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Michael Brenndoerfer

Capital Market Theory - CAPM and the Efficient Frontier › The Risk-Free Asset and Portfolio Combinations

Michael BrenndoerferDecember 14, 20257 min read

Quantitative FinanceAll chapters

Part 9 of 15Portfolio Theory and Investment Analysis

Chapter 47Capital Market Theory - CAPM and the Efficient Frontier › The Risk-Free Asset and Portfolio Combinations

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The introduction of a risk-free asset fundamentally changes the efficient frontier we developed in the previous chapter. Recall that without a risk-free asset, the efficient frontier is a curved hyperbola in mean-standard deviation space. When investors can borrow or lend at the risk-free rate, new portfolio possibilities emerge that dramatically simplify your decision problem.

Risk-Free Asset

A risk-free asset provides a guaranteed return with zero variance. In practice, short-term government securities (like U.S. Treasury bills) approximate the risk-free asset, though even these carry some inflation risk and reinvestment risk.

To understand why the risk-free asset matters so much, consider the fundamental question facing every investor: how should you allocate between safe and risky investments? Before CAPM, this question seemed hopelessly complex because different risky portfolios offered different risk-return tradeoffs. The risk-free asset provides a universal anchor for constructing these allocations.

Consider combining a risk-free asset with return $R_f$ and a risky portfolio $P$ with expected return $E[R_P]$ and standard deviation $\sigma_P$ . If we invest a fraction $w$ in the risky portfolio and $(1-w)$ in the risk-free asset, we can derive the portfolio's characteristics by applying the basic rules of portfolio mathematics. The expected return calculation follows from the linearity of expectations. Since we earn $R_f$ on the safe portion and $E[R_P]$ on the risky portion, the combined expected return is simply the weighted average:

\begin{aligned} E[R_c] &= (1-w)R_f + wE[R_P] && \text{(weighted average)} \\ &= R_f + w(E[R_P] - R_f) && \text{(rearrange terms)} \end{aligned}

where:

$E[R_c]$ : expected return of the combined portfolio
$w$ : weight invested in the risky portfolio
$R_f$ : return on the risk-free asset
$E[R_P]$ : expected return of the risky portfolio

The rearranged form reveals an important insight: the expected return equals the risk-free rate plus a premium that scales with your allocation to the risky portfolio. This premium, $E[R_P] - R_f$ , represents the reward for taking on risk, and you earn more of it as you increase your risky allocation.

The variance calculation is where the magic happens. Normally, portfolio variance depends on the variances of both components plus their covariance. But the risk-free asset, by definition, has zero variance and zero covariance with any other asset. This special property leads to a remarkable simplification:

\begin{aligned} \sigma_c^2 &= w^2\sigma_P^2 + (1-w)^2\sigma_{Rf}^2 + 2w(1-w)Cov(R_P, R_f) && \text{(portfolio variance)} \\ &= w^2\sigma_P^2 + 0 + 0 && \text{(risk-free implies zero variance/covariance)} \\ \sigma_c &= \sqrt{w^2\sigma_P^2} = w\sigma_P && \text{(take square root)} \end{aligned}

where:

$\sigma_c$ : standard deviation of the combined portfolio
$w$ : weight invested in the risky portfolio
$\sigma_P$ : standard deviation of the risky portfolio
$\sigma_{Rf}$ : standard deviation of the risk-free asset (zero)

This result has practical implications. The standard deviation of the combined portfolio is simply the weight in the risky portfolio times the risky portfolio's standard deviation. There are no squared terms or covariance adjustments to worry about. Risk scales linearly with your allocation to risky assets.

Notice that both expected return and standard deviation are linear in $w$ . This means combinations of the risk-free asset and any risky portfolio trace out a straight line in mean-standard deviation space. This linearity is the key geometric insight that underlies all of CAPM. In contrast, combinations of two risky assets typically trace out curved paths because their covariance introduces nonlinearity into the portfolio variance formula.

When $w = 0$ , we hold only the risk-free asset and earn $R_f$ with zero risk. When $w = 1$ , we hold only the risky portfolio. When $w > 1$ , we borrow at the risk-free rate to leverage our position in the risky portfolio. Leveraging amplifies both expected returns and risk proportionally, which is why the straight line extends beyond the risky portfolio's position.

To express the risk-return relationship directly, we substitute $w = \sigma_c / \sigma_P$ from the risk equation into the return equation. This substitution eliminates $w$ and gives us expected return as a function of risk:

\begin{aligned} E[R_c] &= R_f + w(E[R_P] - R_f) \\ &= R_f + \left(\frac{\sigma_c}{\sigma_P}\right)(E[R_P] - R_f) && \text{(substitute } w \text{)} \end{aligned}

Rearranging terms gives the Capital Allocation Line (CAL):

E[R_c] = R_f + \frac{E[R_P] - R_f}{\sigma_P} \sigma_c

where:

$E[R_c]$ : expected return of the combined portfolio
$R_f$ : risk-free rate (intercept)
$\frac{E[R_P] - R_f}{\sigma_P}$ : Sharpe ratio (slope)
$\sigma_c$ : standard deviation of the combined portfolio

This is the equation of a line with intercept $R_f$ and slope $(E[R_P] - R_f)/\sigma_P$ . The slope represents the risk premium per unit of risk, a concept we'll formalize shortly. This slope measures how much additional expected return you receive for each additional unit of risk you accept. A higher slope means a better risk-return tradeoff, which is exactly what you seek.

In[2]:

Code

import numpy as np

# Define risk-free rate and a risky portfolio
rf = 0.03  # 3% risk-free rate
portfolio_return = 0.10  # 10% expected return
portfolio_std = 0.15  # 15% standard deviation

# Generate combinations with different weights
weights = np.linspace(0.0, 2.0, 100)  # From 100% risk-free to 2x leverage
combined_returns = rf + weights * (portfolio_return - rf)
combined_stds = weights * portfolio_std

We can visualize the risk-return trade-off by plotting the Capital Allocation Line.

Out[3]:

Visualization

Line chart showing linear relationship between portfolio risk and return when combining risk-free asset with risky portfolio. — The Capital Allocation Line shows all possible combinations of the risk-free asset and a risky portfolio. Weights between zero and one represent mixed portfolios (lending), and weights greater than one represent leveraged positions (borrowing).

The figure illustrates how the Capital Allocation Line (CAL) extends from the risk-free rate through the risky portfolio and beyond. Investors with low risk tolerance choose portfolios close to the risk-free asset (lending portfolios), while aggressive investors leverage their positions by borrowing at the risk-free rate. The key insight is that the CAL represents all achievable combinations using a particular risky portfolio. Your optimal point on this line depends on your personal risk preferences, but everyone faces the same menu of options defined by this straight line.