Transcript CHAPTER 10

CHAPTER 10
Overcoming VaR's Limitations
INTRODUCTION
• While VaR is the single best way to
measure risk, it does have several
limitations. The most pressing limitations
are the following:
– It assumes that the variances and correlations
between the market-risk factors are stable.
– It does not give a good description of extreme
losses beyond the 99% level.
– It does not account for the additional danger
of holding instruments that are illiquid
INTRODUCTION
• One approach to addressing VaR's limitations is
to measure the risk both with VaR and with other,
completely different methods, such as stress
and scenario testing, as discussed in Chapter 5
• In this chapter, we discuss approaches that can
be used to augment the standard VaR methods
• These methods allow the risk manager to
improve the measurement of VaR, thereby
improving the ability to set capital, measure
performance, and identify excessive risks
INTRODUCTION
• This chapter has three sections
– An approach to letting variances change over
time
– several approaches for assessing extreme
events
– several approaches to quantify liquidity risk.
ALLOWING VARIANCE TO
CHANGE OVER TIME
• The usual approach to constructing the
covariance matrix is to calculate the
variance of the risk factors over the last
few months
• Then we assume that tomorrow's changes
in the risk factors will come from a
distribution that has the same variance as
experienced historically
ALLOWING VARIANCE TO
CHANGE OVER TIME
• We could write this as an equation by saying
that the expected variance tomorrow is the
variance of changes in the factor over the last
few months:
the change of risk factor
on day t
We assume that the mean change is
relatively small and therefore neglect
it from the equation
ALLOWING VARIANCE TO
CHANGE OVER TIME
• Although this approach is simple and
robust
• it is well known by practitioners that the
volatility of the market changes over time
• Sometimes the market is relatively calm,
then a crisis will happen, and the volatility
will jump up
ALLOWING VARIANCE TO
CHANGE OVER TIME
• GARCH is an approach that allows the
estimation of variance to vary quickly with recent
market moves
• GARCH stands for Generalized Autoregressive
Conditional Heteroskedasticity
• The variance on one day is a function of the
variance on the previous day
ALLOWING VARIANCE TO
CHANGE OVER TIME
• GARCH assumes that the variance is
equal to a constant plus a portion of the
previous day's change in the risk factor,
plus a portion of the previous day's
estimated variance
ALLOWING VARIANCE TO
CHANGE OVER TIME
• We can also use GARCH to estimate
the covariance between two factors, x
and y:
APPROACHES FOR ASSESSING
EXTREME EVENTS
• The usual implementations of Parametric and Monte
Carlo VaR assume that the risk factors have a Normal
probability distribution.
• As discussed in the statistics chapter, most markets,
especially poorly developed markets, exhibit many more
extreme movements than would be predicted by a
Normal distribution with the same standard deviation
• The term used to describe probability distributions that
have a kurtosis greater than that of the Normal
distribution is leptokurtosis
• Leptokurtosis can be considered to be a measure of the
fatness of the tails of the distribution
• Measuring the effects of leptokurtosis is important
because risk factors with a high kurtosis pose greater
risks than factors with the same variance but a lower
kurtosis
(a)PDF of Dow Jones Index Return Shock: Linear Model
0.9
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-5 -4.7 -4.4 -4.1 -3.8 -3.5 -3.2 -2.9 -2.6 -2.3 -2 -1.7 -1.4 -1.1 -0.8 -0.5 -0.2 0.12 0.42 0.72 1.02 1.32 1.62 1.92 2.22 2.52 2.82 3.12 3.42 3.72 4.02 4.32 4.62 4.92
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-4.7 -4.4 -4.1 -3.8 -3.5 -3.2 -2.9 -2.6 -2.3
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APPROACHES FOR ASSESSING
EXTREME EVENTS
• We will describe four techniques that are
used to assess the additional risk caused
by leptokurtosis
– Jump Diffusion
– Historical Simulation
– Adjustments to Monte Carlo Simulation
– Extreme Value Theory
Jump Diffusion
• The jump-diffusion model assumes that
tomorrow's random change in the risk factor can
come from one of two Normal distributions
• One distribution describes the typical market
movements
• the other describes crisis movements. In
simplified form
• there is a probability of P that the sample will
come from the typical distribution and a small
probability of (1 — P) that it will come from the
crisis distribution
Jump Diffusion
•The main problem to this approach is that it is difficult to
determine the parameter values
•In the model above, five parameters must be determined
Jump Diffusion
• Normal +Normal=Normal?
• The difference between simple plus and
mixing
• The mixing distribution
– some observations from Dist. 1
– other observations from Dist. 2
Jump Diffusion
-----Distribution 2:
A high Volatility
Distribution
0 .0 2
_____Distribution
1:
A Low Volatility
Distribution
0.02
x21
0.01
x22
x23
…
0 .0 1
x11,x12,x13,x14,..
0 .0 0
0.00
-5
- 4 .2 - 3 .5 - 2 .7
-2
- 1 .2 - 0 .5 0 .2 7 1 .0 2 1 .7 7 2 .5 2 3 .2 7 4 .0 2 4 .7 7
5
4.2 3.4 2.6 1.8
1
0.2 0.62 1.42 2.22 3.02 3.82 4.62
0.02
x21
x22
x23
0.01
---- Distribution 2
___ Distribution 1
x11,x12, .……………… x13,x14
0.00
-5
-4.2 -3.5 -2.7
-2
-1.2 -0.5 0.27 1.02 1.77 2.52 3.27 4.02 4.77
Historical Simulation
• Historical simulation does not require an
assumption for the form of the probability
distribution
• It simply takes the price movements that have
occurred and uses them to revalue the portfolio
directly
• However, historical simulation is strongly
backward looking because the changes in the
risk factors are determined by the last crisis, not
the next crisis.
