Portfolio Theory

Portfolio Theory, pioneered by Harry Markowitz, provides the mathematical framework for constructing investment portfolios to maximize expected return for a given level of market risk, or equivalently, minimize risk for a given expected return.

I. Mean-Variance Optimization (MVO)

Two-Asset Portfolio

The core principle is that the risk of a portfolio is not simply the weighted average of the individual asset risks, but also depends on the correlation between the assets.

For a two-asset portfolio with weights $w_1 = w$ and $w_2 = 1 - w$:

• Expected Return ($\mu_p$):
  $$\mu_p = w\mu_1 + (1 - w)\mu_2$$
• Portfolio Variance ($\sigma_p^2$):
  $$\sigma_p^2 = w^2\sigma_1^2 + (1 - w)^2\sigma_2^2 + 2w(1 - w)\rho\sigma_1\sigma_2$$
  where $\rho$ is the correlation between the two assets. Diversification benefits are maximized when $\rho$ is low or negative.

The Efficient Frontier

The Efficient Frontier is the set of optimal portfolios that offer the highest expected return for a defined level of risk (standard deviation).

• Optimization Problem: For a large number of assets, the problem is to find the weight vector $\mathbf{w}$ that solves:
  $$\min_{\mathbf{w}} \quad \mathbf{w}^\intercal \boldsymbol{\Sigma} \mathbf{w} \quad \text{subject to} \quad \mathbf{w}^\intercal \boldsymbol{\mu} = \mu_p \quad \text{and} \quad \mathbf{w}^\intercal \mathbf{1} = 1$$
  where $\boldsymbol{\Sigma}$ is the covariance matrix of asset returns, and $\boldsymbol{\mu}$ is the vector of expected returns.
• Interpretation: Any portfolio below the Efficient Frontier is sub-optimal, as a higher return could be achieved for the same risk, or lower risk for the same return.

II. Risk-Adjusted Performance and the Market

The Sharpe Ratio

The Sharpe Ratio is the most widely used measure of risk-adjusted return, quantifying the excess return earned per unit of total risk (standard deviation).

where $\mathbb{E}[R_p]$ is the expected portfolio return, $R_f$ is the risk-free rate, and $\sigma_p$ is the portfolio's standard deviation.

Capital Market Line (CML) and Tangency Portfolio

When a risk-free asset is introduced, the optimal investment strategy is to combine the risk-free asset with a single risky portfolio, known as the Tangency Portfolio (or Market Portfolio in the CAPM context).

• CML: The line connecting the risk-free rate to the Tangency Portfolio on the mean-standard deviation plane. All efficient portfolios for an investor are combinations along this line.
• Tangency Portfolio: The portfolio on the Efficient Frontier that has the highest Sharpe Ratio.

III. Asset Pricing Models

These models explain the expected return of an asset based on its exposure to systematic risk factors.

1. Capital Asset Pricing Model (CAPM)

CAPM states that the expected return of an asset is linearly related to its systematic risk ($\beta$) and the expected return of the market portfolio ($R_m$).

• Systematic Risk ($\beta$): Measures the sensitivity of the asset's return to the market's return. It is calculated as $\beta_i = \frac{\text{Cov}(R_i, R_m)}{\text{Var}(R_m)}$.
• Security Market Line (SML): The graphical representation of CAPM, plotting expected return against $\beta$.
• Alpha ($\alpha$): The intercept term in the empirical CAPM regression:
  $$R_i - R_f = \alpha_i + \beta_i (R_m - R_f) + \epsilon_i$$
  $\alpha$ represents the excess return achieved by the asset or portfolio that is not explained by the market risk. It is the primary metric sought by active portfolio managers (alpha generation).

2. Arbitrage Pricing Theory (APT)

APT is a multi-factor model that suggests an asset's expected return is a linear function of its sensitivity to multiple systematic risk factors.

where $\beta_{ij}$ is the sensitivity of asset $i$ to factor $j$, and $\lambda_j$ is the risk premium for factor $j$. Unlike CAPM, APT does not specify the factors; they must be identified empirically.

3. Fama-French 3-Factor Model

An empirical extension of CAPM that incorporates two additional factors found to explain cross-sectional stock returns better than $\beta$ alone:

• SMB (Small Minus Big): The return of a portfolio of small-cap stocks minus the return of a portfolio of large-cap stocks (Size factor).
• HML (High Minus Low): The return of a portfolio of high book-to-market stocks (Value stocks) minus the return of a portfolio of low book-to-market stocks (Growth stocks) (Value factor).

IV. Practical Considerations

• Estimation Error: MVO is highly sensitive to errors in estimating expected returns and the covariance matrix. Small changes in inputs can lead to drastically different, often unstable, optimal portfolios.
• Black-Litterman Model: A practical approach that combines the market equilibrium (CAPM) with an investor's subjective views to produce more stable and intuitive portfolio allocations than pure MVO.
• Risk Parity: An alternative portfolio construction method that focuses on allocating capital such that each asset or risk factor contributes equally to the total portfolio risk, often leading to more diversified and robust portfolios than MVO.