Capital Market Line

Every combination of a risk-free asset and a single risky portfolio traces a straight line in expected return versus standard deviation space, because the risk-free asset has zero variance and zero covariance with the risky holding. Portfolio standard deviation therefore scales linearly with the weight placed in the risky asset, while expected return scales linearly with the same weight. The result is a capital allocation line whose slope is the risky portfolio's Sharpe ratio.

The Capital Market Line is the special case that arises when investors share the same inputs, face the same risk-free rate and all identify the same tangency portfolio. Under those homogeneous-expectations conditions, the tangency portfolio must be the value-weighted market portfolio, since assets can only clear if aggregate demand matches aggregate supply. The steepest attainable allocation line becomes the equilibrium line for all investors.

Stated in words, the expected return of any portfolio on the Capital Market Line equals the risk-free rate plus the market risk premium multiplied by the ratio of that portfolio's standard deviation to the market portfolio's standard deviation. Equivalently, expected return equals the risk-free rate plus the slope of the line times the portfolio's total risk, where the slope is the market's excess return divided by its standard deviation.

In weight terms, if a fraction is invested in the market portfolio and the remainder in the risk-free asset, expected return is the weighted average of the two expected returns, and standard deviation is simply that fraction times the market's standard deviation. Weights below one describe lending portfolios that sit between the risk-free point and the tangency point; weights above one describe borrowing at the risk-free rate to lever the market, extending the line to the right of the tangency portfolio.

The line implies separation: the choice of risky portfolio is identical for everyone and independent of risk tolerance, while risk tolerance determines only how much is allocated between that portfolio and cash. This is the theoretical justification for the core-satellite and index-plus-cash approaches used in practice, and for treating the strategic asset allocation decision separately from the security selection decision.

Market frictions blunt the result. Investors borrow above the risk-free rate, so the segment beyond the tangency point is flatter than the lending segment, producing a kinked line. Investable market proxies exclude human capital, private assets and much of global fixed income, so the observed line is an approximation. Constraints on leverage push risk-seeking investors toward high-beta assets instead, a documented source of the flat empirical relationship between beta and realized return.

Be able to compute the expected return and standard deviation of any lending or borrowing combination, and to solve the reverse problem: given a target expected return or a target volatility, determine the required weight in the market portfolio. Be able to compute the slope as a Sharpe ratio and use it to rank candidate portfolios, recognising that a portfolio plotting above the line would imply an arbitrage-like improvement inconsistent with equilibrium.

Equally important is knowing what the horizontal axis measures. Because portfolios on the line hold the market portfolio in some proportion with cash, they carry no unsystematic risk, so their total risk is entirely systematic. That is why total standard deviation is a valid pricing input here, and why the relationship cannot be applied to individual securities or to undiversified portfolios, which must be evaluated on systematic risk alone.