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Chapter 26
Value at Risk
Introduction
• Risk assessment is the evaluation of
distributions of possible outcomes, with a
focus on the worst that might happen
– Insurance companies, for example,
assess the likelihood of insured events, and the
resulting possible losses for the insurer
– Financial institutions must understand their
portfolio risks in order to determine the capital
buffer needed to support their business
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26-2
Value at Risk
• Value at risk (VaR) is one way to perform
risk assessment for complex portfolios
• In general, computing value at risk means
finding the value of a portfolio such that
there is a specified probability that the
portfolio will be worth at least this much
over a given horizon
– The choice of horizon and probability will depend
on how VaR is to be used
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26-3
Value at Risk (Cont’d)
• There are at least three uses of value
at risk
– Regulators can use VaR to compute capital
requirements for financial institutions
– Managers can use VaR as an input in making
risk-taking and risk-management decisions
– Managers can also use VaR to assess the quality
of the bank’s models
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Value at Risk for One Stock
• Suppose ~xh is the dollar return on a portfolio
over the horizon h, and f (x, h) is the
distribution of returns
• Define the value at risk of the portfolio as the
return, xh(c), such that
Pr( xh  xh (c))  c
• Suppose a portfolio consists of a single stock
and we wish to compute value at risk over the
horizon h
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26-5
Value at Risk for One Stock
(cont’d)
• If we pick a stock price S h and the
distribution of the stock price after h
periods, Sh, is lognormal, then
Sh (c)  S0e
(  0.5 2 ) h hN 1 ( c )
(26.5)
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26-6
Value at Risk for One Stock
(cont’d)
• In practice, it is common to simplify the VaR
calculation by assuming a normal return rather than
a lognormal return. A normal approximation is
Sh  S0 (1  h  z h )
(26.7)
• We could further simplify by ignoring the mean
– Mean is hard to estimate precisely
– For short horizons, the mean is less important than the
diffusion term in an Itô process
Sh  S0 (1  z h )
(26.8)
• Both equations become less reasonable as h grows
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26-7
Value at Risk for One Stock
(cont’d)
• Comparison of three models—lognormal, normal
with mean, and normal without mean
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26-8
VaR for Two or More Stocks
• When we consider a portfolio having two or
more stocks, the distribution of the future
portfolio value is the sum of lognormally
distributed random variables and is
therefore not lognormal
• Since the distribution is no longer
lognormal, we can use the normal
approximation
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26-9
VaR for Two or More Stocks
(cont’d)
• Let the annual mean of the return on stock
i,  i , be i
• The standard deviation of the return on
stock i is i
• The correlation between stocks i and j is ij
• The dollar investment in stock i is Wi
• The value of a portfolio containing n stocks is
n
W  Wi
i 1
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26-10
VaR for Two or More Stocks
(cont’d)
• If there are n assets, the VaR calculation requires
that we specify the standard deviation for each
stock, along with all pairwise correlations
– The return on the portfolio over the horizon h, Rh, is
n
1
Rh 
W

i 1
Wi
i ,h
– Assuming normality, the annualized distribution of the
portfolio return is
1
Rh ~ N 
W
n
 hW
i 1
i
i,

