Portfolio Risk, VaR Budgeting, and Performance Evaluation

A pension allocates $100 million each to four equity managers. Total fund VaR is $18 million. The CIO wants to add a fifth manager and asks: which existing manager should be cut to keep total VaR flat? The wrong answer is "the one with the highest standalone VaR." The right answer needs marginal VaR: the contribution that depends on correlations, not standalone risk.

Undiversified VaR adds individual VaRs as if correlations equaled 1:

$$Va{R}_{\text{undiv}}=\sum _{i}Va{R}_i$$

It is an upper bound and almost always overstates risk. Diversified VaR uses the actual covariance matrix:

$$Va{R}_{\text{div}}=z\cdot \sqrt{w^T\Sigma w}$$

The gap between undiversified and diversified VaR is the diversification benefit. A portfolio with correlations near 1 gets little benefit; a portfolio with negative correlations gets a lot.

KEY: Diversified VaR depends on correlations, not just individual volatilities. Two managers each with $5MVaRcanproduceacombinedVaRanywherefrom$0 (perfect negative correlation) to $10M (perfect positive correlation).

Common mistakes

• Adding standalone VaRs to compute portfolio VaR. Standalone VaRs sum to undiversified VaR, which is an upper bound, usually 30-50% above diversified VaR. Trap: question shows three positions with VaR $5M,$7M, $8M, and the choice "$20M" is the trap. Diversified VaR is lower; the right answer needs the covariance matrix.
• Cutting the manager with highest standalone VaR. The standalone VaR ignores correlations. A high-volatility manager that hedges other positions has low marginal VaR: cutting it INCREASES total portfolio VaR. Trap: candidates pick "the manager with highest standalone VaR" as the cut decision; the correct basis is component VaR.
• Confusing marginal and component VaR. Marginal VaR is per dollar; component VaR is the dollar amount. Component VaRs sum to total VaR; marginal VaRs do not. Trap: answer choices give marginal VaR values that don't sum to total. Candidates who pick the row whose sum matches mismatch the concept.

Bottom line

• Diversified portfolio VaR uses the covariance matrix $=z\cdot \sqrt{w^T\Sigma w}$; undiversified VaR sums standalone VaRs and overstates risk unless correlations equal 1.
• Marginal VaR $=\partial VaR/\partial w_i$, the change in portfolio VaR per dollar added to position $i$. Component VaR = weight times marginal VaR and sums to total VaR (standalone VaR does not).
• Risk budgeting allocates a total VaR limit using component VaR as the accounting unit; equal-dollar allocation does not produce equal-risk allocation.
• Performance measures: Sharpe (total risk), Treynor (systematic), Jensen's α (excess over SML), IR (active return / tracking error), $M^2$ (Sharpe rescaled to benchmark volatility).

Exam shortcut

When the question gives a portfolio VaR question with weights, volatilities, and correlations, ALWAYS use diversified VaR with the covariance matrix. Adding standalone VaRs is the trap answer. When the question asks "which position to cut," the answer is component VaR (or marginal VaR if comparing per-dollar), never standalone VaR.

The full lesson (about 3,320 words, 22 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

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