomly creating many scenarios for future rates
using nonlinear pricing models to estimate the
change in value for each scenario, and then
calculating VaR according to the worst losses
MONTE CARLO SIMULATION VAR
27
 Required:
(1) for each risk factor, specification of a stochastic process
(i.e., distribution and parameters)


assumes that there is a known probability distribution for the risk
factors.
the choice of distributions and parameters such as risk and
correlations can be derived from historical data
(2) valuation models for all assets in the portfolio

fictitious price paths are simulated for all variables of interest.
MONTE CARLO SIMULATION VAR
28
 Monte Carlo simulation has two significant
advantages:
Unlike Parametric VaR, it uses full pricing models and
can therefore capture the effects of nonlinearities
 Unlike Historical VaR, it can generate an infinite number

of scenarios and therefore test many possible future outcomes
MONTE CARLO SIMULATION VAR
29
 Monte Carlo has two important disadvantages:
The calculation of Monte Carlo VaR can take 1000 times
longer than Parametric VaR because the potential price of
the portfolio has to be calculated thousands of times
 The computation speed will lower.
 Unlike Historical VaR, it typically requires the
assumption that the risk factors have a known probability
distribution .

MONTE CARLO SIMULATION VAR
30
 The Monte Carlo approach assumes that there is a
known probability distribution for the risk factors.
 The usual implementation of Monte Carlo assumes
a stable, Normal distribution for the risk factors.
 This is the same assumption used for Parametric
VaR.
 The analysis calculates the covariance matrix for
the risk factors in the same way as Parametric VaR
MONTE CARLO SIMULATION VAR
31
 But unlike Parametric VaR
 Decomposes the covariance matrix and ensures that the
risk factors are correlated in each scenario
 The scenarios start from today's market condition and go
one day forward to give possible values at the end of the
day
 Full, nonlinear pricing models are then used to value the
portfolio under each of the end-of-day scenarios.
 For bonds, nonlinear pricing means using the bondpricing formula rather than duration
 for options, it means using a pricing formula such as
Black-Scholes rather than just using the Greeks.