Advanced D · Portfolio and risk

Multi-strategy Portfolio: merge ledgers first

A portfolio shares capital, liquidity, and limits. Five names do not imply five independent return sources: Trend, Flow, and Event may all lose as risk appetite falls. Merge using consistent valuation times, currencies, and net costs before discussing diversification.

List capital use, net direction, gross notional, financing, and venues for five candidates. Offsetting direction still requires gross notional and margin. Acceptance reconciles strategy equity to portfolio ledgers, with unallocated cash separate—not imaginary invested positions.

Risk Contribution: capital shares differ from risk shares

For weights w and covariance Σ, portfolio variance is wᵀΣw. Strategy i's marginal volatility contribution is (Σw)i/σ_portfolio; total contribution is wi(Σw)i/σ_portfolio. Contributions sum to portfolio volatility. Negative contributions hedge under estimated conditions, without making the strategy itself lossless.

Two uncorrelated teaching strategies have 50% weights and same-period volatilities 10% and 20%. Portfolio volatility is about 11.1803%. Contributions are 2.2361 and 8.9443 percentage points, shares 20% and 80%. At correlation 1, volatility becomes 15%; old contributions no longer apply.

Estimate covariance on one window and recalculate with stressed correlations. Specify windows, units, matrix validity, and estimation errors. Stop on zero volatility, invalid correlation, or nonfinite inputs rather than hiding errors behind zero results.

Beta and Factor Exposure: zero net notional can retain direction

Beta measures sensitivity to a selected market factor. Linear portfolio exposure sums capital-weighted Betas. Factors may include liquidity, volatility, and term structure, with economic explanations and consistent units. Equal-notional related-asset longs/shorts do not guarantee zero Beta.

At weights 0.4 and 0.6 with Betas 1.5 and −0.5, portfolio Beta is 0.3. Add liquidity and inspect common direction. Submit net/gross exposure and factors. Linear models omit option convexity; revalue scenarios separately. Beta neutrality does not waive stress.

Correlation: stress requires different assumptions

Historical correlations depend on windows, synchronization, and regimes. Shared financing, collateral, and exits may be hidden in calm periods but simultaneously destroy liquidity under stress. Low averages cannot replace mechanism review.

Write normal, higher-directional-correlation, and shared-liquidity-shock matrices. Every matrix must be valid; arbitrary cell changes must not leave a non-positive-semidefinite matrix. Submit economics alongside correlations. Unestimable joint tails become deterministic loss scenarios, not fictitious precise probabilities.

Volatility Target: feedback can be procyclical

Basic scale is target divided by forecast volatility, bounded by leverage, turnover, capital, and liquidity. Target 10% and forecast 20% give 0.5 scaling. Low estimates do not permit unlimited size. Fix update frequency, bounds, and minimum adjustments in advance.

Add a volatility jump after calm conditions; compare immediate with rate-limited scaling. Lag may first overshoot risk; collective deleveraging may raise impact. Record targets, realized volatility, changes, fees, and triggers—not smoothed risk plots alone.

Risk Parity: conditions for equal risk

Risk parity aligns contributions with budgets. Two uncorrelated positive-weight strategies may begin with inverse-volatility weights. Correlation, leverage bounds, and negative contributions mean inverse volatility need not achieve parity.

For 10% and 20% volatility, weights 2/3 and 1/3 equalize contributions. Add correlations and capital caps and recheck. Report solver errors, constraints, and sensitivity. Mathematical parity cannot justify illiquid assets.

Kelly: uncertainty matters more than the formula

With independent binary outcomes, fixed payoffs, and repeated bets, Kelly f=p−(1−p)/b. Teaching p=0.4 and b=2 give 0.1; quarter-Kelly is 0.025. This maximizes model long-run log growth, not low drawdown or recommended market size.

Lower estimated wins and raise losses, inspecting changes. Gaps, correlated positions, financing, and errors break assumptions. Use conservative ranges, fractional scales, and hard caps. Negative Kelly means no bet in that direction, not automatic reverse-leverage authorization.

Stress Testing and Scenario Analysis: define shocks before losses

Stress checks vulnerability to specified shocks. Scenarios jointly define prices, correlations, liquidity, financing, and venue states. Linear approximations are a first layer; options, liquidation, and withdrawal freezes require path revaluation and cash constraints.

Cover at least market gaps/correlation rises, Carry financing/basis pressure, unavailable major venues, and stale data/model failures. List inputs, strategy PnL, funding shortfalls, executable exits, and residual exposure. Unvalidated nonlinear terms are unquantified gaps, not zeros.

Acceptance requires pressure losses within advance teaching budgets or executable reductions/stops for breaches, not profitability everywhere. “Low probability” cannot excuse unsurvivable outcomes.

Drawdown Control: triggers need next actions

Drawdown compares equity with historical peaks, including unrealized PnL and fees. Specify observation frequency, thresholds, reduction targets, residual orders, authority, and restart. Doubling risk to recover changes the approved strategy; it is not routine recovery.

Use teaching tiers: first drawdown pauses additions; second executes approved reductions; capital-floor breaches stop and escalate. The capstone handbook fixes thresholds; do not move them outward during operation. Gaps cross thresholds; rules do not guarantee prices.

Replay before triggering, at triggering, and after recovery. Rising prices must not automatically erase pauses. Require causes, balance/order reconciliation, freshness, and authorization.

Project: Portfolio and Risk Report

Use the risk template: capital, net/gross exposures, factors, covariance/contributions, volatility targets, tails, drawdown state, and actions. Date every input, giving units, models, and omitted risks.

Reviewers recalculate capital totals, summed contributions, stress PnL, and drawdowns, then remove a strategy and recalculate. Position, cash, and margin limits must hold together; favorable average risk cannot offset a venue funding shortfall.

VaR and Expected Shortfall: limits of tail summaries

At a fixed horizon/confidence, VaR is a loss quantile and Expected Shortfall (ES) is average loss in that tail. A few worst points dominate high-confidence estimates; disclose coarse resolution in small samples. VaR is not maximum loss, and ES cannot include accidents omitted from the model.

Convert the same net returns into positive losses, sort, specify the quantile convention, and calculate tail means. Add an extreme loss. For ties at discrete quantiles, state whether probability mass is apportioned to tails; do not silently switch algorithms. Submit horizon, confidence, count, loss sign, and quantile method alongside stress and drawdowns. VaR alone does not authorize positions.

Ruin paths and liquidation constraints

Test survival by combining daily returns, costs, margin, and liquidity in one path. Stop additions at capital floors and continue valuing remaining positions. Do not delete negative balances and resume at positive equity. No ruin under resampling means only none within that sample/model.

Compare two risk fractions with the same seed, then change losing-streak structure. Record ruin counts, maximum drawdown, funding gaps, and post-stop exposure. “Maximum tolerable leverage” is an upper bound for specified scenarios/constraints, not safe market leverage. Liquidation feeds back through forced selling, reduced depth, and further triggers; link prices and exit capacity. Linear correlation changes alone cannot describe this loop.