## **Learning Outcomes**

This article explains the role of mean-variance and resampled optimization in risk budgeting for portfolio construction, including:

- Distinguishing clearly between traditional mean-variance optimization (MVO) and resampled MVO in terms of objectives, mechanics, and output portfolios.
- Describing how expected returns, variances, and correlations are used as inputs, and why small changes in these assumptions can shift allocations dramatically.
- Analyzing the problem of estimation error in optimization and linking it to unstable, overly concentrated portfolios that may be impractical for institutional investors.
- Explaining the resampling process—generating simulated input sets, optimizing repeatedly, and averaging weights—to reduce sensitivity to any single return or risk estimate.
- Assessing how resampled optimization typically leads to more diversified, stable portfolios and smoother changes in allocations across different market scenarios.
- Evaluating the implications of each approach for risk budgeting, including alignment with specified volatility targets, risk limits, and regulatory or policy constraints.
- Interpreting the effect of optimization choices on asset-class risk contributions, overall diversification, and the robustness of strategic asset allocation decisions.
- Applying these concepts in a CFA Level 3 exam context to justify portfolio recommendations, identify weaknesses in proposed allocations, and choose appropriate optimization techniques.

## **CFA Level 3 Syllabus**

For the CFA Level 3 exam, you are expected to understand the principles and application of portfolio optimization techniques under risk constraints, with a focus on the following syllabus points:

- Distinguishing between mean-variance optimization (MVO) and resampled mean-variance optimization for portfolio construction
- Assessing the role of input sensitivity and estimation error in MVO
- Recognizing how resampling can address certain shortcomings of traditional optimization
- Applying risk budgeting to asset allocation and analyzing implications for diversification

## **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 is a primary drawback of traditional mean-variance optimization in portfolio construction?
2. How does resampled mean-variance optimization differ from standard mean-variance optimization, and what practical benefit does it provide?
3. Why is risk budgeting important in institutional portfolio construction?
4. True or false: Changing expected return inputs by a small amount has minimal effect on mean-variance-optimized portfolios.

## **Introduction**

Portfolio optimization is central to risk budgeting and asset allocation decisions. Two prominent approaches are mean-variance optimization (MVO) and resampled optimization. This article outlines the mechanics, merit, and limitations of each method, emphasizing their impact on portfolio construction for CFA candidates. Understanding how these frameworks respond to changing input assumptions is essential for risk budgeting, diversification, and achieving robust practical allocations.

> **Key Term: mean-variance optimization (MVO)**
> An optimization method that uses expected returns, risks, and correlations to identify portfolio allocations which maximize return for a target risk, or minimize risk for a target return.
>
> **Key Term: resampled mean-variance optimization**
> A portfolio construction technique that generates multiple simulations of input parameters, optimizes each, and averages the portfolio weights to improve diversification and reduce input sensitivity.
>
> **Key Term: risk budgeting**
> The process of deciding how much risk to allocate to different components of a portfolio, based on desired objectives and constraints.

## **MEAN-VARIANCE OPTIMIZATION AND ITS LIMITATIONS**

Mean-variance optimization (MVO), developed by Markowitz, determines the portfolio allocation that results in the highest expected return for a given level of risk, or vice versa. Portfolio weights are chosen based on forecasts for expected returns, variances (risk), and correlations (covariances).

The main steps:

- Estimate expected returns, variances, and correlations for each asset class or security
- Specify an investor's risk-aversion or risk-budgeting requirements
- Solve for the allocation maximizing expected return per unit of risk, subject to constraints

MVO forms the basis for much strategic asset allocation. Its theoretical appeal is the construction of a mathematically “efficient” frontier of portfolios.

However, MVO is highly sensitive to small changes in its assumptions, especially expected return inputs. Minor adjustments to expected returns can shift optimal allocations dramatically—sometimes resulting in highly concentrated, impractical portfolios. Estimation error is unavoidable, as forward-looking values can only be imperfectly estimated. As a consequence, portfolios may not be stable, intuitive, or robust to alternative scenarios.

