Transcript CHAPTER 6

CHAPTER 6
The Three Common Approaches
for Calculating Value at Risk
INTRODUCTION
• VaR is is a good measure of risk
– it combines information on the sensitivity of the value to changes
in market-risk factors with information on the probable amount of
change in those factors.
– VaR tries to answer the question: how much could we lose today
given our current position and the possible changes in the
market?
– VaR formalizes that question into the calculation of the level of
loss that is so bad that there is only a 1 in 100 chance of there
being a loss worse than the calculated VaR.
– VaR estimates this level by knowing the current value of the
portfolio and calculating the probability distribution of changes in
the value over the next trading day.
– From the probability distribution we can read the confidence
level for the 99-percentile loss.
INTRODUCTION
• To estimate the value's probability distribution, we use
two sets of information. the current position, or holdings,
in the bank's trading portfolio, and an estimate of the
probability distribution of the price changes over the next
day.
• 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 CarloSimulation.
LIMITATIONS SHARED BY ALL
THREEMETHODS
• 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
– This risk-factor approach uses less data than would be required
if we tried to collect historical price information for every security.
LIMITATIONS SHARED BY ALL
THREEMETHODS
• 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
THREEMETHODS
• 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
THREEMETHODS
• 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
THREEMETHODS
• 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
– Nonlinearity refers, to the price change not being at linear
function of the change in the risk factors. This is especially
important for options
• the ability to capture non-Normality
– 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
THREEMETHODS
PARAMETRIC VAR
• 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
• 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
• 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
• 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
PARAMETRIC VAR
• To give an intuitive understanding of Parametric VaR, we
have provided three worked-out examples.
• The examples are fundamentally quite simple, but they
intro-duce the method of calculating Parametric VaR.
• There are a lot of equations, but the underlying math is
mostly algebra rather than complex statistics or calculus
• Three different notations are used in this chapter
– Algebraic
– Summation
– matrix.
PARAMETRIC VAR
• If we have a portfolio of two instruments, the
loss on the portfolio (Lp) will be the sum of the
losses on each instrument:
• The standard deviation of loss on the
portfolio (σp) will be as follows:
Algebraic
Notation
PARAMETRIC VAR
Summation Notation
PARAMETRIC VAR
Matrix Notation
PARAMETRIC VAR
Example One
• 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 x Vs
Example One
• 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 x ΔVs
•
• The standard deviation of the value is the number of shares
multiplied by the standard deviation of the value of each share
•
•
•
σv = N x 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 calculatethe 99 VaR as follows
VaR = 2.32 x N xσs
Example Two
• As a slightly more complex example,
consider a government bond held by a U.K.
bank denominated in British pounds with a
single payment.
• The present value in pounds (PVp) is
simply the value of the cash flow in
pounds (Cp) at time t discounted
according to sterling interest rates for that
maturity, rp:
present value
Example Two
The derivative of
PVp with respect
to rp
Example Two
• To make this example more concrete,
consider a bond paying 100 pounds (Cp)
in 5 years' time (t), with the 5-year
discount rate at 6% (rp), and a standard
deviation in the rate of 0.5% (σr).
• The present value is then 74 pounds, the
sensitivity dr is -352 pounds per 100%
increase in rates, and the VaR is 4.1
pounds
Example Two
Example Three
• The two examples above were simple because
they had only one risk factor
• Now let us consider a multidimensional case: the
same simple bond as before, but now held by a
U.S. bank.
• The U.S. bank is exposed to two risks: changes
due to sterling interest rates and changes due to
the pound-dollar exchange rate.
• The value of the bond in dollars is the value in
pounds multiplied by the FX rate
Example Three
Example Three
Example Three
Example Three
Example Three
Example Three
Example Three
Example Three
Example Three
Example Three
Example Three
Homework
• Now let us consider a U.S. bond, but now held by a
Taiwan’s bank.
• Other conditions are consistent with the precious case.
• Consider this bond paying 100 U.S. dollars (CUSD) in 5
years' time (t)
• Two risks
– Changes due to U.S. interest rates (rUS)
– changes due to the NTD-USD exchange rate (FX)
• Find the current 5-year interest rate (rUS) and the current
NTD-USD exchange rate (FX)
• Find the standard deviation in the interest rate and the
exchange rate
• Find the correlation coefficient between rUS and FX
Using Parametric VaR to Calculate Risk
Sensitivity for Several Positions
• In the example above, we had one security that
was sensitive to two different risk factors.
