The capital market line

What happens when you combine the risk-free asset with a risky portfolio?

The risk-free asset has zero variance and zero covariance with everything else, so it adds no risk of its own. Put a weight w in a risky portfolio and 1 − w in the risk-free asset, and both return and risk move in a straight line with w.

$$E(R_p) = w\,E(R_i) + (1 - w)\,R_f, \qquad \sigma_p = w\,\sigma_i$$

A weight between 0 and 1 means part of your wealth is lent at the risk-free rate. A weight above 1 means you borrowed at the risk-free rate and invested more than your wealth. The line simply continues past the risky portfolio.

Each investor’s line depends on which risky portfolio they believe is best. That is the capital allocation line (CAL): one line per risky portfolio, and the best one is the steepest. So what happens when everyone agrees on which risky portfolio that is?

Why does everyone end up holding the market portfolio?

Suppose every investor uses the same inputs: the same expected returns, standard deviations and correlations. This is homogeneity of expectations . Everyone then draws the same efficient frontier, finds the same steepest line from the risk-free rate, and picks the same tangency portfolio. If every investor holds it, it must contain every risky asset in proportion to its market value. That is the market portfolio .

The capital market line (CML) is the CAL whose risky portfolio is the market portfolio. In practice a broad index stands in for “the market”, because no one can observe every risky asset.

$$E(R_p) = R_f + \frac{E(R_m) - R_f}{\sigma_m}\,\sigma_p$$

The intercept is the risk-free rate and the slope is the extra return per unit of total risk, the market’s own Sharpe ratio.

$$\text{Slope} = \frac{E(R_m) - R_f}{\sigma_m}$$

To land on a given risk level, hold the right share of the market. The rest sits in the risk-free asset (or is borrowed).

$$w = \frac{\sigma_p}{\sigma_m}$$

Slide the weight to move along the line and watch the readouts. Then raise the borrowing rate and see the line bend at the market portfolio. Everyone holds the market; how much of it decides where you sit.

Weight in the market portfolio, w = σp ÷ σmthe rest is lent or borrowed at the risk-free rate

• Lending portfolio Use it w between 0 and 1; part of wealth in the risk-free asset; more risk averse
• The market portfolio Memorize w = 1; all wealth in risky assets
• Borrowing (leveraged) portfolio Use it w above 1; borrow at the risk-free rate and invest more than all wealth in the market

Holding the market and adjusting only the lending or borrowing is a passive portfolio strategy: it lands on the CML. An active portfolio strategy departs from the market weights because the manager believes the market’s prices or forecasts are wrong. The names for the points on the line follow the weight: a lending portfolio has w below 1, a borrowing portfolio has w above 1, and a borrowing portfolio is also called a leveraged portfolio .

Approaches to holding risky assetsthe CML describes the passive case

• Passive Use it hold the market (an index fund), plus lending or borrowing
  • Lands on the CML the efficient set when everyone shares expectations
• Active Use it overweight what you think is cheap, underweight what you think is dear
  • Bets that expectations differ from the market’s can land above or below the CML

What if borrowing costs more than lending?

Most investors cannot borrow at the risk-free rate. Suppose they lend at R_f but borrow at a higher rate R_b. To the left of the market portfolio nothing changes. To the right, the leveraged return is the market return on the whole position, less the cost of the borrowed part.

$$E(R_p) = w\,E(R_m) - (w - 1)\,R_b, \qquad w > 1$$

The line therefore has a kink at the market portfolio and is flatter beyond it. Its slope there is (E(R_m) − R_b) ÷ σ_m.

Go deeper Why the market portfolio should hold everything

The theoretical market portfolio holds every risky asset, not only listed shares: bonds, real estate, even human capital. Nobody can observe all of that, so analysts use a broad index as a proxy. Different proxies give different CMLs, which is one reason the model is hard to test.

The CML’s slope is the best Sharpe ratio available from combining the risk-free asset with risky assets. No other risky portfolio has a steeper line from Rf.

In practice, limits on short selling and on margin borrowing also stop investors from reaching every point on the line.

Exam tip Weight first: w = σp ÷ σm. Then E = Rf + w × (E(Rm) − Rf). If w is above 1, check whether the question gives a borrowing rate.