Transcript Value at Risk - Drake University

Market Risk
and Value at Risk
Finance 286
Drake Fin 286
DRAKE UNIVERSITY
Market Risk
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Macroeconomic changes can create
uncertainty in the earnings of the Financial
Institutions trading portfolio.
Important because of the increased emphasis
on income generated by the trading portfolio.
The trading portfolio (Very liquid i.e. equities,
bonds, derivatives, foreign exchange) is not
the same as the investment portfolio (illiquid
ie loans, deposits, long term capital).
Importance of Market Risk
Measurement
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Management information – Provides info on the risk
exposure taken by traders
Setting Limits – Allows management to limit
positions taken by traders
Resource Allocation – Identifying the risk and return
characteristics of positions
Performance Evaluation – trader compensation –did
high return just mean high risk?
Regulation – May be used in some cases to
determine capital requirements
Measuring Market Risk
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The impact of market risk is difficult to
measure since it combines many sources of
risk.
Intuitively, all of the measures of risk can be
combined into one number representing the
aggregate risk
One way to measure this would be to use a
measure called the value at risk.
Value at Risk
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Value at Risk measures the market value
that may be lost given a change in the
market (for example, a change in interest
rates). that may occur with a corresponding
probability
We are going to apply this to look at market
risk.
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A Simple Example
Position A
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Position B
Payout
Prob
Payout
Prob
-100
0.04
-100
0.04
0
0.96
0
0.96
0
VaR at 95%
confidence
level
0
VaR at 95%
confidence
level
From Dowd, Kevin 2002
A second simple example
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Assume you own a 10% coupon bond that
makes semi annual payments with 5 years until
maturity with a YTM of 9%.
The current value of the bond is then 1039.56
Assume that you believe that the most the yield
will increase in the next day is .2%. The new
value of the bond is 1031.50
The difference would represent the value at
risk.
VAR
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The value at risk therefore depends upon the
price volatility of the bond.
Where should the interest rate assumption
come from?
historical evidence on the possible change in
interest rates.
Calculating VaR
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Three main methods
Variance – Covariance (parametric)
Historical
Monte Carlo Simulation
All measures rely on estimates of the
distribution of possible returns and the
correlation among different asset classes.
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Variance / Covariance Method
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Assumes that returns are normally
distributed.
Using the characteristics of the normal
distribution it is possible to calculate the
chance of a loss and probable size of the loss.
Probability
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Cardano 1565 and Pascal 1654
Pascal was asked to explain how to divide up
the winnings in a game of chance that was
interrupted.
Developed the idea of a frequency distribution
of possible outcomes.
This slide and the next few based in
part on Jorion, 1997
An example
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Assume that you are playing a game based
on the roll of two “fair” dice.
Each one has six possible sides that may land
face up, each face has a separate number, 1
to 6.
The total number of dice combinations is 36,
the probability that any combination of the
two dice occurs is 1/36
Example continued
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The total number shown on the dice ranges
from 2 to 12. Therefore there are a total of
12 possible numbers that may occur as part
of the 36 possible outcomes.
A frequency distribution summarizes the
frequency that any number occurs.
The probability that any number occurs is
based upon the frequency that a given
number may occur.
Establishing the distribution
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Let x be the random variable under
consideration, in this case the total number
shown on the two dice following each role.
The distribution establishes the frequency
each possible outcome occurs and therefore
the probability that it will occur.
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Discrete Distribution
Value
(x i)
Freq
(n i)
Prob
(p i)
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2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
5
4
3
2
1
1
36
2
36
3
36
4
36
5
36
6
36
5
36
4
36
3
36
2
36
1
36
Cumulative Distribution
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The cumulative distribution represents the
summation of the probabilities.
The number 2 occurs 1/36 of the time, the
number 3 occurs 2/36 of the time.
Therefore a number equal to 3 or less will
occur 3/36 of the time.
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Cumulative Distribution
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Value
Prob
(p i)
2
1
36
3
2
36
4
3
36
5
4
36
6
5
36
7
6
36
8
5
36
9
4
36
10
3
36
11
2
36
12
1
36
Cdf
1
36
3
36
6
36
10
36
15
36
21
36
26
36
30
36
33
36
35
36
36
36
Probability Distribution Function
(pdf)
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The probabilities form a pdf. The sum of the
probabilities must sum to 1.
11
p
i 1
i
1
The distribution can be characterized by two
variables, its mean and standard deviation
Mean
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The mean is simply the expected value from
rolling the dice, this is calculated by
multiplying the probabilities by the possible
outcomes (values).
11
252
E ( x)   pi xi 
7
36
i 1
In this case it is also the value with the
highest frequency (mode)
Standard Deviation
The variance of the random variable is
defined as:
11
210
2
V ( x)   pi [ xi  E ( x)] 
36
i 1
The standard deviation is defined as the
square root of the variance.
SD( x)  V ( x)  2.415
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Using the example in VaR
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Assume that the return on your assets is
determined by the number which occurs
following the roll of the dice.
If a 7 occurs, assume that the return for that
day is equal to 0. If the number is less than 7
a loss of 10% occurs for each number less
than 7 (a 6 results in a 10% loss, a 5 results
in a 20% loss etc.)
Similarly if the number is above 7 a gain of
10% occurs.
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Discrete Distribution
Value
2
(x i)
Return -50%
(n i)
Prob
1
(p i)
36
3
4
5
6
-40% -30% -20% -10%
2
36
3
36
4
36
5
36
7
0
6
36
8
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9
10% 20%
5
36
4
36
10
11
12
30%
40%
50%
3
36
2
36
1
36
VaR
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Assume you want to estimate the possible
loss that you might incur with a given
probability.
Given the discrete dist, the most you might
lose is 50% of the value of your portfolio.
VaR combines this idea with a given
probability.
VaR
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Assume that you want to know the largest
loss that may occur in 95% of the rolls.
A 50% loss occurs 1/36 = 2.77% 0f the time.
This implies that 1-.027 =.9722 or 97.22% of
the rolls will not result in a loss of greater
than 40%.
A 40% or greater loss occurs in 3/36=8.33%
of rolls or 91.67% of the rolls will not result
in a loss greater than 30%
Continuous time
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The previous example assumed that there
were a set number of possible outcomes.
It is more likely to think of a continuous set of
possible payoffs.
In this case let the probability density function
be represented by the function f(x)
Discrete vs. Continuous
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Previously we had the sum of the probabilities
equal to 1. This is still the case, however the
summation is now represented as an integral
from negative infinity to positive infinity.
Discrete
11
p
i 1
i
1
Continuous



