# Capital Allocation Between a Risk-Free Asset and a Risky Asset

Investors want to earn the highest return possible for a level of risk that they are willing to take. So how does an investor allocate her capital to maximize her investment utility? The simplest way to examine this is to consider a portfolio consisting of 2 assets: a risk-free asset that has a low rate of return but no risk, and a risky asset that has a higher expected return for a higher risk. Investment risk is measured by the standard deviation of investment returns—the greater the standard deviation, the greater the risk. By varying the relative proportions of the 2 assets, an investor can earn a risk-free of return by investing all of her money in the risk-free asset, or she can potentially earn the maximum return by investing entirely in the risky asset, or she can select a risk-return trade-off that is anywhere between these 2 extremes by selecting varying proportions of the 2 assets.

## Capital Allocation Line (CAL)

The investment quality of a 2-asset portfolio is determined by the proportion of the risky asset to the risk-free asset. If this portfolio consists of a risky asset with a proportion of *y*, then the proportion of the risk-free asset must be *1 – y*.

Portfolio Return = y * Risky Asset Return + (1 – y) × Risk-free Return

One way to adjust the riskiness of a portfolio is by varying the proportion of the risk-free asset and the risky asset. The **investment opportunity set** is the set of all combinations of the risky and risk-free assets, which graphs as a line when plotted as return against risk, which is measured by the standard deviation. The line begins at the intercept with the minimum return and no risk of the risk-free asset when the entire portfolio is invested in the risk-free asset to the maximum return and risk when the entire portfolio is invested in the risky asset. Hence, this **capital allocation line** (**CAL**) is the graph of all possible combinations of the risk-free asset and the risky asset.

The slope of the capital allocation line is equal to the incremental return of the portfolio to the incremental increase of risk. Hence, the slope of the capital allocation line is called the **reward-to-variability ratio** because the expected return increases continually with the increase of risk as measured by the standard deviation.

Slope of CAL | = | Reward-to-Variability Ratio | = | Portfolio Return – Risk-Free Return Standard Deviation of Portfolio |

The risk-free return is subtracted from the portfolio return to yield the proportion of the return due to the risky asset. The increase in the return for an increased risk is the reward for the increase in risk—the **risk premium**.

## Capital Market Line (CML)

The **capital market line** (**CML**) is the capital allocation line formed when the risky asset is a market return rather than a single-asset return.

No investment is totally risk-free, but United States Treasuries come close. Although T-bills are often cited as being closest to the ideal risk-free asset for their short terms and low interest rate risk, they do have reinvestment risk. Another security that is close to the ideal are Treasury-Inflation Protected Securities (TIPS), which pay a fixed interest rate on a principal that is adjusted for inflation. For their term length, which can be 5, 10, or 20 years, there is no reinvestment risk, and the interest rate risk is mitigated by the increasing principal, since some of the change in prevailing interest rates results from changes in inflation.

For the risky asset, many investors choose a mutual fund or an exchange-traded fund based on a market index, which provides some diversification in the risky asset without the need for security analysis. This passive strategy of selecting a market index security or investment for the risky asset is called the **mutual fund theorem**.