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Set the CAPM beside the Arbitrage Pricing Theory (APT), explaining the APT along with the assumptions it makes.

FRM® Part I · Foundations of Risk Management · The Arbitrage Pricing Theory and Multifactor Models of Risk and Return · FRM1.FR.6.a

The idea

The CAPM says expected return depends on one thing, beta to the market portfolio. Expected return = Rf + β × (E(Rm) − Rf). It rests on investors who care only about mean and variance and hold the same market portfolio. The Arbitrage Pricing Theory starts from a different place. It assumes returns follow a linear model of several systematic factors plus an idiosyncratic term. It also assumes enough assets exist for idiosyncratic risk to diversify away, and that no arbitrage persists. From these assumptions, expected return equals the risk-free rate plus the sum of each factor beta times its risk premium. APT does not say what the factors are and does not need mean-variance preferences. The CAPM is its single-factor special case. The relation holds closely for well-diversified portfolios and only approximately for individual assets. Exam point: only systematic factor risk earns a premium, never idiosyncratic risk.

A practice question

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

Which of the following is an assumption of the Arbitrage Pricing Theory?

  1. Idiosyncratic risk can be diversified away in a portfolio of many assets, and arbitrage opportunities do not persist in the market. - correct
  2. All investors share the same beliefs and hold the market portfolio.Why not: This is a CAPM assumption; APT needs neither homogeneous beliefs nor investors holding the market portfolio.
  3. Asset returns are driven by one common factor, the return on the market portfolio.Why not: This describes the single-factor CAPM; APT allows several systematic factors.
  4. Investors choose portfolios solely on the basis of mean and variance of returns.Why not: Mean-variance preferences underpin the CAPM; APT is built on no-arbitrage and does not require them.

APT assumes a linear factor structure for returns, enough assets to diversify away idiosyncratic risk, and no persistent arbitrage. Common beliefs, mean-variance preferences and a single market factor belong to the CAPM.

A harder one

Under a two-factor APT, well-diversified portfolios P and Q have identical exposures to both factors. P has an expected return of 11% and Q has an expected return of 9%. Which response and outcome is consistent with the APT?

  1. Buy Q and sell P, because P's higher expected return signals extra risk that the factors do not capture.Why not: With identical factor exposures there is no extra priced risk; the gap is a mispricing, not compensation.
  2. Take no action, because P's higher return compensates for its idiosyncratic risk.Why not: Idiosyncratic risk is diversified away in well-diversified portfolios and is not priced.
  3. Buy P and sell Q; the buying pressure raises P's price, lowering its expected return, until the gap closes. - correct
  4. Buy P and sell Q; the gap closes as P's price falls and Q's price rises.Why not: Reverses the price effect: buying P raises its price (lowering its expected return) and selling Q lowers its price.

Two portfolios with the same factor betas must have the same expected return, otherwise a zero-investment, zero-factor-risk position in well-diversified portfolios earns an arbitrage profit. Arbitrageurs buy the cheap portfolio and sell the dear one until prices adjust.

The hardest tier

In a single-factor APT world with a risk-free rate of 3%, well-diversified portfolio A has a factor beta of 1.2 and an expected return of 9.0%, and well-diversified portfolio B has a beta of 0.6 and an expected return of 5.4%. An arbitrageur sells USD 100 million of B and invests the proceeds in a mix of A and the risk-free asset that has the same beta as B. The annual riskless profit is closest to:

  1. USD 0.6 million - correct
  2. USD 1.2 millionWhy not: Puts 60% in A (using B's beta as the weight) and 40% in the risk-free asset, which gives a beta of 0.72, not 0.6.
  3. USD 0.9 millionWhy not: Holds 50% in A but forgets the 50% in the risk-free asset, so it understates the replicating return (4.5%).
  4. USD 3.6 millionWhy not: Subtracts B's return from A's return without matching beta; the two portfolios have different risk.

A beta of 0.6 is replicated with 50% A and 50% risk-free: 0.5 x 9.0% + 0.5 x 3% = 6.0%. B earns only 5.4%, so selling B and holding the replica earns 0.6% on USD 100 million = USD 0.6 million with no factor risk. Expected return per unit of beta is 5.0% for A versus 4.0% for B, which shows the mispricing.

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