drake drake university fin 129 market risk and value at risk finance 129
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DrakeDRAKE UNIVERSITY
Fin 129
Market Riskand Value at Risk
Finance 129
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Fin 129Market 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).
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Fin 129
Importance of Market Risk Measurement
Management information – Provides info on the risk exposure taken by tradersSetting Limits – Allows management to limit positions taken by tradersResource Allocation – Identifying the risk and return characteristics of positionsPerformance Evaluation – trader compensation – did high return just mean high risk? Regulation – May be used in some cases to determine capital requirements
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Fin 129Measuring 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 riskOne way to measure this would be to use a measure called the value at risk.
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Fin 129Value 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.
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Fin 129A Simple Example
Position A Position B
Payout Prob Payout Prob
-100 0.04 -100 0.04
0 0.96 0 0.96
VaR at 95% confidence
level0
VaR at 95% confidence
level0
From Dowd, Kevin 2002
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Fin 129A 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 1039.56Assume 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
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Fin 129VAR
The value at risk therefore depends upon the price volatility of the bond.Where should the interest rate assumption come from?
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Fin 129Calculating VaR
Three main methodsVariance – Covariance (parametric)
Historical
Monte Carlo Simulation
All measures rely on estimates of the distribution of possible returns and the correlation among different asset classes.
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Fin 129
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.
This slide and the next few based in part on Jorion, 1997
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Fin 129Probability
Cardano 1565 and Pascal 1654Pascal 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.
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Fin 129An 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
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Fin 129Example 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.
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Fin 129Establishing 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.
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Fin 129Discrete 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
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Fin 129Cumulative 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.
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Fin 129Cumulative Distribution
Value 2 3 4 5 6 7 8 9 10 11 12Prob 1 2 3 4 5 6 5 4 3 2 1 (p i) 36 36 36 36 36 36 36 36 36 36 36
Cdf 1 3 6 10 15 21 26 30 33 35 3636 36 36 36 36 36 36 36 36 36 36
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Fin 129
Probability Distribution Function (pdf)
The probabilities form a pdf. The sum of the probabilities must sum to 1.
111
1
i
ip
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Fin 129Mean
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)
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Fin 129Standard Deviation
The variance of the random variable is defined as:
The standard deviation is defined as the square root of the variance.
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Fin 129Using 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.
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Fin 129Discrete 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
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Fin 129VaR
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.
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Fin 129VaR
Assume that you want to know the largest loss that may occur in 95% of the rolls.A 50% loss occurs 1/36 = 2.77% 0f the time. This implies that 1-.027 =.9722 or 97.22% of the rolls will not result in a loss of greater than 40%.A 40% or greater loss occurs in 3/36=8.33% of rolls or 91.67% of the rolls will not result in a loss greater than 30%
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Fin 129Continuous 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)
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Fin 129Discrete 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
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Fin 129Discrete 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
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Fin 129Discrete vs. Continuous
The variance of X is then found using the same principle as before.
Discrete Continuous
dxxfXExXV )()]([)( 2
N
iii XExpXV
1
2)]([)(
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Fin 129
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.
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Fin 129Linear 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
dxxxfXE )()(
dxxyfYE )()(
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Fin 129Linear Combinations
We can substitute since Y=a+bX, then simplify by rearranging
)(
)()()()()(
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dxxxfbdxxfadxxfbXabXaE
dxxyfYE
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Fin 129Variance
Similarly the variance can be found
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dxxfXbEabxa
dxxfbXaEbXabXaVYV
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Fin 129Standard Deviation
Given the variance it is easy to see that the standard deviation will be
)(XbSD
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Fin 129
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.
)(),( 12212 xfdxxxf
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Fin 129Expectations
)()()()(
),(),(
),(),(
),()()(
2122221111
212111222121211
21212212121121
2121212121
XEXEdxxfxdxxfx
dxdxxxfxxdxdxxxfx
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dxdxxxfxxXXE
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Fin 129Variance
Similarly the variance can be reduced
),(2)()(
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2121
21212
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XXCovXVXV
dxdxxxfXXExx
XXV
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Fin 129A 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.
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Fin 129The 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) NThe distribution function is
2
2)(
2
1
22
1)(
x
exf
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Fin 129
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.
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Fin 129
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.
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Fin 129
Prob Ranges for Normal Dist.
68.26%95.46%99.74%
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Fin 129An Example
Lets define X as a function of a standard normal variable in other words is N(0,1))
X= We showed earlier that
Therefore
)()( XbEabXaE
)()( EE
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Fin 129Variance
We showed that the variance was equal to
Therefore
22 )()( VV
)()( 2 XVbbXaV
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Fin 129An 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
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Fin 129One 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.
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Fin 129VaR
Given the last slide it is easy to see that you would be 97.73% confident that the loss would not exceed -23%.
