dess analyst value at risk

8/12/2019 DESS Analyst Value at Risk

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DrakeDRAKE UNIVERSITYUNIVERSITEDAUVERGNE

Market Riskand Value at Risk

Finance 129


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DrakeDrake University

Market Risk

Macroeconomic changes can createuncertainty in the earnings of the Financial

institutions trading portfolio.Important because of the increased emphasison income generated by the trading portfolio.

The trading portfolio (Very liquid i.e. equities,bonds, derivatives, foreign exchange) is notthe same as the investment portfolio (illiquidie loans, deposits, long term capital).


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Importance of Market RiskMeasurement

Management informationProvides info on the riskexposure taken by traders

Setting LimitsAllows management to limitpositions taken by traders

Resource AllocationIdentifying the risk and returncharacteristics of positions

Performance Evaluationtrader compensationdidhigh return just mean high risk?

RegulationMay be used in some cases todetermine capital requirements


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Measuring Market Risk

The impact of market risk is difficult tomeasure since it combines many sources of

risk.Intuitively all of the measures of risk can becombined into one number representing theaggregate risk

One way to measure this would be to use ameasure called the value at risk.


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Value at Risk

Value at Risk measures the market valuethat may be lost given a change in the

market (for example, a change in interestrates). that may occur with a correspondingprobability

We are going to apply this to look at marketrisk.


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A Simple Example

Position A Position BPayout Prob Payout Prob-100 0.04 -100 0.04

0 0.96 0 0.96VaR at 95%

confidence

level 0VaR at 95%

confidence

level 0From Dowd, Kevin 2002


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A second simple example

Assume you own a 10% coupon bond thatmakes 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 yieldwill increase in the next day is .2%. The new

value of the bond is 1031.50

The differencewould represent the value atrisk.


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VAR

The value at risk therefore depends upon theprice volatility of the bond.

Where should the interest rate assumptioncome from?

historical evidence on the possible change ininterest rates.


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Calculating VaR

Three main methods

VarianceCovariance (parametric)

Historical

Monte Carlo SimulationAll measures rely on estimates of thedistribution of possible returns and the

correlation among different asset classes.


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Variance / Covariance Method

Assumes that returns are normallydistributed.

Using the characteristics of the normaldistribution it is possible to calculate thechance of a loss and probable size of the loss.


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This slide and the next few based in part on Jorion, 1997

Probability

Cardano 1565 and Pascal 1654

Pascal was asked to explain how to divide up

the winnings in a game of chance that wasinterrupted.

Developed the idea of a frequency distribution

of possible outcomes.


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An example

Assume that you are playing a game basedon the roll of two fair dice.

Each one has six possible sides that may landface up, each face has a separate number, 1to 6.

The total number of dice combinations is 36,the probability that any combination of thetwo dice occurs is 1/36


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Example continued

The total number shown on the dice rangesfrom 2 to 12. Therefore there are a total of

12 possible numbers that may occur as partof the 36 possible outcomes.

A frequency distribution summarizes thefrequency that any number occurs.

The probability that any number occurs isbased upon the frequency that a givennumber may occur.


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Establishing the distribution

Let x be the random variable underconsideration, in this case the total number

shown on the two dice following each role.The distribution establishes the frequencyeach possible outcome occurs and thereforethe probability that it will occur.


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Discrete Distribution

Value 2 3 4 5 6 7 8 9 10 11 12(xi)

Freq 1 2 3 4 5 6 5 4 3 2 1(n i)

Prob 1 2 3 4 5 6 5 4 3 2 1

(pi

) 36 36 36 36 36 36 36 36 36 36 36


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Cumulative Distribution

The cumulative distribution represents thesummation of the probabilities.

