Efficient Diversification Lecture Notes

Efficient Diversification Study Notes

6-2 Diversification and Portfolio Risk

• Types of Risk

  • Market/Systematic/Nondiversifiable Risk

  • Risk factors common to the entire economy, affecting all securities to some extent.

  • Unique/Firm-Specific/Nonsystematic/Diversifiable Risk

  • Risk that can be eliminated through diversification, relates to individual firms or specific sectors.

Figures Illustrating Risk

• Figure 6.1: Risk as Function of Number of Stocks in Portfolio

  • The graph illustrates the relationship between the number of stocks in a portfolio and the respective σ values:

  • A: Represents firm-specific risk only.

  • B: Represents the combined market and unique risk.

• Figure 6.2: Risk versus Diversification

  • The average portfolio standard deviation is plotted against the number of stocks in the portfolio, indicating a decrease in risk with more assets.

  • Notable points include:

  • 0% to 100% in portfolio diversification correlating to a percentage change in risk, with a diminishing risk return as more stocks are added along the y-axis.

  • Indicated values are shown for various portfolios ranging from 1 to 1,000 stocks.

6.2 Asset Allocation with Two Risky Assets

• Covariance and Correlation

  • Portfolio risk heavily depends on the covariance between asset returns.

  • Expected Return on Two-Security Portfolio:

• Covariance Calculations:

  • Covariance formula:
    $Cov(r<em>s, r</em>B) = \Sigma p(i)[r<em>s(i) - E(r</em>s)][r<em>B(i) - E(r</em>B)]$

  • Correlation Coefficient:
    $\rho<em>{SB} = \frac{Cov(r</em>s, r<em>B)}{\sigma</em>{r<em>s}\sigma</em>{r_B}}$

  • The correlation coefficient's range is from -1.0 to 1.0.

• Correlation Coefficient Interpretation:

  • If $r = 1.0$, securities are perfectly positively correlated.

  • If $r = -1.0$, securities are perfectly negatively correlated.

Two Asset Portfolio Standard Deviation

• The formula for standard deviation in a two-asset portfolio is: $\sigma<em>p^2 = w</em>A^2\sigma<em>A^2 + w</em>B^2\sigma<em>B^2 + 2w</em>A w<em>B Cov(r</em>A, r_B)$

  • Where:

  • $\ ext{w}$ represents weights of investments in each asset.

General Formula for n-Security Portfolio

• Expected Return

  • $r_p = \text{Weighted average return of n securities}$

• Variance Calculation

  • $s_p^2$ considers all pairwise covariance measures.

Three Key Rules in Asset Allocation

1. Rate of Return (RoR)

   • Weighted average returns on components, considering investment proportions as weights.

2. Expected Rate of Return (ERR)

   • Weighted average of expected returns on components, with proportions as weights.

3. Variance of RoR

   • Underlies the overall risk assessment.

Numerical Example

• Given:

  • Returns: Bonds = 6%, Stocks = 10%

  • Standard Deviations: Bonds = 12%, Stocks = 25%

  • Weights: Bonds = 0.5, Stocks = 0.5

  • Correlation Coefficient = 0

• Calculating Expected Return:

  • $Return = 0.5(6) + 0.5(10) = 8$

• Calculating Standard Deviation:

  • $\text{Standard Deviation} = \sqrt{0.5^2(12^2) + 0.5^2(25^2) + 2(0.5)(0.5)(12)(25)(0)} = 13.87$

Risk-Return Trade-Off

• Investment Opportunity Set:

  • Represents available portfolio risk-return combinations.

  • Mean-Variance Criterion:

  • If $E(r<em>A) \geq E(r</em>B)$ and $\sigma<em>A \leq \sigma</em>B$, then Portfolio A dominates Portfolio B.

  • This forms the basis for optimal decision-making in investment.

Figures Related to Risk-Return Trade-Off

• Figure 6.3: Investment Opportunity Set:

  • Displays the expected return versus standard deviation, showing the minimum-variance portfolio location.

• Figure 6.4: Opportunity Sets & Correlation Coefficients:

  • Represents how changing correlation coefficients of the assets affects expected return trends and standard deviation.

6.3 The Optimal Risky Portfolio with a Risk-Free Asset

• Slope of the Capital Allocation Line (CAL):

  • Represents the Sharpe Ratio of the risky portfolio.

• Optimal Risky Portfolio:

  • Defined as the best combination of risky and safe assets in a portfolio.

  • Identified as the one associated with the “steepest” CAL.

Figures Illustrating Optimal Risky Portfolio

• Figure 6.5: Two Capital Allocation Lines

  • Comparison of stocks and bonds to the expected return against the standard deviation.

• Figure 6.6: Bond, Stock, and T-Bill Optimal Allocation

  • Displays expected return percentages and varying asset proportions for optimal portfolio O.

• Figure 6.7: The Complete Portfolio

  • Outlines E(r_p) along with varying portfolio components and their respective expected returns.

• Figure 6.8: Portfolio Composition: Asset Allocation Solution

  • Details asset allocations suggesting a diversified portfolio.

6.4 Efficient Diversification with Many Risky Assets

• Efficient Frontier of Risky Assets:

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

  • Methods for Portfolio Optimization:

  • Maximize risk premium for any level of standard deviation.

  • Minimize standard deviation for any level of risk premium.

  • Maximize the Sharpe ratio for specified standard deviation or risk premium.

Figures Illustrating Efficient Frontier

• Figure 6.9: Portfolios Constructed with Three Stocks:

  • Forecasts how changes in stock selection can influence expected returns and standard deviation outcomes.

• Figure 6.10: Efficient Frontier: Risky and Individual Assets:

  • Displays the trajectory of both risky portfolios and individual asset performances along the expected return vector.

Choosing Optimal Risky Portfolio

• Concept of Optimal Portfolio:

  • CAL will be tangent to the efficient frontier.

  • Involves a two-task separation property:

  • Determining optimal risky portfolio.

  • Choosing the best mix of risky assets with the risk-free asset, considering personal acceptance of risk.