f ( x)dx  1
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Discrete vs. Continuous
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The expected value of X is then found using
the same principle as before, the sum of the
products of X and the respective probabilities
Discrete
Continuous

n
E ( X )   pi xi
i 1
E( X ) 
 xf ( x)dx

Discrete vs. Continuous
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The variance of X is then found using the
same principle as before.
Discrete
Continuous

N
V ( X )   pi [ xi  E ( X )]
i 1
2
V ( X )   [ x  E ( X )]2 f ( x)dx

Combining Random Variables
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One of the keys to measuring market risk is
the ability to combine the impact of changes
in different variables into one measure, the
value at risk.
First, lets look at a new random variable, that
is the transformation of the original random
variable X.
Let Y=a+bX where a and b are fixed
parameters.
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Linear Combination
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The expected value of Y is then found using
the same principle as before, the sum of the
products of Y and the respective probabilities

E( X ) 
 xf ( x)dx


E (Y ) 
yf
(
x
)
dx


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Linear Combinations
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We can substitute since Y=a+bX, then simplify by
rearranging

E (Y ) 
 yf ( x)dx







E (a  bX )   (a  bX ) f ( x)dx  a  f ( x)dx  b  xf ( x)dx
 a  bE ( X )
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Variance
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Similarly the variance can be found

V (Y )  V (a  bX )   [( a  bX )  E (a  bX )]2 f ( x)dx


  [a  bx  a  bE ( X )] f ( x)dx
2


  b [ x  E ( X )] f ( x)dx  b V ( X )
2

2
2
Standard Deviation
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Given the variance it is easy to see that the standard
deviation will be
bSD(X )
Combinations of Random
Variables
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No let Y be the linear combination of two random
variables X1 and X2 the probability density function
(pdf) is now f(x1,x2)
The marginal distribution presents the distribution
as based upon one variable for example.



