21 - 1
CHAPTER 21 (Hull): RISK MANAGEMENT
Value-at-Risk (VaR)
Value-at-risk (VaR) is defined to be the number a > 0
such that the probability of a loss more than |a| is less
than some probability, e.g., 1%; i.e., in terms of
profit,
Probability{|a| < profit} = 0.99,
or Probability{profit < |a|} = 0.01.
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ASIDE: Note that VAR stands for vector auto-regression.
In 2011, one of the Nobel Prize winners in
Economics was the first to develop VAR techniques.
We can easily re-write this formula in terms of returns,
R, and if returns are normally distributed with mean
 and standard deviation , then it is known that
Probability{  2.576 < R} = 0.995.
The number 2.576 is used for (one-sided) confidence
intervals and hypothesis tests. In statistics, it’s
sometimes called z, for  = 0.5%. To see this, first
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note that R 

 is a standard normal random
variable, so we can re-write this probability as
Probability{2.576 < R 

 } = 0.995;
or in general, Probability{z <
R 

 } = 1  ;
alternatively, Probability{ R 

 < z} = ;
or, Probability{R <  z} = .

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Here is the graph:
z





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Various probabilities can be used. Here’s a useful table:
(one-sided)
Confidence level (1 – )  z
90% 10% 1.282
95% 5% 1.645
97.5% 2.5% 1.960
99% 1% 2.326
99.5% 0.5% 2.576
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Example 1: An investor has $2 million invested in a
bond. The bond has a mean return of 8% and a
volatility of 10%. What is the VaR (using  = 0.5%)
for the bond if we assume that returns are normally
distributed? Assume that the bond will be held for
one year.
Answer: We have  = 8%,  = 0.10, z = z0.005 = 2.576.
    2.576 = 0.08  2.576(0.10) = 0.1776.
I’ll call 17.76% the return VaR. (This is just my
terminology).
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So, in terms of profit, the lower limit is
  17.76%2 million = $355,200;
we’ll call $355,200 the dollar VaR (we always take
absolute value).
In words, there is only a ½ % chance of a loss of more
than $355,200 over a one-year period. This is then
the dollar VaR for  = 0.5%.
The interval 17.76% < R is just a one-sided confidence
interval for the return, R.
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Example 2: Repeat the previous question but with
 = 5%. In this case, z = z0.05 = 1.645.
    1.645 = 0.08  1.645(0.10) = 0.0845.

Holding Period. If the bond will be held for a period
other than one year, we must adjust both the bond’s
mean return and its volatility for the holding period.
For a holding period of  years, where
 =
yearperdaystrading#
daystrading#
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we can compute the volatility of returns (as a
function of ) using the formula
  () =   ,
while the expected return for the holding period is
  () = .
Note that we normally assume that there are 252 trading
days per year for these calculations. (Usually,  = 1
day, or  = 10 trading days.)
In general then, the return VaR is
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  z   ,
where | | represents absolute value.
Example: If in our previous example, we had a 3-
month (90-day) holding period, (let’s say this is 90
trading days) then the volatility for the holding
period is
  () =   = 0.10 90
252
= 5.976%
and the expected return for this holding period is
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  () = · = 8% 90
252
= 2.857%.
The one-sided confidence interval corresponding to a
99.5% level is (here z = z0.005 = 2.576 again)
  z0.005  = 2.857  2.576(5.976) = 12.54%,
so 12.54% is the return VaR, for  = 0.5% and for a
holding period of 90 trading days.
So, in terms of profit/loss: the lower limit is
  12.54%2 million = 250,800,
i.e., the dollar VaR = 250,800, for  = 0.5%.
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ASIDE: VaR of Equity Sector. We know that the
variance of the return on a portfolio of two assets,
security A and security B is given by
 p2 = xA2 A2 + xB2 B2 + 2xAxBcov(RA, RB).
Here xi = niVi/Vp, where ni is the number of shares of
asset i held, Vi is the market price of asset i, and Vp is
the value of the portfolio. Note that
1
1
N
i
i
x

 .
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We’ll also write cov(RA, RB) = AB = AB; so, the
correlation is  =
BA
AB


