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Market Riskand Value at Risk

- Finance 129

Market Risk

- 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

- 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

- 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

- 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.

A Simple Example

From Dowd, Kevin 2002

A second simple example

- 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

- 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

- 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.

Variance / Covariance Method

- 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

- 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.

An example

- 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

- 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

- 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.

Discrete Distribution

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

i) 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 (p

i) 36 36 36 36 36 36 36 36 36 36 36

Cumulative Distribution

- 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.

Cumulative Distribution

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

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

36 36 36 36 36 36

Probability Distribution Function (pdf)

- The probabilities form a pdf. The sum of the

probabilities must sum to 1. - The distribution can be characterized by two

variables, its mean and standard deviation

Mean

- The mean is simply the expected value from

rolling the dice, this is calculated by

multiplying the probabilities by the possible

outcomes (values). - In this case it is also the value with the

highest frequency (mode)

Standard Deviation

- The variance of the random variable is defined

as - The standard deviation is defined as the square

root of the variance.

Using the example in VaR

- 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.

Discrete Distribution

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

i) 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

VaR

- 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

- 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 3/368.33 of

rolls or 91.67 of the rolls will not result in a

loss greater than 30

Continuous time

- 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

- 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 Continuous

Discrete vs. Continuous

- 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

Discrete vs. Continuous

- The variance of X is then found using the same

principle as before. - Discrete Continuous

Combining Random Variables

- 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 YabX where a and b are fixed parameters.

Linear Combination

- 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

Linear Combinations

- We can substitute since YabX, then simplify by

rearranging

Variance

- Similarly the variance can be found

Standard Deviation

- Given the variance it is easy to see that the

standard deviation will be

Combinations of Random Variables

- 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.

Expectations

Variance

- Similarly the variance can be reduced

A special case

- If the two random variables are independent then

the covariance will reduce to zero which implies

that - V(X1X2) V(X1)V(X2)
- However this is only the case if the variables

are independent implying hat there is no gain

from diversification of holding the two

variables.

The Normal Distribution

- 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

Standard Normal Distribution

- 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

- 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

An Example

- 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
- Therefore

Variance

- We showed that the variance was equal to
- Therefore

An Example

- 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

- 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

- Given the lastit is easy to see that you

would be 97.73 confident that the loss would not

exceed -23.

Continuous Time

- Let q represent quantile such that the area to

the right of q represents a given probability of

occurrence. - In our example above -2.00 would produce a

probability of 97.73 for the standard normal

distribution

VAR A second example

- 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.

- 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

- Assume we own seven year zero coupon bonds with a

face value of 1,631,483.00 with a yield of

7.243 - Todays Market Value
- 1,631,483/(1.07243)71,000,000
- 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

Approximations - Duration

- The duration of the bond would be 7 since it is a

zero coupon. - Modified duration is then 7/1.07143 6.527
- The price change would then be
- 1,000,000(-6.57)(.00165) 10,769.55

Approximations - linear

- 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

- 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

- 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

- 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

- Bank for International Settlements (BIS) 1998

market risk capital requirements are based on a

10 day holding period.

Problems with estimation

- 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

- 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

- 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

- 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

Equity DEAR

- 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 equal be

based on the beta of the individual stock

Equity DEAR

- 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

- DEAR (Dollar value )( Equity volatility)
- (1,000,000)(0.033)
- 33,000

VAR and Market Risk

- 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

- 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.

Variance Covariance

variance covariance

VAR for Portfolio

Comparison

- 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 52,090.

Risk Metrics

- 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

- 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

- 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 b thought

of as a normal distribution.

Normal Illustration continued

- 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

- 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 were realized instead of

attempting to estimate it as a normal

distribution.

Back Simulation

- 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

- 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

- 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

- Calculate the historical variance covariance

matrix. - Use the matrix with random draws to simulate

10,000 possible scenarios for each asset.

BIS Standardized Framework

- 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

- 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

- 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

- 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

- 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 the is a vertical offset or

disallowance factor.

Horizontal Offsets

- Within 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 Zones
- Also across zones the short an long positions

must be offset.

VaR Problems

- 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 VaRs

A Simple Example

- Assume a financial institution is facing the

following three possible scenarios and associated

losses - Scenario Probability Loss
- 1 .97 0
- 2 .015 100
- 3 .015 0
- The VaR at the 98 level would equal 0
- This and subsequent examples are based on Meyers

2002

A Simple Example

- 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 VaRs VaRA VaRB 0
- The VaR at the 98 level for A and B combined
- VaR(AB)100

Coherence of risk measures

- Let r(X) and r(Y) be measures of risk associated

with event X and event Y respectively - Subadditvity implies r(XY) lt r(X) r(Y).
- Monotonicity. Implies XgtY then r(X) gt r(Y).
- Positive homogeneityGiven l gt 0 r(lX) lr(X).

- Translation Invariance. Given an additional

constant amount of loss a, r(Xa) r(X)a.

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 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

- 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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Normal Distribution

- How important is the assumption that everything

is normally distributed? - It depends on how and why a distribution differs

from the normal distribution.

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Two explanations of Fat Tails

- 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

- Both explanations have some truth, it is

important to estimate variations from the

underlying assumed distribution.

Measuring Volatilities

- 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

- One solution is to calculate the moving average

of the volatility

Moving Averages

Historical Simulation

- 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

- 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

- JP Morgan uses an Exponentially Weighted Moving

Average. - This method used a decay factor that weights

each days percentage price change. - A simple version of this would be to weight by

the period in which the observation took place.

Risk Metrics

- 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

- 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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Measuring Correlation

- Covariance
- Combines the relationship between the stocks with

the volatility. - () the stocks move together
- (-) The stocks move opposite of each other

Measuring Correlation 2

- Correlation coefficient The covariance is

difficult to compare when looking at different

series. Therefore the correlation coefficient is

used. - The correlation coefficient will range from
- -1 to 1

Timing Errors

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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)

- 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)

- 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

- 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

- 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

- 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

- 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

- 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

- 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 TestingOther Simulation Techniques

- 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

- 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 an 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

- 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

- 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

- 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

- 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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