The number 2 occurs 1/36 of the time, thenumber 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

Value 2 3 4 5 6 7 8 9 10 11 12Prob 1 2 3 4 5 6 5 4 3 2 1

(p i) 36 36 36 36 36 36 36 36 36 36 36

Cdf 1 3 6 10 15 21 26 30 33 35 3636 36 36 36 36 36 36 36 36 36 36


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Probability Distribution Function(pdf)

The probabilities form a pdf. The sum of theprobabilities must sum to 1.

The distribution can be characterized by twovariables, its mean and standard deviation

111

1

i

ip


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Mean

The mean is simply the expected value fromrolling the dice, this is calculated by

multiplying the probabilities by the possibleoutcomes (values).

In this case it is also the value with thehighest frequency (mode)

736

252)(

11

1

i

iixpxE


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Standard Deviation

The variance of the random variable isdefined as:

The standard deviation is defined as the

square root of the variance.

36210)]([)(

11

1

2 i ii xExpxV

415.2)()( xVxSD

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Using the example in VaR

Assume that the return on your assets isdetermined by the number which occursfollowing the roll of the dice.

If a 7 occurs, assume that the return for thatday is equal to 0. If the number is less than 7a loss of 10% occurs for each number less

than 7 (a 6 results in a 10% loss, a 5 resultsin a 20% loss etc.)

Similarly if the number is above 7 a gain of10% occurs.

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Discrete Distribution

Value 2 3 4 5 6 7 8 9 10 11 12(xi)

Return -50% -40% -30% -20% -10% 0 10% 20% 30% 40% 50%(n i)

Prob 1 2 3 4 5 6 5 4 3 2 1(p i) 36 36 36 36 36 36 36 36 36 36 36

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VaR

Assume you want to estimate the possibleloss that you might incur with a given

probability.Given the discrete dist, the most you mightlose is 50% of the value of your portfolio.

VaR combines this idea with a givenprobability.

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VaR

Assume that you want to know the largestloss 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% ofthe rolls will not result in a loss of greaterthan 40%.

A 40% or greater loss occurs in 3/36=8.33%of rolls or 91.67% of the rolls will not resultin a loss greater than 30%

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Continuous time

The previous example assumed that therewere a set number of possible outcomes.

It is more likely to think of a continuous set ofpossible payoffs.

In this case let the probability density function

be represented by the function f(x)

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Discrete vs. Continuous

Previously we had the sum of the probabilitiesequal to 1. This is still the case, however thesummation is now represented as an integralfrom negative infinity to positive infinity.

Discrete Continuous

111

1

i

ip

1)( dxxf

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The expected value of X is then found usingthe same principle as before, the sum of theproducts of X and the respective probabilities

Discrete Continuous

dxxxfXE )()(

n

i

iixpXE1

)(

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The variance of X is then found using thesame principle as before.

Discrete Continuous

dxxfXExXV )()]([)( 2

N

i

ii XExpXV1

2)]([)(

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Combining Random Variables

One of the keys to measuring market risk isthe ability to combine the impact of changes

in different variables into one measure, thevalue at risk.

First, lets look at a new random variable, thatis the transformation of the original randomvariable X.

Let Y=a+bX where a and b are fixedparameters.

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Linear Combination

The expected value of Y is then found usingthe same principle as before, the sum of theproducts of Y and the respective probabilities

dxxxfXE )()(

dxxyfYE )()(

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Linear Combinations

We can substitute since Y=a+bX, then simplify byrearranging

)(

)()()()()(

)()(

XbEa

dxxxfbdxxfadxxfbXabXaE

dxxyfYE

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Variance

Similarly the variance can be found

)()()]([

)()]([

)()]()[()()(

222

2

2

XVbdxxfXExb

dxxfXbEabxa

dxxfbXaEbXabXaVYV

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Standard Deviation

Given the variance it is easy to see that the standarddeviation will be

)(XbSD

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Combinations of RandomVariables

No let Y be the linear combination of two randomvariables X1and X2the probability density function(pdf) is now f(x1,x2)

The marginal distribution presents the distributionas based upon one variable for example.

