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Explain what adding a risk-free asset does, and the capital allocation line

CFA® Level I · Portfolio Management · Portfolio Risk and Return: Part II · PM.2.c

The idea

Introducing a risk-free asset lets an investor combine it with any risky portfolio along a straight line, because the risk-free asset has zero variance and zero covariance with everything. The steepest such line, tangent to the efficient frontier, dominates every other combination: its slope is the Sharpe ratio of the tangency portfolio. If investors share the same expectations and can borrow and lend at the risk-free rate, each holds some mix of the risk-free asset and that same tangency portfolio, varying only the proportions — lending at the risk-free rate for a conservative position, borrowing to move beyond the tangency point for an aggressive one.

Formulas

Capital allocation line

E(Rp) = rf + [(E(Rm) − rf) ÷ σm] × σp

rf
the return on the risk-free asset
E(Rm)
the expected return of the risky portfolio being combined with it
σm
the standard deviation of that risky portfolio
σp
the standard deviation of the combination
E(Rp)
the expected return of the combination

A straight line, because mixing with a risk-free asset has no correlation term to bend it. Left of the risky portfolio is lending at rf, right of it is borrowing to buy more.

Sharpe ratio

Sharpe = (Rp − rf) ÷ σp

Rp
the portfolio's return
rf
the risk-free rate
σp
the standard deviation of the portfolio's return

Return earned per unit of total risk, and the slope of the capital allocation line drawn through that portfolio. The steepest line is the best one available, which is what picking the tangency portfolio means.

A practice question

From the question bank, with the reason each wrong option is wrong.

Why is the capital allocation line straight rather than curved?

  1. Because the risk-free asset has zero variance — correct
  2. Because expected returns are assumed equalWhy not: They are not, and if they were there would be nothing to choose between portfolios.
  3. Because short sales are not permittedWhy not: Short sale rules affect what is reachable on the line, not whether it is straight.

It also has zero covariance with the risky portfolio. With one asset contributing no risk at all, portfolio standard deviation is simply proportional to the weight in the risky portfolio, which makes the relationship linear.

A harder one

What does the slope of the capital allocation line represent?

  1. The risk aversion of the investorWhy not: Aversion decides where on the line the investor sits, not how steep the line is.
  2. The Sharpe ratio of the risky portfolio — correct
  3. The beta of the risky portfolioWhy not: Beta measures sensitivity to the market and belongs to the security market line, which plots against a different axis.

Excess return divided by standard deviation is the slope, so choosing the steepest line is the same as choosing the highest Sharpe ratio.

The hardest tier

How does an investor reach a point on the capital allocation line beyond the tangency portfolio?

  1. By selecting a riskier portfolio from the efficient frontierWhy not: Every such portfolio lies below the line. Moving along the line requires the risk-free asset, not a different risky mix.
  2. By reducing the weight in the tangency portfolioWhy not: That moves back toward the risk-free asset, which is the opposite direction.
  3. By borrowing at the risk-free rate and investing more than their capital in the tangency portfolio — correct

Moving up the line past tangency requires a weight above one in the risky portfolio, funded by borrowing. The composition of the risky portfolio does not change.

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