ROM SIMULATION Exact Moment Simulation using Random Orthogonal Matrices Bachelier Finance Society Meeting Toronto 2010 Henley Business School at Reading Contact Author : d.ledermann@icmacentre.ac.uk Alexander and Ledermann (ICMA Centre) 23rd June 2010 1 / 20
Outline Contents ROM Simulation ROM Simulation Properties Applications in Finance Alexander and Ledermann (ICMA Centre) 23rd June 2010 2 / 20
ROM Simulation Introduction: Motivation Simulating a N(µ n,σ n ) multivariate sample X mn (Monte Carlo): Simulate standard normal Z mn = (z ij ), where z ij = Φ 1 (p ij ), p ij U[0,1] and Φ is the standard normal c.d.f Set X mn = 1 m µ n + Z mn A n where A na n = Σ n and 1 m = (1,...,1) Problem: Error in sample moments M(X mn ) = m 1 1 mx mn V (X mn ) = m 1 (X mn 1 m M(X mn )) (X mn 1 m M(X mn )) That is, particularly when m is small M(X mn ) µ n and V (X mn ) Σ n Alexander and Ledermann (ICMA Centre) 23rd June 2010 3 / 20
ROM Simulation ROM Simulations Solution: Replace Z mn with an L-matrix satisfying L nml mn = I n and 1 ml mn = 0 n e.g Apply Gram-Schmidt (GS) orthogonalisation Exact Mean-Covariance Sample: ROM Simulations where X mn = 1 m µ n + m 1 2Q m L mn R n A n A n A n = Σ n R n is a random orthogonal matrix (ROM) Q m is a permutation satisfying 1 mq m = 1 m Alexander and Ledermann (ICMA Centre) 23rd June 2010 4 / 20
ROM Simulation ROM Simulated Paths Different ROMs, applied to the same L-matrix, lead to different samples: 0.6 0.4 0.2 0 0.2 0.4 0.6 Variable 1 (no ROM) Variable 2 (no ROM) Variable 1 (with ROM) Variable 2 (with ROM) 0.8 0 5 10 15 20 25 Figure: Both simulations based on the same multivariate normal sample. The solid lines show the paths from the first simulation (no ROM) and the dashed lines show the second simulation (with ROM). Path correlation is 0.75. Alexander and Ledermann (ICMA Centre) 23rd June 2010 5 / 20
L-Matrix Types ROM Simulation Parametric: Orthogonalisation of a zero mean parametric sample ROM Simulation (small) adjustment to Monte Carlo Data Specific: Orthogonalise a collection of data (mean deviations) ROM Simulation infinitely many historical samples Deterministic: Orthogonalise linearly independent vectors v j = (0,...,0,1, 1,...,1, 1,0,...,0) GS L k }{{}}{{} mn j 1 2k L 1 mn relates to Helmertian (1876) matrices; k > 1 are new ROM Simulation target higher multivariate moments Alexander and Ledermann (ICMA Centre) 23rd June 2010 6 / 20
Outline ROM Simulation Properties ROM Simulation ROM Simulation Properties Applications in Finance Alexander and Ledermann (ICMA Centre) 23rd June 2010 7 / 20
ROM Simulation Properties Multivariate Skewness and Kurtosis We employ the multivariate measures introduced by Mardia (1970) Skewness: m m τ M (X mn ) = m 2 { (xi µ n)v (X mn ) 1 (x j µ n) } 3 i=1 j=1 Kurtosis: m κ M (X mn ) = m 1 { (xi µ n)v (X mn ) 1 (x i µ n) } 2 i=1 Key Property is invariance under non-singular affine transformations: τ M (X mn ) = τ M (1 m b n + X mn B n ) κ M (X mn ) = κ M (1 m b n + X mnb n ) Alexander and Ledermann (ICMA Centre) 23rd June 2010 8 / 20
ROM Simulation Properties Skewness and Kurtosis of ROM Simulations ROM simulations are (random) affine transformations of L-matrices Parametric: Multivariate normal case E[τ M (L mn )] = n(n + 1)(n + 2)m 1 E[κ M (L mn )] = n(n + 2)(m 1)(m + 1) 1 Data Specific: ROM simulation moments identical to historical data Deterministic: When k = 1 τ M (L 1 mn) = n [ (m 3) + (m n) 1] κ M (L 1 mn) = n [ (m 2) + (m n) 1] When k > 1 Moments available numerically Calibrate m (and k) for moment targeting Alexander and Ledermann (ICMA Centre) 23rd June 2010 9 / 20
ROM Simulation Properties Orthogonal Matrices Recall: ROM simulation equation X mn = 1 m µ + m 1 2Q m L mn R n A n Q m are (cyclic) permutation matrices R n are combinations of the following random orthogonal matrix types (1) Sign Matrices: R n = diag { ( 1) d 1,...,( 1) dn} where d k Bin(1,p k ) (2) Upper Hessenberg Rotations: R n = G n (θ 1 )G n (θ 2 )... G n (θ n 1 ) where G n (θ j ) are Givens (1958) rotations From a random skew-symmetric matrix, satisfying S n = S n, we form (3) Cayley (1846) Rotations: R n = (I n S n ) 1 (I n + S n ) (4) Exponential Rotations: R n = exp(s n ) Alexander and Ledermann (ICMA Centre) 23rd June 2010 10 / 20