2
f ( x1 , x2 )dx2  f ( x1 )
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Expectations
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 
E( X1  X 2 ) 
 
1
2
( x1  x2 ) f ( x1 , x2 )dx1dx2
 
 

 
1
 
x f ( x1 , x2 )dx1dx2 
2 1
 
 
1
2
x2 f ( x1 , x2 )dx1dx2
 

  

  1 x1   2 f ( x1 , x2 )dx1dx2    2 x2   1 x1 f ( x1 , x2 )dx1dx2 

 
   







x f ( x1 )dx1 
1 1


2
x2 f ( x2 )dx2  E ( X 1 )  E ( X 2 )
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Variance
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Similarly the variance can be reduced
V ( X1  X 2 )
 

 
[ x1  x2  E ( X 1  X 2 )] f ( x1 , x2 )dx1dx2
2
1
2
 
 V ( X 1 )  V ( X 2 )  2Cov( X 1 , X 2 )
A special case
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If the two random variables are independent
then the covariance will reduce to zero which
implies that
V(X1+X2) = V(X1)+V(X2)
However this is only the case if the variables
are independent – implying that there is no
gain from diversification of holding the two
variables.
The Normal Distribution
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For many populations of observations as the
number of independent draws increases, the
population will converge to a smooth normal
distribution.
The normal distribution can be characterized by its
mean (the location) and variance (spread) N(m,s2).
The distribution function is
f ( x) 
1
2s
2
e
 1
2

(
x

m
)
 2s 2



Standard Normal Distribution
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The function can be calculated for various
values of mean and variance, however the
process is simplified by looking at a standard
normal distribution with mean of 0 and
variance of 1.
Standard Normal Distribution
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Standard Normal Distributions are symmetric
around the mean. The values of the
distribution are based off of the number of
standard deviations from the mean.
One standard deviation from the mean
produces a confidence interval of roughly
68.26% of the observations.
Prob Ranges for Normal Dist.
68.26%
95.46%
99.74%
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An Example
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Lets define X as a function of a standard normal
variable e (in other words e is N(0,1))
X= m  es
We showed earlier that
E (a  bX )  a  bE ( X )
Therefore
E ( m  se )  m  sE (e )  m
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Variance
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We showed that the variance was equal to
V (a  bX )  b V ( X )
2
Therefore
V (m  se )  s V (e )  s
2
2
An Example
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Assume that we know that the movements in
an exchange rate are normally distributed
with mean of 1% and volatility of 12%.
Given that approximately 95% of the
distribution is within 2 standard deviations of
the mean it is easy to approximate the
highest and lowest return with 95%
confidence
XMIN = 1% - 2(12%) = -23%
XMAX = 1% + 2(12%) = +25%
One sided values
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Similarly you can find the standard deviation
that represents a one sided distribution.
Given that 95.46% of the distribution lies
between -2 and +2 standard deviations of the
mean, it implies that (100% - 95.46)/2 =
2.27% of the distribution is in each tail.
This shows that 95.46% + 2.27% = 97.73%
of the distribution is to the right of this point.
VaR
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Given the last slide it is easy to see that you
would be 97.73% confident that the loss
would not exceed -23%.
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Continuous Time
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Let q represent quantile such that the area to
the right of q represents a given probability of
occurrence.

