Market Risk and Value at Risk

1
Market Riskand Value at Risk

• Finance 129

2
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).

3
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

4
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.

5
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.

6
A Simple Example
From Dowd, Kevin 2002
7
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.

8
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.

9
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.

10
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.

11
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.

12
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

13
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.

14
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.

15
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
16
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.

17
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
18
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

19
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)

20
Standard Deviation

• The variance of the random variable is defined
  as
• The standard deviation is defined as the square
  root of the variance.

21
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.

22
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
23
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.

24
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

25
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)

26
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

27
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

28
Discrete vs. Continuous

• The variance of X is then found using the same
  principle as before.
• Discrete Continuous

29
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.

30
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

31
Linear Combinations

• We can substitute since YabX, then simplify by
  rearranging

32
Variance

• Similarly the variance can be found

33
Standard Deviation

• Given the variance it is easy to see that the
  standard deviation will be

34
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.

35
Expectations
36
Variance

• Similarly the variance can be reduced

37
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.

38
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

39
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.

40
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.

41
Prob Ranges for Normal Dist.
68.26
95.46
99.74
42
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

43
Variance

• We showed that the variance was equal to
• Therefore

44
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

45
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.

46
VaR

• Given the last slide it is easy to see that you
  would be 97.73 confident that the loss would not
  exceed -23.

47
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

48
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

49
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.

50
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

51
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

52
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

53
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.

54
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)

55
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)

56
N

• Bank for International Settlements (BIS) 1998
  market risk capital requirements are based on a
  10 day holding period.

57
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.

58
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..

59
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

60
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

61
FX DEAR

• DEAR (Dollar value )( FX volatility)
• (1,000,000)(.00932)
• 9,320

62
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

63
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

64
Equity DEAR

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

65
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.

66
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.

67
Variance Covariance
68
variance covariance
69
VAR for Portfolio
70
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.

71
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.

72
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.

73
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.

74
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.

75
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.

76
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.

77
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

78
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??

79
Monte Carlo Approach

• Calculate the historical variance covariance
  matrix.
• Use the matrix with random draws to simulate
  10,000 possible scenarios for each asset.

80
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

81
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

82
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.

83
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.

84
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.

85
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.

86
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

87
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

88
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

89
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.

90
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

91
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

92
(No Transcript)
93
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.

94
(No Transcript)
95
(No Transcript)
96
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.

97
Implications

• Both explanations have some truth, it is
  important to estimate variations from the
  underlying assumed distribution.

98
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.

99
Moving Average

• One solution is to calculate the moving average
  of the volatility

100
Moving Averages
101
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.

102
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.

103
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.

104
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)

105
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.

106
(No Transcript)
107
Measuring Correlation

• Covariance
• Combines the relationship between the stocks with
  the volatility.
• () the stocks move together
• (-) The stocks move opposite of each other

108
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

109
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

110
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.

111
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.

112
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)

113
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.

114
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

115
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.

116
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?

117
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.

118
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

119
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.

120
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.

121
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

122
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

123
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

124
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.

125
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.

126
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.

127
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.

128
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.

129
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..

130
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.

131
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.

132
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.

133
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?

134
Stress Test Limits

• Similar to VaR limits should be set on the
  acceptable loss according to stress limit testing
  (and its associated probability).