# 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 risk-return profile that yields the greatest satisfaction? 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 return by investing all 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)

Asset allocation is the apportionment of funds among different types of assets, such as stocks and bonds, having different ranges of expected returns and risk. Capital allocation, on the other hand, is the apportionment of funds between risk-free investments, such as T-bills, and risky assets, such as stocks. The simplest case of capital allocation is the allocation of funds between a risky asset and a risk-free asset. The risk-return profile of this 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, as 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 = the incremental return of the portfolio to the incremental increase in 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 in risk as measured by the standard deviation.

 Slope of CAL = Reward-to-Variability Ratio = Portfolio Return – Risk-Free Return Standard Deviation of Portfolio In the graph below, the capital allocation line (CAL) is plotted with the assumptions that the risk-free rate has a 4% return and zero standard deviation, and the risky asset has an expected return of 12% and a standard deviation of 15%. Note that the intercept of the CAL is at 4%, which is the risk-free rate. This capital allocation line is the investment opportunity set of all possible combinations of the risk-free and risky asset.

The risk-free return is subtracted from the portfolio return to yield the proportion of the return due to the risky asset. The increased return for the increased risk is the reward for accepting the increased 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 sometimes called the mutual fund theorem.