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VaR Monte Carlo Simulation

Capital Market Risk Advisors

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Monte Carlo Simulation

Monte Carlo is most helpful when some or all assets in a portfolio are not amenable to analytical treatment

1 Scenario Generation -produce a large number of future price scenarios 2 Portfolio valuation - for each scenario, compute a portfolio value 3 Summary - report the results of the simulation, either as a portfolio distribution or as a particular risk measure

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Monte Carlo Simulation Scenario generation

Monte Carlo begins with the generation of n normal variables with unit variance and correlation matrix .

factorization, yielding =ATA

Decompose the correlation matrix using the Cholesky

Generate an n × 1 vector Z of independent standard normal variables Let Y = AZ. The elements of Y will each have unit variance with the correlation matrix

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Monte Carlo Simulation Cholesky Decomposition

Suppose

=ATA, where A is an upper triangular matrix, how do

we find A?

s11 s12 s13 a11 0 0 s21 s22 s23 = a21 a22 0 s31 s32 s33 a31 a32 a33 a11 a21 a31 0 a a 22 32 0 0 a33

2 a11a21 a11a31 s11 s12 s13 a11 2 2 s21 s22 s23 = a11a21 a21 + a22 a21a31 + a32a22 s31 s32 s33 a a a a + a a a2 + a2 + a2 11 31 21 31 32 22 31 32 33

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General Result

i-1 2 aii = sii -aik k=1

1/ 2

i-1 1 aij = (sij -aikajk )1/ 2 j = i +1, i + 2,, N aii k=1

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