Portfolio Risk, VaR Budgeting, and Performance Evaluation

Value at risk (VaR) is a loss quantile at a specified confidence and horizon. The analytical calculations here assume linear exposures, a covariance model and zero expected profit or loss over the risk horizon. Under a normal approximation, portfolio VaR is a confidence multiplier times dollar profit-and-loss volatility. Tail dependence, nonlinear positions and changing liquidity require additional analysis.

Let $x_i$ denote dollar positions, $\Sigma$ the covariance matrix of asset returns and $s$ dollar portfolio volatility. Keep dollar positions separate from weights that sum to one:

$$s=\sqrt{x^T\Sigma x},\qquad VaR_p=zs,\qquad VaR_i=z|x_i|\sigma_i.$$

Undiversified VaR sums the individual values. In this linear covariance model, it is an upper bound on diversified VaR and represents no diversification among position profit-and-loss risks. Do not generalize that result to every possible distribution: VaR is not universally subadditive. Correlation changes the portfolio result without changing any standalone volatility.

Marginal VaR measures the local change in total VaR per extra dollar in a position, with other dollar positions fixed. Component VaR attributes total VaR to current holdings:

Common mistakes

• Confusing units. Marginal VaR is per dollar; component VaR is a dollar allocation of risk.
• Using components as finite removal effects. Removing an entire manager changes the remaining risk contributions and requires recomputation.
• Choosing by volatility alone. Without correlations, the highest-volatility holding need not have the highest component contribution.

Bottom line

• Define confidence, horizon, model and units before computing VaR.
• Marginal risk is local; component risk allocates; incremental risk recomputes.
• Risk-minimizing and return-per-risk-maximizing positions can differ.
• Policy and active budgets depend on covariance and need separate monitoring.

Exam shortcut

Label every risk answer with its units. If the question changes a whole position, recompute total risk instead of treating its old component as the removal effect. For performance, first decide whose cash-flow timing is being evaluated, then choose the benchmark and denominator. Check an attribution by summing the effects back to active return.

The full lesson (about 4,055 words, 27 min read) adds 8 worked examples, all 6 common mistakes, a self-check, free in the app.

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