In matrix notation, and for a portfolio of N securities, the
variance of the portfolio can be compactly denoted
by  p2 = xTx, where  is the variance-covariance
matrix for the N securities and x is a column vector
of weights, xT = [x1, x2,...,xN]; that is,
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 p2 = xTx = [x1, x2,...,xN]




















NNN
N
N x
x





1
2
2
112
1
2
1





= 
 
N
i
N
j
jiji xx
1 1
 (where ii = 2i ).
Note that
1
1
N
i
i
x

 .
We then summarize the dollar VaR as Vpzp  , rather
than using Vp|p  zp  |. We (often) ignore p
because (i) p is hard to estimate, and (ii) if we use
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short holding periods, p is not important. For
example,
252
1 = 0.063, while
252
1 = 0.004. As
mentioned above, we may also use different
confidence levels, other than 0.5%.
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Example 2: Suppose you have the following portfolio:
Security # of Shares Price Position Value
Stock A 100,000 20 2,000,000
Stock B 100,000 30 3,000,000
Note that xA = 2/5 = 0.4, and xB = 0.6, and Vp =
5,000,000. We’ll assume that the holding period is
one day.
The variance-covariance matrix is
   =
0 04 0 048
0 048 0 09
. .
. .




=
2
2
A AB
BA B
 
 
 
 
 
.
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Note that A = 0.04= 0.2, B = 0.09 = 0.3, and
 = AB
A B

 
= 0.048
(0.20)(0.30)
= 0.8.
The portfolio variability is
 p2 = (0.4)2(0.04) + (0.6)2(0.09) + 2(0.4)(0.6)(0.048)
= 0.06184,
so that p = 0.2487.
For a 99.5% confidence interval, we use za = z0.005 =
2.576, and so the dollar value at risk is
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  Vpzp  = 5,000,000(2.576)(0.2487) 1
252

= 201,768.
So, there is only a 0.5% chance that the losses will
exceed 201,768 in one day in this equity portfolio.
Similarly the return VaR is
000,000,5
768,201 = 4%.
Note that VaRs are not additive, just as standard
deviations are not additive.
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Note also that there is a benefit to diversification. All
else being equal, the lower the correlation between
the two assets, the lower the portfolio variance, and
so, the lower the VaR for the portfolio. (See Table
15-9, p. 584, or Table 19-0, 2nd Ed.)
Value-at-Risk of Fixed-Income Sector. In the end, we
use exactly the same procedure to calculate the VaR
for a portfolio of bonds.

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VaR for Derivatives
VaR of Options. The return on a call option whose price
is C is
RC =
dC
C
≈
0
0
C
CC  , where  = dt.
We can rearrange this to write it in terms of the call’s
derivative:
RC =
dC
dPA
dP
P
A
A
P
C
A ,
where PA is the price of the underlying asset.
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The first term in this product is the hedge ratio, or the
delta () of the option, and the second term is the
return on the underlying asset. So, we can re-write
the return as:
RC = RA
P
C
A ,
where RA is the return on the underlying asset.
The standard deviation of this return is then
St.dev(RC) = C = A
P
C
A .
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For options, we then summarize the return VaR as
zC  , rather than using Cz   . We
(often) ignore  because (i)  is hard to estimate, and
(ii) if we use short holding periods,  is not
important. For example,
252
1 = 0.063, while
252
1
= 0.004. As mentioned above, we may also use
different confidence levels, other than 0.5%.
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So, given that St.dev(RC) = C = A
P
C
A , when we
multiply this by z (and  ) we have the return VaR
for the option for a holding period of  years:
return VaR = zC  = zA
P
C
A  .
Finally, we calculate the dollar VaR, by multiplying the
return VaR by the call price, C. So, if we want the
dollar VaR for a holding period of  years, with
confidence level , we have
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dollarVaR = zAPA  .
NOTE 1: returns on calls and puts are not normally
distributed, however this approximation may be
reasonable for short holding periods. Extensions to
this approach, particularly using second order
approximations, are possible.
Note 2: if the option is a put option, then the delta is
negative, in which case the formulas become
return VaR = zput  = AA
Pz
put
     ;
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dollarVaR = A Az P      .
Example: Suppose we have a call option on asset A
where PA = 20, and A = 0.20. Assume the hedge
ratio is  = 0.45, the call price is C = $0.75, and the
holding period is one day. Then, for a 99.5%
confidence interval,
dollar VaR = 2.576(0.45)(0.2)(20) 1
252
= 0.2921.
If each option contract is for 1000 shares, then
dollar VaR per contract = 0.29211000 = 292.10.
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For the return VaR for the call we just divide by C:
return VaR = 0.2921
0.75
= 39%.
Note that the return VaR for the underlying stock is
return VaRstock = 2.5760.20
1
252
= 3%.
This is why speculating in options is considered so
risky! Hedging with options is another matter,
because with hedging you have a portfolio of the
option plus some position in the underlying asset,
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and the portfolio’s VaR will be different from the
option’s VaR.

