Arbitrage pricing theory · RiskAtoms Academy

By the end you will be able to

1Explain how the arbitrage pricing theory gets expected returns from the absence of arbitrage alone.
2Price a well-diversified portfolio from its factor exposures and the factor risk premia.
3Build the riskless trade that profits from a mispriced portfolio, and show its profit does not depend on the economy.
4Compare APT with CAPM: what each assumes and when to use which.

Why it matters

CAPM needs a long list of assumptions: every investor holds mean-variance efficient portfolios and agrees on the inputs. In 1976 Stephen Ross asked a smaller question: what must be true of expected returns if nobody can make money for nothing? The answer is a pricing model that needs no market portfolio at all.

Core explanation

No free lunch, one factor at a time

An arbitrage costs nothing, carries no risk and makes money for sure. In a working market such chances vanish almost as soon as they appear, because traders pile in. The law of one price is the simplest case: two things that pay the same must cost the same.

The arbitrage pricing theory (APT) applies that idea to risk. It starts from a factor model like those in L3.1.11: Factor models and Fama-French: each asset's return reacts to a few common factors, plus news of its own. How strongly it reacts is its exposure $b$ to that factor. In a well-diversified portfolio, the own news cancels out (the same diversification as in L3.1.09: CAPM, beta and the security market line), so only factor risk is left.

The key step

Two well-diversified portfolios with the same factor exposures have the same risk. By the law of one price, they must offer the same expected return. Since any exposure can be copied by mixing a few portfolios with the risk-free asset, expected returns must line up on a straight line (one factor) or a flat plane (two factors) in the exposures.

Illustrative dataThree well-diversified portfolios

One factor, expected returns for a year, risk-free rate 3%. Portfolio A has exposure 1.00 and offers 9%; B has 0.50 and offers 6%; C has 0.75 and offers 8%. Made-up numbers.

The risk-free asset and A fix the line. Its slope is the factor risk premium $\lambda = (9\% - 3\%) / 1.0 = 6\%$ . B is on it: $3\% + 0.5 \times 6\% = 6\%$ . C is not. A copy of C, 75% in A and 25% in the risk-free asset, has the same exposure of 0.75 but offers only 7.5%. C pays 0.5% too much for its risk: it is cheap. Buy C, sell the copy, and earn £50,000 a year on £10,000,000 with no money down and no factor risk. As traders do this, C's price rises and its expected return falls back to the line.

ExploreOne factor: every well-diversified portfolio on one line

A and B sit on the line. Move portfolio C off it and see the trade that makes money for nothing.

C's factor exposure b0.75

C's expected return8.0%

Fair return for b7.50%

C offers8.00%

Gap0.50%

Copy C's exposure with 75% in A and 25% in the risk-free asset (a negative figure means borrowing). C is cheap: buy C, sell the copy.

Riskless profit on £10,000,000£50,000

How APT differs from CAPM. CAPM (L3.1.09: CAPM, beta and the security market line) is an equilibrium model: it assumes how investors behave and derives that the market portfolio is efficient. APT assumes much less, only a factor model and no arbitrage, and it does not say what the factors are or how big their premia are. Those come from data.

	CAPM	APT
Starts from	Mean-variance investors in equilibrium	A factor model and no arbitrage
Risk factors	One: the market portfolio	Several, not named by the theory
Needs the market portfolio?	Yes, and it must be efficient	No
Holds for	Every asset	Well-diversified portfolios (most assets, approximately)
Premia	Market risk premium	Estimated from data, one per factor
Testable?	Hard: the true market is unobservable	Only once the factors are chosen

Sort itCAPM, APT or both?

Which model does each statement describe?

Expected return is linear in the asset's exposure to risk.
Only risk that cannot be diversified away earns a premium.
The market portfolio is mean-variance efficient.
Prices follow because traders exploit any free lunch.
The theory itself does not say which risks are priced.

The formal version

The starting point is a factor model for every asset:

The derivation. Take weights $w_1, \dots, w_n$ on $n$ assets, with $n$ large, chosen so that:

\begin{aligned} &\text{(1) it costs nothing:} && \textstyle\sum_i w_i = 0 \\[2pt] &\text{(2) it has no factor risk:} && \textstyle\sum_i w_i\, b_{ik} = 0 \quad \text{for } k = 1, \dots, K \\[2pt] &\text{(3) it is well diversified:} && \textstyle\sum_i w_i\, \varepsilon_i \approx 0 \end{aligned}

Condition (3) works because $\operatorname{var}\big(\sum_i w_i\varepsilon_i\big) = \sum_i w_i^2\sigma_{\varepsilon_i}^2$ , which shrinks towards zero when the weights are spread thinly over many assets. The portfolio's return is then

\begin{aligned} R_w = \sum_i w_i R_i &= \sum_i w_i\,\mathbb{E}[R_i] + \sum_k \Big(\sum_i w_i b_{ik}\Big) F_k + \sum_i w_i \varepsilon_i \\ &= \sum_i w_i\,\mathbb{E}[R_i] + 0 + 0 \end{aligned}

