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Explain the market portfolio and the idea of two-fund separation

CFA® Level I · Portfolio Construction · Portfolio Risk and Return: Part II · PM.2.d

The idea

Two-fund separation is the result that the investment decision splits cleanly in two: which risky portfolio to hold, and how much of total wealth to place in it. The first answer is the same for every investor who shares the same expectations and can borrow and lend at the risk-free rate - the tangency portfolio. Only the second answer depends on risk aversion. If all investors hold the same risky portfolio, it must in equilibrium be the market portfolio, which is the logical step that takes the framework from the capital allocation line to the capital market line and the CAPM.

Formulas

Capital market 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

The capital allocation line drawn through the market portfolio, so E(Rm) and σm are the market's. It plots total risk, which is the difference from the security market line that plots beta.

A practice question

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

In the CAPM, what are the two decisions separated by two-fund separation?

  1. Which securities to buy and when to buy themWhy not: Selection and timing are the active manager's problems. The separation result is about composition against allocation.
  2. Which risky portfolio to hold and how much to allocate to it - correct
  3. Whether to use active or passive management, and which managerWhy not: An implementation choice, which the theory does not address.

Two-fund separation says every investor needs only two building blocks: the risk-free asset and one risky portfolio. So the investment decision splits in two. First, which risky portfolio to hold. This is the same for everyone. Second, how much to put in it versus the risk-free asset. This depends on personal taste for risk. Money in risky portfolio = w x Wealth, where w is chosen by the investor. Example: a cautious investor sets w = 30%, an aggressive one sets w = 130% by borrowing, yet both hold the same risky mix.

A harder one

Which line results from replacing the tangency portfolio with the market portfolio?

  1. The capital market line - correct
  2. The security market lineWhy not: Plots expected return against beta and describes individual assets, not combinations of the market with cash.
  3. The minimum-variance frontierWhy not: Built from risky assets alone and curved; the line in question involves the risk-free asset and is straight.

With shared expectations, everyone holds the same tangency portfolio, and in equilibrium it must be the market portfolio. Swapping the market portfolio into the capital allocation line gives the capital market line. Expected return = Risk-free rate + [(Market return - Risk-free rate) / Market standard deviation] x Portfolio standard deviation. It is straight and applies to efficient portfolios. The security market line is different: it applies to individual assets and plots return against beta.

The hardest tier

Investors hold different expectations about returns. What happens to the result?

  1. All investors still hold one common tangency portfolioWhy not: Homogeneous expectations are what make the portfolio common. Without them, the common portfolio disappears.
  2. The tangency portfolio becomes riskier for everyoneWhy not: There is no longer one tangency portfolio to become riskier; each investor now has their own.
  3. Each investor computes a different tangency portfolio - correct

The tangency portfolio is built from an investor's beliefs about expected returns, risks and correlations (how closely assets move together). If investors hold different beliefs, they draw different efficient frontiers and find different tangency portfolios. So there is no longer one risky portfolio that everyone holds. Example: an investor who expects shares to return 10% and another who expects 6% will choose different risky mixes. Common expectations are as important to the result as the shared risk-free rate.

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