)(),( 12212 xfdxxxf

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Expectations

)()()()(

),(),(

),(),(

),()()(

2122221111

212111222121211

21212212121121

2121212121

XEXEdxxfxdxxfx

dxdxxxfxxdxdxxxfx

dxdxxxfxdxdxxxfx

dxdxxxfxxXXE

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Variance

Similarly the variance can be reduced

),(2)()(

),()]([

)(

2121

2121

2

212121

21

XXCovXVXV

dxdxxxfXXExx

XXV

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A special case

If the two random variables are independentthen the covariance will reduce to zero which

implies thatV(X1+X2) = V(X1)+V(X2)

However this is only the case if the variablesare independentimplying hat there is nogain from diversification of holding the twovariables.

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The Normal Distribution

For many populations of observations as thenumber of independent draws increases, thepopulation will converge to a smooth normal

distribution.

The normal distribution can be characterized by itsmean (the location) and variance (spread) N(m,s2).

The distribution function is

2

2 )(

2

1

22

1)(

ms

s

x

exf

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Standard Normal Distribution

The function can be calculated for variousvalues of mean and variance, however the

process is simplified by looking at a standardnormal distribution with mean of 0 andvariance of 1.

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Standard Normal Distribution

Standard Normal Distributions are symmetricaround the mean. The values of the

distribution are based off of the number ofstandard deviations from the mean.

One standard deviation from the meanproduces a confidence interval of roughly

68.26% of the observations.

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Prob Ranges for Normal Dist.

68.26%95.46%

99.74%

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An Example

Lets define X as a function of a standard normalvariable e (in other words e is N(0,1))

X= m es

We showed earlier that

Therefore

)()( XbEabXaE

mesmsem )()( EE

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Variance

We showed that the variance was equal to

Therefore

22 )()( sessem VV

)()( 2

XVbbXaV

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An Example

Assume that we know that the movements inan exchange rate are normally distributed

with mean of 1% and volatility of 12%.Given that approximately 95% of thedistribution is within 2 standard deviations ofthe mean it is easy to approximate the

highest and lowest return with 95%confidence

XMIN= 1% - 2(12%) = -23%

XMAX= 1% + 2(12%) = +25%

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One sided values

Similarly you can find the standard deviationthat represents a one sided distribution.

Given that 95.46% of the distribution liesbetween -2 and +2 standard deviations of themean, 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.

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VaR

Given the last slide it is easy to see that youwould be 97.73% confident that the loss

would not exceed -23%.


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VAR A second example

Assume that the mean yield change on a bondwas zero basis points and that the standard

deviation of the change was 10 Bp or 0.001Given that 90% of the area under the normaldistribution is within 1.65 standard deviations

on either side of the mean (in other words

between mean-1.65sand mean +1.65s)

There is only a 5% chance that the level ofinterest rates would increase or decrease by

more than 0 + 1.65(0.001) or 16.5 Bp

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Price change associated with16.5Bp change.

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 couldcalculate an estimate based upon duration.

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Example 2

Assume we own seven year zero couponbonds with a face value of $1,631,483.00

with a yield of 7.243%Todays Market Value

$1,631,483/(1.07243)7=$1,000,000

If rates increase 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

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Approximations - Duration

The duration of the bond would be 7 since itis a zero coupon.

Modified duration is then 7/1.07143 = 6.527The price change would then be

1,000,000(-6.57)(.00165) = $10,769.55

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Approximations - linear

Sometimes it is also estimated by figuring thethe 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 aprice decrease of $652.77.

This is a 652.77/1,000,000 = .06528%

change in the price of the bond per basispoint

The value at risk is then

1,000,000(.00065277)(16.5) = $10,770.71

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Precision

The actual calculation of the change shouldbe accomplished by discounting the value of

the bond across the zero coupon yield curve.In our example we only had one cash flow.