• If the portfolio is made up of several securities,
each of which is affected by the same risk factor,
then the sensitivity of the portfolio to the risk
factor is simply the sum of these sensitivities for
the individual positions.
• For example, consider a portfolio holding our
example 100-pound five-year bond and 100
pounds of cash
Using Parametric VaR to Calculate Risk
Sensitivity for Several Positions
Using Parametric VaR to Calculate Risk
Sensitivity for Several Positions
Using Parametric VaR to Calculate Risk
Sensitivity for Several Positions
Using Parametric VaR to Calculate Risk
Sensitivity for Several Positions
Homework
• Now let us consider a bond portfolio with a U.S. bond
and a U.K. bond,
• The bond portfolio is held by a Taiwan’s bank.
• Other conditions are consistent with the precious case.
• Consider the US and UK bonds paying 100 US dollars
(CUSD) and 100 British pound (CBP), respectively, in 5
years' time (t)
• Four risk factors
–
–
–
–
Changes due to U.S. interest rates (rUS)
changes due to the NTD-USD exchange rate (FXUSD)
Changes due to U.K. interest rates (rUK)
changes due to the NTD-BP exchange rate (FXBP)
HISTORICAL-SIMULATION VAR
• 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.
HISTORICAL-SIMULATION VAR
• As an example, let's consider calculating
the VaR for a five-year, zero-coupon bond
paying $100
• We start by looking back at the previous
trading days and noting the five year rate
on each day.
• We then calculate the proportion by which
the rate changed from one day to the next
HISTORICAL-SIMULATION VAR
The change rate of
interest rate for one day
HISTORICAL-SIMULATION VAR
Homework
• Consider a Taiwan bond held by a Taiwan’s bank.
• Other conditions are consistent with the precious
case.
• Consider this bond paying 100 NT dollars (CNTD)
in 5 years' time (t)
• One risk factor
– Changes due to Taiwan interest rates (rTAIWAN)
• Use the historical simulation approach to
calculate the VaR
• Use one-year historical data at least
HISTORICAL-SIMULATION VAR
• 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 Joint-Normal with
stable correlation
HISTORICAL-SIMULATION VAR
• 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
– 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
• 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
• 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
• 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
– Unlike Historical VaR, it typically requires the
assumption that the risk factors have a
Normal or Log-Normal distribution.
MONTE CARLO SIMULATION
VAR
• 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, Joint-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
• 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.
MONTE CARLO SIMULATION
VAR
• From the scenarios, VaR is selected to be the 1percentile worst loss
• For example, if1000 scenarios were created, the
99% VaR would be the tenth-worst result
• Figure 6-4summarizes the Monte Carlo
approach
• Most of the Monte Carlo approach is
conceptually simple. The one mathematically
difficult step is to decompose the covariance
matrix in such a way as to allow us to create
random scenarios with the same correlation as
the historical market data
MONTE CARLO SIMULATION
VAR
MONTE CARLO SIMULATION
VAR
• For example, in the previous example of a Sterling bond
held by a U.S. bank, we assumed a correlation of -0.6
between the interest rate and exchange rate
• In other words, when the interest rate increases, we
would expect that the exchange rate would tend to
decrease.
• One way to think of this is that 60% of the change in the
exchange rate is driven by changes in the interest rate.
The other 40% is driven by independent, random factors.
• The trick is to create random scenarios that properly
capture such relationships
MONTE CARLO SIMULATION
VAR
• If we just have two factors, we can easily create
correlated random numbers in a simple way:
• Please refer to P120
MONTE CARLO SIMULATION
VAR
Example Three
Example Three
MONTE CARLO SIMULATION
VAR
• For the previous bond example, we would
create changes in the risk factors rp and FX by
using the following equations: EXCEL
Homework
• Find the realized data to redo the sample
example in the previous homework
• Three approaches are asked to apply
– Parametric VaR
– Historical VaR
– Monte Carlo VaR
• Compare the differences among them
Homework
• Now let us consider a U.S. bond, but now held
by a Taiwan’s bank.
• Other conditions are consistent with the precious
case.
• Consider this bond paying 100 U.S. dollars
(CUSD) in 5 years' time (t)
• Two risk factors
– Changes due to U.S. interest rates (rUS)
– changes due to the NTD-USD exchange rate (FX)
Quick Quiz
• Four popular limitations for VaR ?
– Calculation speed
– Non-normality
– Nonlinearity
– Too heavily dependent on historical data
• Three common approaches for VaR
– The main advantage/disadvantage for each of
them?
LIMITATIONS SHARED BY ALL
THREEMETHODS