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Fin 129Continuous 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
q
dxxfqXprobc )()(
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Fin 129VAR 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 0.001Given 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-1.65 and mean +1.65)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
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Fin 129
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.
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Fin 129Example 2
Assume we own seven year zero coupon bonds with a face value of $1,631,483.00 with a yield of 7.243%Today’s Market Value
$1,631,483/(1.07243)7=$1,000,000If rates increase to by 16.5Bp to 7.408% the market value is
$1,631,483/(1.07408)7 = $989,295.75
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Fin 129Approximations - Duration
The duration of the bond would be 7 since it is a zero coupon.Modified duration is then 7/1.07243 = 6.527The price change would then be1,000,000(-6.57)(.00165) = $10,769.55
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Fin 129Approximations - 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 pointThe value at risk is then
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Fin 129Precision
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….
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Fin 129DEAR
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 RiskDEAR is often estimated using our linear measure:
(market value)(price sensitivity)(change in yield)Or
(Market value)(Price Volatility)
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Fin 129VAR
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)
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Fin 129N
Bank for International Settlements (BIS) 1998 market risk capital requirements are based on a 10 day holding period.
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Fin 129Problems 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.
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Fin 129
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.
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Fin 129
DEAR of a foreign Exchange Position
Assume the firm has a 1.6 Million trading position in eurosAssume that the current exchange rate is Euro1.60 / $1 or $.0625 / EuroThe $ value of the francs is then E1.6 million ($0.0625/Euro) =$1,000,000
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Fin 129FX 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 1.65FX volatility is then 1.65(56.5bp) = 93.2bp or 0.932%
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Fin 129FX DEAR
DEAR = (Dollar value )( FX volatility)=($1,000,000)(.00932)
=$9,320
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Fin 129Equity 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
22mi
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Fin 129Equity 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 m = 2% then using the same confidence interval,
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Fin 129Equity DEAR
DEAR = (Dollar value )( Equity volatility)
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Fin 129VAR 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.
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Fin 129Aggregation
The aggregation of the DEAR for the three assets can be thought of as the aggregation of three standard deviations.To aggregate we need to consider the covariance among the different asset classes. Consider the Bond, FX position and Equity that we have recently calculated.
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Fin 129Variance Covariance
Seven Year zero
E/$1US Stock Index
Seven Year Zero
1 -.20 .4
E/$1 1 .1
US Stock Index
1
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Fin 129variance covariance
2
1
USz,
USEuro,
Euroz,
2US
2Euro
2Z
)2(
)2(
)2(
)(DEAR)(DEAR)(DEAR
Portfolio
DEAR
USZ
USSwf
EuroZ
DEARDEAR
DEARDEAR
DEARDEAR
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Fin 129VAR for Portfolio
969,39$
)33)(77.10)(4)(.2(
)33)(32.9)(1)(.2(
)32.9)(77.10)(2.(2(
)(33)(9.32)(10.77
Portfolio
DEAR
2
1222
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Fin 129Comparison
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.
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Fin 129Risk Metrics
JP Morgan has the premier service for calculating the value at riskThey currently cover the daily updating and production of over 450 volatility and correlation estimates that can be used in calculating VAR.
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Fin 129
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.
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Fin 129
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.
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Fin 129
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.
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Fin 129Historical Simulation
Similar to the variance covariance approach, the idea is to look at the past history over a given time frame.
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Fin 129Back Simulation
Step 1: Measure exposures. Calculate the total $ valued exposure to each assetsStep 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.
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Fin 129Back Simulation
Step 4 Repeat step 3 using historical data for each of the assets for the last 500 daysStep 5 Rank the days from worst to best. Then decide on a confidence level.
Step 6 calculate the VAR
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Fin 129Historical Simulation
Provides a worst case scenario, where Risk metrics the worst case is a loss of negative infinityProblems:
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??
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Fin 129Monte Carlo Approach
Calculate the historical variance covariance matrix.Use the matrix with random draws to simulate 10,000 possible scenarios for each asset.
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Fin 129
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 BISSubject to audits
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Fin 129Risk Charges
Each asset is given a specific risk charge which represents the risk of the assetFor example US treasury bills have a risk weight of 0 while junk and would have a risk weight of 8%.
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Fin 129Specific 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.
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Fin 129
General Market Risk Charges
Reflect the product of the modified duration and expected interest rate shocks for each maturityRemember this is across different types of assets with the same maturity….
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Fin 129Vertical 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.
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Fin 129Horizontal Offsets
Within ZonesFor each maturity bucket there are differences in maturity creating again the inability to let short and long positions exactly offset each other.Between ZonesAlso across zones the short and long positions must be offset.