ROM Simulation Properties ROM Simulation Densities: Rotation Effects 1000 Deterministic ROM Simulations 800 600 Upper, Hessenberg 400 200 0 4 3 2 1 0 1 2 3 4 800 600 Cayley 400 200 0 4 3 2 1 0 1 2 3 4 600 500 400 Exponential 300 200 100 0 4 3 2 1 0 1 2 3 4 Figure: Histograms for the 5th marginal density of a ROM simulation involving deterministic L-matrices (m = 15, n = 10, k = 2). Over 10,000 observations are used for each simulation. Marginals are compared to scaled normal distributions. Alexander and Ledermann (ICMA Centre) 23rd June 2010 11 / 20
ROM Simulation Properties ROM Simulation Densities: Sign Matrix Effects 1200 Effect of Sign Matrices 1000 Positive Skewness 800 600 400 200 0 4 3 2 1 0 1 2 3 4 1200 1000 800 Negative Skewness 600 400 200 0 4 3 2 1 0 1 2 3 4 Figure: Histograms for the 5th marginal distribution of two ROM simulations involving deterministic L-matrices (m = 15, n = 10, k = 0) and Cayley matrices. In the lower figure sign matrices are used to induce negative skew. Alexander and Ledermann (ICMA Centre) 23rd June 2010 12 / 20
Outline Applications in Finance ROM Simulation ROM Simulation Properties Applications in Finance Alexander and Ledermann (ICMA Centre) 23rd June 2010 13 / 20
Applications in Finance Portfolio Value-at-Risk (VaR) The level VaR(α,h) represents the h-day portfolio loss that we assume is exceeded with probability α If portfolio h-day returns are normally distributed then VaR(α,h) = Φ 1 (α)σ h µ h Portfolio losses are typically non-normal (leptokurtic) Common to simulate losses (returns) and calculate empirical quantiles VaR(α,h) = q α (r h m) where r h m is a vector containing m scenarios for portfolio h-day returns Historical simulation is particularly popular Alexander and Ledermann (ICMA Centre) 23rd June 2010 14 / 20
Applications in Finance An MSCI Country Index Portfolio We consider a portfolio whose assets individually track the n = 45 country indices in the MSCI All Country World Index Portfolio return r π is weighted average of asset returns x = (x 1,...,x n ) r π = π(x) = n w i x i where n w i = 1 i=1 i=1 Correlations between the assets returns are key Multivariate kurtosis is also important Let X mn denote m (simulated) scenarios on the n assets, then X mn π r m where r m is a vector of portfolio scenarios Alexander and Ledermann (ICMA Centre) 23rd June 2010 15 / 20
Applications in Finance Scenario Generation Techniques for VaR Two year historical window target moments µ n, S n, κ M We generate scenarios X mn using six different techniques: (1) - (3) ROM Simulations Type I L-matrices used to target κ M Three types of random orthogonal matrices (4) Multivariate Normal (Monte Carlo and analytic) (5) Multivariate Student-t (Monte Carlo, ν = 6 degrees of freedom) (6) Historical simulation (two years of observations) Note: Limited data are available for historical quantile estimation Estimate portfolio VaR roll window forward and repeat Alexander and Ledermann (ICMA Centre) 23rd June 2010 16 / 20
Applications in Finance Daily VaR for Equally Weighted Portfolio 0.08 MSCI Portfolio : 1% Daily VaR 0.07 0.06 0.05 Hessenberg Cayley Exponential Normal (MC) Student-t (MC) Historical Normal (Analytic) 0.04 0.03 0.02 0.01 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Figure: Evolution of daily VaR, given as a percentage of the portfolio value Alexander and Ledermann (ICMA Centre) 23rd June 2010 17 / 20
Daily VaR Exceedances Applications in Finance 0.1 0.08 0.06 0.04 0.02 0 0.02 0.04 0.06 0.08 0.1 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Figure: Equally weighted daily portfolio returns, plotted with negative VaR Alexander and Ledermann (ICMA Centre) 23rd June 2010 18 / 20
Applications in Finance Proportion of Exceedances for Equally Weighted Portfolio Exceedances Coverage Tests m 1 m 1 /m Uncond. Cond. Hessenberg 24 0.89% 0.36 6.03% Cayley 32 1.18% 0.86 4.45% Exponential 56 2.07% 23.89 47.49% Normal (MC) 73 2.70% 53.80 84.48% Student-t (MC) 41 1.52% 6.27 24.23% Historical 37 1.37% 3.31 18.39% Normal (Analytic) 74 2.73% 55.84 85.82% Table: m is the total number of out-of-sample returns (2647 daily). The 1% critical values are 6.63 for the Unconditional test and 9.21 for the Conditional test Alexander and Ledermann (ICMA Centre) 23rd June 2010 19 / 20
Applications in Finance Summary Exact mean-covariance samples are generated from L-matrices Orthogonal matrices can be used to randomise these samples Different simulation properties associated with different types of orthogonal matrix Target higher order moments with deterministic L-matrices ROM simulation is a useful scenario generation technique Portfolio Value-at-Risk Portfolio allocation optimisations Alexander and Ledermann (ICMA Centre) 23rd June 2010 20 / 20