Arbitrage pricing theory needs three things.
The first is a return-generating process. Every asset's return is a linear function of k common factors plus a term unique to that asset: R_i = E(R_i) + b_i1 F_1 + b_i2 F_2 + ... + b_ik F_k + e_i. Each F_j is a factor surprise with a mean of zero, each b_ij is asset i's sensitivity to factor j, and e_i is firm-specific with a mean of zero and no correlation with the factors or with any other asset's e.
Notice that the factors enter as deviations from expectation. Their expected values are already inside E(R_i), so a factor that comes in exactly as forecast moves nobody's return.
The second is that many well-diversified portfolios exist. A well-diversified portfolio holds enough names, in small enough weights, that the firm-specific terms cancel and the portfolio's return is driven by the factors alone. This is an idealisation, and the active-risk arithmetic measures exactly how badly it fails for a concentrated book.
The third is the no-arbitrage condition. If a combination of assets can be assembled that costs nothing, carries no factor exposure and has a positive expected return, investors will take it in unlimited size, and the prices they push around will destroy it. Since capital is not literally unlimited, the honest form of the assumption is weaker: enough capital pursues such opportunities that they are small and short-lived.
From those three, the pricing relation follows: for any well-diversified portfolio p,
E(R_p) = R_f + b_p1 λ_1 + b_p2 λ_2 + ... + b_pk λ_k
where λ_j is the risk premium for a unit of exposure to factor j, the same for every asset. The proof is the arbitrage argument itself, worked below on three portfolios and two factors.
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What does arbitrage pricing theory not assume?
Now list what the derivation never used. It did not need investors to be mean-variance optimisers. It did not need returns to be normally distributed. It did not need a single-period horizon or a quadratic utility function. Above all it did not need the market portfolio, which means the whole of Roll's critique passes it by.
Against that, it gives up a great deal. It does not tell you how many factors there are, what they are, or what any λ should be. Every one of those has to be supplied from outside, and that supply is the entire practical content of factor modelling.
Read the theory as a constraint rather than a forecast. It says that once you have named your factors, the expected returns of everything exposed to them must be mutually consistent. It says nothing about whether you named the right ones.
An arbitrage constructed, and then closed by prices
The argument is easiest to believe when you build the trade. Take a risk-free rate of 4.00% and two factors, a surprise in industrial production and a surprise in inflation. Three well-diversified portfolios are on offer:
| Portfolio | Exposure to factor 1 | Exposure to factor 2 | Expected return |
|---|---|---|---|
| A | 1.0 | 0.5 | 11.0% |
| B | 2.0 | 1.5 | 17.0% |
| C | 1.5 | 1.0 | 13.0% |
Two portfolios and two unknowns pin down the plane. From A, 1.0 λ_1 + 0.5 λ_2 = 7.0; from B, 2.0 λ_1 + 1.5 λ_2 = 13.0. Solving gives λ_1 = 8.00% and λ_2 = -2.00%. Inflation surprise carries a negative premium, which is exactly what you should expect: an asset that does well when inflation surprises upward is insurance, and insurance is expensive, so it earns less.
Price C off that plane and it should return 4.00 + 1.5(8.00) + 1.0(-2.00) = 14.00%. It is offering 13.00%. One full point of expected return is missing, and that is enough.
Build the replication. Find weights in A and B that reproduce C's exposures: 1.0 w_A + 2.0 w_B = 1.5 and 0.5 w_A + 1.5 w_B = 1.0 give w_A = 0.50 and w_B = 0.50, with nothing left for the risk-free asset. That half-and-half combination has C's exposures exactly and an expected return of 0.5(11.0) + 0.5(17.0) = 14.00%. Sell short $1,000,000 of C and buy $500,000 of A and $500,000 of B. Net investment is zero and net factor exposure is zero on both factors. Expected profit is (14.00 - 13.00)% × $1,000,000 = $10,000, and because the exposures cancel and the portfolios are well diversified, that profit does not depend on what the factors do.
Then watch it close. Suppose C is a claim on an expected terminal value of $113.00 that currently trades at $100.00, which is where the 13.00% came from. For C to offer 14.00%, its price must fall to 113.00 / 1.14 = $99.1228, a drop of 0.8772%. Less than one percent of price movement erases an arbitrage worth $10,000 per million, which is why these gaps do not sit around waiting.
Three cautions before you trust the trade. The profit is riskless only if the firm-specific terms really have vanished, so a "well-diversified portfolio" that is eleven names is not one. The exposures have to be measured, and measurement error in b shows up as residual factor risk in a trade you believed was hedged. And short selling has to be available at the assumed cost, which in India's single-stock market is a real constraint rather than a footnote.