Introduction
The Capital Allocation Line (CAL) is a key concept in portfolio theory, representing all possible combinations of a risk-free asset and a risky asset or portfolio of risky assets. It shows the trade-off between expected return and risk (standard deviation) for different allocations, helping investors determine the optimal mix based on their risk tolerance. When considering two risky assets, the CAL extends naturally, illustrating how a combination of risky assets and a risk-free asset can be leveraged for optimal return.
Understanding the Capital Allocation Line
- Definition:
- The CAL is a straight line on the risk-return graph.
- The y-intercept represents the risk-free rate R_f.
- The slope represents the reward-to-risk ratio (Sharpe ratio) of the risky portfolio:
\text{Slope} = \frac{E[R_p] - R_f}{\sigma_p}
Where E[R_p] is expected return of the risky portfolio and \sigma_p is its standard deviation.
- Equation of CAL:
E[R_c] = R_f + \frac{E[R_p] - R_f}{\sigma_p} \cdot \sigma_c
Where:
- E[R_c] = expected return of the combined portfolio
- \sigma_c = standard deviation of the combined portfolio (depends on risky asset allocation)
Two Risky Assets
1. Portfolio of Two Risky Assets
Consider two risky assets, A and B, with:
- Expected returns: E[R_A], E[R_B]
- Standard deviations: \sigma_A, \sigma_B
- Correlation coefficient: \rho_{AB}
The expected return of the portfolio:
E[R_p] = w_A E[R_A] + w_B E[R_B]
Where w_A + w_B = 1
2. Combining with a Risk-Free Asset
- When a risk-free asset is added, the combined portfolio expected return:
Combined standard deviation:
\sigma_c = (1 - w_f) \sigma_p
Where w_f is the weight in the risk-free asset.
This produces the Capital Allocation Line: a straight line connecting the risk-free asset and the risky portfolio.
Example Scenario
Assume:
- Asset A: E[R_A] = 10%, \sigma_A = 15%
- Asset B: E[R_B] = 15%, \sigma_B = 20%
- Correlation: \rho_{AB} = 0.3
- Risk-free rate: R_f = 3%
- Choose a portfolio mix: w_A = 0.6, w_B = 0.4
Expected return:
E[R_p] = 0.6 \cdot 0.10 + 0.4 \cdot 0.15 = 0.06 + 0.06 = 0.12 = 12%Portfolio standard deviation:
\sigma_p = \sqrt{0.6^2 \cdot 0.15^2 + 0.4^2 \cdot 0.20^2 + 2 \cdot 0.6 \cdot 0.4 \cdot 0.15 \cdot 0.20 \cdot 0.3}
- Combining with risk-free asset (50% in risk-free, 50% in portfolio):
- Expected return: E[R_c] = 0.5 \cdot 0.03 + 0.5 \cdot 0.12 = 0.015 + 0.06 = 0.075 = 7.5%
- Standard deviation: \sigma_c = 0.5 \cdot 13.73% \approx 6.87%
Interpretation: By mixing the risk-free asset with a two-asset portfolio, investors can adjust their risk-return profile along the CAL.
Benefits of the CAL with Two Assets
- Efficient Diversification: Combining two risky assets reduces portfolio volatility due to imperfect correlation.
- Flexible Risk Allocation: Weighting between the risky portfolio and risk-free asset allows customization of risk exposure.
- Sharpe Ratio Optimization: Identifies the tangency portfolio, where the CAL has the highest slope, maximizing return per unit of risk.
Practical Considerations
- Correlation Matters: Lower correlation between assets reduces portfolio risk and shifts CAL upward.
- Leverage Options: Borrowing at the risk-free rate allows investment beyond 100% in the risky portfolio, extending the CAL.
- Monitoring: Asset correlations, volatility, and returns change over time, so allocations should be periodically reviewed.
Conclusion
The Capital Allocation Line for two assets illustrates the trade-off between expected return and risk when combining a risk-free asset with a portfolio of two risky assets. Using diversification principles, investors can construct a portfolio that lies on the CAL, achieving an optimal balance of risk and return. Understanding the CAL, portfolio standard deviation, and expected return formulas allows investors to tailor allocations according to their risk tolerance and investment objectives, while maximizing potential returns.
