A single portfolio volatility number answers "how much risk?" but not the question that matters to a portfolio manager: "where is it coming from, and is that where I want it?" This note builds the standard answer machinery: the marginal contribution to risk (what happens to risk if I trade), the contribution to risk (how today's risk is allocated across positions), and the Euler-theorem fact that makes the latter add up exactly to total risk — the licence for risk budgeting. The same homogeneity argument extends to Value at Risk, so tail-risk budgets inherit the whole framework. A complete worked example closes the note; it is the exam's favourite calculation from this part of the course.
Learning objectives. By the end of this note you should be able to:
- derive and interpret it (including the systematic/specific split and group MCR);
- derive the Euler decomposition from linear homogeneity;
- distinguish MCR (dynamic) from CR (static) and define risk budgeting;
- state the conditional-expectation caveat on interpreting CR;
- extend contributions to VaR under normality;
- execute a full MCR/CR/PCR table by hand.
In this lesson
- 4.1 Marginal contribution to risk (MCR)
- 4.2 Contribution to risk (CR) and percentage contribution (PCR)
- 4.3 Contribution to Value at Risk
- 4.4 Constraints and implementation as risk topics
- 4.5 Worked example (exam style)
The full lesson, with worked examples and exercise solutions, is for members.
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