`VaR Monte Carlo SimulationCapital Market Risk Advisors© CMRAMonte Carlo SimulationMonte Carlo is most helpful when some or all assets in a portfolio are not amenable to analytical treatment1 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© CMRA2Monte Carlo Simulation Scenario generationMonte Carlo begins with the generation of n normal variables with unit variance and correlation matrix .factorization, yielding =ATADecompose the correlation matrix  using the CholeskyGenerate 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© CMRA3Monte Carlo Simulation Cholesky DecompositionSuppose=ATA, where A is an upper triangular matrix, how dowe 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   © CMRA4General Result  i-1  2 aii = sii -aik   k=1   1/ 2i-1 1 aij = (sij -aikajk )1/ 2 j = i +1, i + 2,, N aii k=1© CMRA5`

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