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ASIDES: Monte Carlo Simulation. The above example
becomes unrealistic for longer holding periods,
because the delta changes over time—both because
the price of the underlying asset changes, and
because the time to maturity of the option changes.
Monte Carlo simulation is more appropriate.
It turns out that Monte Carlo results show that VaR—
particularly VaR with the normal distribution—is an
inappropriate measure of risk for options.

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ASIDE #1: Modeling Correlation
If you want to do a Monte Carlo simulation for a
portfolio, you will need to generate returns that are
correlated with each other. Here’s the general
procedure for this type of problem.
Call the variance-covariance matrix of returns ; so, 
is an nn matrix, in general. The vector of mean
returns is denoted , where T = (1, 2, … , n).
For variance-covariance matrices, it is generally
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possible to find a sort of “square-root” of , X,
which satisfies
   = XTX.
The book calls this the Cholesky factorization, and
programs like Mathematica, Matlab, and Gauss have
built-in Cholesky factorization codes.
Now, to generate correlated returns, first generate i.i.d.
standard normal random variables, (one series of
random numbers for each stock in the portfolio).
Call these series (zA1, zA2,...,zAN) and (zB1, zB2,...,zBN).
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(Again, there can be n of these series in general.)
Next, form the matrix
Z =
z z z
z z z
A A AN
B B BN
1 2
1 2






.
If the returns simulated are meant to be over the time
interval t, then the series of correlated returns is
given by
R = t XTZ + t 
R R R
R R R
A A AN
B B BN
1 2
1 2






.
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You can now use these returns to simulate the
corresponding prices.

ASIDE #2: MORE ON VALUE-AT-RISK
NOTE 1: VaR can also be calculated using distributions
other than the normal distribution, such as jump
diffusions, GARCH, or the student’s t distribution.
NOTE 2: A problem with VaR: Suppose we have a
portfolio, p, consisting of two assets, 1 and 2.
Normally,
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Riskp < Risk1 + Risk2
due to the benefit of diversification. (This property
is known as sub-additivity.) However, it is possible
for
VaRp > VaR1 + VaR2;
i.e., sub-additivity sometimes fails for VaR. In other
words, VaR doesn’t deal with aggregation or
diversification well. So, VaR may not be a
“coherent” measure of risk. It is for this reason that
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some have proposed an alternative called (tail)
conditional value-at-risk, CVaR:
CVaR = the conditional expectation of the return given
that the return is less than the (one-sided) VaR
= E[R|R < |VaR|] = { | |}
[ 1 ]
{ | |}
R VaRE R
P R VaR



 
.
(Note that CVaR > 0.)
What does P{R < |VaR|} equal to?
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If we use R =
0
0
S
SST  , then we can re-write this in terms
of the expectation ]1[ }{ XST TSE  , where
X = S0(1 – VaR). [Check on your own.] From our
knowledge of the Black-Scholes formula, we can
calculate this expectation:
]1[ }{ XST TSE  = S0e
TN(d1),
where
d1 =
21
0 2ln( / ) ( )( )S X T t
T t
 

  

,
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and  is the stock’s (actual) expected return (with
continuous compounding).