Systematic
Alpha Research
& Risk Models

In plain English: At month T, look at how each stock performed over the PAST 12 months (not including last month). Stocks that went up the most are "momentum winners" — put them in the LONG portfolio. Stocks that fell the most are "momentum losers" — put them in the SHORT portfolio. Each month you rebalance and collect the spread.

Why IC matters more than backtest returns: A strategy can show positive returns by LUCK. IC tests whether the signal GENUINELY ranks stocks correctly. If IC is positive consistently, the factor has real predictive power — not just a backtest artifact. Every serious quant uses IC to validate signals before putting capital at risk.

What N(d₂) means: The risk-neutral probability that the option expires in-the-money. If d₂ = 0, N(0) = 0.5 — 50% chance. If d₂ = 2, N(2) ≈ 0.977 — 97.7% chance the call expires ITM.

S·N(d₁): Expected value of receiving the stock if you exercise.
K·e^(-rT)·N(d₂): Present value of paying strike K, probability-weighted.

Delta (Δ): If stock goes up ₹1, call gains Δ rupees. ATM call ≈ 0.5.
Gamma (Γ): How fast delta changes. Highest ATM near expiry. Long gamma = benefit from big moves.
Vega (ν): Option value per 1% rise in vol. Long options always positive vega.
Theta (θ): Daily time decay — how much value lost each day. Negative for buyers.
Rho (ρ): Sensitivity to interest rate. Matters more for long-dated options.

In plain English: XGBoost builds trees one by one. Each new tree tries to fix the errors of the previous ones. The regularisation term Ω prevents overfitting — critical in finance where data is scarce and noisy.

Why better than random forest for finance: Gradient boosting focuses on hard examples. In return prediction, the "hard" stocks — the ones with ambiguous signals — get more attention in each subsequent tree.

Why walk-forward beats train/test split: A single split gives one OOS estimate — which could be lucky or unlucky. Walk-forward gives many OOS periods, testing whether the model works across different market regimes (bull, bear, sideways, high-vol, low-vol). If IC is positive across all regimes, the model is robust.

Bias-Variance tradeoff: In finance, overfitting is the dominant failure mode. OOS R² of 4.3% sounds small but is actually meaningful — finance signal-to-noise is extremely low.

VaR limitation — why CVaR is better: VaR tells you NOTHING about losses beyond the threshold. Two portfolios can have identical VaR but very different tail risk. CVaR (Expected Shortfall) asks: IF we breach VaR, how bad is it on average? CVaR is a coherent risk measure — VaR is not (it can violate sub-additivity). Basel III now requires CVaR, not VaR.

Portfolio VaR formula: σ_p = √(w'Σw) where Σ is the covariance matrix and w is the weight vector. This is why correlation matters — two assets with high individual VaR but negative correlation can have LOW portfolio VaR. The correlation heatmap shows where diversification benefit exists in your portfolio.

Cointegration vs correlation: Two stocks can be highly correlated (move together short-term) but not cointegrated (diverge permanently long-term). Cointegration is stronger — it means the SPREAD between the two prices is stationary (mean-reverting). This is what makes pairs trading work.

HDFC Bank and ICICI Bank are both large private sector banks exposed to the same Indian credit cycle, so their prices are cointegrated.

Why half-life matters: Half-life tells you how quickly the spread reverts to its mean. 18 days means: if the spread is 1 std dev away from mean today, in 18 days it will be 0.5 std dev away — on average.

Trading rule: Open position at |Z| > 2. Close at |Z| < 0.5. Stop-loss at |Z| > 3 (in case cointegration breaks down).

Why Nelson-Siegel: Only 4 parameters describe the entire yield curve across all maturities. β₀ is the long-run interest rate. β₁ determines whether the curve is upward or downward sloping. β₂ controls the hump.

RBI current curve: β₀=7.1, β₁=-0.8 (normal upward slope), β₂=1.2 (slight hump around 5Y tenor).

Duration in plain English: If interest rates rise by 1%, a bond with duration 7.1 years loses approximately 7.1% in value. That's why long-dated bonds are riskier in a rising rate environment.

Convexity adjustment: Duration is a linear approximation. Convexity corrects for the curvature — bonds gain MORE when rates fall than they lose when rates rise by the same amount. This is positive convexity, a desirable property.

The key insight (Markowitz 1952): What matters is not just each asset's own risk, but how assets move TOGETHER (covariance). Two risky assets with negative correlation can form a portfolio with LOWER risk than either asset alone. This is the math behind diversification.

Practical limitation: MVO is extremely sensitive to expected return estimates (μ), which are nearly impossible to estimate precisely. Small errors in μ lead to wildly different portfolio weights.

Risk Parity vs MVO: MVO requires expected returns (hard to estimate). Risk parity only requires the covariance matrix (more stable).

Bridgewater All Weather: The most famous risk parity fund. Holds bonds, equities, gold, and commodities weighted so each contributes equal risk. Uses leverage to bring bonds to equity-equivalent risk level.

Weakness: Underperforms when bonds and equities fall together — as in 2022 when rate rises hurt both asset classes simultaneously.