Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
1
Historical Simulation,
Value-at-Risk,
and Expected Shortfall
Elements of
Financial Risk Management
Chapter 2
Peter Christoffersen

Overview
Objectives
• Introduce the most commonly used method for
computing VaR, namely Historical Simulation
and discuss the pros and cons of this method.
• Discuss the pros and cons of the V aR risk measure
• Consider the Expected Shortfall, ES, alternative.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Chapter is organized as follows:
• Introduction of the historical simulation (HS)
method and its pros and cons.
• Introduction of the weighted historical simulation
(WHS). We then compare HS and WHS during
the 1987 crash.
• Comparison of the performance of HS and
RiskMetrics during the 2008-2009 financial crisis.
• Then we simulate artificial return data and assess
the HS VaR on this data.
• Compare the VaR risk measure with ES.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Defining Historical Simulation
• Let today be day t. Consider a portfolio of n
assets. If we today own Ni,t units or shares of asset
i then the value of the portfolio today is
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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• We use today’s portfolio holdings but historical
asset prices to compute yesterday’s pseudo portfolio
value as

Defining Historical Simulation
• This is a pseudo value because the units of each
asset held typically changes over time. The
pseudo log return can now be defined as
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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• Consider the availability of a past sequence of m
daily hypothetical portfolio returns, calculated
using past prices of the underlying assets of the
portfolio, but using today’s portfolio weights, call
it {RPF,t+1-t}m
t=1

Defining Historical Simulation
• Distribution of RPF,t+1 is captured by the histogram of
{RPF,t+1-t}m
t=1
• The VaR with coverage rate, p is calculated as 100pth
percentile of the sequence of past portfolio returns.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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• Sort the returns in {RPF,t+1-t}m
t=1 in ascending order
• Choose VaRP
t+1 such that only 100p% of the
observations are smaller than the VaRP
t+1
• Use linear interpolation to calculate the exact VaR
number.

Pros and Cons of HS
Pros
• the ease with which it is implemented.
• its model-free nature.
Cons
• It is very easy to implement. No numerical
optimization has to be performed.
• It is model-free. It does not rely on any particular
parametric model such as a RiskMetrics model.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Issues with model free nature of HS
How large should m be?
• If m is too large, then the most recent observations
will carry very little weight, and the VaR will tend to
look very smooth over time.
• If m is too small, then the sample may not include
enough large losses to enable the risk manager to
calculate VaR with any precision.
• To calculate 1% VaRs with any degree of precision
for the next 10 days, HS technique needs a large m
value
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Figure 2.1:
VaRs from HS with 250 and 1,000 Return Days
Jul 1, 2008 - Dec 31, 2010
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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0.0200
0.0300
0.0400
0.0500
0.0600
0.0700
0.0800
0.0900
Jun-08 Jul-08 Sep-08 Oct-08 Dec-08 Feb-09 Mar-09 May-09 Jul-09 Aug-09 Oct-09 Dec-09 Jan-10
Historical
Simulation
VaR
HS-VaR(250)
HS-VaR(1000)

Weighted Historical Simulation
• WHS relieves the tension in the choice of m
• It assigns relatively more weight to the most recent
observations and relatively less weight to the returns further
in the past
• It is implemented as follows –
• Sample of m past hypothetical returns, {RPF,t+1-t}m
t=1
is assigned probability weights declining exponentially
through the past as follows
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Weighted Historical Simulation
– Today’s observation is assigned the weight 1 =
(1- ) / (1- m)
– t goes to zero as t gets large, and that the weights
t for t = 1,2,...,m sum to 1
– Typical value for  is between 0.95 and 0.99
• The observations along with their assigned weights
are sorted in ascending order.
• The 100p% VaR is calculated by accumulating the
weights of the ascending returns until 100p% is
reached.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Pros and Cons of WHS
Pros
• Once  is chosen, WHS does not require
estimation and becomes easy to implement
• It’s weighting function builds dynamics into the
WHS technique
• The weighting function also makes the choice of
m somewhat less crucial.
• WHS responds quickly to large losses
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Pros and Cons of WHS
Cons
• No guidance is given on how to choose η
• Effect on the weighting scheme of positive versus
negative past returns
• If we are short the market, a market crash has no
impact on our VaR. WHS does not respond to
large gains
• the multiday VaR requires a large amount of past
daily return data, which is not easy to obtain.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Advantages of Risk Metrics model
• It can pick up the increase in market variance from
the crash regardless of whether the crash meant a
gain or a loss
• In this model, returns are squared and losses and
gains are treated as having the same impact on
tomorrow’s variance and therefore on the portfolio
risk.
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Figure 2.2 A:
Historical Simulation VaR and Daily Losses from
Long S&P500 Position, October 1987
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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-15%
-10%
-5%
0%
5%
10%
15%
20%
25%
01-Oct 06-Oct 11-Oct 16-Oct 21-Oct 26-Oct 31-Oct
HS
VaR
and
Loss
Loss Date
Return VaR (HS)

