## **Learning Outcomes**

This article explains marginal contribution to risk and risk contributions in portfolio management for the CFA Level 3 exam, including:

- calculating marginal contribution to risk (MCR) for individual assets and asset classes using portfolio volatility, correlations, and position weights;
- deriving each asset’s risk contribution (rc) from its weight and MCR, and reconciling the sum of risk contributions with total portfolio risk;
- interpreting MCR and rc to assess the relative importance of different positions, sectors, and strategies in overall portfolio risk allocation;
- constructing and adjusting risk budgets that allocate absolute or active risk across assets, asset classes, or factors in line with investor objectives and constraints;
- comparing risk budgeting with traditional mean–variance optimization, including the intuition behind risk parity portfolios and equal risk contribution allocations;
- evaluating how changes in volatility, correlation, borrowing, and investment constraints alter marginal contributions to risk and observed risk contributions through time;
- applying the marginal risk framework to monitoring, attribution, and rebalancing decisions so that realised portfolio risk stays aligned with the intended risk budget.

## **CFA Level 3 Syllabus**

For the CFA Level 3 exam, you are expected to understand how marginal contribution to risk and risk budgeting are used in portfolio management, with a focus on the following syllabus points:

- Defining marginal contribution to risk and explaining how it is calculated for assets and asset classes
- Calculating risk contributions (rc) and interpreting their significance in portfolio risk allocation
- Applying risk budgeting concepts to equity, fixed income, and multi-asset portfolios
- Explaining the incorporation of risk budgeting into strategic asset allocation and monitoring processes
- Assessing the effects of active risk, active share, and constraints on risk budgeting and portfolio structure

## **Test Your Knowledge**

_Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision._

1. What does marginal contribution to risk mean in the context of a portfolio?
2. How does risk budgeting differ from traditional mean-variance optimization?
3. If a portfolio consists of equities and bonds, what happens to each asset's risk contribution if correlation between them increases?
4. True or false? A portfolio where each asset's excess return per unit of marginal contribution to risk equals the information ratio is risk-budget optimal.

## **Introduction**

Risk budgeting is a practical method for allocating risk efficiently within a portfolio rather than allocating only capital. This article explains how marginal contribution to risk is calculated, how each asset’s risk contribution (rc) is determined, and how these measures are integrated in portfolio construction. You will see how risk budgeting helps manage portfolio risk, supports investment objectives, and informs rebalancing disciplines for CFA Level 3 exams.

> **Key Term: Marginal Contribution to Risk (MCR)**
> The incremental change in total portfolio risk (typically measured by standard deviation or volatility) resulting from a marginal increase in the weight of a specific asset or asset class in the portfolio.
>
> **Key Term: Risk Contribution (rc)**
> The portion of total portfolio risk attributable to each asset or asset class, usually calculated as the weight of the asset multiplied by its marginal contribution to risk.

## Marginal Contribution to Risk: The Core Concept

Marginal contribution to risk (MCR) quantifies the sensitivity of total portfolio risk to changes in the weight of a particular position. In a multi-asset portfolio, it is the partial derivative of portfolio risk (standard deviation) with respect to the asset’s weight.

For an asset $i$ in a portfolio of $N$ assets, the marginal contribution to risk is:

$$
\text{MCR}\_i = \frac{\partial \sigma_P}{\partial w_i}
$$

where:

- $\sigma_P$ is the portfolio standard deviation
- $w_i$ is the weight of asset $i$

## Risk Contribution (rc)

Once the MCR for all assets is calculated, risk contribution (rc) for each asset is:

$$
\text{rc}\_i = w_i \times \text{MCR}\_i
$$

This answer represents how much each asset contributes to total portfolio risk. Summing all risk contributions across assets equals the total portfolio risk.

### **Worked Example 1.1**

**A portfolio is 60% equities and 40% bonds. The portfolio volatility is 9%. The marginal contribution to risk for equities is 10%, and for bonds 7%. What are the risk contributions?**

> **Answer:**
>
> - Equities: $rc\_{eq} = 0.60 \times 10\% = 6.0\%$
> - Bonds: $rc\_{bond} = 0.40 \times 7\% = 2.8\%$
> - Total risk allocated: $6.0\% + 2.8\% = 8.8\%$ (Small rounding differences possible.)

## Practical Use in Risk Budgeting

Risk budgeting allocates desired proportions of total risk to various portfolio components—assets, sectors, or risk factors—rather than only specifying percentage capital allocation. This technique supports portfolio objectives and fits regulatory, liquidity, or operational constraints.

