16-12-2020
Value-at-Risk (VaR)
Sankarshan Basu
Professor of Finance
Indian Institute of Management
Bangalore
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?”
VaR and Regulatory Capital
• Regulators base the capital they require
banks to keep on VaR
• The market-risk capital has traditionally been
calculated from a 10-day VaR estimated
where the confidence level is 99%
• Credit risk and operational risk capital are
based on a one-year 99.9% VaR
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Advantages of VaR
• It 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?”
VaR vs. Expected Shortfall
• VaR is the loss level that will not be exceeded with a specified
probability
• Expected shortfall (ES) is the expected loss given that the loss
is greater than the VaR level (also called C-VaR and Tail Loss)
• Regulators have indicated that they plan to move from using
VaR to using ES for determining market risk capital
• Two portfolios with the same VaR can have very different
expected shortfalls
Distributions with the Same VaR but
Different Expected Shortfalls
VaR
VaR
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Normal Distribution Assumption
• When losses (gains) are normally distributed with
mean m and standard deviation s
VaR m s N 1 ( X )
e Y 2
2
ES m s
2 (1 X )
1
where Y is N (X).
Changing the Time Horizon
• If losses in successive days are independent,
normally distributed, and have a mean of zero
T - day VaR 1 - day VaR T
T - day ES 1 - day ES T
VaR - Calculation
• Historical simulation
• Model building approach
• Monte Carlo simulation
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Historical Simulation
• Historical simulation involves using past data in a
very direct way as a guide to what might happen in
the future.
• The first step is to identify the market variables
affecting the portfolio.
– These will typically be exchange rates, equity prices,
interest rates, and so on.
• We then collect data on the movements in these
market variables over the most recent 500 days.
– This provides us with 500 alternative scenarios for what
can happen between today and tomorrow.
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Historical Simulation – contd.
• For each scenario we calculate the dollar change in the value
of the portfolio between today and tomorrow.
– This defines a probability distribution for daily changes in the value of
our portfolio.
• The fifth-worst daily change is the first percentile of the
distribution.
• The estimate of VaR is the loss when we are at this first
percentile point.
• Assuming that the last 500 days are a good guide to what
could happen during the next day, we are 99% certain that we
will not take a loss greater than our VaR estimate.
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Historical Simulation – contd.
• Suppose we use m days of historical data.
• Let vi be the value of a variable on day i
• There are m-1 simulation trials
• The ith trial assumes that the value of the market
variable tomorrow (i.e., on day m+1) is
vi
vm
vi 1
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Historical Simulation – contd.
0.1343*(0.1162/0.1159)
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The Model-Building Approach
• The main alternative to historical simulation is to
make assumptions about the probability
distributions of the returns on the market variables
• This is known as the model building approach (or
sometimes the variance-covariance approach)
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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 – contd.
• The standard deviation of the change in the
portfolio in 1 day is $200,000
• The standard deviation of the change in 10
days is
200,000 10 $632,500
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Microsoft Example – contd.
• We assume that the expected change in the
value of the portfolio is zero (This is OK for
short time periods)
• We assume that the change in the value of the
portfolio is normally distributed
• Since N(–2.326)=0.01, the VaR is
2.326 632,500 $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 SD per 10 days is
50,000 10 $158,144
• The VaR is
158,100 2.326 $367,800
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Portfolio
• Now consider a portfolio consisting of both
Microsoft and AT&T
• 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
s X Y s 2X sY2 2rs X s Y
• In this case sX = 200,000 and sY = 50,000 and
r = 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=$219,000
• What is the incremental effect of the AT&T
holding on VaR?
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ES Results (10-day 99%)
• Microsoft shares: $1,687,000
• AT&T shares: $421,700
• Portfolio: $1,857,600
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The Linear Model
We assume
• 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
n n
Variance of Portfolio Return rij wi w j s i s j
i 1 j 1
wi is weight of ith asset in portfolio
s i2 is variance of return on ith asset
in portfolio
rij is correlation between returns of ith
and jth assets
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VaR for Bond Portfolios
Task: To calculate VaR of a T-Bond with a principal of $1
mn. and coupon of 10% per annum payable semi-
annually maturing in 0.8 years
Method: Cash flow mapping and Variance-covariance
method for calculating VaR
The bond cash flows have to be converted to equivalent
3-,6- and 12-month cash flows using cash flow
mapping since zero rates, bond price volatilities and
return correlations are available for these maturities
only.
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Example: VaR for Bond Portfolios
DATA:
3-m 6-m 1-year
Zero rate 5.50 6.00 7.00
(% with annual Compounding)
Bond price volatility 0.06 0.10 0.20
(% per day)
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Example: VaR for Bond Portfolios
DATA:
Correlations 3-m 6-m 1-year
between daily returnsbond bond bond
3-month bond 1.0 0.9 0.6
6-month bond 0.9 1.0 0.7
1-year bond 0.6 0.7 1.0
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Example: VaR for Bond Portfolios
The T-bond cash flows are:
Time Amount
0.3-year $ 50,000
0.8-year $1,050,000
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Example: VaR for Bond Portfolios
0.8-year T-Bond
= 0.3-year ZCB ($50,000)
+ 0.8-year ZCB ($1,050,000)
0.3-year ZCB is mapped into equivalent
positions in 3-m and 6-m ZCBs and 0.8-year
ZCB is mapped into equivalent positions in
6-m and 1-year ZCBs
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Example: VaR for Bond Portfolios
The cash flow mapping procedure
The present value and variance of
equivalent positions should remain the same
As the original cash flow.