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DEAR

Since we assumed that the yield change wasassociated with a daily movement in rates, wehave 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)

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VAR

Given the DEAR you can calculate the Valueat Risk for a given time frame.

VAR = DEAR(N)0.5

Where N = number of days

(Assumes constant daily variance and noautocorrelation in shocks)

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N

Bank for International Settlements (BIS) 1998market risk capital requirements are based on

a 10 day holding period.

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Problems with estimation

Fat TailsMany securities have returns thatare not normally distributed, they have fat

tails This will cause an underestimation ofthe risk when a normal distribution is used.

Do recent market events change thedistribution? Risk Metrics weights recent

observations higher when calculatingstandard Dev.

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Interest Rate Risk vs.Market Risk

Market risk is more broad, but Interest RateRisk is a component of Market Risk.

Market risk should include the interaction ofother economic variables such as exchangerates.

Therefore, we need to think about thepossibility of an adverse event in theexchange rate market and equity marketsetc.. Not just a change in interest rates..

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DEAR of a foreign Exchange Position

Assume the firm has Swf 1.6 Million tradingposition in swiss francs

Assume that the current exchange rate isSwf1.60 / $1 or $.0625 / Swf

The $ value of the francs is then

Swf1.6 million ($0.0625/Swf) =$1,000,000

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FX DEAR

Given a standard deviation in the exchangerate 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 thatwill not occur more than 5% of the time soagain we can look at 1.65s

FX volatility is then 1.65(56.5bp) = 93.2bp or0.932%

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FX DEAR

DEAR = (Dollar value )( FX volatility)

=($1,000,000)(.00932)

=$9,320

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Equity DEAR

The return on equities can be split intosystematic and unsystematic risk.

We know that the unsystematic risk can bediversified away.

The undiversifiable market risk will equal bebased on the beta of the individual stock

22

mis

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Equity DEAR

If the portfolio of assets has a beta of 1 thenthe market risk of the portfolio will also have

a beta of 1 and the standard deviation of theportfolio can be estimated by the standarddeviation of the market.

Let sm= 2% then using the same confidence

interval, the volatility of the market will be

1.65(2%) = 3.3%

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UV GN


Equity DEAR

DEAR = (Dollar value )( Equity volatility)

=($1,000,000)(0.033)=$33,000

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VAR and Market Risk

The market risk should then estimate thepossible change from all three of the assetclasses.

This DOES NOT just equal the summation ofthe three estimates of DEAR because thecovariance of the returns on the differentassets must be accounted for.

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Aggregation

The aggregation of the DEAR for the threeassets can be thought of as the aggregationof three standard deviations.

To aggregate we need to consider thecovariance among the different asset classes.

Consider the Bond, FX position and Equitythat we have recently calculated.

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Variance Covariance

Seven Yearzero

Swf/$1US StockIndex

Seven YearZero

1 -.20 .4

Swf/$1 1 .1

US StockIndex

1

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variance covariance

2

1

USz,

USSwf,

Swfz,

2

US

2

Swf

2

Z

)2(

)2(

)2()(DEAR)(DEAR)(DEAR

Portfolio

DEAR

USZ

USSwf

SwfZ

DEARDEAR

DEARDEAR

DEARDEAR

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VAR for Portfolio

969,39$

)33)(77.10)(4)(.2(

)33)(32.9)(1)(.2(

)32.9)(77.10)(2.(2(

)(33)(9.32)(10.77

Portfolio

DEAR

2

1222

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Comparison

If the simple aggregation of the threepositions occurred then the DEAR would havebeen estimated to be $53,090. It is easy toshow that the if all three assets were perfectlycorrelated (so that each of their correlationcoefficients was 1 with the other assets) you

would calculate a loss of $52,090.

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Risk Metrics

JP Morgan has the premier service forcalculating the value at risk

They currently cover the daily updating andproduction of over 450 volatility andcorrelation estimates that can be used incalculating VAR.

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Normal Distribution Assumption

Risk Metrics is based on the assumption thatall asset returns are normally distributed.