c  prob( X  q) 
 f ( x)dx
q
In our example above -2.00 would produce a
probability of 97.73% for the standard normal
distribution
VAR A second example
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Assume that the mean yield change on a bond
was zero basis points and that the standard
deviation of the change was 10 Bp or 0.001
Given that 90% of the area under the normal
distribution is within 1.65 standard deviations
on either side of the mean (in other words
between mean-1.65s and mean +1.65s)
There is only a 5% chance that the level of
interest rates would increase or decrease by
more than 0 + 1.65(0.001) or 16.5 Bp
Price change associated with
16.5Bp change.
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You could directly calculate the price change,
by changing the yield to maturity by 16.5 Bp.
Given the duration of the bond you also could
calculate an estimate based upon duration.
Example 2
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Assume we own seven year zero coupon
bonds with a face value of $1,631,483.00
with a yield of 7.243%
Today’s Market Value
$1,631,483/(1.07243)7=$1,000,000
If rates increase to by 16.5Bp to 7.408% the
market value is
$1,631,483/(1.07408)7 = $989,295.75
Which is a value decrease of $10,704.25
Approximations - Duration
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The duration of the bond would be 7 since it
is a zero coupon.
Modified duration is then 7/1.07243 = 6.527
The price change would then be
1,000,000(-6.57)(.00165) = $10,769.55
Approximations - linear
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Sometimes it is also estimated by figuring the
the change in price per basis point.
If rates increase by one basis point to 7.253%
the value of the bond is $999,347.23 or a
price decrease of $652.77.
This is a 652.77/1,000,000 = .06528%
change in the price of the bond per basis
point
The value at risk is then
1,000,000(.00065277)(16.5) = $10,770.71
Precision
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The actual calculation of the change should
be accomplished by discounting the value of
the bond across the zero coupon yield curve.
In our example we only had one cash flow….
DEAR
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Since we assumed that the yield change was
associated with a daily movement in rates, we
have calculated a daily measure of risk for the
bond.
DEAR = Daily Earnings at Risk
DEAR is often estimated using our linear
measure:
(market value)(price sensitivity)(change in yield)
Or
(Market value)(Price Volatility)
VAR
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Given the DEAR you can calculate the Value
at Risk for a given time frame.
VAR = DEAR(N)0.5
Where N = number of days
(Assumes constant daily variance and no
autocorrelation in shocks)
N
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Bank for International Settlements (BIS) 1998
market risk capital requirements are based on
a 10 day holding period.
Problems with estimation
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Fat Tails – Many securities have returns that
are not normally distributed, they have “fat
tails” This will cause an underestimation of
the risk when a normal distribution is used.
Do recent market events change the
distribution? Risk Metrics weights recent
observations higher when calculating
standard Dev.
Interest Rate Risk vs.
Market Risk
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Market risk is more broad, but Interest Rate
Risk is a component of Market Risk.
Market risk should include the interaction of
other economic variables such as exchange
rates.
Therefore, we need to think about the
possibility of an adverse event in the
exchange rate market and equity markets
etc.. Not just a change in interest rates..
DEAR of a foreign Exchange Position
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Assume the firm has Swf 1.6 Million trading
position in swiss francs
Assume that the current exchange rate is
Swf1.60 / $1 or $.0625 / Swf
The $ value of the francs is then
Swf1.6 million ($0.0625/Swf) =$1,000,000
FX DEAR
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Given a standard deviation in the exchange
rate of 56.5Bp and the assumption of a
normal distribution it is easy to find the DEAR.
We want to look at an adverse outcome that
will not occur more than 5% of the time so
again we can look at 1.65s
FX volatility is then 1.65(56.5bp) = 93.2bp or
0.932%
FX DEAR
DEAR = (Dollar value )( FX volatility)
=($1,000,000)(.00932)
=$9,320
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Equity DEAR
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The return on equities can be split into
systematic and unsystematic risk.
We know that the unsystematic risk can be
diversified away.
The undiversifiable market risk will be based
on the beta of the individual stock
 s
2
i
2
m
Equity DEAR
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If the portfolio of assets has a beta of 1 then
the market risk of the portfolio will also have
a beta of 1 and the standard deviation of the
portfolio can be estimated by the standard
deviation of the market.
Let sm = 2% then using the same confidence
interval, the volatility of the market will be
1.65(2%) = 3.3%
Equity DEAR
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DEAR = (Dollar value )( Equity volatility)
=($1,000,000)(0.033)
=$33,000
VAR and Market Risk
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The market risk should then estimate the
possible change from all three of the asset
classes.
This DOES NOT just equal the summation of
the three estimates of DEAR because the
covariance of the returns on the different
assets must be accounted for.
Aggregation
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The aggregation of the DEAR for the three
assets can be thought of as the aggregation
of three standard deviations.
To aggregate we need to consider the
covariance among the different asset classes.
Consider the Bond, FX position and Equity
that we have recently calculated.
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Variance Covariance
Seven Year
Zero
Swf/$1
US Stock
Index
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Seven Year
zero
Swf/$1
US Stock
Index
1
-.20
.4
1
.1
1
variance covariance
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(DEAR Z ) 2  (DEAR Swf ) 2  (DEAR US ) 2 


DEAR
 (2  z,Swf DEARZ DEARSwf )



Portfolio
 (2 Swf, US DEARSwf DEARUS )


 (2  z, US DEARZ DEARUS )

