Markowitz Portfolio Theory and Sharpe Optimization

Markowitz Portfolio Selection

The Markowitz model focuses on the relationship between portfolio risk, expected returns, and the covariance and correlation between assets. Diversification reduces the variability of expected returns. The model identifies combinations that provide the maximum return for a given level of risk. A portfolio is considered efficient when it offers the highest return for a specific risk level or the minimum risk for a target return.

Core Investment Hypotheses

• Rational Investor Behavior: Investors prefer higher returns for a given risk level.
• Risk Aversion: Investors prefer lower risk for a given return level.

The Three-Step Optimization Process

1. Determine the Efficient Frontier

The efficient frontier consists of all efficient combinations of risk and return. This is obtained using a quadratic programming model that seeks to maximize returns for a known risk level.

2. Define Utility Curves

Utility curves are determined by the investor's conduct toward risk. Their form must reflect rational behavior, where utility increases as we move toward higher return levels for each level of risk.

3. Define the Optimal Portfolio

The optimal portfolio represents the most useful combination of assets for a specific investor, selected from the efficient frontier.

Sharpe Solution for Portfolio Estimation

The number of estimates required by the Markowitz model to obtain efficient portfolios is very high, creating a computational problem:

• N: Expected returns
• N: Variances
• N(N-1)/2: Covariances

Total estimates: N(N+3)/2 (e.g., 65 estimates for 10 assets). Sharpe proposed a model to reduce these calculations by relating asset returns to market indices rather than calculating individual covariances for every pair of assets.

The Sharpe Model Formula

The return of an asset is defined as: Ri = ai + Bi(I) + Ei

• Ri: Return of the asset.
• ai: Parameter indicating return independent of the market.
• Bi: Parameter indicating sensitivity to market changes.
• I: Market index performance (RMT).
• Ei: Residual variations independent of the market.

Risk Components

Risk is divided into two categories:

• Systematic Risk: Related to the market index.
• Specific Risk: Associated with individual asset features, which can be reduced or eliminated through diversification.

Chart A: Total Risk Reduction

The Sharpe model significantly reduces the number of required estimates to 3N + 2 (e.g., 32 estimates for 10 assets), consisting of:

• N: Parameters for ai
• N: Parameters for bi
• N: Variances
• 1: Index return
• 1: Index variance