This is not a valid assumption for many asstsfor example call optionsthe most aninvestor can loose is the price of the calloption. The upside is large, this implies a

large positive skew.

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Normal Assumption Illustration

Assume that a financial institution has a largenumber of individual loans. Each loan can bethought of as a binomial distribution, the loaneither repays in full or there is default.

The sum of a large number of binomialdistributions converges to a normal

distribution assuming that the binomial areindependent.

Therefore the portfolio of loans could b

thought of as a normal distribution.

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Normal Illustration continued

However, it is unlikely that the loans are trulyindependent. In a recession it is more likelythat many defaults will occur.

This invalidates the normal distributionassumption.

The alternative to the assumption is to use ahistorical back simulation.


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Back Simulation

Step 1: Measure exposures. Calculate thetotal $ valued exposure to each assets

Step 2: Measure sensitivity. Measure thesensitivity of each asset to a 1% change ineach of the other assets. This number is thedelta.

Step 3: Measure Risk. Look at the annual %change of each asset for the past day andfigure out the change in aggregate exposure

that day.

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Back Simulation

Step 4 Repeat step 3 using historical data foreach of the assets for the last 500 days

Step 5 Rank the days from worst to best.Then decide on a confidence level. If youwant a 5% probability look at the return with95% of the returns better and 5% of the

return worse.Step 6 calculate the VAR

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Historical Simulation

Provides a worst case scenario, where Riskmetrics the worst case is a loss of negativeinfinity

Problems:

The 500 observations is a limited amount, thusthere is a low degree of confidence that it

actually represents a 5% probability. Shouldwe change the number of days??

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Monte Carlo Approach

Calculate the historical variance covariancematrix.

Use the matrix with random draws to simulate10,000 possible scenarios for each asset.

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BIS Standardized Framework

Bank of International Settlements proposed astructured framework to measure the marketrisk of its member banks and the offsettingcapital required to manage the risk.

Two options

Standardized Framework (reviewed below)

Firm Specific Internal Framework

Must be approved by BIS

Subject to audits

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Risk Charges

Each asset is given a specific risk chargewhich represents the risk of the asset

For example US treasury bills have a riskweight of 0 while junk and would have a riskweight of 8%.

Multiplying the value of the outstandingposition by the risk charges provides capitalrisk charge for each asset.

Summing provides a total risk charge

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Specific Risk Charges

Specific Risk charges are intended to measurethe risk of a decline in liquidity or credit riskof the trading portfolio.

Using these produces a specific capitalrequirement for each asset.

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General Market Risk Charges

Reflect the product of the modified durationand expected interest rate shocks for eachmaturity

Remember this is across different types ofassets with the same maturity.

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Vertical Offsets

Since each position has both long and shortpositions for different assets, it is assumedthat they do not perfectly offset each other.

In other words a 10 year T-Bond and a highyield bond with a 10 year maturity.

To counter act this the is a vertical offset ordisallowance factor.

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Horizontal Offsets

Within Zones

For each maturity bucket there are

differences in maturity creating again theinability to let short and long positions exactlyoffset each other.

Between Zones

Also across zones the short an long positionsmust be offset.

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VaR Problems

Artzner (1997), (1999) has shown that VaR isnot a coherent measure of risk.

For Example it does not posses the propertyof subadditvity. In other words the combined

portfolio VaR of two positions can be greaterthan the sum of the individual VaRs

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A Si l E l *


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A Simple Example*

Assume a financial institution is facing thefollowing three possible scenarios andassociated losses

Scenario Probability Loss1 .97 02 .015 100

3 .015 0

The VaR at the 98% level would equal = 0

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A Simple Example

Assume you the previous financial institution and itscompetitor facing the same three possible scenarios

Scenario Probability Loss A Loss B Loss A & B

1 .97 0 0 02 .015 100 0 1003 .015 0 100 100

The VaR at the 98% level for A or B alone is 0The Sum of the individual VaRs = VaRA+ VaRB= 0

The VaR at the 98% level for A and B combinedVaR(A+B)=100

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Coherence of risk measures

Let (X) and (Y) be measures of riskassociated with event X and event Yrespectively

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 additionalconstant amount of loss a, (X+a) = (X)+a.

