An investor’s optimal portfolio occurs at a point at which his capital allocation line (which shows a combination of the risk-free asset and the optimal risky portfolio) is tangent to his indifference curve. The capital market line is a special case of capital allocation line based on the capital asset pricing model.

## Capital allocation line

Adding a risk-free asset to the portfolio of risky assets improves the risk-return tradeoff because generally, a risk-free asset has a low correlation with risky assets. This is because even though the expected return on a portfolio is the weighted average of individual asset returns, its variance (and standard deviation) may be lower when assets have less than perfect correlation.

Expected return on a portfolio of N assets equals the weighted average return of the constituent securities, and covariance between any two assets equals their correlation coefficient multiplied by the standard deviation of each asset. Similarly, the covariance of an asset with itself equals its variance. Hence, using these relationships allow us to simplify the above equations.

## Capital market line

A capital market line (CML) is a capital allocation line (CAL) which plots a combination of the risk-free asset and the market portfolio.

Risk-free asset is an asset with no (default, inflation, liquidity, interest rate) risk. US treasury bills are used as proxies for risk-free return.

Even though the market can be defined broadly to include all assets of value, investment practitioners usually define it narrowly with reference to a specific asset class, such as the US large-cap stocks, etc. In this reading, when we mean market return, we mean return on the broad equity-market index, such as S & P 500.

The CML intersects the y-axis at the risk-free rate and is tangent to the efficient frontier at the market portfolio. Any return above the CML is not achievable and any return below the CML is inefficient.

The expected return on a portfolio on CML equals the weighted average return.

\[ E(R_P)=w_1×R_f+(1-w_1)×E(R_m) \]Since a risk-free asset has a standard deviation of zero and its correlation with the market portfolio is also zero, the variance of a portfolio on CML is given by the following equation:

\[ σ_p=(1-w_1)σ_m \]If we combine the aforementioned equations, we get the following equation for CML:

\[ E(R_P)=R_f+\frac{(E(R_m)-R_f)}{σ_m}×σ_p \]The slope of the equation is called the market price of risk. It shows that we move up the CML, both the expected return and risk (standard deviation) increase.

## Leveraged portfolios

When an investor’s allocation to the risk-free asset is positive, the portfolios are called lending portfolios. However, if an investor is willing to take more risk, it can borrow at the risk-free rate and invest the proceeds in the market portfolio. This would result in the investor having greater than 100% exposure to the market and his returns would be magnified. Such portfolios are called leveraged portfolios.

In reality, however, an investor would not be able to borrow at the risk-free rate. This would cause the CML to have a downward bend as soon as an investor’s exposure to the market portfolio crosses 100%. This is because if the borrowing rate is greater than the lending, the slope of the CML would decrease.