The Capital Market Line and Separation Theorem

Mean-variance optimisation without a riskless asset produces a curved efficient frontier — every investor with a different risk tolerance chooses a different point on that curve, holding a different risky portfolio. Adding a riskless asset transforms this picture completely. The efficient frontier straightens into a line. Every investor — regardless of risk tolerance — holds the same risky portfolio (the tangency portfolio), combined with lending or borrowing at the riskless rate to suit their individual risk preference. This is the Separation Theorem: the investment decision (which risky portfolio to hold) is separated from the financing decision (how much to borrow or lend).

The line connecting the riskless rate to the tangency portfolio is the Capital Market Line (CML). Under CAPM assumptions — homogeneous expectations plus a riskless asset available to all — the tangency portfolio is the market portfolio, and every investor lies somewhere on the CML. The slope of the CML is the market Sharpe ratio: the maximum reward per unit of standard deviation available to any investor in equilibrium. This is arguably the most consequential geometry in modern finance.

Learning objectives. By the end of this note you should be able to: