Robust and Rich Roy Welsch Xinfeng Zhou Massachusetts Institute of Technology

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Robust and Rich Roy WelschXinfeng
ZhouMassachusetts Institute of Technology

• email rwelsch_at_mit.edu
• International Conference on Robust Statistics
• Technical University of Lisbon
• 17 July 2006

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Notation

• N number of risky assets
• Ri return of the ith asset in the portfolio
• wi weight of the ith asset in the portfolio,
• ?i expected return of the ith asset
• ? covariance matrix of the returns of N assets
• T number of time periods for estimating ?.
• Note wi could be negative for short sales.

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Mean-Variance Portfolio Optimization

• Portfolio return
• Rp w1R1 w2R2 . . . wnRn
• Expected portfolio return
• E(Rp) wT?
• Variance of the portfolio return
• Var(Rp) wT?w
• Mean-variance portfolio optimization minimizes
  the variance of a portfolio return for a given
  level of expected return ?p
• subject to wT? ?p, wTe 1
• where e is the n ? 1 column vector with all
  elements 1.

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Problems with Mean-Variance

• Static just one period.
• Sensitive to inputs which are, in turn, subject
  to random errors in the estimation of expected
  return and variance which are usually obtained
  from historical return data.
• This sensitivity often leads to extreme portfolio
  weights and dramatic swings in weights with only
  minor changes in expected returns or the
  covariance matrix. This can lead to frequent
  rebalancing and excessive transaction costs.
• For stable covariance estimation, we prefer long
  historical time series (the number of assets, N,
  far smaller than the number of time periods, T).
  However, old historical data may not reflect
  current market dynamics.
• Underlying multivariate normal assumption may not
  be right.

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Some Solutions

• Factor models (CAPM, etc.), Bayesian shrinkage,
  GARCH models.
• Regularization (penalty) methods.
• Robust estimation of the expected return and the
  covariance matrix. We will focus on this.
• Combinations of the above methods.

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Fast-MCD

• The minimum covariance determinant (MCD) proposed
  by Rousseeuw (1985) looks for the covariance
  matrix of h data points (T / 2 ? h lt T) with the
  smallest determinant. The breakdown is (T ? h) /
  T. The resulting covariance matrix is biased
  (and can be adjusted to be unbiased), but this
  multiplicative factor has no effect on portfolio
  weight allocation. MCD is not feasible for N gt
  20 in our situation. Fast-MCD proposed by
  Rousseeuw and Van Driessen (1999) makes large N
  (51 in our data) feasible. MCD retains affine
  equivariance.

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Pairwise Robust Covariance

• If the affine equivariance assumption is
  dropped, faster robust pairwise covariance
  estimators are available.
• Khan et.al. (2005) compared several approaches
  to robust pairwise covariance estimation while
  investigating ways to make least-angle regression
  (LARS) (Efron, et. al., 2003) robust. They found
  a two-step, two-dimensional Winsorization method
  to be effective and fast. We use a modified form
  of their idea with adjustment to insure a
  positive definite covariance matrix.

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Huber Winsorization
For each (time) vector of returns xi, i 1, . .
. , N, the transformation is used to
shrink outliers towards the median with the
Huber function Hc min max(c, x), c, c gt 0.
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Bivariate Winsorization

• Huber Winsorization fails to take the orientation
  of the bivariate data into consideration.
  Bivariate Winsorization (after centering) sets
• with xt (xti, xtj)T and D(xt) the Mahalanobis
  distance based on some initial ?0, ?0 and
  constant c.

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Iterated Bivariate Winsorization

• For each pair of variables xi, xj compute
• Let xt xti, xtjT.
• For each ?k, ?k, calculate the Mahalanobis
  distance for each return pair
• and weight

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• Update ?k1, ?k1
• until convergence.
• All pairwise covariances are combined to form an
  initial covariance matrix. This is converted to
  positive semi-definite using a method due to
  Maronna and Zamar (2002).
• We call this I2D-Winsor.

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Fast 2-D Winsorization

• Iteration is expensive and Khan, et. al. (2005)
  proposed taking one bivariate Winsorization step
  from an improved starting ?0. They start with
  univariate Huber Winsorization but use two tuning
  constants, c1 and c2. The constant c1 (chosen to
  be 2) is used in the two quadrants with the most
  data (n1) and the second constant c2 n2 / n1
  with n2 T n1 is used in the remaining two
  quadrants. This pulls the Huber Winsorization
  boundary in where there is less data and a higher
  chance of data not following the ellipsoidal
  pattern for the main part of the data.

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Fast 2-D

• The classical correlation is computed on this
  Winsorized data and all pairwise correlations
  form the full initial correlation matrix which,
  if necessary, is made positive definite. Then one
  step of bivariate Winsorization is used and this
  new matrix is again made positive definite. We
  call this F2D-Winsor.

