The CAPM and APT: Beta, Alpha, Factor Risk, and No-Arbitrage Pricing

Two investments can have the same volatility and still require different expected returns. The important distinction is which risks remain when investors combine assets into portfolios. CAPM prices exposure to the market portfolio; APT relates expected returns to exposures to several common factors under a no-arbitrage framework.

Learning goals and conventions

You should finish able to distinguish the CML from the SML, calculate beta and alpha, choose a project discount rate, separate a factor regression from a pricing restriction, and construct a factor-matched portfolio to understand APT.

Use simple total returns, a common holding period, and consistent currency units. Annual examples use annual inputs; regression examples explicitly labeled monthly use monthly inputs. Expected returns are forecasts, not promised outcomes. Unless stated otherwise, the risk-free return is known at the start of the modeled period.

Symbol	Meaning
$R_i$, $R_M$	Asset and market returns
$r_f$	Risk-free return for the same period
$\mu_i$, $\mu_M$	Expected asset and market returns
$\sigma_i$, $\sigma_M$	Return standard deviations
$\beta_i$	Market exposure in CAPM
$b_{ik}$	Asset $i$‘s loading on factor $k$
$\lambda_k$	Expected compensation per unit of factor exposure
$\alpha_i$	Intercept or pricing error, depending on the specified model

1. CAPM begins with an efficient market portfolio

In the standard CAPM, investors share mean–variance beliefs, have a common horizon and opportunity set, trade frictionlessly, and can borrow and lend at a common risk-free rate. Market clearing identifies their common tangency portfolio with the market-value-weighted portfolio of risky assets.

Let fraction $x\ge0$ of wealth be invested in the market and the balance in the risk-free asset:

$$\mu_P=r_f+x(\mu_M-r_f),\qquad \sigma_P=x\sigma_M.$$

Eliminating $x$ gives the capital market line (CML):

$$\mu_P=r_f+\frac{\sigma_P}{\sigma_M}(\mu_M-r_f).$$

It applies to efficient positive-market-exposure combinations in this idealized model. It does not assign a required return to an arbitrary individual stock using that stock’s total volatility.

Worked CML example

Suppose $r_f=3\%$, market expected return is 9%, and market volatility is 15%. An efficient portfolio with volatility 10% has $x=10/15=2/3$ and expected return

$$3\%+\frac{10\%}{15\%}(6\%)=7\%.$$

A stock with 10% volatility need not have a 7% required return. Its covariance with the market, rather than volatility alone, determines its CAPM return.

2. Beta and the security market line

Market beta is

$$\beta_i=\frac{\operatorname{Cov}(R_i,R_M)}{\operatorname{Var}(R_M)} =\rho_{iM}\frac{\sigma_i}{\sigma_M}.$$

CAPM states

$$\boxed{\mu_i=r_f+\beta_i(\mu_M-r_f)}.$$

This security market line (SML) applies within CAPM equilibrium to individual securities and arbitrary portfolios, including inefficient portfolios. It relates expected return to market covariance exposure. Total volatility still matters for an investor’s overall risk; it is simply not the sole priced risk measure in this model.

Worked example: high volatility, low beta

A stock has volatility 30%, market correlation 0.25, and market volatility 15%. Then

$$\beta_i=0.25\frac{0.30}{0.15}=0.5.$$

With $r_f=3\%$ and market premium 6%, required return is 6%. Much of its standalone uncertainty need not be market exposure.

Beta	CAPM expected return at 3% risk-free and 6% market premium
-0.5	0%
0	3%
0.5	6%
1	9%
1.5	12%

A negative-beta asset may be valuable as a hedge and have expected return below the risk-free rate. A beta-zero risky asset is not risk-free: it can retain substantial idiosyncratic volatility. The risk-free asset has beta zero, but the converse is false.

Portfolio beta is linear

For fixed wealth weights summing to one:

$$\beta_P=\sum_iw_i\beta_i.$$

A portfolio with 40% in beta-0.5 equity, 50% in beta-1.4 equity, and 10% in the risk-free asset has beta $0.4(0.5)+0.5(1.4)=0.9$. CAPM expected return is $3\%+0.9(6\%)=8.4\%$.

