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Multi-Regime Portfolio Optimization

Overview

  • Identification of Current Regime
  • MCMC simulated over a given period of time (e.g. one month)
    • Expected Return of each asset
    • Monte-Carlo simulation⇒VaR of any virtual portfolio (Gaussian mixture)
  • Maximization of Expected Return / VaR
  • Rebalancing based on dynamic criterion (profit taking) and/or signal strength
  • Benchmark: Same with 1 regime = Markowitz

Spectral Embedding and Clustering

Clustering by Tree Algorithm

Regime Sequence

  • Assume that Y1Ym have mixed joint distribution

\(\ P = \pi_1P_1+…+\pi_qP_q\) with \(\pi_1>…>\pi_q\)

  • Fat tails can be measured as the ratio of risk under P1 risk under other regimes.
  • An optimizer that only accounts for some regimes will reduce the risk under those regimes, but increase the risk under other regimes, hence increasing fat-tailedness

⇒Regime changes have an aggravated impact on portfolio risk

 

Dynamics in the Market Variable Space

  • Finite Number of Regimes R1,…,Rm
    • Covariance Matrix Gk
    • Drift Vector mk
    • Rk ~ N(mk,Gk)
  • Transition Probability Matrix
    • P(t,t+1) = (phk)
    • phk = Probability of transitioning from Rh to Rk
    • ∀h, \(\Sigma^m_{k=1}p_{hk}=1\)
    • P(t,t+n) = Pn
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  • Euler Scheme:
    • Discretized Time t0,…,tn
    • Pick new regime R(ti+1) according to P applied to current regime R(ti)
    • Simulate Market Evolution following R(ti+1)
  • Gaussian Mixture ⇒ “Fat Tails”
  • Asymptotically Gaussian
    • \(\mu_∞=\Sigma^m_{k=1}\pi_k\mu_k\)
    • \(\Gamma_∞=\Sigma^m_{k=1}\pi_k\Gamma_k\)
    • P’ \(\pi=\pi\), \(\pi=(\pi_1,…,\pi_m)\)
  1. Determine Breakpoints(\({t_1,…,t_n}\)) and “homogenous” periods \(J_k=[t_{k-1},t_k]\)
  2. Calibrate a Gaussian distribution\(N(\mu_k,\Gamma_k)\) over consecutive periods \(J_k\)
  3. Clusterize the set of \(\Phi=(\mu_k,\Gamma_k)\), using some information distance (K-L, Tsallis, Hellinger…)
  4. Issue: these distributions lie in a high dimensional space n(n + 3)/2⇒No recurrence
    1. Reduce dimension by describing the market with fewer indices
    2. Project \(\Phi_k\) onto a lower dimensional space, approximately preserving distances, using Spectral Embedding
      • Common practice: focus only on volatility
    3. The projection now has some recurrence.
  5. Estimate Transition Probabilities
    1. Baum-Welch algorithm: too imprecise
    2. SVM or EM provide more accurate results
    3. Depends on time spent within a regime
  6. Crisis Prediction: Mild regime that is likely to transit to a wild one

Multi-Regime Simulation vs. Observed

Pink: Multi-Guassian Simulation (MCMC)

Green: Actual Returns

Joint distribution of 4 risk factors:

SP500

SP Sector Financials

SP Sector Oil Companies

MSCI World

Simulation on US Market