Chapter 6 Efficient Diversification

Chapter 6: Efficient Diversification

6.1 Diversification and Portfolio Risk

  • Types of Risks

    • Market/Systematic Risk:

      • Risk factors that pertain to the whole economy.

      • Influenced by the business cycle, inflation, interest rates, and exchange rates.

    • Unique/Firm-Specific Risk:

      • Idiosyncratic risks associated with individual firms.

      • Examples include success in research and development, management style, and philosophy.

      • This risk can be eliminated through diversification and is assumed to be independent.

6.1 Diversification and Portfolio Risk (continued)

  • Covariance and Correlation:

    • These metrics assess how two securities move together and affect portfolio risk.

    • They are crucial for determining the expected return and variance in a portfolio with multiple assets.

    • The equation for expected return on a two-security portfolio is given by:

      • E(rp) = W1r1 + W2r2

        • Where:

          • E(rp) = expected return of the portfolio.

          • W1/W2 = Proportions of funds in security 1/2.

          • r1/r2 = Expected returns on securities 1/2.

6.2 Asset Allocation with Two Risky Assets

  • Portfolio Risk Dependence

    • Portfolio risk is determined by the covariance between returns of assets.

    • Higher covariance means less benefit from diversification.

Scenarios for Stock and Bond Funds

  • Probability and Returns:

    • Severe recession (0.05): Stock -37%, Bond -9%

    • Mild recession (0.25): Stock -11%, Bond 15%

    • Normal growth (0.40): Stock 14%, Bond 8%

    • Boom (0.30): Stock 30%, Bond -5%

    • Expected Returns:

      • Stock Fund: 10%

      • Bond Fund: 5%

Variance and Standard Deviation Calculations

  • For Each Fund:

    • Calculate the variance and standard deviation based on the different scenarios and expected returns.

Portfolio Composition Example

  • Portfolio Mixed Investment:

    • Scenario with 40% in stock fund and 60% in bond fund also requires variance and standard deviation calculations based on returns and probabilities.

Covariance and Correlation Calculations

  • Covariance Formula:

    • Covariance 𝐶𝑜𝑣(𝑟1, 𝑟2) = Σ [ P(i) * (r1(i) - E(r1))(r2(i) - E(r2))]

  • Correlation Coefficient:

    • ρ = 𝐶𝑜𝑣(𝑟1, 𝑟2) / (σ1 * σ2)

Use of Historical Data

  • Variability and covariability change slowly and need actual returns for estimations.

  • Focus on deviations of returns from their average.

6.3 The Optimal Risky Portfolio with a Risk-Free Asset

  • Optimal Risky Portfolio:

    • Characterized by the steepest Capital Allocation Line (CAL).

    • Optimal portfolio weight calculations must account for both risky and risk-free assets.

Efficient Frontier of Risky Assets

  • Represents portfolios that maximize expected return at each level of risk.

  • It can be achieved through methods to maximize risk premium, minimize risk, or maximize Sharpe ratio.

Single-Index Stock Market

  • Key Components:

    • Beta (β): Reflects a stock's sensitivity to market movements.

    • Alpha (α): Expected return beyond what is predicted by market indices.

    • The excess return formula: R_i = β_i * R_m + α_i + e_i where:

      • R_i = return on stock

      • R_m = return on market

Statistical Interpretation

  • Security Characteristic Line (SCL):

    • Represents the relationship between predicted excess returns and market excess returns.

    • Analyzing regression helps in estimating risks and returns accurately.

Diversification in Single-Index Security Market

  • Nonsystematic Risks:

    • Achievable through optimal portfolio weights while accounting for expected returns.

6.6 Risk of Long-Term Investments

  • Stock risk diminishes with a long horizon; risk premium grows, indicating a smaller standard deviation with time.

  • Sharpe ratio increases as the investment horizon increases.