1 n n
   hWW
i j
2  i j ij
W i1 j 1

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(26.9)
26-11
Var for Nonlinear Portfolios
• If a portfolio contains options as well as stocks,
it is more complicated to compute the distribution
of returns
– The sum of the lognormally distributed stock prices
is not lognormal
– The option price distribution is complicated
• There are two approaches to handling nonlinearity
– Delta approximation: we can create a linear approximation
to the option price by using the option delta
– Monte Carlo simulation: we can value the option using an
appropriate option pricing formula and then perform Monte
Carlo simulation to obtain the return distribution
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26-12
Delta Approximation
~ , we can approximate the
• If the return on stock i is 
i
return on the option as  i~i , where i is the option delta
• The expected return on the stock and option portfolio over
the horizon h is then
n
1
Rp 
W
 S (  N  )
i 1
i i
i
i
i
(26.10)
– The term i + Nii measures the exposure to stock i
• The variance of the return is
1 n n
2
 p  2  Si S j (i  Ni i )( j  N j  j ) i j ij
W i1 j 1
(26.11)
• With this mean and variance, we can mimic the n-stock
analysis
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26-13
Delta Approximation (cont’d)
• Comparison of exact portfolio value with a
delta approximation
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Monte Carlo Simulation
• Monte Carlo simulation works well in situations
where we need a two-tailed approach to VaR (e.g.,
straddle)
– Simulation produces the distribution of portfolio values
• To use Monte Carlo simulation
– We randomly draw a set of stock prices
– Once we have the portfolio values corresponding to each
draw of random prices, we sort the resulting portfolio
values in ascending order
– The 5% lower tail of portfolio values, for example, is used
to compute the 95% value at risk
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26-15
Monte Carlo Simulation (cont’d)
• Example 26.6
– Consider the 1-week 95% value at risk of an atthe-money written straddle on 100,000 shares of
a single stock
– Assume that S = $100, K = $100,  = 30%, r =
8%, T = 30 days, and  = 0
– The initial value of the straddle is $685,776
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26-16
Monte Carlo Simulation (cont’d)
• First, we randomly draw a set of z ~ N(0,1), and
construct the stock price as
S h  S0e
(  0.5 2 ) h  h z
(26.13)
• Next, we compute the Black-Scholes call and put
prices using each stock price, which gives us a
distribution of straddle values
• We then sort the resulting straddle values in
ascending order
• The 5% value is used to compute the 95%
value at risk
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26-17
Monte Carlo Simulation (cont’d)
• Histogram of values resulting
from 100,000 random
simulations of the value
of the straddle
• The 95% value at risk is $943,028  ($685,776) =
$257,252
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Monte Carlo Simulation (cont’d)
• Note that the value of the portfolio never
exceeds about $597,000
– If a call and put are written on the same stock,
stock price moves can never induce the two to
appreciate together. The same effect limits a loss
• When options are written on different
stocks, it is possible for both to gain or lose
simultaneously. As a result, the distribution
of prices has a greater variance and
increased value at risk
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26-19
Monte Carlo Simulation (cont’d)
• Example 26.7: Histogram of values of a portfolio
that contains a written put and call having different,
correlated underlying stocks
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26-20
VaR for Bonds
• The risk of a bond and other interest-rate sensitive
claims can be measured as the risk of a portfolio of
zero-coupon bonds
– Suppose a zero-coupon bond matures at time T, has price
P(T), and that the annualized yield volatility of the bond is
T
– For a zero-coupon bond, duration equals maturity. Thus, if
the yield changes by , the percentage change in the bond
price will be approximately T
– Using this linear approximation based on duration, over
the horizon h the bond has a 95% chance of being worth
more than
P(T ) 1   T T h  (1.645) 
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26-21
VaR for Bonds (cont’d)
• Now suppose that instead of a single bond
we have a portfolio of zerocoupon bonds
– As with a portfolio of stocks, we can use the
delta approximation, only instead of correlated
stock returns we have correlated bond yields
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26-22
VaR for Bonds (cont’d)
• In general, if we are analyzing the risk of an
instrument with multiple cash flows, the first step is
to find the equivalent portfolio of zero-coupon
bonds
– For example, a 10-year bond with semiannual coupons = a
portfolio of 20 zero-coupon bonds
• Every interest rate claim is decomposed in this way
into interest rate “buckets” containing the claim’s
constituent zero-coupon bonds
• A set of bonds and swaps reduces to a portfolio of
long and short positions in zero-coupon bonds
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26-23
VaR for Bonds (cont’d)
• We need volatilities and correlations for all
the bonds
• Volatility and yields are tracked only at
certain benchmark maturities
– The goal is to find an interpolation procedure to
express any hypothetical zero-coupon bond in
terms of the benchmark zero-coupon bonds
– This procedure in which cash flows are allocated
to benchmark claims is called cashflow
mapping
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26-24
VaR for Bonds (cont’d)
• Suppose that we wish to assess the risk of a 12year zero-coupon bond, given information on the
10-year and 15-year zero-coupon bonds
– We can use simple linear interpolation to obtain the yield
and yield volatility for the 12-year bond from those of the
10-year and 15-year bonds
– If the yield and volatility of the t-year bond are yt and t,
the
y12  (0.6  y10 )  (0.4  y15 )
(26.14)
 12  (0.6   10 )  (0.4   15 )
(26.15)
– These interpolations enable us to determine the price and
volatility of $1 paid in year 12. Note, however, that they do
not provide correlations between the 12-year zero and the
adjacent benchmark bonds
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26-25
VaR for Bonds (cont’d)
• The price is
e y12 12
• To find the combination of the 10- and 15-year zero-coupon
bonds have the same volatility as the hypothetical 12-year
bond, we must solve
 212  ( 2 210 )  [(1  )2  210 ]  [210.15 (1  )1015]
(26.16)
– where  equals the fraction allocated to the 10-year bond
– Since this is a quadratic equation, there are two solutions for .
Typically, only one of the two solutions will be economically
appealing
– Given the weights, value at risk for the 12-year bond can be
computed in the same way as VaR for a bond portfolio
• Similarly, mapping can be applied to any claim with multiple
cash flows
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26-26
Estimating Volatility
• Volatility is the key input in any VaR calculation
• In most examples, return volatility is assumed to
be constant and returns are independent over time
• We require correlation estimates in order to
compute volatilities of portfolios
• Assessing return correlation is complicated
– Over horizons as short as a day, returns may be negatively
correlated due to factors such as bid-ask bounce
– With commodities, return independence is not reasonable
for long horizons due to supply and demand responses
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26-27
Bootstrapping Return
Distributions
• It is possible to use observed past returns to create an
empirical probability distribution of returns that can then be
used for simulation. This procedure is called bootstrapping
– The idea of bootstrapping is to sample, with replacement, from
observed historical returns under the assumption that future
returns will be drawn from the same distribution
• Advantages and disadvantages of bootstrapping
– The advantage is that, since bootstrapping is not based on a
particular assumed distribution, it is consistent with any
distribution
of returns
– The disadvantage is that key features of the data might be lost
when the data are reshuffled. There is also the question of how
to bootstrap multiple series in such a way that correlations
are preserved
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26-28
Issues With VaR
• One of the problems with VaR is that small
changes in the VaR probability can cause
VaR to change by a large amount
– Suppose you are comparing activity C, which
generates a $1 loss with a 1.1% probability,
with activity D, which generates a $1m loss with
a 0.9% probability
– Any reasonable rule would require more capital
for activity D, but a 1% VaR would be greater for
C than for D
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26-29
Issues With VaR (cont’d)
• One approach to improving VaR is to
compute the average loss conditional upon
the VaR loss being exceeded. This is called
the Tail VaR
– The 1% Tail VaR for activity C would be $1, and
the 1% Tail VaR for activity D would be
$900,000 (instead of the VaR of $0)
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26-30
Issues With VaR (cont’d)
• It is argued that a reasonable risk measure should
have certain properties, among them
subadditivity
– If (X) is the risk measure associated with activity X, then
 is subadditive if for two activities X and Y
(X  Y )  (X)  (Y )
(26.21)
– This says that the risk measure (i.e., the capital required) for
the two activities combined should be less than for the two
activities separately
• VaR is not subadditive
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