That return is known in advance and the portfolio cost nothing. If it were positive, everyone would buy it in unlimited size; if negative, sell it. So with no arbitrage it must be zero:

\sum_i w_i = 0 \;\text{ and }\; \sum_i w_i\, b_{ik} = 0 \text{ for all } k \quad\Longrightarrow\quad \sum_i w_i\,\mathbb{E}[R_i] = 0

In words: every weight vector at right angles to the vector of ones and to each exposure vector is also at right angles to the vector of expected returns. That can only happen if the expected returns are a combination of those vectors:

\mathbb{E}[R_i] = \lambda_0 + b_{i1}\lambda_1 + b_{i2}\lambda_2 + \dots + b_{iK}\lambda_K

Reading the lambdas. The risk-free asset has no factor exposure, so $\lambda_0 = r_f$ . A factor portfolio $P_k$ with exposure 1 to factor $k$ and 0 to the others earns $\mathbb{E}[R_{P_k}] = r_f + \lambda_k$ , so $\lambda_k = \mathbb{E}[R_{P_k}] - r_f$ : the premium for one unit of factor $k$ risk.

\boxed{\;\mathbb{E}[R_i] = r_f + \sum_{k=1}^{K} b_{ik}\,\lambda_k\;}

CAPM as a special case

With one factor and that factor the market return, the exposure is beta (

b_i = \beta_i

) and the market itself is the factor portfolio, so

\lambda = \mathbb{E}[R_m] - r_f

. The APT equation becomes

\mathbb{E}[R_i] = r_f + \beta_i(\mathbb{E}[R_m] - r_f)

: the security market line of L3.1.09: CAPM, beta and the security market line.

Finding the premia with two factors. Two well-diversified portfolios A and B give two equations in $\lambda_1, \lambda_2$ . Solve them by elimination (Cramer's rule):

\begin{aligned} b_{A1}\lambda_1 + b_{A2}\lambda_2 &= \mathbb{E}[R_A] - r_f \\ b_{B1}\lambda_1 + b_{B2}\lambda_2 &= \mathbb{E}[R_B] - r_f \\[4pt] \lambda_1 &= \frac{(\mathbb{E}[R_A] - r_f)\,b_{B2} - b_{A2}\,(\mathbb{E}[R_B] - r_f)}{b_{A1}b_{B2} - b_{A2}b_{B1}} \\[4pt] \lambda_2 &= \frac{b_{A1}\,(\mathbb{E}[R_B] - r_f) - b_{B1}\,(\mathbb{E}[R_A] - r_f)}{b_{A1}b_{B2} - b_{A2}b_{B1}} \end{aligned}

The same algebra gives the weights of a copy: to match exposures $(b_1, b_2)$ , solve $w_A b_{A1} + w_B b_{B1} = b_1$ and $w_A b_{A2} + w_B b_{B2} = b_2$ , then put $1 - w_A - w_B$ in the risk-free asset.

Try it

Portfolio D is mispriced. Move the two weights until your copy has D's exposures, and watch the trade's P&L in the three economies. While any factor risk is left, the rows differ; once the exposures match, every row shows the same profit. Then change D's expected return to its fair value and see the profit disappear.

Try itBuild the arbitrage

Portfolio D has exposures b₁ = 0.80 (growth) and b₂ = 1.00 (inflation). Mix A and B (the rest goes in the risk-free asset) so your copy has the same exposures.

Weight in A50%

Weight in B50%

D's expected return9.0%

Economy next year	Growth surprise	Inflation surprise	Trade P&L
Boom	2%	1%	+£40,000
As expected	0%	0%	+£50,000
Stagflation	−3%	2%	+£65,000

Your copy

Risk-free weight0%

b₁ (D needs 0.80)0.750

b₂ (D needs 1.00)1.000

Expected return9.50%

Portfolio D

Offers9.00%

APT fair return9.80%

Not yet hedged: the trade still carries −0.05 of growth risk and 0.00 of inflation risk, so its P&L swings with the economy.

Worked example

A two-factor arbitrage

Illustrative dataTwo factors, three portfolios

Factor 1 is the surprise in GDP growth, factor 2 the surprise in inflation. Risk-free rate 3%. All three portfolios are well diversified. Made-up numbers.

