PolyModel and Risk Management

By definition, a “model” is aimed at stressing a fund or an asset and anticipate its response to the stress

A PolyModel is a Collection of Single-Factor (Nonlinear) Models

• Step 1: Identify a LARGE Set of Factors
  • LONG HISTORY (20+ Yrs incl. crises)
  • As many factors as potential risk sources ⇒Several 100’s
• Step 2: Scan Factors One at a Time
  • Focus on EXTREME MOVES⇒Nonlinear Models
  • Select only factors with a strong statistical relationship to the fund Þ Score = –Ln(p-value)
  • A risk factor is selected if its Score is at least ½ highest score across all factors
• Step 3: Stress Selected Factors
  • 99%-Confidence Worst Impact of each Selected Factor
  • Single Factor \(StressVAR=√(Worst Impact)^2+(Specific Risk)^2\)
  • Global STRESS VaR = MAX(Single Factor Stress VaR)

• Handle 100’s of Market Factors
• Model rare events (“Black Swans”)
• More accurate when needed, than when not needed!
  • Tail concentration effect
• Suited for risk measurement and stress scenarios
  • Prediction from individual factors can be merged
  • Risk measure = STRESS VaR (worst case) includes hidden risks
• Can be aggregated for a portfolio
  • Risk contributions involve extreme correlations
  • Superior allocation and optimization

• Risk Factor: X(t)
• Invested Asset: Y(t)
• Bayesian Computation of E(Y|X)
  • Prior regime probabilities: ergodic limit \(\pi_k\)(uninformed Jeffreys prior)
  • Given the value of X, compute the implied (posterior) probabilities

    \(p_k(X)\)

  • \(E_{R_k}(Y|X)=a_kX+b_k\) linear for one regime

  • \(E(Y|X)=\Sigma^m_{k=1}p_k(X)E_{R_k}(Y|X)=f(X)\) with nonlinear f in multi-regimes

  • The more probabilities are sensitive to X, the more nonlinear is f ⇒PolyModels used as Crisis indicator

• HMM is a dynamic model that can be simulated
• PolyModels are a statistical analysis of the dynamics

Question: Fund risks and future returns?

Input: PolyModel (instead of series of returns)

• Uses market information⇒Sensitivity to risk factors

Answer:

• Selects funds that have a convex (antifragile) response to dominant factors
• Rejects funds that have a concave (fragile) response to dominant factors

Advanced methods (to be tested)

• Deep learning
• Adversarial learning

Learning to “guess the future”: contradicts Market Efficiency Hypothesis

Learning risks with behavior under stress: Use sensitivity to factors as input

Unsupervised Learning

• Train the system on all past situations to predict returns

⇒Efficient Market Hypothesis!!!

• • No expected return, most likely lured by Black Swan risk
• Predict volatility (or another risk measure): possible, but doesn’t produce a trading signal

Supervised Learning

• Build a Risk Profile for each asset: list of responses to stress tests
• Train the allocation system (across asset classes and within each asset class) on the basis of the Risk profile
  • Performance, volatility are not predicting future regime change
  • Downturn Indicator: Left skew, fragility