Figure 2.2 B:
Historical Simulation VaR and Daily Losses from
Short S&P500 Position, October 1987
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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-15%
-10%
-5%
0%
5%
10%
15%
20%
25%
01-Oct 06-Oct 11-Oct 16-Oct 21-Oct 26-Oct 31-Oct
HS
VaR
and
Loss
Loss Date
Return VaR (WHS)

Figure 2.3 A:
Historical Simulation VaR and Daily Losses from
Short S&P500 Position, October 1987
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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-25%
-20%
-15%
-10%
-5%
0%
5%
10%
15%
01-Oct 06-Oct 11-Oct 16-Oct 21-Oct 26-Oct 31-Oct
HS
VaR
and
Loss
Loss Date
Return VaR (HS)

Figure 2.3 B:
Weighted Historical Simulation VaR and Daily Losses
from Short S&P500 Position, October 1987
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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-25%
-20%
-15%
-10%
-5%
0%
5%
10%
15%
01-Oct 06-Oct 11-Oct 16-Oct 21-Oct 26-Oct 31-Oct
WHS
VaR
and
Loss
Loss Date
Return VaR (WHS)

Evidence from the 2008-2009 Crisis
• We consider the daily closing prices for a total
return index of the S&P 500 starting in July 2008
and ending in December 2009.
• The index lost almost half its value between July
2008 and the market bottom in March 2009.
• The recovery in the index starting in March 2009
continued through the end of 2009.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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1,000
1,200
1,400
1,600
1,800
2,000
2,200
Jul-08 Sep-08 Nov-08 Jan-09 Mar-09 May-09 Jul-09 Sep-09 Nov-09
S&P500-Closing
Price
S&P500
Figure 2.4: S&P 500 Total Return Index:
2008-2009 Crisis Period

Evidence from the 2008-2009 Crisis
• The 10-day 1% HS VaR is computed from the 1-day
VaR by simply multiplying it by
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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• Alternative to HS is the RiskMetrics variance
model
• 10-day, 1% VaR computed from the Risk- Metrics
model is as follows:

Evidence from the 2008-2009 Crisis
• where the variance dynamics are driven by
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Difference between the HS and the RM VaRs
• The HS VaR rises much more slowly as the crisis
gets underway in the fall of 2008
• The HS VaR stays at its highest point for almost a
year during which the volatility in the market has
declined considerably
• HS VaR will detect the brewing crisis quite slowly
and will enforce excessive caution after volatility
drops in the market

Figure 2.5: 10-day, 1% VaR from Historical
Simulation and RiskMetrics During the
2008-2009 Crisis Period
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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0%
5%
10%
15%
20%
25%
30%
35%
40%
Jul-08 Aug-08 Sep-08 Oct-08 Nov-08Dec-08 Jan-09 Feb-09Mar-09 Apr-09May-09 Jun-09 Jul-09 Aug-09 Sep-09 Oct-09 Nov-09Dec-09
Value
at
Risk
(VaR)
RiskMetrics HS

Evidence from the 2008-2009 Crisis
• The units in figure above refer to the least percent of
capital that would be lost over the next 10 days in
the 1% worst outcomes.
• Let’s put some dollar figures on this effect
• Assume that each day a trader has a 10-day, 1%
dollar VaR limit of $100,000
• Thus each day he is therefore allowed to invest
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Evidence from the 2008-2009 Crisis
• Let’s assume that the trader each day invests the
maximum amount possible in the S&P 500
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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• The daily P/L is computed as

Figure 2.6: Cumulative P/L from Traders with HS
and RM VaRs
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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-400,000
-350,000
-300,000
-250,000
-200,000
-150,000
-100,000
-50,000
0
50,000
100,000
150,000
Jul-08 Aug-08 Sep-08 Oct-08 Nov-08 Dec-08 Feb-09 Mar-09 Apr-09 May-09 Jun-09 Jul-09 Aug-09 Oct-09 Nov-09 Dec-09
Cumulati
ve
P/L
P/L from RM VaR
P/L from HS VaR