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Example: VaR for Bond Portfolios
First, the 0.3-yr and 0.8-yr zero rates and
volatilities are worked out by interpolation.
0.3-yr 0.8-yr
Zero rate (%) 5.60 6.60
Bond price volatility 0.068 0.16
(% per day)
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Example: VaR for Bond Portfolios
The present values of T-Bond cash flows are as
follows:
PV(0.3-yr ZCB)
= 50,000/1.0560.3 = 49,189.32
PV(0.8-yr ZCB)
= 1,050,000/1.0660.8 = 997,662
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Example: VaR for Bond Portfolios
Cash flow mapping of 0.3-yr ZCB
Suppose, we allocate of the PV to the 3-m bond and
(1- ) to the 6-m bond.
Then, matching variances, we obtain,
0.000682 = 0.00062 2+0.0012(1- )2
+2*0.9*0.0006*0.001* *(1 - )
= 0.760259
Position in 3-m ZCB = 0.760259*49,189.32 = $37,397
Position in 6-m ZCB = 0.239741*49,189.32 = $11,793
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Example: VaR for Bond Portfolios
Cash flow mapping of 0.8-yr ZCB
Suppose, we allocate of the PV to the 6-m bond and
(1- ) to the 1-yr bond.
Then, matching variances, we obtain,
0.00162 = 0.0012 2+0.0022(1- )2
+2*0.7*0.001*0.002* *(- )
= 0.320337
Position in 6-m ZCB = 0.320337*997,662 = $319,589
Position in 1-yr ZCB = 0.679663*997,662 = $678,074
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Example: VaR for Bond Portfolios
Cash flow mapping of 0.8-yr T-Bond
$50,000 $1,050,000 Total
(0.3-yr) (0.8-yr) ($)
3-m bond 37,397 37,397
6-m bond 11,793 319,589 331,382
1-yr bond 678,074 678,074
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Example: VaR for Bond Portfolios
Calculation of VaR
The variance of the portfolio of 3-m, 6-m and 1-
yr ZCBs can be worked out using the
volatilities, correlations and value weights.
Variance = $2,628,518 and
Std. Dev. = $1621.3
So, 10-day 99% VaR = 1621.3*√10*2.33 =
$11,946
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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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Approximate VaR for Options using The Linear
Model
• Consider a portfolio of options on a single
stock price, S.
• Let δ be the delta of the position and Δx, the
percentage change in stock price in 1 day.
Then,
P S
and x
S S
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Approximate VaR for Options using The
Linear Model
• As an approximation
P S S x
• Similarly when there are many underlying market
variables
n n
P S x x where S
i i i i i i i i
i 1 i 1
and i is the delta of the portfolio with respect to the ith
asset.
• VaR can then be calculated using linear model.
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Approximate VaR for Options using The Linear
Model– 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 1,000x1 30 20,000x2
• where x1 and x2 are the percentage changes in the
two stock prices and ∆P is the resultant change in the
value of the portfolio.
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Approximate VaR for Options using The
Linear Model– Example
• Assuming that the daily volatility of Microsoft is 2%
and the daily volatility of AT&T is 1%, and the
correlation between the daily changes is 0.3, the
standard deviation of ∆P is
(120,000 * 0.02) 2 (600,000 * 0.01) 2 2 * (120,000 * 0.02) * (600,000 * 0.01) * 0.3
7,099
• Because N(-1.65) = 0.05,
5-day 95% VaR =1.65*√5*7,099 = $26,193
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Shortcoming of the Linear Model for
Options
• The linear model fails to capture skewness in
the probability distribution of the portfolio
value.
• It does not take into account gamma of the
portfolio.
– Gamma measures the curvature of the
relationship between the portfolio value and an
underlying market variable.
• For a more accurate estimate of VaR for
options quadratic model should be used.
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Quadratic Model
For a portfolio including options dependent
on a single stock price it is approximately true
that
1
P S (S ) 2
2
This becomes (using Δx),
1 2
P S x S ( x ) 2
2
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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 trial the 1 percentile
is the 10th worst case.
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Model Building vs Historical Simulation
Model building approach can be used for
investment portfolios where there are no
derivatives, but it does not usually work when
for portfolios where
• there are derivatives
• positions are close to delta neutral
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Improving VaR/ES estimates
• VaR/ES calculation is backward looking. But some
modifications are possible.
a) Volatility updating can lead to more extreme
outcomes considered when market is highly
volatile.
b) Extreme value theory provides a way of
extending the tails of the loss distribution
obtained from historical data.
c) Calculating stressed VaR / ES considering the
impact of a particularly difficult period of 250
days that has occurred in the past.
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Volatility Updating
• Use a volatility updating scheme to calculate values
of each variable under historical simulation as
follows.
• Value of market variable under ith scenario
v (vi vi 1)s n 1 / s i
vn i 1
vi 1
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Extreme Value Theory (EVT)
• EVT is the term used to describe the science of
estimating the tails of a distribution.
• EVT can be used to improve VaR estimates and to
help in situations where analysts want to estimate
VaR with a very high confidence level.
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