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Coherent Measures of Risk

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 lossconditioned 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 UNIVERSITEDAUVERGNE

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Tail VaR*

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 33 1 2 3

4 5 4 9

5 3 3 6

VaR60% 4 4 9

TVaR60% 4.5 4.5 9

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Normal Distribution


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Normal Distribution

How important is the assumption thateverything is normally distributed?

It depends on how and why a distributiondiffers from the normal distribution.

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S&P 500 Monthly Returns vs. Normal Dist

-30

20

70

120

170

220

- 0 . 4 7 5 - 0 . 4 2 5 - 0 . 3 7 5 - 0 . 3 2 5 - 0 . 2 7 5 - 0 . 2 2 5 - 0 . 1 7 5 - 0 . 1 2 5 - 0 . 0 7 5 - 0 . 0 2 5 0 . 0 2 5 0 . 0 7 5 0 . 1 2 5 0 . 1 7 5 0 . 2 2 5 0 . 2 7 5 0 . 3 2 5 0 . 3 7 5 0 . 4 2 5 0 . 4 7 5

Returns

Observations

S&P

Nor mal


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D kTwo explanations of Fat Tails


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Two explanations of Fat Tails

The true distribution is stationary andcontains fat tails.

In this case normal distribution would beinappropriate

The distribution does change through time.

Large or small observations are outliers drawnfrom a distribution that is temporarily out ofalignment.

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D kImplications


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Implications

Both explanations have some truth, it isimportant to estimate variations from theunderlying assumed distribution.

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D kMeasuring Volatilities


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Measuring Volatilities

Given that the normality assumption is centralto the measurement of the volatility andcovariance estimates, it is possible to attemptto adjust for differences from normality.

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D kMoving Average


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Moving Average

One solution is to calculate the moving average ofthe volatility

M

rM

i

it

1

2

s

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D kMoving Averages


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Moving Averages

Moving Avergages of Volatility S&P 500 Monthly Return

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0 100 200 300 400 500 600 700 800 900

1 year

2 year

5 year

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D kHistorical Simulation


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Historical Simulation

Another approach is to take the daily pricereturns and sort them in order of highest tolowest.

The volatility is then found based off of aconfidence interval.

Ignores the normality assumption! But

causes issues surrounding window ofobservations.

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D kNonconstant Volatilities


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Nonconstant Volatilities

So far we have assumed that volatility isconstant over time however this may not bethe case.

It is often the case that clustering of returnsis observed (successive increases ordecreases in returns), this implies that thereturns are not independent of each other aswould be required if they were normallydistributed.

If this is the case, each observation should

not be equally weighted. UNIVERSITEDAUVERGNE

D kRiskMetrics


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RiskMetrics

JP Morgan uses an Exponentially WeightedMoving Average.

This method used a decay factor thatweights each days percentage price change.

A simple version of this would be to weight bythe period in which the observation took

place.


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DrakeDecay Factors


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Decay Factors

JP Morgan uses a decay factor of .94 for dailyvolatility estimates and .97 for monthlyvolatility estimates

The choice of .94 for daily observationsemphasizes that they are focused on veryrecent observations.

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Drake


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Decay Factors

0

0.01

0.02

0.03

0.04

0.05

0.06

0 20 40 60 80 100 120 140 160

Days

Weighting

0.94

0.97

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DrakeMeasuring Correlation


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Measuring Correlation

Covariance:

Combines the relationship between the stocks

with the volatility.(+) the stocks move together

(-) The stocks move opposite of each other

iBBiAAi PkkkkABCov ))(()(

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DrakeMeasuring Correlation 2


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Measuring Correlation 2

Correlation coefficient: The covariance isdifficult to compare when looking at different

series. Therefore the correlation coefficient isused.

