Capital Allocation Line

The linearity follows directly from the statistics. A portfolio combining weight w in a risky portfolio P and (1 − w) in the risk-free asset has an expected return equal to the risk-free rate plus w times the risk premium of P. Because the risk-free asset has zero variance and, by construction, zero covariance with anything, the portfolio standard deviation collapses to simply w multiplied by the standard deviation of P. There is no diversification term to bend the relationship.

Solving the second equation for w gives w equal to the portfolio's standard deviation divided by that of P, and substituting into the return equation produces expected return equal to the risk-free rate plus the risk premium of P divided by its standard deviation, all multiplied by the chosen standard deviation. That coefficient is the Sharpe ratio, and it is constant, so the locus is a ray in mean–standard-deviation space.

Weights above one represent borrowing at the risk-free rate and investing more than initial capital in the risky portfolio. Mathematically the line simply extends beyond the point representing P, offering higher expected return and proportionally higher volatility with no change in slope. This is the analytical basis for margin investing and for target-volatility and risk-parity mandates that scale exposure up or down rather than reselecting securities.

In practice, few investors borrow at the Treasury rate. When the borrowing rate exceeds the lending rate, the allocation line kinks at the tangency point: the segment to the left retains the steeper slope based on the lending rate, while the leveraged segment has a flatter slope reflecting the higher cost of funds. Candidates should recognise that a leveraged investor therefore faces a different, and generally less attractive, tangency portfolio than an unleveraged one.

The capital allocation line is the mechanism behind two-fund separation. Given a set of risky assets, the investor first identifies the risky portfolio that maximises the Sharpe ratio, which is the portfolio at which a ray from the risk-free rate touches the efficient frontier of risky assets. That choice is independent of risk aversion. Only in the second step does preference enter, when the investor selects a point along the line by choosing how much cash to hold or how much leverage to take.

Graphically, the optimal portfolio sits where the investor's highest attainable indifference curve is tangent to the allocation line. A more risk-averse investor, with steeper indifference curves, settles nearer the risk-free intercept; a risk-tolerant investor moves further out. When every investor holds the same risky portfolio and expectations are homogeneous, that portfolio becomes the market portfolio and the line becomes the capital market line.

Expect to compute a combined portfolio's expected return and standard deviation from a stated weight, and to reverse the calculation: given a target volatility, divide it by the risky portfolio's standard deviation to obtain the required risky weight, then read off the expected return from the line equation. Weights exceeding one should be interpreted as borrowing, and a negative risk-free weight identified accordingly.

Equally testable is ranking. Two candidate risky portfolios generate two allocation lines, and the one with the higher Sharpe ratio dominates at every level of risk, regardless of which portfolio has the higher raw expected return. Questions frequently exploit this by presenting a high-return, high-volatility portfolio that produces the flatter line. Finally, be prepared to distinguish the line drawn to an arbitrary portfolio from the efficient frontier itself and from the capital market line.