Chapter 4: Introduction to Modern Portfolio Theory

Formulated by Harry Markowitz in his landmark 1952 paper, Modern Portfolio Theory (MPT) revolutionized financial economics by demonstrating mathematically that an investment should not be judged solely on its standalone risk and return characteristics, but on how it co-moves with other assets in a diversified portfolio.

Core Assumptions of Modern Portfolio Theory

Rational Risk-Averse Investors: Investors prefer higher expected return to lower expected return for a given level of risk, and prefer lower risk to higher risk for a given level of return.
Normal Distribution of Returns: Security returns are sufficiently described by their first two statistical moments: Mean (Expected Return) and Variance (Standard Deviation).
Single-Period Investment Horizon: All investors evaluate portfolio decisions over a uniform, single holding period.
Frictionless Capital Markets: There are no transaction costs, brokerage fees, or bid-ask spreads, and assets are infinitely divisible.
Homogeneous Expectations: All investors have identical expectations regarding the expected returns, variances, and covariances of all available securities.

Classification of Investor Risk Attitudes

Risk-Averse Investor: Demands additional expected return (a risk premium) to accept additional variance. The utility function is strictly concave ($U''(W) < 0$).
Risk-Neutral Investor: Evaluates investments solely based on expected return, completely indifferent to variance. The utility function is linear ($U''(W) = 0$).
Risk-Seeking Investor: Prefers gambles with high variance even if expected return is lower. The utility function is convex ($U''(W) > 0$).

The mathematical power of Modern Portfolio Theory lies in the formulation of portfolio expected return and portfolio variance.

1. Expected Return of an Individual Security and Portfolio

The expected return of an individual security is the probability-weighted sum of its potential returns across $m$ possible economic states:

E(R_i) = \sum_{s=1}^{m} p_s \cdot R_{is}

For a portfolio comprising $n$ assets with portfolio weights $w_i$ (where $\sum_{i=1}^n w_i = 1$), the Portfolio Expected Return is the linear weighted average of individual expected returns:

E(R_p) = \sum_{i=1}^{n} w_i \cdot E(R_i)

2. Variance of an Individual Security and Covariance

The variance of return measures standalone dispersion around the mean:

\sigma_i^2 = \sum_{s=1}^{m} p_s \cdot [R_{is} - E(R_i)]^2

The Covariance between security $i$ and security $j$ measures how the two assets co-vary:

\text{Cov}(R_i, R_j) = \sigma_{ij} = \sum_{s=1}^{m} p_s \cdot [R_{is} - E(R_i)] \cdot [R_{js} - E(R_j)]

The Correlation Coefficient (

\rho$_{ij}$) normalizes covariance between -1 and +1:</p> <p>

$\rho$ {ij} = \frac{\sigma{ij}}{\sigma_i \cdot \sigma_j} \quad (-1 \le $\rho$ _{ij} \le +1)

</p> <h3>3. Portfolio Variance of a Two-Asset Portfolio</h3> <p>For a portfolio of two securities $A$ and $B$:</p> <p>

$\sigma_p$ ^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \text{Cov}(R_A, R_B) = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \sigma_A \sigma_B $\rho$ _{AB}

</p> <h3>IMPACT OF CORRELATION ON PORTFOLIO RISK</h3> <table><thead><tr><th style="text-align:left">CORRELATION ($\rho$_{AB})</th><th style="text-align:left">PORTFOLIO VARIANCE RESULT</th><th style="text-align:left">DIVERSIFICATION BENEFIT</th></tr></thead><tbody><tr><td style="text-align:left"><strong>$\rho$ = +1.0</strong></td><td style="text-align:left">$\sigma_p$ = $w_A \sigma_A$ + $w_B \sigma_B$</td><td style="text-align:left"><strong>Zero Diversification:</strong> Portfolio</td></tr><tr><td style="text-align:left">(Perfect Positive)</td><td style="text-align:left">(Linear weighted standard dev).</td><td style="text-align:left">risk is simple weighted average.</td></tr><tr><td style="text-align:left"><strong>-1.0 &lt; $\rho$ &lt; +1.0</strong></td><td style="text-align:left">$\sigma_p$ &lt; $w_A \sigma_A$ + $w_B \sigma_B$</td><td style="text-align:left"><strong>Partial Diversification:</strong> Total</td></tr><tr><td style="text-align:left">(General Imperfect Corr)</td><td style="text-align:left">(Portfolio risk is strictly lower).</td><td style="text-align:left">risk is lower than weighted sum!</td></tr><tr><td style="text-align:left"><strong>$\rho$ = -1.0</strong></td><td style="text-align:left">$\sigma_p$ =</td><td style="text-align:left">$w_A \sigma_A$ - $w_B \sigma_B$</td></tr><tr><td style="text-align:left">(Perfect Negative)</td><td style="text-align:left">(Can construct a risk-free portfolio with positive expected return!).</td><td style="text-align:left">Portfolio variance reaches zero when w_A = \sigma_B/(\sigma_A+\sigma_B)</td></tr></tbody></table> <p>24</p> <p>

By varying the weights $w_i$ across all available risky assets in the investment universe, an analyst generates an entire opportunity set of portfolios—the Feasible Set.

The Markowitz Efficient Frontier

The Efficient Frontier represents the set of optimal portfolios that offer:

The maximum expected return for a given level of portfolio risk (variance); or
The minimum risk for a given target level of expected return.

Markowitz Modern Portfolio Theory: Efficient Frontier Architecture

Portfolio Zone / Boundary	Geometric Position on Graph	Financial Definition & Optimization Rules
Efficient Frontier (Upper Envelope)	Hyperbolic upper boundary curve from GMVP upwards	Represents the set of Pareto-optimal portfolios offering the maximum expected return $E(R_p)$ for any given level of portfolio risk ($\sigma_p$).
Global Minimum Variance Portfolio (GMVP)	Westernmost inflection vertex on the curve	The unique portfolio composition providing the absolute lowest possible volatility ($\sigma_{\min}$) across all risky asset combinations.
Inefficient Feasible Region	Lower shaded interior region below the frontier	Sub-optimal portfolios that are dominated by portfolios directly above them having identical risk but higher expected returns.
Infeasible Region	Any coordinate point lying northwest above the curve	Statistically unobtainable combinations of ultra-high expected return and low risk given prevailing asset correlations and covariances.

Global Minimum Variance Portfolio (GMVP): The unique portfolio on the efficient frontier that possesses the lowest absolute standard deviation among all possible combinations of risky assets.
Dominance Principle: Any portfolio lying below the efficient frontier is strictly suboptimal ("dominated") because an investor could achieve higher return for identical risk, or lower risk for identical return, by moving to the frontier.

The Portfolio Optimization Process

To identify the efficient frontier mathematically, a quadratic programming optimization is solved:

\min_{w} \frac{1}{2} \sum_{i=1}^{n} \sum_{j=1}^{n} w_i w_j \sigma_{ij} \quad \text{subject to} \quad \sum_{i=1}^{n} w_i E(R_i) = E(R^*) \quad \text{and} \quad \sum_{i=1}^{n} w_i = 1

Where $E(R^*)$ is the target expected portfolio return.

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