Multi-Regime Portfolio Optimization

• Assume that Y₁…Y_m 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,…,R_m
  • Covariance Matrix G_k
  • Drift Vector m_k
  • R_k ~ N(m_k,G_k)
• Transition Probability Matrix
  • P(t,t+1) = (p_hk)
  • p_hk = Probability of transitioning from R_h to R_k
  • ∀h, \(\Sigma^m_{k=1}p_{hk}=1\)
  • P(t,t+n) = Pⁿ

• Euler Scheme:
  
  • Discretized Time t₀,…,t_n
  • Pick new regime R(t_i+1) according to P applied to current regime R(t_i)
  • Simulate Market Evolution following R(t_i+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