iml value-at-risk backtesting (casault) · introduction sasiml references vardefinition vartheory...
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IntroductionSAS IML
References
Using IML to Calculate Value at Risk (VaR)SAS’s ‘functional’ sibling
Sébastien Casault
ENMAX Energy Corporation
5 October, 2016
Sébastien Casault Using IML to Calculate Value at Risk (VaR)
![Page 2: IML Value-At-Risk Backtesting (Casault) · Introduction SASIML References VaRDefinition VaRTheory Maththeory Var-covartwocommodityportfoliocalculation Simplified var r covar (parametric)](https://reader031.vdocuments.us/reader031/viewer/2022022520/5b1e32577f8b9a901f8b58c7/html5/thumbnails/2.jpg)
IntroductionSAS IML
References
VaR DefinitionVaR TheoryMath theory
Introduction
DefinitionValue at Risk (VaR) is a measure of the risk of investments. Itestimates how much a set of investments might lose, given normalmarket conditions, in a set time period such as a day. – Wikipedia
Value at Risk is often part of an organization’s controls andperformance metrics. It’s a simple number that can be used toquickly gauge both the dynamics of and exposure to a market.
Sébastien Casault Using IML to Calculate Value at Risk (VaR)
![Page 3: IML Value-At-Risk Backtesting (Casault) · Introduction SASIML References VaRDefinition VaRTheory Maththeory Var-covartwocommodityportfoliocalculation Simplified var r covar (parametric)](https://reader031.vdocuments.us/reader031/viewer/2022022520/5b1e32577f8b9a901f8b58c7/html5/thumbnails/3.jpg)
IntroductionSAS IML
References
VaR DefinitionVaR TheoryMath theory
Graphical representation
$0
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Gaussian
VaRi = wiPiσiZα
It’s a measure of maximum* potential loses assuming that theunderlying commodity behaves normally.Things get a little trickier when we have a portfolio ofinstruments – diversification reduces risk.
Sébastien Casault Using IML to Calculate Value at Risk (VaR)
![Page 4: IML Value-At-Risk Backtesting (Casault) · Introduction SASIML References VaRDefinition VaRTheory Maththeory Var-covartwocommodityportfoliocalculation Simplified var r covar (parametric)](https://reader031.vdocuments.us/reader031/viewer/2022022520/5b1e32577f8b9a901f8b58c7/html5/thumbnails/4.jpg)
IntroductionSAS IML
References
VaR DefinitionVaR TheoryMath theory
Var-covar two commodity portfolio calculationSimplified var‐covar (parametric) Value at Risk (VaR) calculation
Portfolio inputs Power GasPrice 35.00$ 2.55$ Confidence level 0.975Position 500 5,000 critical z 1.960Volatility 8.98% 4.60%CorrelationExposure 17,500.00$ 12,730.00$ VaR 3,080.08$ 1,147.72$
Portfolio exposure 30,230.00$
Var‐covar matrix, ∑ Matrix MathPosition (x') ∑ Position (x)
Power Gas 17,500.00$ 12,730.00$ 0.00806404 0.00082616 17,500.00$ Power 0.0081 0.0008 0.00082616 0.002116 12,730.00$ Gas 0.0008 0.0021
Position (x') ∑ xPortfolio 17,500.00$ 12,730.00$ 151.64$ Variance 3,180,611.77$ 41.39$ Volatility 1,783.43$ VaR 3,495.45$ x' ∑ x
3,180,612$
20%
Sébastien Casault Using IML to Calculate Value at Risk (VaR)
![Page 5: IML Value-At-Risk Backtesting (Casault) · Introduction SASIML References VaRDefinition VaRTheory Maththeory Var-covartwocommodityportfoliocalculation Simplified var r covar (parametric)](https://reader031.vdocuments.us/reader031/viewer/2022022520/5b1e32577f8b9a901f8b58c7/html5/thumbnails/5.jpg)
IntroductionSAS IML
References
VaR DefinitionVaR TheoryMath theory
Matrix multiplication
(AB)ij =m∑
k=1AikBkj
Sébastien Casault Using IML to Calculate Value at Risk (VaR)
![Page 6: IML Value-At-Risk Backtesting (Casault) · Introduction SASIML References VaRDefinition VaRTheory Maththeory Var-covartwocommodityportfoliocalculation Simplified var r covar (parametric)](https://reader031.vdocuments.us/reader031/viewer/2022022520/5b1e32577f8b9a901f8b58c7/html5/thumbnails/6.jpg)
IntroductionSAS IML
References
VaR DefinitionVaR TheoryMath theory
Base SAS matrix multiplication
1 Reshape matrix to ‘long’ format
Data step with array to cycle through columns
2 Perform product of two matrices
PROC SQL to perform pairwise sum(product)
3 Use PROC Transpose to reshape the results
Sébastien Casault Using IML to Calculate Value at Risk (VaR)
![Page 7: IML Value-At-Risk Backtesting (Casault) · Introduction SASIML References VaRDefinition VaRTheory Maththeory Var-covartwocommodityportfoliocalculation Simplified var r covar (parametric)](https://reader031.vdocuments.us/reader031/viewer/2022022520/5b1e32577f8b9a901f8b58c7/html5/thumbnails/7.jpg)
IntroductionSAS IML
References
IntroductionExampleInterfacing with imlVaR calculation
The Interactive Matrix Language
x11 x12 x13 . . . x1nx21 x22 x23 . . . x2n...
