Modern Portfolio Theory (MPT) Explained with Examples

Modern Portfolio Theory (MPT), introduced by Harry Markowitz in 1952, is a framework for constructing portfolios that maximize expected return for a given level of risk, or equivalently minimize risk for a given expected return. The central idea is diversification — by combining assets that are not perfectly correlated, investors can reduce overall portfolio risk.

Core Concepts and Formulas

1. Expected Portfolio Return:
   E[R_p] = Σ (wᵢ × E[Rᵢ])
   Portfolio return is the weighted average of individual asset returns.
2. Portfolio Variance (Risk):
   Var(R_p) = wᵀ Σ w
   Risk depends on both individual volatilities and covariances between assets.
3. Two-Asset Case:
   Var(R_p) = w²σ₁² + (1-w)²σ₂² + 2w(1-w)ρσ₁σ₂
   This shows how correlation (ρ) reduces risk through diversification.
4. Efficient Frontier:
   A curve representing the set of optimal portfolios with the highest expected return for each risk level.
5. Tangency Portfolio and Capital Market Line (CML):
   With a risk-free asset, the best risk-return tradeoff is achieved by combining the risk-free rate with the tangency portfolio.

Numerical Two-Asset Example

Suppose we have two assets:

• Asset A: Expected return = 8%, Volatility = 12%
• Asset B: Expected return = 12%, Volatility = 20%
• Correlation = 0.30

The minimum-variance portfolio weight for Asset A is calculated as:

w* = (σB² − ρσAσB) / (σA² + σB² − 2ρσAσB) ≈ 0.82

This means:

• 82% in Asset A, 18% in Asset B
• Portfolio expected return ≈ 8.72%
• Portfolio volatility ≈ 11.45% (lower than either asset alone)

Takeaway: Diversification reduces risk without lowering return proportionally.

Multi-Asset Example

In practice, portfolios consist of many assets. By estimating expected returns and the covariance matrix of returns, investors can calculate:

• The minimum-variance portfolio (lowest possible risk)
• The tangency portfolio (highest Sharpe ratio)
• The Capital Market Line (CML) that investors can use depending on their risk tolerance

For example, in a four-asset portfolio with a 3% risk-free rate, we may find:

• Minimum-variance portfolio: Expected return 6.4%, Volatility 7.2%
• Tangency portfolio: Expected return 8.3%, Volatility 8.9%, Sharpe ≈ 0.60

Limitations of MPT

• Estimation risk: Small errors in expected returns or covariances can lead to unstable results.
• Assumption of normality: MPT assumes risk is fully captured by variance, which may ignore extreme events.
• Practical constraints: Transaction costs, no-short-selling rules, and position limits often change the optimal portfolio.

Conclusion

Modern Portfolio Theory remains a cornerstone of investment management. It demonstrates the importance of diversification and provides a mathematical way to balance risk and return. While it has limitations, extensions such as the Black–Litterman model and robust optimization address many of these challenges. Investors and fund managers still rely on its insights when building portfolios today.