Adjustments to Monte Carlo
Simulation
• Usually, Monte Carlo simulation uses Normal
distributions
• However, it is also possible to carry out Monte
Carlo evaluation using leptokurtic distributions,
such as jump diffusion or the Student's T
distribution
• It is relatively easy to create such distributions
for single risk factors, but more difficult to ensure
that the correlations between the factors are
correct
Extreme Value Theory
• Extreme Value Theory (EVT) takes a different
approach to calculating VaR
• EVT concentrates on estimating the shape of
only the tail of a probability distribution
• Given this shape, we can find estimates for
losses associated with very small probabilities,
such as the 99.9% VaR
• A typical shape used is the Generalized Pareto
Distribution that has the following form:
Extreme Value Theory
• Here, a, b, and c are variables that are chosen
so the function fits the data in the tail
• The main problem with the approach is that it
is only easily applicable to single risk factors
• It is also, by definition, difficult to parameterize
because there are few observations of
extreme events
LIQUIDITY RISK
• The Importance of Measuring Liquidity Risk
– Liquidity risks can increase a bank's losses; therefore,
they should be included in the calculation of VaR and
economic capital
– There are two kinds of liquidity risk: liquidity risk in
trading, and liquidity risk in funding (also known as
funding risk)
– The funding risk is the possibility that the bank will run
out of liquid cash to pay its debts
– This funding risk is usually considered in the
framework for asset liability management and will be
discussed in later chapters
– This chapter discusses liquidity risk in trading
LIQUIDITY RISK
– The liquidity risk in trading is the risk that a
trader will be unable to quickly sell a security
at a fair price. This could happen if few people
normally trade the given security e.g., if it was
the equity for a small company
– It could also happen if the general market is in
crisis and few people are interested in buying
new securities
LIQUIDITY RISK
• We can view these two possible loss
mechanisms as two extreme manifestations of
the same problem
– In one extreme, the trader sells immediately at an
unusually low price
– In the other extreme, the trader slowly sells at the
current fair price, but risks suffering additional losses
• It is important to recognize the liquidity risk
because it can add significantly to losses
• Furthermore, if liquidity risk is not included in the
risk measurement, it gives incentives to traders
to buy illiquid securities
Quantifying Liquidity
• The close-out time
– is the time required to bring the position to a state
where the bank can make no further loss from the
position
– It is the time taken to either sell or hedge the
instrument
– The number of days can be based on the size of the
position held by the trader compared with the daily
traded volume
Quantifying Liquidity
• F is a factor that gives the percentage of the daily
volume that can be sold into the market without
significantly shifting the price
• If F were set equal to 10%, it would imply that 10% of
the daily volume can be sold each day without
significantly shifting the market
• The Daily Volume can be the average daily volume or
the volume in a crisis period
• The volume in a crisis period could be approximated as
the average volume minus a number of standard
deviations
Quantifying Liquidity
• Another alternative to quantifying the liquidity risk is to
measure the average bid-ask spread relative to the mid
price
• The bid is the price that investors are willing to bid (or
pay) to own the security.
• The ask is the price that owners of the security are
asking to sell the security
• The mid is halfway between the bid and ask
• If the bid and ask are close to the mid, it implies that
there are many market participants who agree on the fair
value of the security and are willing to trade close to that
price
Quantifying Liquidity
• If the bid-ask spread is wide, it means that few
investors are willing to buy the security at the
price the sellers think is fair
• If a trader wanted to sell the security immediately,
the trader would have to lower the ask price to
equal the bid rather than wait for some investor
to agree that the high ask price was fair
• Both the close-out time and the bid-ask spread
can be used to quantify liquidity risk
• We will explore how in the following sections.
Using Close-Out Time to Quantify
Liquidity Risk
• The most common approach to assessing the
liquidity risk is to use the "square-root-of-T”
adjustment for VaR
• This is also known as "close-out-adjusted VaR“
• The result of the approach is that the VaR for a
position that takes T days to close is assumed to
equal the VaR for an equivalent liquid position
that could be closed out in one day times the
square root of T
Using Close-Out Time to Quantify
Liquidity Risk
• The approach assumes that the position will
be held for T days
• then on the last day, it will be sold completely
• It uses the reasoning that the losses over T
days will be the sum of losses over the
individual days
is the cumulative loss over T days
is the loss on day 2
Using Close-Out Time to Quantify
Liquidity Risk
• We can assume with reasonable accuracy that
losses are independent and identically
distributed (IID)
• meaning losses are not correlated day to day,
and the standard deviation of losses is the
same each day
• The variance of the loss over T days is
therefore the sum of the variance of the losses
on the individual days
Using Close-Out Time to Quantify
Liquidity Risk
• If we assume that the variance of the losses on
each day is the same, then the sum equals T
times the variance on the first day
Using Close-Out Time to Quantify
Liquidity Risk
• A slightly refined approach is to assume that the
position is closed out linearly over T days
• In this case, the variance of the loss decreases
linearly each day
Using Close-Out Time to Quantify
Liquidity Risk
Using Close-Out Time to Quantify
Liquidity Risk
• To illustrate the difference between this
and the simple square-root-of-T
adjustment
• consider a closeout period of 10 days
• The square-root-of-T method gives:
Using Close-Out Time to Quantify
Liquidity Risk
• Whereas the linear close-out gives a measure of
VaR that is significantly smaller
Using the Bid-Ask Spread to
Assess Liquidity Risk
• The closeout adjustments discussed above
assumed that the trader was taking one extreme
course of action by gradually closing out the
position at the mid price and refusing to give any
discount
• The other extreme is to assume that the trader
will sell out immediately by giving a discount that
brings the price down to the bid price. This
discount is an additional loss
• Please refer to Page 163