Capital Allocation Line Explorer
A Capital Allocation Line (CAL) describes all portfolios formed by combining a single risky portfolio P with a risk-free asset F. Because the risk-free asset has zero variance and zero correlation with everything, the combination is special: expected return and standard deviation both move linearly with the fraction \(y\) invested in the risky portfolio.
$$E[R_c] = R_f + y \cdot \bigl(E[R_P] - R_f\bigr) \qquad \sigma_c = y \cdot \sigma_P$$Eliminating \(y\) gives the equation of the CAL in \((\sigma, E[R])\) space: a straight line through \((0, R_f)\) with slope equal to the Sharpe ratio, \(SR = (E[R_P] - R_f) / \sigma_P\). A steeper line means more expected return per unit of risk — it is always better to invest along a higher-Sharpe CAL. The investor's own risk aversion determines where on the line they sit, but the slope is set by the risky portfolio choice. When \(y > 1\) the investor is borrowing at the risk-free rate to lever the risky portfolio.
The visualization
Adjust the parameters for Portfolio P (orange). The comparison Portfolio Q (blue) has fixed characteristics so you can see what a lower or higher Sharpe ratio looks like. Use the weight slider to move your complete portfolio along the CAL.
Risk-Free Asset
Risky Portfolio P (adjustable)
Complete Portfolio Allocation
Portfolio P Results
Sharpe ratio: SR = (E[RP] − Rf) / σP (slope of the CAL)
Lending (0 ≤ y ≤ 1): part of wealth earns Rf risk-free
Borrowing / leverage (y > 1): more than 100% in the risky portfolio
Key takeaways
- The slope of the CAL is the Sharpe ratio. Higher Sharpe ratio means more expected return per unit of risk — a steeper, more attractive line. Every investor prefers to invest along the highest available CAL.
- Risk preference determines where on the CAL you sit, not which CAL you choose. A more risk-averse investor holds more in the risk-free asset (lower y); a more risk-tolerant investor holds more in the risky portfolio (higher y). Both can use the same risky portfolio.
- Leverage (y > 1) amplifies both return and risk proportionally. Borrowing to buy more of the risky portfolio shifts the complete portfolio to the right along the same line.
- The tangency portfolio is special. When the risky portfolio is the tangency portfolio (the highest-Sharpe portfolio on the mean-variance frontier), the CAL becomes the Capital Market Line (CML) — the best possible risk-return trade-off available.
- Portfolio Q illustrates the Sharpe ratio comparison. If Q has a higher Sharpe ratio than P, its CAL lies above P's CAL everywhere — every investor, regardless of risk aversion, prefers Q over P as their risky portfolio.