chapter 6

Lý thuyết

6.1: Diversification and Portfolio Risk

• Concept: Diversification reduces risk by combining assets with returns that are not perfectly correlated.
• Types of Risk:
  • Systematic Risk (Market Risk): Risk that affects the entire market or economy as a whole (VD: suy thoái, lạm phát, thiên tai,…). It’s also known as undiversifiable risk because it can’t be eliminated by diversifying your investments. Cannot be diversified away.
  • unique risk/firm-specific risk/nonsystematic risk/diversifiable risk: Risk specific to a particular company or industry. It’s also known as diversifiable risk because it can be reduce or eliminated through diversification. (VD: cty truyền thông ngu, thay đổi quy định vs hàng hoá, ….) -> Portfolio Risk: Determined by the weights of the assets, their variances, and their correlations to maximize return and reduce risk.

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6.2: Asset Allocation with Two Risky Assets

• Risk and Return Trade-Off: Combining two risky assets produces a portfolio with risk and return characteristics dependent on their covariance.
• Covariance and Correlation:
  • Covariance measures how two assets move together.
  • Correlation (\(\rho\)) is a standardized version of covariance, ranging between -1 and 1.
• Efficient Frontier for Two Assets:
  • Portfolios with low or negative correlations offer significant diversification benefits.

Using Historical Data to Estimate Means, Standard Deviations, Covariance, and Correlation

1. Portfolio Expected Return: \[ E(r_P) = w_1E(r_1) + w_2E(r_2) \] Where \(w_1, w_2\) are weights of assets in the portfolio, and \(E(r_1), E(r_2)\) are their expected returns.

2. Portfolio Variance (Two Assets): \[ \sigma_P^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\rho_{1,2}\sigma_1\sigma_2 \] Where \(\sigma_1, \sigma_2\) are the standard deviations, and \(\rho_{1,2}\) is the correlation coefficient between the two assets.

3. Covariance: \[ \text{Cov}(r_1, r_2) = \rho_{1,2} \sigma_1 \sigma_2 \]

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6.3: The Optimal Risky Portfolio with a Risk-Free Asset

• Introduction: Investors allocate wealth between a risk-free asset and an optimal risky portfolio to maximize return for a given level of risk.

• Capital Allocation Line (CAL):

  • A straight line representing the risk-return trade-off of combining a risk-free asset with a portfolio of risky assets.
  • The slope of CAL is the Sharpe Ratio.

  \[ \text{Sharpe Ratio} = \frac{E(r_P) - r_f}{\sigma_P} \]

  Where \(r_f\) is the risk-free rate, \(E(r_P)\) is the portfolio’s expected return, and \(\sigma_P\) is the portfolio’s standard deviation.

• Optimal Portfolio:

  • Maximizes the Sharpe Ratio by choosing weights that balance risk and return.

Sharpe ratio maximizing portfolio weights with two risky assets (B and S) and a risk-free asset:

Formula for \(w_B\) (weight of asset B in the optimal portfolio):

\[ w_B = \frac{[E(r_B) - r_f] \sigma_S^2 - [E(r_S) - r_f] \sigma_B \sigma_S \rho_{BS}}{[E(r_B) - r_f] \sigma_S^2 + [E(r_S) - r_f] \sigma_B^2 - [E(r_B) - r_f + E(r_S) - r_f] \sigma_B \sigma_S \rho_{BS}} \]

\[ w_S = 1 - w_B \]

Where:

- \(E(r_B)\): Expected return of asset \(B\),

- \(E(r_S)\): Expected return of asset \(S\),

- \(r_f\): Risk-free rate,

- \(\sigma_B\): Standard deviation of asset \(B\),

- \(\sigma_S\): Standard deviation of asset \(S\),

- \(\rho_{BS}\): Correlation coefficient between \(B\) and \(S\).

