## 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
*Y*_{1}…*Y*have mixed joint distribution_{m}

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

- Fat tails can be measured as the ratio of risk under P
_{1}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
*R*_{1},…,*R*_{m}- Covariance Matrix G
_{k} - Drift Vector
*m*_{k} *R*~_{k}*N*(*m*,_{k}*G*)_{k}

- Covariance Matrix G
- Transition Probability Matrix
*P*(*t*,*t*+1) = (*p*)_{hk}*p*= Probability of transitioning from_{hk}*R*to_{h}*R*_{k}- ∀h, \(\Sigma^m_{k=1}p_{hk}=1\)
*P*(*t*,*t*+*n*) =*P*^{n}

- Euler Scheme:
- Discretized Time
*t*_{0},…,*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})

- Discretized Time
- 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’ \(\pi=\pi\), \(\pi=(\pi_1,…,\pi_m)\)

- Determine Breakpoints(\({t_1,…,t_n}\)) and “homogenous” periods \(J_k=[t_{k-1},t_k]\)
- Calibrate a Gaussian distribution\(N(\mu_k,\Gamma_k)\) over consecutive periods \(J_k\)
- Clusterize the set of \(\Phi=(\mu_k,\Gamma_k)\), using some information distance (K-L, Tsallis, Hellinger…)
- Issue: these distributions lie in a high dimensional space
*n*(*n*+ 3)/2⇒No recurrence- Reduce dimension by describing the market with fewer indices
- Project \(\Phi_k\) onto a lower dimensional space, approximately preserving distances, using
*Spectral Embedding*- Common practice: focus only on volatility

- The projection now has some
*recurrence*.

- Estimate Transition Probabilities
- Baum-Welch algorithm: too imprecise
- SVM or EM provide more accurate results
- Depends on time spent within a regime

- 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