Mean-Variance Analysis: Definition, Examples, and Calculation Techniques

Comprehensive guide on mean-variance analysis, including definitions, examples, calculation methods, historical context, and applications in real-world scenarios.

Mean-variance analysis is a critical financial concept used in portfolio theory and investment management. It involves evaluating investment portfolios by analyzing their expected returns and associated risks, quantified as variance or standard deviation. This approach enables investors to balance their desire for higher returns against the potential risks.

Core Concepts in Mean-Variance Analysis

Expected Return (Mean)

The expected return is the weighted average of possible returns, with the weights being the probabilities of each outcome. The formula for the expected return \( E(R) \) of a portfolio is:

$$ E(R) = \sum_{i=1}^{n} p_i R_i $$

Where:

• \( p_i \) is the probability of the ith outcome
• \( R_i \) is the return associated with the ith outcome

Portfolio Variance

Portfolio variance measures the dispersion of returns for a given portfolio and provides insight into the risk level. The variance \( \sigma^2 \) of a portfolio can be calculated as:

$$ \sigma^2 = \sum_{i=1}^{n} p_i (R_i - E(R))^2 $$

Covariance and Correlation

Covariance indicates how two assets move together, which helps in diversification. It’s calculated as:

$$ \text{Cov}(R_i, R_j) = E[(R_i - E(R_i))(R_j - E(R_j))] $$

Correlation standardizes covariance by dividing it by the product of the two assets’ standard deviations:

$$ \rho_{ij} = \frac{\text{Cov}(R_i, R_j)}{\sigma_i \sigma_j} $$

Portfolio Optimization

Using the mean-variance framework, investors aim to construct a portfolio with an optimal balance of expected return and risk. The goal is to achieve the highest possible return for a given level of risk.

Historical Context and Development

Mean-variance analysis was introduced by Harry Markowitz in his seminal 1952 paper “Portfolio Selection.” Markowitz’s work laid the foundation for modern portfolio theory (MPT), for which he was awarded the Nobel Prize in Economic Sciences in 1990.

Practical Applications

Investment Strategies

Mean-variance analysis is a cornerstone for various investment strategies, including constructing the efficient frontier and the Capital Market Line (CML).

Risk Management

It aids in identifying the risk-return trade-off and helps investors decide which portfolios align with their risk tolerance and investment goals.

Example Calculation

Consider a portfolio with the following assets:

• Asset A: Expected Return = 10%, Variance = 0.04
• Asset B: Expected Return = 15%, Variance = 0.09

Assume equal weights and no correlation between the assets. The portfolio’s expected return \( E(R_p) \) and variance \( \sigma^2_p \) can be calculated as:

$$ E(R_p) = 0.5(10\%) + 0.5(15\%) = 12.5\% $$

$$ \sigma^2_p = 0.5^2 \cdot 0.04 + 0.5^2 \cdot 0.09 = 0.01 + 0.0225 = 0.0325 $$

Comparing Related Concepts

Mean-Variance vs. Mean-Absolute Deviation Analysis

While mean-variance focuses on variance as a risk measure, mean-absolute deviation (MAD) analysis considers the average absolute deviation from the mean.

Modern Portfolio Theory (MPT)

MPT extends mean-variance analysis by incorporating multiple assets and emphasizing diversification to construct efficient portfolios.

FAQs

What is the key advantage of mean-variance analysis?

It helps investors achieve a balanced portfolio by optimizing the trade-off between expected returns and risk.

Can mean-variance analysis be used for all types of investments?

It is most effective for traditional investments like stocks and bonds but may be less suitable for alternative investments with different risk-return profiles.

What are the limitations of mean-variance analysis?

It assumes that returns are normally distributed and that investors are rational, both of which may not always hold true in real markets.

References

1. Markowitz, H. (1952). Portfolio Selection. Journal of Finance.
2. Bodie, Z., Kane, A., & Marcus, A. J. (2014). Investments. McGraw-Hill Education.

Summary

Mean-variance analysis is an essential financial tool for assessing and balancing the trade-off between risk and expected return in investment portfolios. By understanding its concepts, historical context, and practical applications, investors can make informed decisions to optimize their portfolios and achieve their investment objectives.