1
2
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VAR for Portfolio
(10.77)  (9.32)  (33)

DEAR  (2(.2)(10.77)(9.32)

Portfolio  (2)(.1)(9.32)(33)

 (2)(.4)(10.77)(33)
2
2
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2
1
2


  $39,969



Comparison
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If the simple aggregation of the three
positions occurred then the DEAR would have
been estimated to be $53,090. It is easy to
show that the if all three assets were perfectly
correlated (so that each of their correlation
coefficients was 1 with the other assets) you
would calculate a loss of $53,090.
Risk Metrics
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JP Morgan has the premier service for
calculating the value at risk
They currently cover the daily updating and
production of over 450 volatility and
correlation estimates that can be used in
calculating VAR.
Normal Distribution Assumption
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Risk Metrics is based on the assumption that
all asset returns are normally distributed.
This is not a valid assumption for many assts
for example call options – the most an
investor can loose is the price of the call
option. The upside is large, this implies a
large positive skew.
Normal Assumption Illustration
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Assume that a financial institution has a large
number of individual loans. Each loan can be
thought of as a binomial distribution, the loan
either repays in full or there is default.
The sum of a large number of binomial
distributions converges to a normal
distribution assuming that the binomial are
independent.
Therefore the portfolio of loans could be
thought of as a normal distribution.
Normal Illustration continued
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However, it is unlikely that the loans are truly
independent. In a recession it is more likely
that many defaults will occur.
This invalidates the normal distribution
assumption.
The alternative to the assumption is to use a
historical back simulation.
Historical Simulation
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Similar to the variance covariance approach,
the idea is to look at the past history over a
given time frame.
However, this approach looks at the actual
distribution that was realized instead of
attempting to estimate it as a normal
distribution.
Back Simulation
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Step 1: Measure exposures. Calculate the
total $ valued exposure to each assets
Step 2: Measure sensitivity. Measure the
sensitivity of each asset to a 1% change in
each of the other assets. This number is the
delta.
Step 3: Measure Risk. Look at the annual %
change of each asset for the past day and
figure out the change in aggregate exposure
that day.
Back Simulation
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Step 4 Repeat step 3 using historical data for
each of the assets for the last 500 days
Step 5 Rank the days from worst to best.
Then decide on a confidence level. If you
want a 5% probability look at the return with
95% of the returns better and 5% of the
return worse.
Step 6 calculate the VAR
Historical Simulation
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Provides a worst case scenario, where Risk
metrics the worst case is a loss of negative
infinity
Problems:
The 500 observations is a limited amount, thus
there is a low degree of confidence that it
actually represents a 5% probability. Should
we change the number of days??
Monte Carlo Approach
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Calculate the historical variance covariance
matrix.
Use the matrix with random draws to simulate
10,000 possible scenarios for each asset.
BIS Standardized Framework
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Bank of International Settlements proposed a
structured framework to measure the market
risk of its member banks and the offsetting
capital required to manage the risk.
Two options
Standardized Framework (reviewed below)
Firm Specific Internal Framework
Must be approved by BIS
Subject to audits
Risk Charges
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Each asset is given a specific risk charge
which represents the risk of the asset
For example US treasury bills have a risk
weight of 0 while junk and would have a risk
weight of 8%.
Multiplying the value of the outstanding
position by the risk charges provides capital
risk charge for each asset.
Summing provides a total risk charge
Specific Risk Charges
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Specific Risk charges are intended to measure
the risk of a decline in liquidity or credit risk
of the trading portfolio.
Using these produces a specific capital
requirement for each asset.
General Market Risk Charges
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Reflect the product of the modified duration
and expected interest rate shocks for each
maturity
Remember this is across different types of
assets with the same maturity….
Vertical Offsets
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Since each position has both long and short
positions for different assets, it is assumed
that they do not perfectly offset each other.
In other words a 10 year T-Bond and a high
yield bond with a 10 year maturity.
To counter act this there is a vertical offset or
disallowance factor.
Horizontal Offsets
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Within Time Zones
For each maturity bucket there are
differences in maturity creating again the
inability to let short and long positions exactly