### Steps in Risk Budgeting

1. **Specify the overall portfolio risk budget.** For example, the target active risk (tracking error) or volatility.
2. **Assign target risk contributions.** Choose risk targets for each portfolio part (e.g., equities 60%, bonds 40% of risk).
3. **Calculate marginal contributions and risk contributions** for all assets.
4. **Structure the weights** so that actual risk contributions match targets.

### **Worked Example 1.2**

**A multi-asset portfolio has a volatility target of 8%. The manager wants each of four equally weighted assets (A, B, C, D) to contribute 25% of risk. How is this achieved?**

> **Answer:**
> The portfolio weights and asset correlations should be set so that, after calculation, each asset's $rc_i$ equals 2% (which is 25% of the total 8%). Usually, this requires iterative optimization balancing weights and correlations.

## Why Not Just Allocate Capital

Allocating risk instead of capital accounts for the fact that assets’ volatilities and correlations differ widely:

- A 10% weight in an equity may add more to overall risk than 20% in government bonds.
- Risk budgeting allows comparison of disparate assets based on their risk—not just their size in the portfolio.

## Portfolio Incorporation and Risk Budgeting

Risk budgeting is integrated into portfolio construction by aligning the sum of individual asset risk contributions with the portfolio’s target risk and the investor’s objective. This process may be implemented for absolute risk or active risk (when managing versus a benchmark), depending on mandate and constraints.

> **Key Term: Active Risk (Tracking Error)**
> The standard deviation of portfolio returns in excess of benchmark returns. A key measure for risk budgeting relative to an index.
>
> **Key Term: Risk Budget Optimality**
> A portfolio is risk-budget optimal when the ratio of an asset’s expected excess return to its marginal contribution to risk is the same for all assets and equals the portfolio’s information ratio.

## Risk Parity versus Traditional Optimization

Risk parity is a special form of risk budgeting, where assets are weighted so that all contribute equally to total risk. Unlike mean-variance optimization—which allocates based on return and correlation forecasts—risk parity is return-agnostic and focuses solely on spreading risk evenly, irrespective of return expectations.

### **Worked Example 1.3**

**Suppose a risk parity portfolio is constructed from three assets (US stocks, developed market stocks, government bonds), each assigned the same risk contribution. What is the likely effect on asset weights compared to a 60/40 portfolio?**

> **Answer:**
> Government bonds are typically less volatile than equities. To achieve equal risk contributions, the portfolio must overweight low-volatility assets (bonds) relative to traditional market-cap or fixed mixes.

### **Revision Tip**

_If a question asks for marginal contribution to risk, always check whether portfolio risk is measured in absolute or active (relative) terms. Focus calculations on the appropriate volatility measure for the exam question._

## How Constraints and Correlations Affect Risk Contributions

Constraints such as limits on weights, regulatory rules, and liquidity needs may change the marginal contribution to risk and thus the allocation resulting from a risk budget. As correlations among assets rise, risk contributions become more similar, potentially concentrating risk in fewer sources.

### **Exam Warning**

> It is a common exam mistake to assume that equally weighting assets also equalizes their risk contributions. In diversified portfolios, risk contributions are rarely equal when assets vary significantly in volatility or correlation.

## Monitoring and Rebalancing: Applying the Marginal Risk Framework

Ongoing risk monitoring should track whether actual risk contributions equal the intended risk budget. If not, rebalancing is necessary. This process ensures the risk allocation continues to align with the strategic objective, especially during market moves.

## **Summary**

Marginal contribution to risk measures how each asset changes total portfolio risk with small weight changes. Risk contributions (rc) enable allocation and adjustment of actual portfolio risk, not just capital. Risk budgeting sets target contributions for portfolio parts and builds or rebalances the portfolio accordingly. This approach is widely used for absolute risk, active risk, and in constructing risk parity portfolios. Applying this framework helps CFA candidates answer quantitative and scenario-based questions precisely and avoids common calculation pitfalls in the exam.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Explain marginal contribution to risk (MCR) and how to calculate it
- Calculate and interpret risk contributions (rc) for portfolio assets
- Describe the process of setting and monitoring risk budgets
- Distinguish risk budgeting and risk parity from traditional optimization
- Integrate risk budgeting into portfolio objectives and rebalancing
- Recognize the role of constraints and asset correlation in altering risk contributions

## **Key Terms and Concepts**

- Marginal Contribution to Risk (MCR)
- Risk Contribution (rc)
- Active Risk (Tracking Error)
- Risk Budget Optimality

Risk budgeting and integration - Marginal contribution to risk and rc