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Practical Issues

Practical Issues. Stephen Mildenhall CNA Re. FFT Roadmap. exponentiation. Addition Exp(x+y) Easy?. Multiplication Exp(x)Exp(y) Hard?. logarithm. FFT Roadmap. Random Variable: X Fourier Transform: t  E(exp(itX)). Random Variables X, Y X + Y. t  E(exp(itX)) t  E(exp(itY))

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Practical Issues

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  1. Practical Issues Stephen MildenhallCNA Re

  2. FFT Roadmap exponentiation Addition Exp(x+y) Easy? Multiplication Exp(x)Exp(y) Hard? logarithm

  3. FFT Roadmap Random Variable: XFourier Transform: t  E(exp(itX)) Random Variables X, Y X + Y t  E(exp(itX)) t  E(exp(itY)) tE(exp(itY)).E(exp(itY)) MGF or Fourier Transform Same information

  4. FFT Roadmap • Fourier Transform is a continuous function • Discrete Fourier Transform (DFT) is a sample of the Fourier Transform • Contains approximately the same information • Two-way street, convert X to DFT(X) to X • Computes sums (convolutions) • DFT is just matrix multiplication DFT(X)= M X • Fast Fourier Transform is a very, very fast way of computing DFT(X) • Available in MALT, Excel, Matlab, SAS, C/C++ • Black-box or learn to program

  5. FFT Roadmap DFT(X), DFT(Y)(n x 1 vectors of complex numbers, computed very quickly using FFT) Random variables X, Y (n x 1 vectors of real numbers, sum=1) DFT(X) x DFT(Y)(component-by-component product) Inverse DFT = distribution of X+Y (!)

  6. n X Y X+Y 0 0.2 0 0 1 0.2 1 0.2 2 0.2 0 0.2 3 0.2 0 0.2 4 0.2 0 0.2 5 0 0 0.2 6 0 0 0 7 0 0 0 n Re(FTX) Im(FTX) Re(FTY) Im(FTY) Re(FTX+Y) Im(FTX+Y) 0 1.0000 0.0000 1.0000 0.0000 1.0000 0.0000 1 0.0000 -0.4828 0.7071 -0.7071 -0.3414 -0.3414 2 0.2000 0.0000 0.0000 -1.0000 0.0000 -0.2000 3 0.0000 -0.0828 -0.7071 -0.7071 -0.0586 0.0586 4 0.2000 0.0000 -1.0000 0.0000 -0.2000 0.0000 3 0.0000 0.0828 -0.7071 0.7071 -0.0586 -0.0586 2 0.2000 0.0000 0.0000 1.0000 0.0000 0.2000 1 0.0000 0.4828 0.7071 0.7071 -0.3414 0.3414 Step-by-Step Convolution Calculation Distributions FFTs

  7. MALT Site • http://www.mynl.com/MALT/home.html • Written in response to COOS Call Program, 2000-01 • Windows 2000/NT, IE 5 required • Download and install COM objects required • Can script or use in Excel/VBA

  8. Independent Clayton Gumbel Venter HRT Warning to ERM Practice • A correlation matrix is not enough to describe the correlations at the tails

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