Capital Allocation Line (CAL): \[ E(r_C) = r_f + \frac{E(r_P) - r_f}{\sigma_P} \cdot \sigma_C \] Where \(\sigma_C\) is the standard deviation of the complete portfolio. —

6.4: Efficient Diversification with Many Risky Assets

• Efficient Frontier: A curve showing the best risk-return trade-offs from a portfolio of multiple risky assets.
• Separation Property:
  • Portfolio construction can be separated into:
    • Identifying the optimal risky portfolio.
    • Allocating wealth between the risk-free asset and the optimal portfolio based on investor preferences.
• Risk of Long-Term Investments:
  • Evaluates how compounding affects risk and return for long-term investments.

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6.5: Single-Index Model

• Purpose: Simplifies the covariance calculations by assuming asset returns are driven by a common market factor (e.g., an index).
• Components of Return:
  • Systematic Component: Related to market movements.
  • Idiosyncratic Component: Unique to the asset.
• Benefits:
  • Reduces complexity of managing large portfolios.
  • Focuses on market-wide factors for risk and return. \[ \sigma_A^2 = \beta^2 \sigma_M^2 + \sigma_{\text{residual}}^2 \] —

6.6: Risk of Long-Term Investments

• Misconception: Long-term investments do not necessarily reduce risk due to uncertainty in compounding.
• Clarification: While diversification mitigates some risks, investors need to consider how risks compound over time.

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Bài tập

1. In forming a portfolio of two risky assets, what must be true of the correlation coefficient between their returns if there are to be gains from diversification? Explain. (LO 6-1)

• To achieve gains from diversification, the correlation coefficient between the returns of the two assets must be less than 1.
• A lower or negative correlation increases the diversification benefit by reducing portfolio risk.

2. When adding a risky asset to a portfolio of many risky assets, which property of the asset is more important, its standard deviation or its covariance with the other assets? Explain. (LO 6-1) covariance
3. A portfolio’s expected return is 12%, its standard deviation is 20%, and the risk-free rate is 4%. Which of the following would make for the greatest increase in the portfolio’s Sharpe ratio? (LO 6-3)

1. An increase of 1% in expected return.
2. A decrease of 1% in the risk-free rate.
3. A decrease of 1% in its standard deviation.

Sharpe Ratio: \[ \text{Sharpe Ratio} = \frac{E(r_P) - r_f}{\sigma_P} \] Where \(r_f\) is the risk-free rate, \(E(r_P)\) is the portfolio’s expected return, and \(\sigma_P\) is the portfolio’s standard deviation.

4. An investor ponders various allocations to the optimal risky portfolio and risk-free T-bills to construct his complete portfolio. How would the Sharpe ratio of the complete portfolio be affected by this choice? (LO 6-3) không ảnh hưởng

\[ \sigma_A^2 = \beta^2 \sigma_M^2 + \sigma_{\text{residual}}^2 \]

case 1 \[ \sigma_A^2 = (1.65)^2 \times (0.2)^2 + (0.3)^2 \] \[ \sigma_A^2 = (2.7225) \times (0.04) + (0.09) = 0.1089 + 0.09 = 0.1989 \] case 2 \[ \sigma_A^2 = (1.5)^2 \times (0.2)^2 + (0.33)^2 \] \[ \sigma_A^2 = (2.25) \times (0.04) + (0.1089) = 0.09 + 0.1089 = 0.1989 \]

không khác j \[ \sigma_P^2 = w_M^2 \sigma_M^2 + w_A^2 \sigma_A^2 + 2w_Mw_A \rho_{M,A} \sigma_M \sigma_A \]

10. Reward-to-Volatility Ratio (Sharpe Ratio)

1. Portfolio Standard Deviation: Solve for \(\sigma_P\) using the CAL formula: \[ E(r_P) = r_f + \frac{\sigma_P}{\sigma_O} (E(r_O) - r_f) \] Where \(\sigma_O\) and \(E(r_O)\) are the standard deviation and expected return of the optimal risky portfolio.

Substitute: \[ w_r = \frac{12\% - 5\%}{14\% - 5\%} = \frac{7\%}{9\%} = 0.7778 \, \text{or } 77.78\% \, \text{in the risky portfolio}. \]

2. Proportions of T-Bill and Risky Assets: Calculate the weights:
   • Proportion in T-Bills (\(w_f\)).
   • Proportions in risky assets (\(1 - w_f\)).