...... . . . ...
xm1 xm2 xm3 . . . xmn
IML is a matrix programming language similar to R and MATLABthat enables you to use natural mathematical syntax to writecustom algorithms and to compute statistics that are not built intoany SAS procedure:
1 iterative-type statistical procedures;2 doing operations on row and column of data sets;3 e.g., matrix multiplication.
Sébastien Casault Using IML to Calculate Value at Risk (VaR)
![Page 8: IML Value-At-Risk Backtesting (Casault) · Introduction SASIML References VaRDefinition VaRTheory Maththeory Var-covartwocommodityportfoliocalculation Simplified var r covar (parametric)](https://reader031.vdocuments.us/reader031/viewer/2022022520/5b1e32577f8b9a901f8b58c7/html5/thumbnails/8.jpg)
IntroductionSAS IML
References
IntroductionExampleInterfacing with imlVaR calculation
Matrix multiplication example
proc iml;print "---------- SAS/IML Results --------------";x = 1:3;m = {1 4 7, 2 5 8, 3 6 9};q = m * t(x);
print("------------- R Results -----------------")x <- rep(1:3) # vector of sequence 1,2,3m <- matrix(1:9, nrow=3) # 3 x 3 matrixq <- m %*% x # matrix multiplicationprint(q)
Sébastien Casault Using IML to Calculate Value at Risk (VaR)
![Page 9: IML Value-At-Risk Backtesting (Casault) · Introduction SASIML References VaRDefinition VaRTheory Maththeory Var-covartwocommodityportfoliocalculation Simplified var r covar (parametric)](https://reader031.vdocuments.us/reader031/viewer/2022022520/5b1e32577f8b9a901f8b58c7/html5/thumbnails/9.jpg)
IntroductionSAS IML
References
IntroductionExampleInterfacing with imlVaR calculation
Pushing and pulling data
Pull datasets
proc iml;use WORK.correlationMatric;
read all into correls[];close WORK.correlationMatric;
Push datasets
create portVaR var {variables};append;
close portVaR;
Sébastien Casault Using IML to Calculate Value at Risk (VaR)
![Page 10: IML Value-At-Risk Backtesting (Casault) · Introduction SASIML References VaRDefinition VaRTheory Maththeory Var-covartwocommodityportfoliocalculation Simplified var r covar (parametric)](https://reader031.vdocuments.us/reader031/viewer/2022022520/5b1e32577f8b9a901f8b58c7/html5/thumbnails/10.jpg)
IntroductionSAS IML
References
IntroductionExampleInterfacing with imlVaR calculation
VaR calculation
zScore = probit(.975);
DO i = 1 to items;indivVaR[i] = volatility[i] * portfolio[i] * zScore;
END;/* indivVaR = volatility # portfolio * zScore; */
portVaR = sqrt(t(indivVaR) * correls * indivVaR);
Sébastien Casault Using IML to Calculate Value at Risk (VaR)
![Page 11: IML Value-At-Risk Backtesting (Casault) · Introduction SASIML References VaRDefinition VaRTheory Maththeory Var-covartwocommodityportfoliocalculation Simplified var r covar (parametric)](https://reader031.vdocuments.us/reader031/viewer/2022022520/5b1e32577f8b9a901f8b58c7/html5/thumbnails/11.jpg)
IntroductionSAS IML
References
IntroductionExampleInterfacing with imlVaR calculation
VaR backtesting
Inputs: historical daily Profits & Losses and calculatedVaR.
This example is lifted from the SAS website and shows—over oneyear—that the PnL was greater than VaR about once a month, asexpected for a calculation using a 95th percentile
Sébastien Casault Using IML to Calculate Value at Risk (VaR)
![Page 12: IML Value-At-Risk Backtesting (Casault) · Introduction SASIML References VaRDefinition VaRTheory Maththeory Var-covartwocommodityportfoliocalculation Simplified var r covar (parametric)](https://reader031.vdocuments.us/reader031/viewer/2022022520/5b1e32577f8b9a901f8b58c7/html5/thumbnails/12.jpg)
IntroductionSAS IML
References
Bibliography
1 SAS/IML(R) 14.1 User’s Guide2 https://en.wikipedia.org/wiki/Value_at_risk
Sébastien Casault Using IML to Calculate Value at Risk (VaR)