The risk-free asset, the market portfolio and the CML

Adding a risk-free asset to risky assets

A portfolio of many risky assets has an expected return equal to the weighted average of the assets' expected returns, while its risk depends on the weights, each asset's risk and every pairwise covariance. As long as assets are not perfectly correlated, the portfolio can be less risky than the assets inside it.

Now add a risk-free asset: no default, inflation, liquidity or interest rate risk. Short-term government bills are the usual proxy. Its standard deviation is zero, so its covariance (and correlation) with any risky asset is zero.

Because the risk-free asset contributes no risk and no covariance, every mix of it with a risky portfolio lies on a straight line in expected return vs standard deviation space. This line is the capital allocation line (CAL). Combining the two often gives a better risk-return trade-off than holding either one alone.

Different risky portfolios give different CALs. The best one is the CAL with the highest slope, because it offers the most expected return for each unit of risk. Where on that line an investor sits depends on risk aversion: the optimal portfolio is where the CAL touches the investor's highest indifference curve.

Think of the risk-free asset as a volume knob. The risky portfolio sets the tune (the slope); the knob only decides how loud you play it.

Is there one optimal risky portfolio for everyone?

An asset's value is the present value of its expected cash flows, discounted at a required return that reflects its risk. Analysts forecast cash flows and choose discount rates differently, so they reach different values for the same share.

If one analyst values a share above its price, she overweights it; another who values it below the price may hold none. With heterogeneous expectations, every investor can end up with a different optimal risky portfolio, and no unique optimal portfolio exists.

Capital market theory therefore assumes homogeneous expectations: all investors use the same inputs, reach the same valuations and so hold the same optimal risky portfolio. That portfolio must be the market portfolio, with each asset weighted by its market value divided by the value of all assets.

Even when expectations differ, market prices reflect the view of the marginal, informed investor, so the market portfolio still serves as the natural benchmark against which other portfolios are judged.

Passive vs active, and what 'the market' means

If markets are informationally efficient, the price already reflects all public information, and investors cannot expect to earn more than the required return. Spending money on your own valuations is then pointless; you simply accept market prices.

Portfolios built on the assumption that market prices are unbiased are passive portfolios, usually index funds that replicate a market-cap weighted index at low cost. Active portfolios reflect the manager's own estimates: assets judged undervalued get a weight above their index weight, others a zero weight, or a negative weight if short selling is allowed.

In theory the market includes all risky assets, anything with value: shares, bonds, real estate, even human capital. But not every asset is tradable (a national monument, human capital), and not every tradable asset is investable for every investor (some share classes are reserved for certain investors).

In practice analysts use a broad local stock index as a proxy for the market. That means 'market return' and 'market risk premium' usually refer to that equity index, not to all risky assets.

The capital market line

The CML is the special CAL where the risky portfolio is the market portfolio M. Graphically, M is where a line from $R_f$ is tangent to the Markowitz efficient frontier, so the CML is the best attainable CAL.

Put weight $w_1$ in the risk-free asset and $1 - w_1$ in the market. Because the risk-free asset has zero variance and zero covariance, the portfolio's standard deviation collapses to $(1 - w_1)\sigma_m$, and expected return is the weighted average.

Substituting gives a line: intercept $R_f$ and slope $[E(R_m) - R_f]/\sigma_m$, called the market price of risk. The slope is positive because the market is expected to earn more than the risk-free rate; moving up the line raises both risk and expected return.

Any portfolio above the CML cannot be built from the available assets, so it is unachievable. Any portfolio below it (including single securities and other inefficient portfolios) is dominated: some point on the CML offers more return for the same risk.