Evidence from the 2008-2009 Crisis
Performance difference between HS and RM VaRs
• The RM trader will lose less in the fall of 2008 and
earn much more in 2009.
• The HS trader takes more losses in the fall of 2008
and is not allowed to invest sufficiently in the market
in 2009
• The HS VaR reacts too slowly to increases in
volatility as well as to decreases in volatility.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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The Probability of Breaching the HS VaR
• Assume that the S&P 500 market returns are
generated by a time series process with dynamic
volatility and normal innovations
• Assume that innovation to S&P 500 returns each
day is drawn from the normal distribution with
mean zero and variance equal to
• We can write:
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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The Probability of Breaching the HS VaR
• Simulate 1,250 return observations from above equation
• Starting on day 251, compute each day the 1-day, 1%
VaR using Historical Simulation
• Compute the true probability that we will observe a loss
larger than the HS VaR we have computed
• This is the probability of a VaR breach
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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Figure 2.7: Actual Probability of Loosing More than
the 1% HS VaR When Returns Have Dynamics
Variance
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
1 101 201 301 401 501 601 701 801 901
Probability
of
VaR
Breach

The Probability of Breaching the HS VaR
• where is the cumulative density function for a
standard normal random variable.
• If the HS VaR model had been accurate then this
plot should show a roughly flat line at 1%
• Here we see numbers as high as 16% and numbers
very close to 0%
• The HS VaR will overestimate risk when true
market volatility is low, which will generate a low
probability of a VaR breach
• HS will underestimate risk when true volatility is
high in which case the VaR breach volatility will be
high
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VaR with Extreme Coverage Rates
• The tail of the portfolio return distribution conveys
information about the future losses.
• Reporting the entire tail of the return distribution
corresponds to reporting VaRs for many different
coverage rates
• Here p ranges from 0.01% to 2.5% in increments
• When using HS with a 250-day sample it is not
possible to compute the VaR when
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Figure 2.8: Relative Difference between Non-
Normal (Excess Kurtosis=3) and Normal VaR
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0
10
20
30
40
50
60
70
80
0 0.005 0.01 0.015 0.02 0.025
Relative
VaR
Difference
VaR Coverage Rate, p

VaR with Extreme Coverage Rates
• Note that (from the above figure) as p gets close to
zero the nonnormal VaR gets much larger than the
normal VaR
• When p = 0.025 there is almost no difference
between the two VaRs even though the underlying
distributions are different
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Expected Shortfall
• VaR is concerned only with the percentage of losses that
exceed the VaR and not the magnitude of these losses.
• Expected Shortfall (ES), or TailVaR accounts for the
magnitude of large losses as well as their probability of
occurring
• Mathematically ES is defined as
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Expected Shortfall
• The negative signs in front of the expectation and the
VaR are needed because the ES and the VaR are
defined as positive numbers
• The ES tells us the expected value of tomorrow’s
loss, conditional on it being worse than the VaR
• The Expected Shortfall computes the average of the
tail outcomes weighted by their probabilities
• ES tells us the expected loss given that we actually
get a loss from the 1% tail
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Expected Shortfall
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• To compute ES we need the distribution of a normal
variable conditional on it being below the VaR
• The truncated standard normal distribution is defined
from the standard normal distribution as

Expected Shortfall
• () denotes the density function and () the
cumulative density function of the standard
normal distribution
• In the normal distribution case ES can be derived
as
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Expected Shortfall
• In the normal case we know that
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• The relative difference between ES and VaR is
• Thus, we have

Expected Shortfall
• For example, when p =0.01, we have
and the relative difference is then
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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• In the normal case, as p gets close to zero, the ratio of
the ES to the VaR goes to 1
• From the below figure, the blue line shows that when
excess kurtosis is zero, the relative difference between
the ES and VaR is 15%

Expected Shortfall
• The blue line also shows that for moderately large
values of excess kurtosis, the relative difference
between ES and VaR is above 30%
• From the figure, the relative difference between VaR
and ES is larger when p is larger and thus further
from zero
• When p is close to zero VaR and ES will both
capture the fat tails in the distribution
• When p is far from zero, only the ES will capture the
fat tails in the return distribution
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Figure 2.9: ES vs VaR as a Function of Kurtosis
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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0
5
10
15
20
25
30
35
40
45
50
0 1 2 3 4 5 6
(ES
-
VaR
)
/
ES
Excess Kurtosis
p=1% p=5%

Summary
• VaR is the most popular risk measure in use
• HS is the most often used methodology to
compute VaR
• VaR has some shortcomings and using HS to
compute VaR has serious problems as well
• We need to use risk measures that capture the
degree of fatness in the tail of the return
distribution
• We need risk models that properly account for the
dynamics in variance and models that can be used
across different return horizons
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
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