offset each other.
Between Time Zones
Also across zones the short and long positions
must be offset.
VaR Problems
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Artzner (1997), (1999) has shown that VaR is
not a coherent measure of risk.
For Example it does not posses the property
of subadditvity. In other words the combined
portfolio VaR of two positions can be greater
than the sum of the individual VaR’s
A Simple Example*
Assume a financial institution is facing the
following three possible scenarios and
associated losses
Scenario
1
2
3
Probability
.97
.015
.015
Loss
0
100
0
The VaR at the 98% level would equal = 0
*This and subsequent examples are based on Meyers 2002
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A Simple Example
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Assume you the previous financial institution and its
competitor facing the same three possible scenarios
Scenario Probability Loss A Loss B Loss A & B
1
.97
0
0
0
2
.015
100
0
100
3
.015
0
100
100
The VaR at the 98% level for A or B alone is 0
The Sum of the individual VaR’s = VaRA + VaRB = 0
The VaR at the 98% level for A and B combined
VaR(A+B)=100
Coherence of risk measures
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Let (X) and (Y) be measures of risk
associated with event X and event Y
respectively
Subadditvity implies (X+Y) < (X) + (Y).
Monotonicity. Implies X>Y then (X) > (Y).
Positive homogeneity:Given l > 0 (lX) =
l(X).
Translation Invariance. Given an additional
constant amount of loss a, (X+a) = (X)+a.
Coherent Measures of Risk
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Artzner (1997, 1999) Acerbi and Tasche
(2001a,2001b), Yamai and Yoshiba (2001a,
2001b) have pointed to Conditional Value at Risk
or Tail Value at Risk as coherent measures.
CVaR and TVaR measure the expected loss
conditioned upon the loss being above the VaR
level.
Lien and Tse (2000, 2001) Lien and Root (2003)
have adopted a more general method looking at
the expected shortfall
Tail VaR*
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TVaRa (X) = Average of the top (1-a)% loss
For comparison let VaRa(X) = the (1-a)% loss
* Meyers 2002 The Actuarial Review
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Scenario
X1
X2
X1+X2
1
4
5
9
2
2
1
3
3
1
2
3
4
5
4
9
5
3
3
6
VaR60%
4
4
9
TVaR60%
4.5
4.5
9
Normal Distribution
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How important is the assumption that
everything is normally distributed?
It depends on how and why a distribution
differs from the normal distribution.
Fin 286
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S&P 500 Monthly Returns vs. Normal Dist
220
Observations
170
120
S&P
Nor mal
70
20
-0. 475
-0. 425
-0. 375
-0. 325
-0. 275
-0. 225
-0. 175
-0. 125
-0. 075
-0. 025
0. 025
-30
Returns
0. 075
0. 125
0. 175
0. 225
0. 275
0. 325
0. 375
0. 425
0. 475
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Bond returns vs. Normal
250
Observations
200
LT Corp
150
LT Govt
Norm
100
50
0
-0.09
-0.07
-0.05
-0.03
-0.01
0.01
Returns
0.03
0.05
0.07
0.09
Two explanations of “Fat Tails”
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The true distribution is stationary and
contains fat tails.
In this case normal distribution would be
inappropriate
The distribution does change through time.
Large or small observations are outliers drawn
from a distribution that is temporarily out of
alignment.
Implications
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Both explanations have some truth, it is
important to estimate variations from the
underlying assumed distribution.
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Measuring Volatilities
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Given that the normality assumption is central
to the measurement of the volatility and
covariance estimates, it is possible to attempt
to adjust for differences from normality.
Moving Average
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One solution is to calculate the moving average of
the volatility
M
s
r
i 1
2
t i
M
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Moving Averages
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Moving Avergages of Volatility S&P 500 Monthly Return
0.018
0.016
0.014
0.012
1 year
2 year
5 year
0.01
0.008
0.006
0.004
0.002
0
0
100
200
300
400
500
600
700
800
900
Historical Simulation
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Another approach is to take the daily price
returns and sort them in order of highest to
lowest.
The volatility is then found based off of a
confidence interval.
Ignores the normality assumption! But
causes issues surrounding window of
observations.
Nonconstant Volatilities
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So far we have assumed that volatility is
constant over time however this may not be
the case.
It is often the case that clustering of returns
is observed (successive increases or
decreases in returns), this implies that the
returns are not independent of each other as
would be required if they were normally
distributed.
If this is the case, each observation should
not be equally weighted.
RiskMetrics
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JP Morgan uses an Exponentially Weighted
Moving Average.
This method used a decay factor that
weight’s each days percentage price change.
A simple version of this would be to weight by
the period in which the observation took
place.
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Risk Metrics
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t 1
s  (1  l ) l ( X t  m )