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Historical Data

• Daily returns on 51 MSCI US Industry sector
  indexes from 01/03/1995 to 02/07/2005 (2600 days
  of data). Broader than the SP 500. We need to
  find the weights to use on each of the n 51
  indexes in our portfolio.
• Rebalance as follows Estimate sector weights
  using most recent T 100 daily returns,
  rebalance every 5 trading days. With 2600 days
  there are 500 rebalances. Trading costs (when
  used) are 5 cents for each 100 bought or sold.
• We use the following constraints
• wT? ?p, wTe 1, ?1 ? wi ? 1.
• The market portfolio consists of all individual
  stocks (about 700) in the 51 indexes weighted by
  market capitalization.

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Financial Performance Measures

• mean the sample mean of weekly ex-post returns.
• STD the sample standard deviation of weekly
  ex-post returns.
• Information ratio (annualized)
• ?-VaR (? 5, 1) the sample ?-quantile of the
  weekly ex-post returns distribution.
• ?-CVaR (? 5, 1) the sample conditional mean
  of the weekly ex-post returns distribution, given
  the returns are below the ?th quantile.
• Max DD the maximum drawdown, which is the
  maximum loss in a week.
• CRet cumulative return.
• Turnover weekly asset turnover, defined as the
  mean of the absolute
• weight changes for 500 updates.
• CRet_cost cumulative return with transaction
  costs.
• IRcost information ratio with transaction costs.

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Winsorization Results
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Contamination Models

• MCD
• Each row (time observation) either from F0 or H.
  Implies either a bad day on the market (all
  stocks) or a high correlation among stocks. In
  fact, rarely true.
• Pairwise
• Pairwise correlation permits a more flexible
  error model. Unusual market returns only explain
  a small part of observed outliers. Industrial
  factors and idiosyncratic risk specific to
  individual stocks or groups of stocks explain a
  majority of the outlying data.

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Too Much Turnover?

• The mean-variance portfolio optimization problem
  can be re-expressed as
• subject to w?? ?p and w?e 1. One way to
  possibly reduce turnover would be to penalize
  deviations from the market weights, mj and, at
  the same time, look for sparse solutions that do
  not invest any funds in some securities. The
  LASSO (Tibshirani, 1996) does exactly this. More
  robust loss functions such as L1 and Huber may
  also be used instead of least-squares, but did
  not change the results significantly.

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Penalization

• To implement this we solve (Laupréte, 2001)
• and use 5-fold cross-validation to find ? based
  on prediction error for the out-of-sample data.
  The recently developed LARS (least-angle
  regression) algorithm (Efron, et. al. 2004)
  greatly speeds up computations for the Lasso
  since solutions for all ? can be found in about
  the same time as one least-squares regression.
  This removes the need for a (non-specific) grid
  search on ?.

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Penalty Results

• We end up with slightly better performance and
  dramatically lower turnover.

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Run Times

• 500 Rebalancings
• V 40 seconds
• F2D-Winsor 35 minutes
• I2D-Winsor 3 hours
• FAST-MCD 10 hours
• V1 4 hours

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Next Steps

• Combine robust covariance and penalty approaches.
• Use individual stocks (about 700) instead of 51
  sector index funds. Then T 100 lt lt N 700.
• Fast algorithms.

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References

• Alqallaf, F.A., et al., (2002) Scalable robust
  covariance and correlation estimates for data
  mining. Proceedings of the eighth ACM SIGKDD
  international conference on Knowledge discovery
  and data mining, Statistical methods, 1423.
• Efron, B., Hastie, T., Johnstone, I., Tibshirani,
  R., (2003) Least Angle Regression, Annals of
  Statistics, 32, 407499.
• Khan, J., Van Aelst, S. and Zamar, R., (2005)
  Robust linear model selection based on Least
  Angle Regression, Technical Report, Department of
  Statistics, University of British Columbia.
• Laupréte, G.J., Portfolio risk minimization under
  departures from normality. MIT PhD Thesis, 2001.
• Maronna, R.A. and R.H. Zamar, (2002) Robust
  estimates of location and dispersion for
  high-dimensional datasets, Technometrics, 44(4),
  307317.

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• Rousseeuw, P.J. and K. Van Driessen, (1999) A
  fast algorithm for the minimum covariance
  determinant estimator, Technometrics, 41(3),
  212223.
• Tibshirani, R., (1996) Regression Shrinkage and
  Selection via the Lasso, Journal of the Royal
  Statistical Society, Series B, 5, 267288.
• Zhou, X., (2006) Application of Robust Statistics
  to Asset Allocation Models, MIT, M.S. Thesis.