Chapter 22
Value at Risk and Expected
Shortfall
Options, Futures, and Other Derivatives,
11th Edition, Copyright © John C. Hull 2021
1
The Question Being Asked in VaR
“What loss level is such that we are X%
confident it will not be exceeded in N
business days?”
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VaR vs. Expected Shortfall
VaR is the loss level that will not be exceeded
with a specified probability
Expected Shortfall (or C-VaR) is the expected
loss given that the loss is greater than the VaR
level
Although expected shortfall is theoretically
more appealing, it is VaR that is used by
regulators in setting bank capital requirements
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VaR and ES
VaR captures an important aspect of risk
in a single number
It is easy to understand
It asks the simple question: “How bad can
things get?”
ES answers the question: “If things do get
bad, just how bad will they be”
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Historical Simulation to Calculate the
One-Day VaR or ES
Create a database of the daily movements in
all market variables.
The first simulation trial assumes that the
percentage changes in all market variables
are as on the first day
The second simulation trial assumes that the
percentage changes in all market variables
are as on the second day
and so on
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Historical Simulation continued
Suppose we use 501 days of historical data
(Day 0 to Day 500)
Let vi be the value of a variable on day i
There are 500 simulation trials
The ith trial assumes that the value of the
market variable tomorrow is
vi
v500
vi −1
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Example : Calculation of 1-day, 99%
VaR or ES for a Portfolio on July 8,
2020 (Table 22.1)
Total Return Index
Value ($000s)
S&P 500
4,000
FTSE 100
3,000
CAC 40
1,000
Nikkei 225
2,000
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Total Return Indices After
Adjusting for Exchange Rates
(Table 22.2)
Day
Date
S&P 500
FTSE 100
CAC 40
Nikkei 225
0
May 9, 2018
5,292.90
8,830.23
16,910.33
322.40
1
May 10, 2018
5,343.70
8,926.56
16,915.41
321.24
2
May 11, 2018
5,354.69
8,982.76
17,065.64
326.20
3
May 14, 2018
5,359.66
8,999.31
17,121.67
328.03
…
……
…..
…..
……
……
499 July 7, 2020
6,445.59
7,269.36
15,784.97
345.40
500 July 8, 2020
6,496.14
7,255.04
15,540.44
342.01
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Scenarios Generated (Table 22.3)
5,343.70
6,496.14 ×
5,292.90
Scenario
DJIA
FTSE 100
CAC 40
Nikkei 225
Portfolio
Value ($000s)
Loss
($000s)
1
6,558.49
7,334.19
15,545.12
340.79
10,064.257
−64.257
2
6,509.50
7,300.72
15,678.46
347.28
10,066.822
−66.822
3
6,502.17
7,268.41
15,591.47
343.93
10,023.722
−23.722
…
…….
…….
…….
……..
…….
……..
499
6,425.90
7,293.40
15,543.89
341.21
9,968.126
31.874
500
6,547.09
7,240.75
15,299.71
338.66
9,990.361
9.639
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Ranked Losses (Table 22.4, page 499)
Scenario Number
Loss ($000s)
427
922.484
429
858.423
424
653.541
415
490.215
482
422.291
440
362.733
426
360.532
99% oneday VaR
99% one day ES is average of the five worst
losses or $669,391
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The N-day VaR or ES
The N-day VaR (ES) for market risk is usually
assumed to be N times the one-day VaR (ES)
In our example the 10-day VaR would be
calculated as 10 × $422,291 = $1,335,401.
This assumption is only perfectly theoretically
correct if daily changes are normally distributed
with zero mean and independent
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Stressed VaR and Stressed ES
Stressed VaR and stressed ES calculations
are based on historical data for a stressed
period in the past (e.g. the year 2008) rather
than on data from the most recent past (as in
our example)
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The Model-Building Approach