Relationship	Risk axis	Scope
CML	Total standard deviation	Efficient market/risk-free combinations
SML	Market beta	All assets and portfolios under CAPM equilibrium

3. Estimating beta from excess returns

Estimate the time-series regression

$$R_{i,t}-R_{f,t}=\alpha_i+\beta_i(R_{M,t}-R_{f,t})+\varepsilon_{i,t}.$$

The OLS slope with an intercept is

$$\widehat\beta_i= \frac{\widehat{\operatorname{Cov}}(R_i-R_f,R_M-R_f)} {\widehat{\operatorname{Var}}(R_M-R_f)}.$$

Use excess returns consistently when historical risk-free returns vary. The market index is an observed proxy for the theoretical market portfolio. Currency, total-return construction, sampling frequency, and observation dates must match.

Worked estimation example

Suppose monthly market excess-return variance is 0.0025 and covariance with the asset’s excess return is 0.0030:

$$\widehat\beta=0.0030/0.0025=1.2.$$

If average monthly market excess return is 0.50% and average asset excess return is 0.75%, the intercept is

$$\widehat\alpha=0.75\%-1.2(0.50\%)=0.15\%\text{ per month}.$$

Beta 1.2 does not mean the stock will rise 1.2% every time the index rises 1%. It describes estimated linear sensitivity, with residual variation around that relation. Betas can change with leverage and business conditions, and estimates have sampling uncertainty.

4. Jensen’s alpha and manager comparison

In the CAPM population relationship, alpha is zero relative to the true market benchmark. In a fitted historical regression, estimated alpha measures average excess return unexplained by the chosen market exposure.

Consider two managers with the same annual average return of 10%, with risk-free return 2% and average market excess return 6%:

Manager	Beta	CAPM benchmark return	Average return minus benchmark
A	1.0	8.0%	+2.0 percentage points
B	1.4	10.4%	-0.4 percentage points

Identical average returns do not imply identical risk-adjusted performance. But an alpha estimate is benchmark-dependent evidence, not automatic proof of skill. Missing factors, fees, stale pricing, survivorship bias, and chance can affect it.

If the monthly alpha estimate above is 0.15% with standard error 0.12%, its $t$-statistic is 1.25. It is positive but not strong conventional evidence against zero. Serial correlation and heteroskedasticity may require robust inference. Always state the period and whether returns are gross or net of fees; multiplying a monthly alpha by 12 is an arithmetic annualization convention, not an exact compounded alpha calculation.

5. CAPM as an opportunity cost of capital

The required return for exposure with beta $\beta$ is

$$r=r_f+\beta\,\pi_M,\qquad \pi_M=\mathbb E[R_M]-r_f.$$

The beta must match the cash-flow claim. Project asset risk, levered shareholder risk, and the parent company’s historical equity beta are not interchangeable.

Worked unlevered project

A project costs 100 today and pays expected 60 at each of the next two year-ends. Assume unlevered cash flows, constant project asset beta 0.8, risk-free rate 3%, and annual market premium 5%, with financing effects excluded.

$$r_A=0.03+0.8(0.05)=7\%,$$ $$NPV=-100+\frac{60}{1.07}+\frac{60}{1.07^2}\approx8.48.$$

It adds value under these assumptions. A positive model NPV does not guarantee favorable realized cash flows. Different project risks should lead to different opportunity costs rather than one indiscriminate company-wide hurdle rate.

Use cost of equity for cash flows to shareholders; use an appropriate WACC for free cash flow to the firm under consistent financing assumptions, or value financing separately. Keep nominal cash flows with nominal rates and currency and maturity conventions aligned. Multiperiod use of one constant beta and rate is itself an assumption.

6. What historical evidence does and does not establish

Some empirical studies find a beta–average-return relation flatter than the standard SML predicts; results depend on the sample, benchmark, estimation method, and portfolio construction. Size sorts investigate return differences associated with market capitalization; they are not the same experiment as beta sorts.

