Arbitrage Pricing Theory

CAPM says expected return depends on exactly one thing, sensitivity to the market portfolio, and it gets there by assuming every investor holds a mean-variance optimal portfolio, a strong behavioral assumption that is hard to defend and even harder to test, because nobody can observe "the market portfolio" of literally everything. Arbitrage Pricing Theory (APT) asks a much weaker question: forget what investors actually do, what must expected returns look like just so that nobody can build a portfolio that costs nothing, carries no risk, and makes free money? That question alone, with almost no assumptions about investor behavior, turns out to pin down a pricing relationship, and it allows more than one factor to matter.

The vending machine that's priced wrong

Imagine two vending machines in the same building sell the identical candy bar, one for $1.00 and one for $1.20. This has nothing to do with anyone's risk preferences or beliefs about candy; it is simply a mispricing that anyone can exploit by buying low and selling high, risklessly and repeatedly, until the prices converge. APT applies the same logic to portfolios of assets instead of candy bars: if you can build two portfolios with identical exposure to every source of systematic risk but different expected returns, that is a mispricing, an arbitrage, and rational capital will pile in and erase it. The only expected-return relationships that can survive are ones where no such riskless, costless combination exists. Notice this argument needs nothing about how investors feel about risk, no assumption that everyone holds the market, just the discipline that a genuine free lunch cannot persist.

From no-arbitrage to a pricing equation

Suppose returns are driven by $k$ common factors plus an asset-specific term:

$r_i = E[r_i] + \beta_{i1} f_1 + \beta_{i2} f_2 + \cdots + \beta_{ik} f_k + \epsilon_i.$

In words: an asset's realized return equals what you expected, plus its sensitivity to each of $k$ shared risk factors ($\beta_{ij}$, how much asset $i$ moves per unit move in factor $j$) times how that factor actually moved, plus a leftover, asset-specific wiggle $\epsilon_i$ that is (by construction) uncorrelated across enough different assets to diversify away in a large portfolio.

The no-arbitrage argument works like this: build a large, well-diversified portfolio, so the $\epsilon_i$ terms cancel out (diversification kills idiosyncratic risk, it does nothing to factor risk). Now you have a portfolio whose return is driven only by its factor exposures. If two such diversified portfolios have the same factor exposures but different expected returns, you can go long the higher-returning one and short the lower-returning one, funded at zero net cost, and end up with a riskless positive expected return, a genuine arbitrage. Ruling that out forces expected returns to be a linear function of factor exposures alone:

$E[r_i] = r_f + \beta_{i1}\lambda_1 + \beta_{i2}\lambda_2 + \cdots + \beta_{ik}\lambda_k.$

In words: an asset's expected return above the risk-free rate is the sum, over every risk factor, of that asset's sensitivity to the factor times the factor risk premium $\lambda_j$, the extra expected return the market pays per unit of exposure to that factor. Unlike CAPM, APT does not tell you what the factors are or how many there are, only that if returns really are driven by $k$ common factors, expected returns must be linear in the exposures to those factors, no ifs.

Worked example: pricing with two factors

Suppose the market has settled into two priced factors: a market factor with risk premium $\lambda_1 = 6\%$ and an inflation-surprise factor with risk premium $\lambda_2 = -2\%$ (assets that do badly when inflation surprises to the upside demand extra compensation, so their premium contribution is negative when they have positive inflation beta, i.e., they benefit from inflation and thus need less compensation — here take the sign as given by the estimated premium). Risk-free rate is 3%.

Stock M has $\beta_1 = 1.2$, $\beta_2 = 0.5$: $E[r_M] = 3\% + 1.2(6\%) + 0.5(-2\%) = 3 + 7.2 - 1.0 = 9.2\%$.

Stock N has $\beta_1 = 0.8$, $\beta_2 = -1.5$: $E[r_N] = 3\% + 0.8(6\%) + (-1.5)(-2\%) = 3 + 4.8 + 3.0 = 10.8\%$.

N has lower market exposure than M but a higher required return, because its negative inflation beta (it does badly precisely when inflation surprises upward) means investors demand extra compensation for that specific risk, a distinction CAPM's single factor cannot express at all.

Worked example: catching a mispricing

A third stock, P, has the same factor exposures as M ($\beta_1 = 1.2$, $\beta_2 = 0.5$) but trades at a price implying an expected return of only 7.5%, versus the 9.2% APT says it should offer. An arbitrageur shorts M (funding the trade) and goes long P... wait, P is underpriced relative to its risk (offers too little return for its risk, meaning its price is too high), so the correct trade is short P, long M: identical factor exposure, and M pays 1.7 percentage points more for the same risk. Scaled to a $50m position, that gap is worth capturing risklessly (ignoring transaction costs and estimation error) until buying pressure on M and selling pressure on P closes the 1.7% gap.

What this means in practice

• APT is the theoretical justification for multi-factor risk and return models, including Barra-style and statistical factor models, which is why factor investing can claim a rigorous no-arbitrage pedigree rather than being pure data mining.
• It does not specify the factors. Real-world implementation (macro factors, statistical factors extracted via PCA, or fundamental style factors like value and momentum) is an empirical choice APT is silent on; see Fama-MacBeth Regression for how factor premia $\lambda_j$ are actually estimated from data.
• Diversification is doing the real work. The pricing relationship only holds exactly for well-diversified portfolios; individual, concentrated stocks can deviate from it without creating an arbitrage, because you cannot diversify away a single stock's idiosyncratic risk by holding just that one stock.

Practice

1. Stock Q has $\beta_1 = 1.0$, $\beta_2 = 0$ under the same two-factor premia above. What is its APT expected return, and why does it match a single-factor CAPM answer exactly?
2. Explain in your own words why diversification is necessary for the no-arbitrage argument to work, using the $\epsilon_i$ term in the factor model.
3. If a five-factor statistical model and a three-factor fundamental model both satisfy APT's linear pricing relationship on the same data, what does that tell you, and what doesn't it tell you, about which model to actually use?