Learning Outcomes
After reading this article, you will be able to explain and compare mean-variance optimization and resampled optimization in the context of portfolio construction for risk budgeting. You will understand how input assumptions influence output, recognize the rationale and practical impact of resampled optimization, and apply these principles to risk allocation decisions, portfolio diversification, and asset allocation processes as required by the CFA Level 3 exam.
CFA Level 3 Syllabus
For CFA Level 3, you are required to understand the principles and application of portfolio optimization techniques under risk constraints. You should focus on the following key topics:
- 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.
- What is a primary drawback of traditional mean-variance optimization in portfolio construction?
- How does resampled mean-variance optimization differ from standard mean-variance optimization, and what practical benefit does it provide?
- Why is risk budgeting important in institutional portfolio construction?
- 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