Fama and French (1992) report that size and book-to-market help describe cross-sectional average returns in their studied sample, while market beta alone performs poorly. This is historical evidence about a specified sample and model, not a guarantee that any value strategy will outperform in every period.

The other references in the supplied outline require narrower interpretation:

• Chordia, Roll, and Subrahmanyam (2000), Commonality in Liquidity, documents common movements in stock liquidity. Common liquidity variation motivates systematic-liquidity-risk research; by itself it does not establish that every illiquidity measure carries a particular return premium.
• Lo and Wang (2006), Trading Volume: Implications of an Intertemporal Capital Asset Pricing Model, develops a model of volume and returns with heterogeneous investors. It should not be reduced to a claim that high trading volume always predicts high returns or is automatically a priced APT factor.

A variable can explain realized return variation without commanding an expected-return premium. Conversely, identifying a characteristic associated with average returns is not yet identifying a common risk shock.

Testing CAPM also jointly tests the chosen market proxy and measurement assumptions. A rejected regression restriction does not uniquely identify why the model failed. Its disciplined benchmark can remain useful even when its empirical restrictions are incomplete.

7. APT: separate the return-generating model from pricing

Stephen Ross’s 1976 APT paper develops an alternative route to expected-return restrictions using factor structure, diversification, and absence of arbitrage.

Start with a centered-shock representation:

$$R_i=\mu_i+\sum_{k=1}^{K}b_{ik}Z_k+\varepsilon_i,$$

where $\mathbb E[Z_k]=0$ and $\mathbb E[\varepsilon_i]=0$. The $Z_k$ are unexpected common shocks. The coefficients $b_{ik}$ describe exposure to those shocks. Residuals must satisfy suitable conditions so broadly diversified portfolios can make residual risk small.

APT’s pricing conclusion is approximately

$$\boxed{\mu_i-r_f=\sum_{k=1}^{K}b_{ik}\lambda_k}$$

for well-diversified portfolios in the standard setting, with qualifications for residual pricing errors and the limiting argument. Here $\lambda_k$ is the expected premium for one unit of exposure to factor $k$, not the realized shock itself.

Why factor units matter

If a macro shock is measured in percentage points of unexpected inflation, its loading and premium must be defined in compatible units. Multiplying a factor’s numerical scale by 100 divides its loading by 100; the product representing expected compensation must remain unchanged.

APT does not require all factors to be directly observed macroeconomic quantities. Economic shocks, statistical factors, and traded factor-mimicking portfolios are possible constructions, with different interpretation and identification challenges.

8. A regression is not automatically an APT model

For traded excess-return factors $F_{k,t}$, practitioners often estimate

$$R_{i,t}-R_{f,t}=\alpha_i+\sum_kb_{ik}F_{k,t}+\varepsilon_{i,t}.$$

If those factors span the relevant priced risks, then the pricing restriction is $\alpha_i=0$ and $\lambda_k=\mathbb E[F_k]$. This gives

$$\mu_i-r_f=\sum_kb_{ik}\mathbb E[F_k].$$

Do not silently transfer this interpretation to centered macro shocks: centered shocks have zero mean, but their risk premia need not be zero. For nontraded factors, the premia generally require a separate cross-sectional estimation or a consistent factor-mimicking construction.

Quantity	Question it answers
Realized factor shock or return	What happened this period?
Factor loading	How exposed is this asset to that variation?
Factor risk premium	What expected compensation is associated with a unit of exposure?
Regression residual	What variation is not explained by the included factors?
Pricing alpha	What expected return remains unexplained by the pricing restrictions?

A high regression $R^2$ is evidence about time-series fit, not proof that the model correctly prices expected returns.

9. Worked two-factor required returns

Assume annual risk-free return 3%, market factor premium 5%, and a second systematic factor premium 2%. Treat the factors as consistently normalized traded excess-return factors for this example.

