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)

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)

IntroductionSAS IML

References

VaR DefinitionVaR TheoryMath theory

Graphical representation

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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)

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)

IntroductionSAS IML

References

VaR DefinitionVaR TheoryMath theory

Matrix multiplication

(AB)ij =m∑

k=1AikBkj

Sébastien Casault Using IML to Calculate Value at Risk (VaR)

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)

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)

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)

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)

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)

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)

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)