Mean-Variance Optimization – an Overview

Mean-variance optimization (“MVO”) forms the foundation for most modern asset allocation methods. MVO works by shifting the weights of asset classes within a portfolio until an optimal mix is found. An optimal mix is found when the portfolio has the best risk-to-return profile. Said another way, the objective of asset-only mean-variance optimization is to maximize the expected return of the asset mix after accounting for a penalty that depends on risk aversion and the expected variance of the asset mix.

A fundamental tenet of MVO lies in the concept of diversification benefit, which refers to the fact that provided that two assets are less than perfectly correlated, adding them together in a portfolio will increase the portfolio standard deviation at a slower rate than it would increase the return.>

For MVO to work, the investor must have the expected assets’ returns, risks, and correlation with other assets.

With these inputs plus a risk aversion parameter, a fundamental equation can be used to calculate an investor’s utility for a given asset mix:

$$ U_m = E(r_m) – 0.005 \lambda \sigma_m^2 $$

Where:

\(U_m\) = Investor’s utility for a given asset mix.

\(E(r_m)\) = Expected return for the given asset mix.

\(\lambda\) = Investor’s degree of risk-aversion.

\(\sigma_m^2\)= Expected variance of the given asset mix.

Utility, as it is used in this context, is meant to represent ‘usefulness’ to an individual investor. Based on various investor preferences (different \(\lambda\)), the same mix could provide different utility for different investors. The mix with the highest utility will be the optimal mix for the investor. 

The risk aversion parameter runs from 0 for an utterly risk-indifferent investor, to 10 for an utterly risk-averse investor. Higher values for this parameter will lower the expected utility of any given asset as shown by multiplying a negative number (-0.005) by the risk aversion parameter. Higher values call for higher penalties on risky assets in the form of lower utility.

The value of 0.005 in the utility equation above is based on the assumption that \(E(R_m)\) and \(\sigma_m\) are expressed as percentages rather than as decimals. If those quantities were expressed as decimals, the 0.005 would change to 0.5.

Mean-Variance Optimization – Critiques

While MVO is an excellent place to begin to search for the optimal asset allocation for an investor, it is not a fool-proof method. For this reason, it is often said to form the foundation of the asset allocation process, but it is often paired with other methods in professional applications. The following criticisms are common to MVO, and may also pertain to other optimization models:

• The asset allocations are sensitive to small changes in inputs.
• The asset allocations tend to be highly concentrated in a subset of available asset classes.
• Many investors are concerned about a lot more than the mean and variance of returns.
• Although asset allocations may appear diversified across assets, sources of risk may not be diversified.
• Most portfolios exist to pay for a liability or consumption series, and MVO allocations are not directly connected to what influences the value of the liability or the consumption series.
• MVO is a single-period framework that does not consider trading/rebalancing costs and taxes.

Despite its potential shortcomings, MVO is typically used as the basis for other methods. The following methods attempt to build from MVO and address some of the most common criticisms.

Reverse Optimization

The reverse optimization process is mostly as it sounds: we reverse the traditional MVO method. Rather than starting with expected asset metrics and using those to arrive at an optimal asset allocation, we begin with the optimal asset allocation per the global market portfolio and then calculate the expected portfolio return and risk.

The Black-Litterman Model is an extension of reverse optimization but includes investor expectations for asset class returns, volatility, and correlations. As resampled allocations are derived, the process is repeated, and the allocations are again updated to include forecasts, allowing for some degree of active management in the portfolio allocation process.

Mean-variance optimization has been made more useful with the Black-Litterman model. A reverse-optimized return can be combined elegantly with investors’ forecasts of expected returns.

By improving the consistency between each asset class’s expected return and its contribution to systematic risk when used with a mean-variance framework or related framework, Black–Litterman expected returns often lead to well-diversified asset allocations. The reverse-optimization process uses market capitalization as a basis of asset allocations but still reflects the investor’s unique forecasts of expected returns.

Question

Trojan Capital hired a new ultra-high-net-worth investor who made a fortune playing high-stakes poker. Trojan is planning on beginning with MVO to create an asset allocation for the portfolio. The portfolio has an expected return of 10.0% and a variance of 4.0%. If the investor’s risk-aversion parameter is very low at 2, the investor’s utility is closest to:

1. -10.00%.
2. 6.00%.
3. 9.84%.

Solution

The correct answer is B:

If \(E(R_m)\) = 0.10, \(\lambda\) = 2, and \(\sigma\) = 0.20 (variance is 0.04), then \(U_m\) is 0.06, or 6%

$$ 0.06 = 0.10 – 0.5(2)(0.04) $$

A is incorrect. It uses standard deviation in place of variance.

C is incorrect. It uses \(0.04^2\) rather than the square root of 0.04.  

Asset Allocation: Learning Module 4: Principles of Asset Allocation;

Los 4(a) Describe and evaluate the use of mean-variance optimization in asset allocation

Los 4(b) Recommend and justify an asset allocation using mean–variance optimization