Asset	Market loading	Second-factor loading	Required excess return	Required total return
A	1.2	0.5	$1.2(5\%)+0.5(2\%)=7\%$	10%
B	0.8	-0.4	$0.8(5\%)-0.4(2\%)=3.2\%$	6.2%
C	1.0	0	5%	8%

A 50/50 portfolio of A and B has loadings 1.0 and 0.05. Its required excess return is $1.0(5\%)+0.05(2\%)=5.1\%$, so required return is 8.1%. The same result follows by averaging 10% and 6.2%.

Negative loading on a positively priced factor reduces required return because the exposure can hedge that factor’s adverse risks. A factor’s sign is a convention: reversing its sign reverses both loadings and its premium, leaving the products unchanged.

10. The no-arbitrage intuition: match factors and financing

Suppose two diversified portfolios X and Y have identical factor exposures, negligible residual risk, and the same initial cost. If X had a higher expected return, buying X and shorting Y would cancel the common shocks while leaving a positive payoff difference in the exact idealization.

Worked factor-matching example

Assume two pure factor portfolios have loading vectors $(1,0)$ and $(0,1)$ and required total returns 8% and 5%, with risk-free return 3%. Their factor premia are therefore 5% and 2%.

A portfolio X with loadings $(0.6,0.4)$ should have expected return

$$3\%+0.6(5\%)+0.4(2\%)=6.8\%.$$

A 60/40 mix of the two factor portfolios has the same exposures and return 6.8%. If X were instead priced to yield 8.8%, and residual risk were exactly absent, long X and short that replicating mix would have zero initial net investment, zero factor exposure, and a certain 2% payoff per unit long notional in this one-period model.

If residual risk is merely small, this finite trade is not a literal riskless arbitrage. APT uses conditions on a large asset universe and diversification to constrain persistent pricing discrepancies through an asymptotic no-arbitrage argument. Estimated expected-return differences can also be wrong, and real trades face costs and financing constraints.

Matching factor loadings is therefore a theoretical pricing argument, not a promise that a pair of similar regressions offers free money.

11. CAPM versus APT: what changes?

Feature	CAPM	APT
Main route	Mean–variance market equilibrium	Factor structure, diversification, and no arbitrage
Priced exposure	Covariance with the market	Loadings on common factors
Factor identification	The theoretical market portfolio	Not uniquely specified by the theory
Core difficulty	Market proxy, expectations, equilibrium assumptions	Factor choice, residual diversification, premium estimation
Pricing statement	All assets in the model’s equilibrium	Approximate restrictions with standard diversification qualifications

APT does not need CAPM’s particular utility and market-portfolio assumptions, but it is not assumption-free. A factor structure, adequate diversification, sensible risk conditions, and no-arbitrage restrictions do the work. It is not simply “CAPM with extra regressors,” nor was it derived just by fixing one historical CAPM test.

Its linear relation is linear in factor exposures. It does not require the factors to be uncorrelated, or imply that prices, payoffs, and every economic relationship are linear. Nonlinear instruments may require nonlinear factors or local approximations; a poorly specified linear model can omit important risks.

APT does not choose the factors or guarantee that each candidate factor is priced. Identification, economic reasoning, out-of-sample evidence, transaction costs, and stability matter more than trying many variables until a fitted alpha disappears.

12. A practical implementation checklist

1. Specify the cash-flow claim, investment horizon, currency, and return convention.
2. Choose a market proxy or factor definitions and state whether factors are traded returns or centered shocks.
3. Estimate loadings using information available at the time; inspect uncertainty and stability.
4. Estimate expected premiums separately from realized shocks and check factor scaling.
5. Evaluate pricing errors, not just time-series fit, with appropriate statistical inference.
6. Test sensitivity to factors, sample periods, costs, and financing; compare with simple benchmarks.

A pricing model organizes required-return assumptions. It does not remove the need to forecast cash flows or assess the uncertainty of its inputs.

