Understanding Two-Asset Mean-Variance Analysis

What is Mean-Variance Analysis?

Mean-Variance Analysis, pioneered by Harry Markowitz, is a foundational concept in Modern Portfolio Theory (MPT). It helps investors construct portfolios that aim to maximize expected return for a given level of risk (measured by variance or standard deviation), or alternatively, minimize risk for a given level of expected return.

Key Concepts:

• Expected Return (Mean): The anticipated profit or loss from an investment. For a portfolio, it's the weighted average of the expected returns of its individual assets.
  Portfolio Return (Rp) = wA * RA + wB * RB
  (where w = weight, R = expected return)
• Volatility (Standard Deviation/Variance): A statistical measure of the dispersion of returns for an asset or portfolio. Higher volatility implies greater risk. Portfolio variance depends not just on individual asset volatilities but also on how their returns move together (correlation).
  Portfolio Variance (σp2) = wA2σA2 + wB2σB2 + 2wAwBσAσBρAB
  (where σ = standard deviation, ρ_AB = correlation coefficient between A and B)
  Portfolio Volatility (σ_p) is the square root of Portfolio Variance.
• Correlation Coefficient (ρ_AB): Measures the degree to which the returns of two assets move in relation to each other. It ranges from -1 (perfect negative correlation) to +1 (perfect positive correlation). A value of 0 means no linear relationship.

Diversification Benefit:

Combining assets that are not perfectly positively correlated (i.e., ρ_AB < 1) can reduce overall portfolio risk (volatility) without necessarily sacrificing expected return. This is the core idea behind diversification. The lower the correlation (especially if negative), the greater the potential diversification benefit.

Efficient Frontier (for Two Assets):

When you plot all possible risk-return combinations for a two-asset portfolio (by varying their weights), you trace out a curve. The **Efficient Frontier** is the upper part of this curve, starting from the Minimum Variance Portfolio. It represents the set of portfolios that offer the highest possible expected return for each given level of risk (volatility), or the lowest risk for each level of expected return.

Minimum Variance Portfolio (MVP):

This is the specific portfolio on the efficient frontier that has the lowest possible volatility (risk). Its weights can be calculated analytically:
wA (MVP) = (σB2 - σAσBρAB) / (σA2 + σB2 - 2σAσBρAB)
wB (MVP) = 1 - wA (MVP)

Sharpe Ratio (Optional):

If a risk-free rate is considered, the Sharpe Ratio measures the risk-adjusted return of a portfolio. It's calculated as:
Sharpe Ratio = (Portfolio Return - Risk-Free Rate) / Portfolio Volatility
A higher Sharpe Ratio is generally better, indicating a better return for the amount of risk taken. The portfolio on the efficient frontier that has the highest Sharpe Ratio is often considered the "optimal" risky portfolio if combining with a risk-free asset (this is the tangency portfolio).

Limitations:

• Input Sensitivity: The results are highly dependent on the accuracy of the input estimates for expected returns, volatilities, and correlation. These inputs are often based on historical data which may not predict future performance.
• Assumptions: The model often assumes returns are normally distributed, which may not always be true in real markets (e.g., "fat tails" or extreme events).
• Static Model: This is a single-period model. In reality, correlations and expected returns change over time.
• Ignores Other Risks: Mean-variance analysis primarily focuses on volatility as risk and doesn't explicitly account for other types like liquidity risk, credit risk, or black swan events.

Conclusion: The two-asset mean-variance analyzer is a powerful educational tool to understand the fundamental trade-off between risk and return and the benefits of diversification. For real-world multi-asset portfolio optimization, more sophisticated tools and considerations are typically used by financial professionals.