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Fin 129VaR 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
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Fin 129A Simple Example*
Assume a financial institution is facing the following three possible scenarios and associated losses
Scenario Probability Loss1 .97 02 .015 1003 .015 0
The VaR at the 98% level would equal = 0
*This and subsequent examples are based on Meyers 2002
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Fin 129A Simple Example
Assume the previous financial institution and its competitor facing the same three possible scenarios
Scenario ProbabilityLoss ALoss B Loss A & B1 .97 0 0 02 .015 100 0 1003 .015 0 100 100
The VaR at the 98% level for A or B alone is 0The Sum of the individual VaR’s = VaRA + VaRB = 0
The VaR at the 98% level for A and B combined VaR(A+B)=100
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Fin 129Coherence of risk measures
Let (X) and (Y) be measures of risk associated with event X and event Y respectivelySubadditvity implies (X+Y) < (X) + (Y). Monotonicity. Implies X>Y then (X) >(Y).Positive homogeneity:Given > 0 (X) = (X). Translation Invariance. Given an additional constant amount of loss , (X+) = (X)+.
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Fin 129Coherent 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
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Fin 129Tail VaR*
TVaR (X) = Average of the top (1-)% loss
For comparison let VaR(X) = the (1-)% loss
* Meyers 2002 The Actuarial Review
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Fin 129
Scenario
X1 X2 X1+X2
1 4 5 9
2 2 1 3
3 1 2 3
4 5 4 9
5 3 3 6
VaR60% 4 4 9
TVaR60% 4.5 4.5 9
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Fin 129Normal 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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Fin 129S&P 500 Monthly Returns vs. Normal Dist
-30
20
70
120
170
220
-0.475 -0.425 -0.375 -0.325 -0.275 -0.225 -0.175 -0.125 -0.075 -0.025 0.025 0.075 0.125 0.175 0.225 0.275 0.325 0.375 0.425 0.475
Returns
Obs
erva
tions S&P
Normal
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Fin 129Bond returns vs. Normal
0
50
100
150
200
250
-0.09 -0.07 -0.05 -0.03 -0.01 0.01 0.03 0.05 0.07 0.09
Returns
Obse
rvat
ions LT Corp
LT Govt
Norm
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Fin 129
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.
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Fin 129Implications
Both explanations have some truth, it is important to estimate variations from the underlying assumed distribution.
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Fin 129Measuring 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.
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Fin 129Moving Average
One solution is to calculate the moving average of the volatility
M
rM
iit
1
2
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Fin 129Moving Averages
Moving Avergages of Volatility S&P 500 Monthly Return
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0 100 200 300 400 500 600 700 800 900
1 year
2 year
5 year
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Fin 129Historical 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.
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Fin 129Nonconstant 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.
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Fin 129RiskMetrics
JP Morgan uses an Exponentially Weighted Moving Average.This method used a decay factor that weight’s each days percentage price change.A simple version of this would be to weight by the period in which the observation took place.
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Fin 129Risk Metrics
Wheren is the number of days used to derive the
volatilityIs the mean value of the distribution
(assumed to be zero for most VaR estimates)
1
2)()1(t
ntt
t X
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Fin 129Decay Factors
JP Morgan uses a decay factor of .94 for daily volatility estimates and .97 for monthly volatility estimatesThe choice of .94 for daily observations emphasizes that they are focused on very recent observations.
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Fin 129
Decay Factors
0
0.01
0.02
0.03
0.04
0.05
0.06
0 20 40 60 80 100 120 140 160
Days
Wei
ghtin
g
0.94
0.97
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Fin 129Measuring Correlation
Covariance:
Combines the relationship between the stocks with the volatility.
(+) the stocks move together (-) The stocks move opposite of each other
iBBiAAi PkkkkABCov ))(()(
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Fin 129Measuring 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
)/()( BAAB ABCovr
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Fin 129Timing 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”
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Fin 129Size 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.
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Fin 129Correlations 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.
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Fin 129Reducing 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)
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Fin 129A 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.
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Fin 129Back 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.
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Fin 129Basle 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.
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Fin 129Back 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?
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Fin 129Back 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 rejectedType II error – the model has been erroneously accepted. Basle specifies a type one error test.
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Fin 129One 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).
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Fin 129Approximations
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.
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Fin 129
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.
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Fin 129
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 skewThe 1 year LIBOR sterling rate shows nothing close to normal behaviorBasic model work about as well as more advanced mathematical models
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Fin 129Basle Tests
Requires that the VaR model must calculate VaR with a 99% confidence and be tested over at least 250 days.Table 4.6Low 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”
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Fin 129Basle
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
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Fin 129Stress 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.
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Fin 129Stress 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.
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Fin 129Stress 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.
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Fin 129
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.
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Fin 129
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.
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Fin 129Managing 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..
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Fin 129Managing 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.
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Fin 129Setting 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.
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Fin 129VaR Limits
Setting limits at the trading unit levelAllows 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 tradersVaR is not familiar to most traders (they d o not work with it daily and may not understand how different choices impact VaR.
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Fin 129
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?
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Fin 129Stress 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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