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

CFA® Level I · Portfolio Construction · 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.

Which portfolio does the steepest capital allocation line touch?

The global minimum-variance portfolioWhy not: A line drawn to it is less steep than one drawn to the tangency point, so it is dominated.
The tangency portfolio - correct
The highest-return portfolio on the frontierWhy not: Its extra return comes with more than proportionate risk, giving a flatter line.

A capital allocation line joins the risk-free rate to a risky portfolio. Its slope shows the extra return earned per unit of risk. Slope = (Expected return - Risk-free rate) / Standard deviation. This is the Sharpe ratio. The steepest line goes to the risky portfolio with the highest Sharpe ratio, which is the tangency portfolio, where the line just touches the frontier. If all investors share the same views and the same risk-free rate, each holds some mix of this portfolio and the risk-free asset. They differ only in the proportions.

A harder one

What does the slope of the capital allocation line represent?

The risk aversion of the investorWhy not: Aversion decides where on the line the investor sits, not how steep the line is.
The Sharpe ratio of the risky portfolio - correct
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.

A slope tells you how much extra return you get for each extra unit of risk. For the capital allocation line, that is the Sharpe ratio of the risky portfolio. Slope = (Expected return of risky portfolio - Risk-free rate) / Standard deviation of risky portfolio. Standard deviation is the usual measure of risk. Example: (8% - 2%) / 15% = 0.40. The investor's own risk aversion only decides where on the line they sit, not how steep it is. Beta belongs to a different line.

The hardest tier

Imani Clarke has $200,000 to invest. She can borrow or lend at the risk-free rate of 3.0%. The tangency portfolio has an expected return of 9.5% and a standard deviation of 16%. She borrows $50,000 at the risk-free rate and invests the full $250,000 in the tangency portfolio. What are the expected return and standard deviation of her position, measured on her own $200,000?

7.9% and 12.0%Why not: This treats the $50,000 as lent at the risk-free rate rather than borrowed, giving 75% in the tangency portfolio and 25% in the risk-free asset. Borrowing does the opposite and takes her beyond 100% in the risky portfolio.
11.1% and 20.0% - correct
11.9% and 20.0%Why not: The risk is right but the borrowing cost has been ignored. Her return on $200,000 must be reduced by the 3.0% interest on the $50,000 she borrowed, which is a weight of -25% in the risk-free asset.

She puts 250,000 into the risky portfolio but has only 200,000 of her own, so her weight there is 250,000 / 200,000 = 125%. Her weight in the risk-free asset is -25%, because a negative weight means borrowing. Expected return = 1.25 x 9.5% + (-0.25) x 3.0% = 11.875% - 0.75% = 11.125%, or 11.1%. Risk = 1.25 x 16% = 20.0%, since the risk-free asset adds no risk. Ignoring the borrowing cost gives 11.9%. Treating the loan as lending gives 7.9% and 12.0%.

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