The correlation coefficient will range from

-1 to +1

)/()( BAAB ABCovr ss

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DrakeTiming Errors


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Timing Errors

To get a meaningful correlation the pricechanges of the two assets should be taken atthe exact same time.

This becomes more difficult with a highernumber 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 thecorrelations look normal

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DrakeSize of portfolio


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Size of portfolio

Many institutions do not consider it practicalto calculate the correlation between each pairof assets.

Consider attempting to look at a portfolio thatconsisted of 15 different currencies. For eachcurrency there are asset exposures in various

maturities.To be complete assume that the yield curvefor each currency is broken down into 12

maturities UNIVERSITEDAUVERGNE

DrakeCorrelations continued


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Correlations continued

The combination of 12 maturities and 15currencies would produce 15 x 12 = 180separate movements of interest rates that

should be investigated.

Since for each one the correlation with eachof the others should be considered, this would

imply 180 x 180 = 16,110 separatecorrelations that would need to bemaintained.

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DrakeReducing the work


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Reducing the work

One possible solution to this would bereducing the number of necessarycorrelations 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 oneassert and short in another to take advantageof changes in the spread)

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DrakeA compromise


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A compromise

Most VaR can be accomplished by developinga hierarchy of correlations based on theamount of each type of trading. It also will

depend upon the aggregation in the portfoliounder consideration. As the aggregationincreases, fewer correlations are necessary.

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DrakeBack Testing


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Back Testing

To look at the performance of a VaR model,can be investigated by back testing.

Back testing is simply looking at the loss on aportfolio compared to the previous days VaRestimate.

Over time the number of days that the VaR

was exceed by the loss should be roughlysimilar to the amount specified by theconfidence in the model

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DrakeBasle Accords


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Basle Accords

To use VaR to measure risk the Basle accordsspecify that banks wishing to use VaR mustundertake two different types of back testing.

Hypotheticalfreeze the portfolio and testthe performance of the VaR model over aperiod of time

Trading OutcomeAllow the portfolio tochange (as it does in actual trading) andcompare the performance to the previous

days VaR UNIVERSITEDAUVERGNE

DrakeBack Testing Continued


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Back Testing Continued

Assume that we look at a 1000 day windowof previous results. A 95% confidenceinterval implies that the VaR level should have

been exceed 50 times.

Should the model be rejected if the it is foundthat the VaR level was exceeded 55 times?

70 times? 100 times?

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DrakeBack test results


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Back test results

Whether or not the actual number ofexceptions differs significantly from theexpectation can be tested using the Z score

for a binomial distribution.

Type I errorthe model has beenerroneously rejected

Type II errorthe model has beenerroneously accepted.

Basle specifies a type one error test.

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DrakeOne tail versus two tail


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One tail versus two tail

Basle does not care if the VaR modeloverestimates the amount of loss and thenumber of exceptions is low ( implies a one

tail test)The bank, however, does care if the numberof exceptions is low and it is keeping toomuch capital (implies a two tail test).

Excess Capital

Trading performance based upon economiccapital

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DrakeApproximations


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Approximations

Given a two tail 95% confidence test and 1000days of back testing the bank would accept 39to 61 days that the loss exceeded the VaR

level.

However this implies a 90% confidence for theone tail test so Basle would not be satisfied.

Given a two tail test and a 99% confidencelevel the bank would accept 6 to 14 days thatthe loss exceeded the trading level, under the

same test Basle would accept 0 to 14 days UNIVERSITEDAUVERGNE

DrakeEmpirical Analysis of VaR

(B t 1998)


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DrakeDrake University(Best 1998)

Whether or not the lack of normality is not aproblem was discussed by Best 1998(Implementing VaR)

Five years of daily price movements for 15assets from Jan 1992 to Dec 1996. Thesample process deliberately chose assets that

may be non normal.VaR Was calculated for each asset individuallyand for the entire group as a portfolio.

