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Capital Allocation and the CML

Adding a risk-free asset to the risky frontier transforms the investment opportunity set. Mixing the risk-free asset with a risky portfolio traces a straight line, the capital allocation line (CAL), whose slope is that risky portfolio’s Sharpe ratio, the reward per unit of total risk. The steepest possible CAL is tangent to the efficient frontier at the optimal risky portfolio, and when that tangency portfolio is the market portfolio the line is called the Capital Market Line (CML). Every investor then holds the same risky mix and dials risk up or down purely by their split between it and the risk-free asset.

Try it yourself

Capital allocation line

Add a risk-free asset and the steepest line from it to the risky frontier is the capital allocation line, tangent at the portfolio with the highest Sharpe ratio. Its slope is the reward per unit of total risk.

Current portfolio (wA = 0.50)11.0% · σ 13.5%
2%5%9%12%15%0%6%12%18%25%Risk σ (%)Expected return E(R) (%)ABRfTMVPFrontierCurrentCAL (Sharpe)
MVP risk σ 11.3%MVP weight w* 0.83Sharpe (current) 0.59
Weight on A, w0.50
Correlation ρ0.20
E(R) of A8.0%
σ of A12.0%
E(R) of B14.0%
σ of B22.0%
Risk-free rate Rf3.0%
The steepest line from Rf touches the frontier at the tangency portfolio (Sharpe 0.60). Every investor holds that one risky mix and slides along the line by lending or borrowing at Rf.
Try this. Set ρ = +1 — the frontier collapses to a straight line, so there is no diversification benefit. Now set ρ = −1 — it bends sharply and the right mix can push risk almost to zero.

Why it matters

A risk-free asset lets you blend safety with a single risky portfolio along a straight line, and you want the steepest such line because steeper means more return per unit of risk. The slope is the Sharpe ratio, so picking the best risky portfolio is the same as maximising Sharpe. The remarkable result is separation. Everyone picks the identical tangency portfolio, and personal risk tolerance only decides how much you lend or borrow at the risk-free rate. When that shared portfolio is the whole market, the line earns the name Capital Market Line and becomes the benchmark for efficiently diversified investing.

Formulas

Sharpe ratio (slope of the CAL)
S=E(Rp)−RfσpS = \dfrac{E(R_p) - R_f}{\sigma_p}
Excess return over the risk-free rate RfR_f per unit of total risk σp\sigma_p. The optimal risky portfolio is the one with the highest Sharpe ratio.
Capital allocation line
E(Rc)=Rf+E(Rp)−Rfσp σcE(R_c) = R_f + \dfrac{E(R_p) - R_f}{\sigma_p}\,\sigma_c
A combined portfolio mixing the risk-free asset with risky portfolio pp. Its expected return rises linearly in its total risk σc\sigma_c.
Capital Market Line
E(Rc)=Rf+E(Rm)−Rfσm σcE(R_c) = R_f + \dfrac{E(R_m) - R_f}{\sigma_m}\,\sigma_c
The special CAL through the market portfolio mm. It prices efficient portfolios by their total risk σc\sigma_c, not by beta.

Worked examples

Scenario

The risk-free rate is 3%. A risky portfolio earns 11% with 20% risk. Find its Sharpe ratio and the expected return of a fifty-fifty mix with the risk-free asset.

Solution

The Sharpe ratio is (11%−3%)/20%=0.40(11\%-3\%)/20\%=0.40. Putting half in the risk-free asset gives total risk σc=0.5(20%)=10%\sigma_c=0.5(20\%)=10\% and return E(Rc)=3%+0.40(10%)=7%E(R_c)=3\%+0.40(10\%)=7\%. The combined portfolio sits exactly halfway along the CAL between the risk-free point and the risky portfolio.

Scenario

Two risky portfolios are available, A with Sharpe 0.5 and B with Sharpe 0.3. Which should anchor the risky mix every investor holds?

Solution

Portfolio A. Because the CAL through A is steeper, it delivers more expected return at every risk level than the CAL through B. Under the separation principle all investors hold A as their risky portfolio and adjust only the risk-free weight, regardless of personal risk tolerance.

Common mistakes

  • ✗The CAL and CML describe the same line for any investor. The CML is the specific CAL drawn through the market portfolio, so a CAL through a suboptimal risky portfolio is flatter and not the CML.
  • ✗A higher expected return alone identifies the best risky portfolio. The right criterion is the Sharpe ratio, the slope of the CAL, so a lower-return portfolio with much less risk can be superior.
  • ✗Adding the risk-free asset changes the shape of the risky frontier. It adds a straight line from the risk-free point, while the curved risky frontier itself is unchanged.
  • ✗The CML prices individual stocks by their total risk. The CML applies to efficient, fully diversified portfolios, whereas individual securities are priced by systematic risk through the SML.

Revision bullets

  • •A risk-free asset plus a risky portfolio traces a straight capital allocation line
  • •The slope of the CAL is the Sharpe ratio, reward per unit of total risk
  • •The optimal risky portfolio maximises the Sharpe ratio, the steepest CAL
  • •Separation: all investors hold the same risky mix and vary only the risk-free weight
  • •The CML is the CAL through the market portfolio, pricing efficient portfolios by total risk

Quick check

The slope of the capital allocation line equals

The Capital Market Line differs from a general capital allocation line because the CML

Connected topics

Return & RiskMean-VarianceEfficient FrontierCAPM & SMLPerformance Eval

Sources

  1. Brailsford, Heaney & Bilson (2015), Ch. 8-9
    Brailsford, T., Heaney, R., & Bilson, C. Investments: Concepts and Applications. 5th ed. Cengage Learning Australia, 2015.
    Develops the capital allocation line, the Sharpe ratio, and the Capital Market Line.
  2. Sharpe (1964)
    Sharpe, W. F. Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. Journal of Finance, 19(3), 425-442, 1964.
    Establishes the risk-free asset, the market portfolio, and the resulting capital market line.
How to cite this page
Dr. Phil's Quant Lab. (2026). Capital Allocation and the CML. Derivatives Atlas. https://phucnguyenvan.com/concept/im-capital-allocation
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