Capital Allocation Line for Three Assets

Introduction

The Capital Allocation Line (CAL) illustrates all possible combinations of a risk-free asset and a risky portfolio, showing the trade-off between expected return and risk (standard deviation). When dealing with three risky assets, the CAL becomes a powerful tool to visualize diversification benefits and determine the optimal allocation between risky assets and the risk-free asset to maximize returns for a given level of risk.

Three Risky Assets Portfolio

1. Definitions

Consider three risky assets: A, B, and C, with:

• Expected returns: E[R_A], E[R_B], E[R_C]
• Standard deviations: \sigma_A, \sigma_B, \sigma_C
• Pairwise correlations: \rho_{AB}, \rho_{AC}, \rho_{BC}

Weights of the assets in the portfolio: w_A, w_B, w_C with w_A + w_B + w_C = 1.

2. Portfolio Expected Return

The expected return of the three-asset portfolio:

E[R_p] = w_A E[R_A] + w_B E[R_B] + w_C E[R_C]

3. Portfolio Standard Deviation

The portfolio variance is:
\sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + w_C^2 \sigma_C^2 + 2 w_A w_B \sigma_A \sigma_B \rho_{AB} + 2 w_A w_C \sigma_A \sigma_C \rho_{AC} + 2 w_B w_C \sigma_B \sigma_C \rho_{BC}
Portfolio standard deviation:

\sigma_p = \sqrt{\sigma_p^2}

4. Combining with Risk-Free Asset

• Let w_f = weight in risk-free asset.
• Expected return of the combined portfolio:

E[R_c] = w_f R_f + (1 - w_f) E[R_p]

Portfolio standard deviation:

\sigma_c = (1 - w_f) \sigma_p

The resulting CAL is a straight line connecting the risk-free asset and the risky portfolio.

Example Scenario

Suppose we have:

• Asset A: E[R_A] = 10%, \sigma_A = 15%
• Asset B: E[R_B] = 12%, \sigma_B = 18%
• Asset C: E[R_C] = 8%, \sigma_C = 10%
• Correlations: \rho_{AB} = 0.2, \rho_{AC} = 0.1, \rho_{BC} = 0.3
• Risk-free rate: R_f = 3%

Step 1: Choose weights

w_A = 0.4, w_B = 0.4, w_C = 0.2

Expected portfolio return:

E[R_p] = 0.4 \cdot 0.10 + 0.4 \cdot 0.12 + 0.2 \cdot 0.08 = 0.04 + 0.048 + 0.016 = 0.104 = 10.4%

Portfolio variance:
\sigma_p^2 = 0.4^2 \cdot 0.15^2 + 0.4^2 \cdot 0.18^2 + 0.2^2 \cdot 0.10^2 + 2 \cdot 0.4 \cdot 0.4 \cdot 0.15 \cdot 0.18 \cdot 0.2 + 2 \cdot 0.4 \cdot 0.2 \cdot 0.15 \cdot 0.10 \cdot 0.1 + 2 \cdot 0.4 \cdot 0.2 \cdot 0.18 \cdot 0.10 \cdot 0.3
\sigma_p^2 = 0.0036 + 0.005184 + 0.0004 + 0.00432 + 0.00012 + 0.000432 = 0.013656

\sigma_p \approx 11.69%

Step 2: Allocate to risk-free asset

• 50% in risk-free, 50% in risky portfolio:
  • Expected return: E[R_c] = 0.5 \cdot 0.03 + 0.5 \cdot 0.104 = 0.015 + 0.052 = 6.7%
  • Standard deviation: \sigma_c = 0.5 \cdot 11.69% \approx 5.85%

Interpretation: The CAL shows a straight line from 3% (risk-free) to 10.4% (risky portfolio) with decreasing risk as more is allocated to the risk-free asset.

Benefits of Three-Asset CAL

1. Diversification: Adding a third asset reduces portfolio risk through imperfect correlations.
2. Flexibility: Allows fine-tuning of risk-return profile along the CAL.
3. Optimization: Identifies the tangency portfolio with the highest Sharpe ratio, which maximizes return per unit of risk.

Practical Considerations

• Correlation and Volatility: Portfolio standard deviation depends on correlations; lower correlations reduce risk.
• Rebalancing: Weights should be adjusted over time as returns, volatility, and correlations change.
• Leverage: Borrowing at the risk-free rate can extend the CAL for more aggressive strategies.

Conclusion

The Capital Allocation Line for three assets illustrates how a combination of three risky assets with a risk-free asset can optimize the trade-off between expected return and risk. By calculating expected portfolio return and standard deviation, investors can determine the optimal mix, achieve diversification benefits, and position their portfolio along the CAL according to their risk tolerance. This framework is a cornerstone of modern portfolio theory and effective investment decision-making.