Modern Portfolio Theory and Mean-Variance Optimization › Summary

This chapter developed MPT as a mathematical framework for analyzing diversification and risk-return tradeoffs.

The key concepts we covered include:

• Portfolio return and risk: Portfolio expected return is the weighted average of asset returns ($\mu_p = \mathbf{w}^\top \boldsymbol{\mu}$), while portfolio variance depends on the full covariance structure ($\sigma_p^2 = \mathbf{w}^\top \Sigma \mathbf{w}$).

• The efficient frontier: Feasible portfolios for which no other feasible portfolio offers at least as high an expected return and no more risk, with at least one strict improvement. A mean-variance investor who prefers higher expected return and lower risk should avoid dominated portfolios.

• Global minimum variance portfolio: The portfolio with the lowest possible variance, given by $\mathbf{w}_{\text{GMV}} = \Sigma^{-1}\mathbf{1} / (\mathbf{1}^\top \Sigma^{-1}\mathbf{1})$.

• Two-fund separation: In the stated unrestricted-weight model, two distinct efficient portfolios span the minimum-variance curve. Their combination is efficient only on the upper branch, at or above the GMV return.

• Diversification benefits and limits: At fixed long-only weights, lowering correlation cannot increase variance; a combination need not have lower risk than both individual assets. In the identical-variance, common-positive-correlation example, the limiting variance is $\rho\sigma^2$, rather than zero.

• Practical optimization: Quadratic programming enables efficient frontier computation with realistic constraints on short selling, position sizes, and sector exposures.

• Estimation error: Mean-variance optimization is highly sensitive to input estimation errors, particularly in expected returns. This sensitivity motivates the regularization techniques and robust methods covered in later chapters.

The framework developed here sets up the Capital Asset Pricing Model in the next chapter, where we'll see how equilibrium considerations determine the pricing of risk in the market portfolio.