The main alternative to historical simulation is
to make assumptions about the probability
distributions of the return on the market
variables and calculate the probability
distribution of the change in the value of the
portfolio analytically
This is known as the model building approach
or the variance-covariance approach
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Daily Volatilities
In option pricing we measure volatility “per
year”
In VaR and ES calculations we measure
volatility “per day”
 y ear
 day =
252
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Daily Volatility continued
Theoretically day is the standard deviation of
the continuously compounded return in one
day
In practice we assume that it is the standard
deviation of the percentage change in one
day
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Microsoft Example
We have a position worth $10 million in
Microsoft shares
The volatility of Microsoft is 2% per day
(about 32% per year)
We use N=10 and X=99
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Microsoft Example continued
The standard deviation of the change in the
portfolio in 1 day is $200,000
Assume that the expected change is zero
(OK for short time periods) and the probability
distribution of the change is
The 1-day 99% VaR is
200,000  2.326 = $465,300
The 10-day 99% VaR is
10  465 ,300 = 1,471,300
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AT&T Example
Consider a position of $5 million in AT&T
The daily volatility of AT&T is 1% (approx
16% per year)
The 10-day 99% VaR is
10  2.326  50,000 = 367 ,800
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Portfolio
Now consider a portfolio consisting of both
Microsoft and AT&T
Assume that the returns of AT&T and
Microsoft are bivariate normal
Suppose that the correlation between the
returns is 0.3
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S.D. of Portfolio
A standard result in statistics states that
 X +Y =  2X + Y2 + 2 X  Y
In this case X = 200,000 and Y = 50,000 and
 = 0.3. The standard deviation of the change
in the portfolio value in one day is therefore
220,200
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VaR for Portfolio
The 10-day 99% VaR for the portfolio is
220,200  10  2.326 = $1,620,100
The benefits of diversification are
(1,471,300+367,800)–1,620,100=$219,00
What is the incremental effect of the AT&T
holding on VaR?
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ES for the Model Building Approach
(equation 22.1)
When the loss over the time horizon has a normal
distribution with mean  and standard deviation  the
ES is
−Y 2
2
e
ES =  + 
2 (1 − X )
where X is the confidence level and Y is the Xth
percentile of a standard normal distribution
For the Microsoft + AT&T portfolio, ES is $1,856,100
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The Linear Model
This assumes
• The daily change in the value of a portfolio is
linearly related to the daily returns from
market variables
• The returns from the market variables are
normally distributed
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Markowitz Result for Variance of
Return on Portfolio
Variance of Portfolio Return =
n
n
  w w  
ij
i
j
i
j
i =1 j =1
wi is weight of ith instrument in portfolio
σ i2 is variance of return on ith instrument
in portfolio
ρij is correlation between returns of ith
and jth instrument s
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VaR Result for Variance of Portfolio
Value (i = wiP) equation 22.3
n
P =   i xi
i =1
n
n
 =  ij  i  j i  j
2
P
i =1 j =1
or
n
 =   i2 i2 + 2 ij  i  j i  j
2
P
i =1
i j
i is the daily volatility of ith instrument (i.e., SD of daily return)
 P is the SD of the change in the portfolio value per day
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Covariance Matrix (vari = covii)
(Table 22.6)
 var1