> **Key Term: estimation error**
> The difference between the input value used for optimization and the actual (unknown) future value.

### **Worked Example 1.1**

**Suppose an asset allocator estimates the expected return on US equities rises by 0.5% (from 6.0% to 6.5%). How might this affect the resulting MVO portfolio?**

> **Answer:** A small increase in US equities’ expected return could cause the optimizer to allocate a much larger proportion, or even all, of the portfolio to US equities. This illustrates the optimizer's disproportionate sensitivity to return assumptions.

## **RESAMPLED MEAN-VARIANCE OPTIMIZATION**

Resampled mean-variance optimization seeks to address the frailty of traditional optimization. The process involves:

- Generating many alternative sets of inputs using statistical sampling around central estimates for returns and covariances
- Optimizing the portfolio for each simulated input set
- Averaging the resulting portfolio weights to produce final recommended allocations

The main benefit: Resampling “smooths out” the effect of estimation error. The resulting portfolios are generally more diversified and less concentrated in a few risky assets than pure MVO portfolios. Portfolios change more gradually in response to assumptions, thus more stable in practice.

> **Key Term: simulation**
> Producing multiple random realizations of expected returns and risks for input into the optimization process.

### **Worked Example 1.2**

**Describe the effect of resampled optimization on a global portfolio's allocations compared to classic MVO.**

> **Answer:** The resampled approach will often allocate moderate weights to most or all asset classes even if MVO would concentrate all capital in one. This more balanced weighting results in greater diversification, reducing the risk of extreme underperformance if estimated inputs turn out to be incorrect.

## **RISK BUDGETING AND PRACTICAL PORTFOLIO CONSTRUCTION**

Risk budgeting defines how much risk is assigned to each asset or strategy, aligning with institutional objectives, regulatory limits, or investment preferences. Risk budgeting can be implemented using either traditional MVO or resampled optimization. The allocation method chosen to determine strategic risk exposures has significant implications for the resulting diversification, liquidity needs, and monitoring requirements.

A key aspect: Resampled portfolios are typically more diversified by design, aligning with the desire for robust risk allocation across asset classes. However, even with resampling, assumptions on the total risk budget and relative return expectations remain critical.

### **Worked Example 1.3**

**An institution sets its overall portfolio volatility target at 10% per annum. Under MVO, the optimized allocation puts 70% into emerging markets, 25% into bonds, and 5% into real estate. What is a likely result if a resampled approach is used instead?**

> **Answer:** The resampled method would likely produce a portfolio with smaller, more balanced weights across multiple asset classes, reflecting reduced sensitivity to any single assumption and a greater safety margin for risk budgeting.

### **Exam Warning**

> While resampled optimization can mitigate over-concentration from input errors, it does not guarantee superior performance out-of-sample. Remember that empirical performance will ultimately depend on the actual realized returns and risks, not just those simulated.

## **Summary**

Both mean-variance and resampled optimization are valuable tools in portfolio construction and risk budgeting. Mean-variance optimization provides a theoretically efficient portfolio but can result in extreme allocations due to input sensitivity. Resampled optimization responds by averaging across simulations of possible input scenarios, resulting in more diversified, stable, and practical allocations. For CFA Level 3, you must be able to explain the mechanics of each approach, compare their implications for risk budgeting, identify common pitfalls, and select the technique better aligned to robust decision making under uncertainty.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Define and describe mean-variance optimization (MVO) for risk budgeting and asset allocation
- Explain the problem of input sensitivity and estimation error in MVO
- Compare the benefits and drawbacks of resampled mean-variance optimization
- Recognize how resampled optimization leads to greater diversification
- Evaluate implications of each approach for practical risk budgeting in portfolios

## **Key Terms and Concepts**

- mean-variance optimization (MVO)
- resampled mean-variance optimization
- risk budgeting
- estimation error
- simulation

Risk budgeting and portfolio optimization - Mean-variance vs resampled optimization