t
2
t n
Where
l is the decay factor
n is the number of days used to derive the
volatility
m Is the mean value of the distribution (assumed to
be zero for most VaR estimates)
Decay Factors
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JP Morgan uses a decay factor of .94 for daily
volatility estimates and .97 for monthly
volatility estimates
The choice of .94 for daily observations
emphasizes that they are focused on very
recent observations.
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Decay Factors
0.06
Weighting
0.05
0.04
0.94
0.97
0.03
0.02
0.01
0
0
20
40
60
80
Days
100
120
140
160
Measuring Correlation
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Covariance:
Cov( AB)   (k Ai  k A )( k Bi  k B ) Pi
Combines the relationship between the stocks
with the volatility.
(+) the stocks move together
(-) The stocks move opposite of each other
Measuring Correlation 2
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Correlation coefficient: The covariance is
difficult to compare when looking at different
series. Therefore the correlation coefficient is
used.
rAB  Cov( AB) /(s As B )
The correlation coefficient will range from
-1 to +1
Timing Errors
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To get a meaningful correlation the price
changes of the two assets should be taken at
the exact same time.
This becomes more difficult with a higher
number of assets that are tracked.
With two assets it is fairly easy to look at a
scatter plot of the assets returns to see if the
correlations look “normal”
Size of portfolio
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Many institutions do not consider it practical
to calculate the correlation between each pair
of assets.
Consider attempting to look at a portfolio that
consisted of 15 different currencies. For each
currency there are asset exposures in various
maturities.
To be complete assume that the yield curve
for each currency is broken down into 12
maturities.
Correlations continued
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The combination of 12 maturities and 15
currencies would produce 15 x 12 = 180
separate movements of interest rates that
should be investigated.
Since for each one the correlation with each
of the others should be considered, this would
imply 180 x 180 = 16,110 separate
correlations that would need to be
maintained.
Reducing the work
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One possible solution to this would be
reducing the number of necessary
correlations by looking at the mid point of
each yield curve.
This works IF
There is not extensive cross asset trading
(hedging with similar assets for example)
There is limited spread trading (long in one
assert and short in another to take advantage
of changes in the spread)
A compromise
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Most VaR can be accomplished by developing
a hierarchy of correlations based on the
amount of each type of trading. It also will
depend upon the aggregation in the portfolio
under consideration. As the aggregation
increases, fewer correlations are necessary.
Back Testing
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To look at the performance of a VaR model,
can be investigated by back testing.
Back testing is simply looking at the loss on a
portfolio compared to the previous days VaR
estimate.
Over time the number of days that the VaR
was exceed by the loss should be roughly
similar to the amount specified by the
confidence in the model
Basle Accords
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To use VaR to measure risk the Basle accords
specify that banks wishing to use VaR must
undertake two different types of back testing.
Hypothetical – freeze the portfolio and test
the performance of the VaR model over a
period of time
Trading Outcome – Allow the portfolio to
change (as it does in actual trading) and
compare the performance to the previous
days VaR.
Back Testing Continued
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Assume that we look at a 1000 day window
of previous results. A 95% confidence
interval implies that the VaR level should have
been exceed 50 times.
Should the model be rejected if the it is found
that the VaR level was exceeded 55 times?
70 times? 100 times?
Back test results
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Whether or not the actual number of
exceptions differs significantly from the
expectation can be tested using the Z score
for a binomial distribution.
Type I error – the model has been
erroneously rejected
Type II error – the model has been
erroneously accepted.
Basle specifies a type one error test.
One tail versus two tail
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Basle does not care if the VaR model
overestimates the amount of loss and the
number of exceptions is low ( implies a one
tail test)
The bank, however, does care if the number
of exceptions is low and it is keeping too
much capital (implies a two tail test).
Excess Capital
Trading performance based upon economic
capital
Approximations
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Given a two tail 95% confidence test and 1000
days of back testing the bank would accept 39
to 61 days that the loss exceeded the VaR
level.
However this implies a 90% confidence for the
one tail test so Basle would not be satisfied.
Given a two tail test and a 99% confidence
level the bank would accept 6 to 14 days that
the loss exceeded the trading level, under the
same test Basle would accept 0 to 14 days.
Empirical Analysis of VaR
(Best 1998)