 cov 21
C =  cov 31

 
 cov
n1

cov12
cov13
var2
cov 23
cov 32
var3


cov n 2
cov n 3
 cov1n 

 cov 2 n 
 cov 3n 


 

 varn 
covij = ij i j where i and j are the SDs of the daily returns
of variables i and j, and ij is the correlation between them
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Alternative Expressions for P2
(equation 22.4)
n
n
 =  cov ij  i  j
2
P
i =1 j =1
 2P = α T Cα
where α is the column vector whose ith
element is α i and α T is its transpose
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Alternatives for Handling
Interest Rates
Duration approach: Linear relation between
P and y but assumes parallel shifts
Cash flow mapping: Cash flows are mapped
to standard maturities and variables are zerocoupon bond prices with the standard
maturities
Principal components analysis: 2 or 3
independent shifts with their own volatilities
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When Linear Model Can be Used
Portfolio of stocks
Portfolio of bonds
Forward contract on foreign currency
Interest-rate swap
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The Linear Model and Options
Consider a portfolio of options dependent on
a single stock price, S. If  is the delta of the
option, then it is approximately true that
P

S
Define
S
x =
S
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Linear Model and Options
continued
Then
P   S = S x
Similarly when there are many underlying
market variables
P 
 S  x
i
i
i
i
where i is the delta of the portfolio with
respect to the ith asset
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Example
Consider an investment in options on Microsoft and
AT&T. Suppose the stock prices are 120 and 30
respectively and the deltas of the portfolio with
respect to the two stock prices are 1,000 and 20,000
respectively
As an approximation
P = 120  1,000 x1 + 30  20,000 x2
where x1 and x2 are the percentage changes in the
two stock prices
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But the distribution of the daily return
on an option is not normal
The linear model fails to capture skewness in the
probability distribution of the portfolio value.
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Impact of gamma (Figure 22.3)
Positive Gamma
Negative Gamma
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Translation of Asset Price Change to Price
Change for Long Call (Figure 22.4)
Long
Call
Asset Price
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Translation of Asset Price Change to Price
Change for Short Call (Figure 22.5)
Asset Price
Short
Call
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Quadratic Model (equation 22.7)
For a portfolio dependent on a single stock
price it is approximately true that
1
2
P = S +  (S )
2
this becomes
1 2
2
P = S x + S  ( x )
2
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Quadratic Model continued
(equation 22.8)
With many market variables we get an
expression of the form
n
n
n
1
P =  S i i xi +  S i S j  ij xi x j
i =1
i =1 j =1 2
where
2
P
i =
Si
 P
 ij =
S i S j
But this is much more difficult to work with
than the linear model
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Monte Carlo Simulation
To calculate VaR using MC simulation we
• Value portfolio today
• Sample once from the multivariate distributions
of the xi
• Use the xi to determine market variables at
end of one day
• Revalue the portfolio at the end of day
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Monte Carlo Simulation continued
Calculate P
Repeat many times to build up a probability
distribution for P
VaR is the appropriate fractile of the
distribution times square root of N
For example, with 1,000 trials the 1 percentile
is the 10th worst case.
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Speeding up Calculations with
the Partial Simulation Approach
Use the approximate delta/gamma
relationship between P and the xi to
calculate the change in value of the portfolio
This can also be used to speed up the
historical simulation approach
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Comparison of Approaches
Model building approach assumes normal
distributions for market variables. It tends to
give poor results for low delta portfolios
Historical simulation lets historical data
determine distributions, but is computationally
slower
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Back-Testing
Tests how well VaR estimates would have
performed in the past
We could ask the question: How often was
the actual 1-day loss greater than the
99%/1- day VaR?
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Principal Components Analysis
for U.S. Treasury Rates
The first factor is a roughly parallel shift
(87.3% of variance in data explained)
The second factor is a twist (another 8.3%
of variance explained)
The third factor is a bowing (another 2.1%
of variation explained)
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The First Three Principal
Components (Figure 22.6)
Factor Loading
0.800
0.600
0.400
0.200
Maturity (years)
0.000
-0.200
0
5
10
15
20
25
30
35
-0.400
-0.600
PC1
PC2
PC3
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Standard Deviation of Factor
Scores (bp) Table 22.10
PC1
PC2
PC3
PC4
…..
11.54
3.55
1.78
1.25
….
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Using PCA to Calculate VaR (Table
22.11)
Example: Sensitivity of portfolio to 1 bp rate
move ($m)
1 yr 2 yr 3 yr 4 yr 5 yr
+10 +4
-8
-7
+2
Sensitivity to first factor is from factor loadings:
10×0.210 + 4×0.286 − 8×0.386 − 7 ×0.430 +2 ×0.428
= −1.99
Similarly sensitivity to second factor = − 3.06
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Using PCA to calculate VaR
continued
As an approximation
P = −1.99 f1 − 3.06 f 2
The factors are independent in a PCA
The standard deviation of P is
1.992 × 11.542 + 3.062 × 3.552 =25.45
The 1 day 99% VaR is 25.45 × 2.326 = 59.2
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