Portfolio Construction & Risk

Excess return over a benchmark after accounting for beta; a measure of skill or edge in backtesting.

Sensitivity of strategy returns to a benchmark; measures systematic (market) risk in backtesting.

Standard deviation of returns; the standard measure of dispersion and risk in backtesting.

Mean–variance portfolio theory: optimal diversification by balancing expected return and variance.

The set of minimum-variance portfolios for each level of return, derived in closed form with Lagrange multipliers, giving the two-fund theorem, the global minimum-variance portfolio, the tangency portfolio, and the capital market line.

The equilibrium that prices every asset by its beta to the market, derived from the tangency portfolio, giving the security market line, plus the assumptions it rests on and the anomalies that broke it.

Why the textbook optimizer is an "error-maximizing machine", the mathematics of weight instability, extreme sensitivity to expected-return estimates, corner solutions, and the shrinkage, resampling, and constraint fixes practitioners actually use.

How to estimate the covariance matrix that every portfolio optimizer inverts, the sample estimator and its bias, the curse of dimensionality when N approaches T, eigenvalue spreading and ill-conditioning, and why the naive estimate is dangerous to invert.

Pulling estimates (e.g. mean, covariance) toward a prior or global average to reduce estimation error.

The optimal convex combination of the noisy sample covariance and a structured target, the shrinkage target, the closed-form optimal intensity that minimizes expected Frobenius loss, and why it dominates the sample matrix out of sample.

The Bayesian fix for unstable mean-variance portfolios, reverse-optimize the market to get equilibrium returns as a prior, express subjective views with confidences, and blend them into posterior expected returns via the master formula.

Allocating so that every asset contributes equal risk rather than equal capital, derived from Euler's decomposition of portfolio volatility, the equal-risk-contribution condition, the role of leverage, and the comparison to 60/40.

Portfolio construction that uses a hierarchical clustering of assets to allocate risk more evenly and robustly.

Decomposing asset risk into a few systematic factors plus idiosyncratic noise, the Barra-style structure, why it cuts the covariance matrix from O(N squared) to O(NK) parameters, factor-covariance and specific-risk estimation, and risk attribution.

The bet size that maximizes long-run geometric growth, derived from log-utility, extended to continuous and multi-asset cases, with the estimation-error and drawdown reasons practitioners bet a fraction of it.

Rules for how much capital to allocate per trade or asset (equal weight, risk parity, Kelly, etc.).

Scaling portfolio exposure so that realized volatility stays near a target (e.g. 10% annual).

Return per unit of total risk (volatility); the standard risk-adjusted performance metric for backtesting.

Return per unit of downside volatility; penalizes only bad volatility in backtesting.

Active return per unit of tracking error; measures excess performance vs a benchmark in backtesting.

Annualized return divided by maximum drawdown; emphasizes drawdown risk in backtesting.

Largest peak-to-trough decline in cumulative equity; a key risk metric for backtesting and live performance.

The loss quantile that dominates risk reporting and Basel capital, its definition, the three estimation methods (historical, parametric, Monte Carlo), how to backtest it with the Kupiec and Christoffersen tests, and its fatal flaws.

The average loss in the tail beyond VaR, its definition as a conditional tail expectation, why it fixes VaR's blindness to tail depth and its lack of subadditivity, the Rockafellar-Uryasev convex formulation, and its adoption by Basel.

The Artzner et al. axioms a sensible risk measure must satisfy, monotonicity, translation invariance, positive homogeneity, and subadditivity, a proof that VaR violates subadditivity, and why expected shortfall satisfies all four.

The AUM ceiling beyond which a strategy stops making money, how market impact and turnover erode alpha, the square-root impact law, the capacity formula that balances gross alpha against trading costs, and why fast strategies capacity-cap first.