13. Practice questions

1. With risk-free return 2%, market expected return 8%, and market volatility 12%, find the expected return of an efficient portfolio with volatility 9%.
2. A stock has volatility 24%, correlation 0.3 with the market, and market volatility 12%. Find beta and its CAPM return using question 1’s return inputs.
3. A portfolio invests 30% in beta-0.6 equity, 50% in beta-1.5 equity, and 20% risk-free. Find beta and required return under the same CAPM inputs.
4. Average monthly fund excess return is 0.9%, market excess return averages 0.6%, and estimated beta is 1.1. Find alpha. If its standard error is 0.15%, find the $t$-statistic.
5. A project costs 100 and pays expected 114 in one year. Project asset beta is 1.0, risk-free rate is 3%, and market premium is 5%. Ignoring financing and tax effects, find NPV.
6. In a two-factor model, risk-free return is 2%, premiums are 4% and 3%, and an asset’s loadings are 0.5 and 1.2. Find required total return.
7. Two equal-cost, exactly factor-replicated portfolios have identical loadings and no residual risk, but their expected returns differ by 1%. Explain the theoretical trade. What changes if residual risk is nonzero?
8. Why can a beta-zero stock be risky? Why does a macro shock with mean zero not imply its risk premium is zero? Does adding factors until $R^2$ is high validate APT?

14. Answers and interpretation

1. $2\%+(9/12)(6\%)=6.5\%$. This uses the efficient CML, not a standalone-stock volatility rule.
2. Beta is $0.3(24/12)=0.6$. Required return is $2\%+0.6(6\%)=5.6\%$.
3. Beta is $0.3(0.6)+0.5(1.5)=0.93$. Required return is $2\%+0.93(6\%)=7.58\%$.
4. Alpha is $0.9\%-1.1(0.6\%)=0.24\%$ per month. Its $t$-statistic is 1.6, which is not strong conventional evidence of a nonzero population alpha by itself.
5. Required return is 8%. NPV is $-100+114/1.08\approx5.56$.
6. Required excess return is $0.5(4\%)+1.2(3\%)=5.6\%$, so required total return is 7.6%.
7. Buy the higher-return portfolio and short the lower-return one in matched notionals. With exact cancellation, zero residual risk, and feasible costless trading, the payoff difference is certain. Nonzero residual risk makes a finite position risky; diversification and asymptotic conditions must be analyzed instead of calling it an exact arbitrage.
8. Beta zero removes market covariance, not all return dispersion. A shock’s zero expectation describes its surprise component; its premium describes expected compensation for bearing exposure. High $R^2$ measures fitted variation and does not establish the cross-sectional expected-return restrictions.

Further reading

• Andrew W. Lo, MIT OpenCourseWare, The CAPM and APT: the course lectures and slides.
• Stephen A. Ross (1976), The Arbitrage Theory of Capital Asset Pricing, Journal of Economic Theory 13, 341–360; an academic-hosted copy provides the original argument.
• Eugene F. Fama and Kenneth R. French (1992), The Cross-Section of Expected Stock Returns, Journal of Finance 47, 427–465.
• Tarun Chordia, Richard Roll, and Avanidhar Subrahmanyam (2000), Commonality in Liquidity, Journal of Financial Economics 56, 3–28.
• Andrew W. Lo and Jiang Wang (2006), Trading Volume: Implications of an Intertemporal Capital Asset Pricing Model, Journal of Finance 61, 2805–2840.

Feynman study lab

Explain it simply

CAPM links expected return to exposure to market risk; APT allows multiple priced sources of risk.

Concrete example. With risk-free return 2%, market premium 5% and beta 1.2, CAPM expected return is 8%.

Teach it back without looking

Close the lesson and explain the idea aloud to a beginner in two minutes. Define each technical term you use, walk through the example, and state one assumption or limitation. Then answer:

Does a beta of 1.2 mean the asset is 20% more volatile than the market?

Check your explanation after you try

No. Beta measures covariance with the market relative to market variance. Total volatility includes asset-specific risk too.

Find the gap and try again

If you hesitated, relied on a formula you could not explain, or missed the self-check, write down the exact gap. Revisit the relevant section of this lesson and work through its example. Close the notes, explain it again in simpler language, then change one number or assumption and predict what changes. Recalculate or check the lesson to test your prediction. Try once more tomorrow from memory.