Capital Allocation Line (CAL) and Optimal Portfolio
October 17, 2024
What is the Capital Allocation Line and an Optimal Portfolio?
The Capital Allocation Line (CAL) represents the risk and return trade-off for a portfolio that combines the risk-free asset and risky assets. It aims to help investors assess their risk profile and it illustrates the level of potential returns that can be expected for the different levels of risk. It is shown as a line chart where one axis measures standard deviation (as a measure of risk) and the other expected returns. The slope of the CAL allows investors to pick points along the line depending on their risk appetite. For example, a conservative investor may have higher exposure to the risk-free asset which typically results in lower overall portfolio risk, but also limits the potential return. The opposite, where a higher allocation to riskier asset classes such as equities would potentially lead to higher return is typical for more adventurous investors.
The optimal portfolio is the point where the line touches the efficient frontier, representing the best risk-return outcome. This portfolio is determined as the market portfolio under the Capital Market Line framework.
Key Learning Points
- The Capital Allocation Line (CAL) demonstrates the relationship between risk and return of a portfolio that holds both the risk-free asset and risky assets
- It helps investors determine their desired level of risk and also allocate capital more efficiently
- The slope of the CAL is known as the “reward-to-variability” and is measured by the Sharpe ratio
- The Capital Market Line (CML) is a variation of the CAL, which considers a market portfolio (such as index fund) instead of the risky asset component of CAL (for example stocks)
Capital Allocation Line Formula
The formula to calculate the CAL is shown below.
Where:
ErP = Expected return of the portfolio
Er(rs) = Expected return of the risky asset
W = Weight of the risky asset in the portfolio
Er(rf) = Expected return of the risk-free asset
As the standard deviation of a risk-free asset is zero, only the risky component of the portfolio determines its risk.
Capital Allocation Line Example
A portfolio is constructed of two assets – a Treasury bill with expected return of 4% and 0% risk (as this is the risk-free) and a stock with expected return of 11% and standard deviation of 22%.
First, we calculate the expected return of the portfolio by using the below formula.
Then, to calculate the risk of the portfolio we need to consider only the standard deviation of the stock since the T-bill is risk-free (hence 0%).
Now, if we assume that a conservative investor allocates 75% of the total assets into the risk-free and 25% to the risky asset, the portfolio’s expected return would be as follows:
ErP = (4% x 75%) + (11% x 25%) = 3% + 2.75% = 5.75%
Calculating the portfolio risk:
Risk of portfolio = 25% × 22% = 5.5%
Capital Market Line vs. Capital Allocation Line
The Capital Market Line (CML) is a specific version of the CAL that is applied to a market portfolio (this can be a domestic, global or any other market index) instead of the line that makes up the allocation between the risk-free asset and a risky asset portfolio. Therefore, the risk portfolio under the CML is the market portfolio. For example, for a US-based equity investor that can be the Russell 3000 Index (a market cap-weighted stock index that seeks to be a benchmark of the entire US stock market). The slope of the CML is the Sharpe ratio of the market portfolio as portfolios that fall on the line optimize the risk and return trade-off. For example, if an investor constructs a portfolio of the risk-free asset and a diversified stock index fund, the end portfolio would lie on the CML.
The below table summarizes the key differences between the CAL and the CML.
| Capital Allocation Line (CAL) | Capital Market Line (CML) | |
| Components | 1. Risk-free asset 2. Risky portfolio |
1. Risk-free asset 2. Market portfolio |
| Focus | Any risky portfolio | The market portfolio |
| Risk-return relationship | Varies by the chosen risky portfolio | Represents the optimal risk-return combinations |
| Slope | The Sharpe ratio of an individual portfolio | The Sharpe ratio of the market portfolio |
| Relevance | Individual portfolios | Market portfolios |
Capital Allocation Line Slope
The slope of the CAL shows the additional return that should be expected for taking on additional risk (i.e. the risk premium). A flatter line is an indication of lower risk premium while a steeper line signals higher risk premium. The slope of the line is known as the “reward-to-variability” ratio or also referred as the Sharpe ratio, which is a measure of the excess return achieved per unit of risk. The asset with the higher Sharpe Ratio is considered superior.
To determine whether a portfolio that is comprised of both risk-free and risky assets is optimal we need to look at the below chart.
Optimal portfolios, where an investor can achieve the best rate of return based on the given level of risk, are located at the tangent of the efficient frontier and the CAL as the slope is the highest at that point. Typically, an efficient portfolio is diversified and is comprised of several assets and asset classes.
How to Draw Capital Allocation Line in Excel?
Below is a task that requires drawing a CAL in Excel.
Data Source: Investing.com
The full solution is available in the free download Capital Allocation Line (CAL) Workout.
Conclusion
To sum up, the Capital Allocation Line (CAL) shows all possible combinations of portfolios constructed of risk-free and risky assets, helping investors to determine their desired level of risk and return based on their risk tolerance. Its slope represents the incremental return for each additional unit of risk (as per the Sharpe ratio). The CAL is a fundamental tool that helps investors to achieve optimal portfolios.