1. Market Risk and Value at Risk
Finance 129

2. Market Risk
Macroeconomic changes can create uncertainty in the earnings of the Financial Institutions
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. VaR at 95% confidence level
A Simple Example
Position A
Position B
Payout
Prob
-100
0.04
0.96
VaR at 95% confidence level
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
Assume that you believe that the most the yield will increase in the next day is .2%. The new value of the bond is

8. VAR
The value at risk therefore depends upon the price volatility of the bond.
Where should the interest rate assumption come from?

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. 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 was interrupted.
Developed the idea of a frequency distribution of possible outcomes.
This slide and the next few based in part on Jorion, 1997

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
(x i)
Freq
(n i)
Prob
(p i)

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
Prob
(p i)
Cdf

18. Probability Distribution Function (pdf)
The probabilities form a pdf. The sum of the probabilities must sum to 1.

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

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 =.9722 or 97.22% of the rolls will not result in a loss of greater than 40%.
A 40% or greater loss occurs in 3/36=8.33% of rolls or 91.67% of the rolls will not result in a loss greater than 30%

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.

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 Y=a+bX, 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(X1+X2) = V(X1)+V(X2)
However this is only the case if the variables are independent – implying that there is no gain from diversification of holding the two variables.

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

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% )/2 = 2.27% of the distribution is in each tail.
This shows that 95.46% % = 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 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 (0.001) 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, with a yield of 7.243%
Today’s Market Value
$1,631,483/( )7=$1,000,000
If rates increase to by 16.5Bp to 7.408% the market value is
$1,631,483/( )7 = $989,295.75

51. Approximations - Duration
The duration of the bond would be 7 since it is a zero coupon.
Modified duration is then 7/ = 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, or a price decrease of $
This is a /1,000,000 = % change in the price of the bond per basis point
The value at risk is then

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.

59. DEAR of a foreign Exchange Position
Assume the firm has a 1.6 Million trading position in euros
Assume that the current exchange rate is Euro1.60 / $1 or $.0625 / Euro
The $ value of the francs is then
E1.6 million ($0.0625/Euro) =$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. DEAR = (Dollar value )( FX volatility)
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 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,

64. DEAR = (Dollar value )( Equity volatility)
Equity DEAR
DEAR = (Dollar value )( Equity volatility)

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 Seven Year zero E/$1 US Stock Index
-.20 .4 .1

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 $53,090.

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.

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.

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.

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.
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%.

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.

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 and 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 VaR’s

87. *This and subsequent examples are based on Meyers 2002
A Simple Example*
Assume a financial institution is facing the following three possible scenarios and associated losses
Scenario Probability Loss
The VaR at the 98% level would equal = 0
*This and subsequent examples are based on Meyers 2002

88. A Simple Example
Assume the previous financial institution and its competitor facing the same three possible scenarios
Scenario Probability Loss A Loss B Loss A & B
The VaR at the 98% level for A or B alone is 0
The Sum of the individual VaR’s = VaRA + VaRB = 0
The VaR at the 98% level for A and B combined
VaR(A+B)=100

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(X+Y) < r(X) + r(Y).
Monotonicity. Implies X>Y then r(X) > r(Y).
Positive homogeneity:Given l > 0 r(lX) = lr(X).
Translation Invariance. Given an additional constant amount of loss a, r(X+a) = 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. Scenario X1 X2
X1+X2 1 4 5 9 2 3 6
VaR60%
TVaR60%
4.5

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.

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.

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.

103. RiskMetrics JP Morgan uses an Exponentially Weighted Moving Average.
This method used a decay factor that weight’s each days percentage price change.
A simple version of this would be to weight by the period in which the observation took place.

104. Risk Metrics
Where l
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.

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.

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 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).

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 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 Testing Other 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 a 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).