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Whether or not the lack of normality is not a
problem was discussed by Best 1998
(Implementing VaR)
Five years of daily price movements for 15
assets from Jan 1992 to Dec 1996. The
sample process deliberately chose assets that
may be non normal.
VaR Was calculated for each asset individually
and for the entire group as a portfolio.
Figures 4. Empirical Analysis of VaR
(Best 1998)
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All Assets have fatter tails than expected
under a normal distribution.
Japanese 3-5 year bonds show significant
negative skew
The 1 year LIBOR sterling rate shows nothing
close to normal behavior
Basic model work about as well as more
advanced mathematical models
Basle Tests
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Requires that the VaR model must calculate
VaR with a 99% confidence and be tested
over at least 250 days.
Table 4.6
Low observation periods perform poorly while
high observation periods do much better.
Clusters of returns cause problem for the
ability of short term models to perform, this
assumes that the data has a longer “memory”
Basle
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The Basle requirements supplement VaR by
Requiring that the bank originally hold 3 times
the amount specified by the VaR model.
This is the product of a desire to produce
safety and soundness in the industry
Stress Testing
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Value at Risk should be supplemented with
stress testing which looks at the worst
possible outcomes.
This is a natural extension of the historical
simulation approach to calculating variance.
VaR ignores the size of the possible loss, if
the VaR limit is exceeded, stress testing
attempts to account for this.
Stress Testing
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Stress Testing is basically a large scenario
analysis. The difficulty is identifying the
appropriate scenarios.
The key is to identify variables that would
provide a significant loss in excess of the VaR
level and investigate the probability of those
events occurring.
Stress Tests
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Some events are difficult to predict, for
example, terrorism, natural disasters, political
changes in foreign economies.
In these cases it is best to look at similar past
events and see the impact on various assets.
Stress testing does allow for estimates of
losses above the VaR level.
You can also look for the impact of clusters of
returns using stress testing.
Stress Testing with
Historical Simulation
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The most straightforward approach is to look
at changes in returns.
For example what is the largest loss that
occurred for an asset over the past 100 days
(or 250 days or…)
This can be combined with similar outcomes
for other assets to produce a worst case
scenario result.
Stress Testing
Other Simulation Techniques
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Monte Carlo simulation can also be employed
to look at the possible bad outcomes based
on past volatility and correlation.
The key is that changes in price and return
that are greater than those implied by a three
standard deviation change need to be
investigated.
Using simulation it is also possible to ask what
happens it correlations change, or volatility
changes of a given asset or assets.
Managing Risk with VaR
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The Institution must first determine its
tolerance for risk.
This can be expressed as a monetary amount
or as a percentage of an assets value.
Ultimately VaR expresses a monetary amount
of loss that the institution is willing to suffer
and a given frequency determined by the
timing confidence level..
Managing Risk with VaR
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The tolerance for loss most likely increases
with the time frame. The institution may be
willing to suffer a greater loss one time each
year (or each 2 years or 5 years), but that is
different than one day VaR.
For Example, given a 95% confidence level
and 100 trading days, the one day VaR would
occur approximately once a month.
Setting Limits
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The VaR and tolerance for risk can be used to
set limits that keep the institution in an
acceptable risk position.
Limits need to balance the ability of the
traders to conduct business and the risk
tolerance of the institution. Some risk needs
to be accepted for the return to be earned.
VaR Limits
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Setting limits at the trading unit level
Allows trading management to balance the
limit across traders and trading activities.
Requires management to be experts in the
calculation of VaR and its relationship with
trading practices.
Limits for individual traders
VaR is not familiar to most traders (they d o
not work with it daily and may not understand
how different choices impact VaR.
VaR and changes in volatility
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One objection of many traders is that a
change in the volatility (especially if it is
calculated based on moving averages) can
cause a change in VaR on a given position.
Therefore they can be penalized for a position
even if they have not made any trading
decisions.
Is the objection a valid reason to not use
VaR?
Stress Test Limits
Similar to VaR limits should be set on the
acceptable loss according to stress limit
testing (and its associated probability).
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