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DrakeFigures 4. Empirical Analysis of VaR

(Best 1998)


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DrakeDrake University(Best 1998)

All Assets have fatter tails than expectedunder a normal distribution.

Japanese 3-5 year bonds show significantnegative skew

The 1 year LIBOR sterling rate shows nothingclose to normal behavior

Basic model work about as well as moreadvanced mathematical models

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DrakeBasle Tests


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Requires that the VaR model must calculateVaR with a 99% confidence and be testedover at least 250 days.

Table 4.6

Low observation periods perform poorly whilehigh observation periods do much better.

Clusters of returns cause problem for theability of short term models to perform, thisassumes that the data has a longer memory

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DrakeBasle


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a eDrake University

The Basle requirements supplement VaR byRequiring that the bank originally hold 3 timesthe amount specified by the VaR model.

This is the product of a desire to producesafety and soundness in the industry


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DrakeStress Testing


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Drake University

g

Stress Testing is basically a large scenarioanalysis. The difficulty is identifying theappropriate scenarios.

The key is to identify variables that wouldprovide a significant loss in excess of the VaRlevel and investigate the probability of those

events occurring.

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DrakeStress Tests


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Drake University

Some events are difficult to predict, forexample, terrorism, natural disasters, politicalchanges in foreign economies.

In these cases it is best to look at similar pastevents 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 ofreturns using stress testing.

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DrakeStress Testing withHistorical Simulation


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Drake UniversityHistorical Simulation

The most straightforward approach is to lookat changes in returns.

For example what is the largest loss thatoccurred 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 casescenario result.

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DrakeStress Testing

Other Simulation Techniques


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Drake UniversityOther Simulation Techniques

Monte Carlo simulation can also be employedto look at the possible bad outcomes basedon past volatility and correlation.

The key is that changes in price and returnthat are greater than those implied by a threestandard deviation change need to beinvestigated.

Using simulation it is also possible to ask whathappens it correlations change, or volatilitychanges of a given asset or assets.

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DrakeManaging Risk with VaR


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Drake University

g g

The Institution must first determine itstolerance for risk.

This can be expressed as a monetary amountor as a percentage of an assets value.

Ultimately VaR expresses an monetaryamount of loss that the institution is willing to

suffer and a given frequency determined bythe timing confidence level..

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DrakeManaging Risk with VaR


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Drake University

g g

The tolerance for loss most likely increaseswith the time frame. The institution may bewilling to suffer a greater loss one time each

year (or each 2 years or 5 years), but that isdifferent than one day VaR.

For Example, given a 95% confidence level

and 100 trading days, the one day VaR wouldoccur approximately once a month.

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DrakeSetting Limits


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Drake University

g

The VaR and tolerance for risk can be used toset limits that keep the institution in anacceptable risk position.

Limits need to balance the ability of thetraders to conduct business and the risktolerance of the institution. Some risk needs

to be accepted for the return to be earned.

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DrakeVaR Limits


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Drake University

Setting limits at the trading unit level

Allows trading management to balance thelimit across traders and trading activities.

Requires management to be experts in thecalculation of VaR and its relationship withtrading practices.

Limits for individual tradersVaR is not familiar to most traders (they d onot work with it daily and may not understandhow different choices impact VaR.

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DrakeVaR and changes in volatility


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Drake University

One objection of many traders is that achange in the volatility (especially if it iscalculated based on moving averages) can

cause a change in VaR on a given position.Therefore they can be penalized for a positioneven if they have not made any trading

decisions.Is the objection a valid reason to not use

VaR?

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DrakeStress Test Limits


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Drake University

Similar to VaR limits should be set on theacceptable loss according to stress limittesting (and its associated probability).

dess analyst value at risk

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