Portfolio	b₁ (growth)	b₂ (inflation)	Expected return
A	1.00	0.50	10%
B	0.50	1.50	9%
D	0.80	1.00	9%

Price D and trade it

Step 1 of 6

1
Write the two pricing equations
$\begin{aligned} 1.0\,\lambda_1 + 0.5\,\lambda_2 &= 10\% - 3\% = 7\% \\ 0.5\,\lambda_1 + 1.5\,\lambda_2 &= 9\% - 3\% = 6\% \end{aligned}$

In practice the profit is not quite riskless. Real portfolios keep a little specific risk, exposures are estimated with error, and trading has costs. That is why APT holds only approximately, and why mispricings can persist for a while.

Regulatory lens · looking ahead

No rule applies directly. The no-arbitrage argument used here is the foundation of derivative pricing: a derivative is valued by building a copy of it from traded assets, as in M5.1: Curves and interest rate derivatives and M6.3: Option pricing and volatility. Factor models with several macroeconomic factors also drive the scenarios in bank and insurer stress tests (M4.9: Supervision, stress testing and disclosure).

Spark Atoms

Spark AtomWhich factors?

In 1986 Nai-Fu Chen, Richard Roll and Stephen Ross tested macroeconomic factors as the APT's risks. Surprises in industrial production, in inflation and in the spread between low-grade and high-grade corporate bond yields, and changes in the slope of the yield curve, all appeared to be priced in US shares.

Spark AtomArbitrage needs capital

Andrei Shleifer and Robert Vishny argued in 1997 that real arbitrage is done by specialists using other people's capital. If a mispricing widens before it closes, their investors may withdraw funds, forcing them to sell at a loss. LTCM in 1998 (L2.1.06: Landmark failures: LTCM and 2008) is the textbook case: its trades looked like arbitrage, and still lost billions.

Key takeaways

APT starts from a factor model and assumes only that there is no arbitrage.
For well-diversified portfolios, expected return = risk-free rate + Σ exposure × factor premium.
Each premium λₖ is the excess return of a portfolio with one unit of exposure to factor k and none to the others.
A portfolio off the line can be copied with other portfolios and the risk-free asset; buying the cheap one and selling the dear one locks in a profit with no factor risk.
CAPM is the one-factor case with the market as the factor. APT needs fewer assumptions but does not name the factors.

Reaction

Check what stuck

Question 1 · objective 2

One factor drives returns. The risk-free rate is 4% and the factor premium is 5%. What expected return should a well-diversified portfolio with exposure b = 1.4 offer, in %?

%

Question 2 · objective 3

With a risk-free rate of 3% and a factor premium of 6%, a well-diversified portfolio with b = 0.6 offers 8%. How much riskless profit, in pounds, does the arbitrage make on £5 million in a year?

£

Question 3 · objective 2

Two factors have premia λ₁ = 4% and λ₂ = 1%, and the risk-free rate is 2%. What expected return should a portfolio with exposures (1.2, 0.5) offer, in %?

%

Question 4 · objective 1

Why does the APT argument need well-diversified portfolios?

ABecause only diversified portfolios have factor exposuresBBecause a single asset's own news would make the 'arbitrage' risky; in a diversified portfolio that news cancels outCBecause regulators forbid trading single assetsDBecause diversified portfolios always have a beta of 1

Question 5 · objective 4

Which statement about APT and CAPM is right?

AAPT needs the market portfolio to be mean-variance efficientBCAPM is the special case of APT with one factor, the market, and λ = E[Rm] − rfCAPT says which factors to useDAPT and CAPM can never give the same expected return

Glossary

Arbitrage: A set of trades that costs nothing to set up, carries no risk, and earns a positive amount for certain.
Law of one price: Two things with the same payoffs must have the same price; otherwise an arbitrage exists.
Factor: A common source of surprises that moves many assets at once, such as unexpected GDP growth or inflation.
Factor exposure: How much an asset's return moves for a one-unit surprise in a factor; also called a factor loading or beta.
Factor risk premium: The extra expected return the market pays for one unit of exposure to a factor.
Well-diversified portfolio: A portfolio spread over so many assets that their own news cancels out and only factor risk is left.
Copy (replicating) portfolio: A mix of other portfolios and the risk-free asset with the same factor exposures as the one being priced.

Deep Dives & Origins

Deep Dives

Z. Bodie, A. Kane and A. J. Marcus, Investments, McGraw-Hill, the chapter on arbitrage pricing theory and multifactor models. Intermediate.
J. H. Cochrane, Asset Pricing, revised ed., Princeton University Press, 2005, the chapter on factor pricing models. Advanced.

Origins

S. A. Ross, “The Arbitrage Theory of Capital Asset Pricing”, Journal of Economic Theory 13(3), 1976.
G. Huberman, “A Simple Approach to Arbitrage Pricing Theory”, Journal of Economic Theory 28(1), 1982.
N.-F. Chen, R. Roll and S. A. Ross, “Economic Forces and the Stock Market”, Journal of Business 59(3), 1986.
A. Shleifer and R. W. Vishny, “The Limits of Arbitrage”, Journal of Finance 52(1), 1997.

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