Portfolio Theory in Financial Economics

Introduction to Portfolio Theory

Definition: Portfolio theory deals with risk, return preferences, and opportunities.
Objective: Maximize portfolio return and minimize risk.

Given assets A, B, and C with known expected returns and volatilities:
- Preference: Investors prefer asset B over A if B has the same volatility but higher expected return.
- Preference: Investors prefer asset C over A if C has the same expected return but lower volatility.

Scenario: Portfolio with assets A and B where A has higher expected return and volatility than B.
Volatility of Portfolio:
- Formula: $\sigma_p = \sqrt{w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \sigma_A \sigma_B \rho_{AB}}$
- Positive Correlation: $\rho_{AB} = 1$ — Volatility is the weighted sum of individual volatilities.
- Negative Correlation: $\rho_{AB} = -1$ — Volatility can reduce to zero, represented on the y-axis in risk-return space.
- Zero Correlation: $\rho_{AB} = 0$ — Volatility is less than the weighted sum of individual volatilities due to zero covariance.

Two Assets: Perfect Positive Correlation: Straight line between assets A and B.
Two Assets: Perfect Negative Correlation: Point on y-axis where volatility is zero.
Two Assets: Zero Correlation: Curved boundary between the straight lines.
Multiple Assets ($n \geq 3$): Forms a boundary area called the mean-variance frontier.

Definition: Part of the mean-variance frontier above the minimum variance portfolio.
Properties:
- Represents portfolios with the highest expected return for a given risk.
- Based on the Two-Fund Separation Theorem: Entire efficient frontier can be constructed from any two efficient portfolios.

Variance Formula for $n$ Assets:
- $\sigma_p^2 = \frac{1}{n} \sigma^2 + \frac{n-1}{n} \sigma_{c}^2$
- As $n$ approaches infinity, portfolio variance reduces to the covariance between assets.
Diversification Effect: Unique risks diversify away, leaving only market/systematic risk.

Portfolio theory helps in understanding risk-return trade-offs and constructing optimal portfolios.
Efficient frontier contains portfolios with maximal returns for given risk levels.
Diversifying assets within a portfolio reduces individual asset risks.

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