9. Factor Pricing Models

A factor pricing model replaces the consumption model's expression for marginal utility growth with a linear model, seeking variables \(f\) for which the following is a sensible approximation:

The essence of asset pricing: there are special "bad states" in which investors especially want their portfolios not to do badly, and will trade off average return to ensure it. The factors are variables indicating these bad states have occurred. Any variable that forecasts returns ("shifts in the investment opportunity set") or forecasts macro variables is a candidate (term premium, dividend/price ratio, stock returns). Should factors be unpredictable? Roughly yes: with a constant real interest rate, marginal utility growth should be unpredictable ("consumption is a random walk" in the quadratic-utility permanent-income model). The real rate is not constant but varies little, so factors proxying for marginal utility need not be totally unpredictable but should not be highly predictable (else the model counterfactually predicts large interest-rate variation). The practical lesson: choose the right units — GNP growth not level, portfolio returns not prices or price/dividend ratios.

9.1 Capital Asset Pricing Model (CAPM)

The CAPM (Sharpe 1964, Lintner 1965) ties \(m\) to the wealth-portfolio return \(R^W\): \(m_{t+1}=a+bR^W_{t+1}\), equivalent to \(E(R^i)=\gamma+\beta_{i,R^W}[E(R^W)-\gamma]\). \(a,b\) are pinned down by pricing any two assets (e.g. \(R^W\) and a risk-free rate). Four classic derivations follow — each must find "which factor proxies for marginal utility" and "linearity between \(m\) and the factor."

① Two-period quadratic utility. Two-period investors with no labor income and quadratic preferences imply the CAPM. Quadratic utility \(U=-\tfrac12(c^*-c_t)^2-\tfrac12\beta E(c^*-c_{t+1})^2\) makes marginal utility linear in consumption (the linearity goal); the two-period assumption sets second-period consumption = wealth, letting us substitute the wealth return for consumption (the factor-naming goal):

② Exponential utility + normal distributions. \(u(c)=-e^{-\alpha c}\) (\(\alpha\) = coefficient of absolute risk aversion) plus normal returns also delivers the CAPM, and its linear demand curves are widely used in incomplete-markets / asymmetric-information models (Grossman-Stiglitz 1980). Under normality \(Eu(c)=-e^{-\alpha E(c)+(\alpha^2/2)\sigma^2(c)}\); maximizing over the amount \(y\) in risky assets gives \(y=\Sigma^{-1}(E(R)-R^f)/\alpha\) (independent of wealth — hence absolute not relative risk aversion). Inverting the first-order condition:

This version is especially interesting because it ties the market price of risk directly to the risk-aversion coefficient.

③ Quadratic value function and dynamic programming. The two-period assumption is unpalatable — but we can let investors live forever if the environment is i.i.d. over time, making the value function quadratic in place of second-period quadratic utility (Fama 1970). With an objective over current consumption and next period's wealth \(U=u(c_t)+\beta E_tV(W_{t+1})\), the first-order condition gives a discount factor using the marginal value of wealth in place of marginal utility of consumption: \(m_{t+1}=\beta V'(W_{t+1})/u'(c_t)\). If the value function is quadratic \(V(W)=-\tfrac{\eta}{2}(W-W^*)^2\), then \(V'\) is linear and again \(m_{t+1}=a_t+b_tR^W_{t+1}\). The value function turns an infinite-horizon problem into a two-period one:

The key fact: in this environment, quadratic utility yields a quadratic value function (found by the functional-equation method: guess quadratic, solve the two-period problem, verify it stays quadratic). But "in this environment" is not innocuous — it relies on constant interest rates, i.i.d. returns, no risky labor income, superficially plausible but in fact almost surely false. The two assumptions each do a job: ① the value function depends only on wealth ⟹ names the factor (other variables entering it would enter \(m\) — the door to the ICAPM); ② the value function is quadratic ⟹ globally linear marginal value ⟹ \(m\) linear in the factor.

④ Log utility. \(u(c)=\ln c\) is a more plausible special case. Define the wealth portfolio as a claim to all future consumption; under log utility its price is proportional to consumption itself, \(p^W_t=\tfrac{\beta}{1-\beta}c_t\), so the wealth-portfolio return is proportional to consumption growth and the discount factor equals the inverse of the wealth-portfolio return:

Log utility is the only assumption here — no constant interest rates, i.i.d. returns, or absence of labor income. Its special property is that "income effects offset substitution effects" (in asset terms, "discount-rate effects offset cashflow effects").

Linearizing any model. The log-utility CAPM got the right variable (\(R^W\)) but a nonlinear form. Three standard tricks linearize \(m=g(f)\) into \(m=a+bf\): (i) Taylor expansion (around the conditional mean); (ii) continuous time (diffusions are locally normal, exact linearization over short intervals); (iii) normal distributions + Stein's lemma: if \(f,R\) are bivariate normal, \(g\) differentiable with \(E|g'(f)|<\infty\), then

Stein's lemma lets us swap a covariance with \(g(f)\) for a covariance with \(f\), exactly yielding a discrete-time linear \(m\) and \(E(R^i)=R^f+\beta_{i,f}\lambda_f\). A warning: Stein's lemma cannot be applied to the log-utility CAPM, because \(R^W\) cannot be normally distributed (\(E[1/R^{W2}]\) does not exist; under log utility consumption/wealth cannot be non-positive). This warns us that the continuous-time approximation (9.15) must not be applied over long horizons or to discount a dividend stream — \(a-bR^W\) is a worse and worse approximation to \(1/R^W\) (the former can go negative, the latter cannot). Rubinstein's (1976) point was actually to advocate the nonlinear \(m=1/R^W\) for arbitrage-free multiperiod discounting.

9.2 Intertemporal Capital Asset Pricing Model (ICAPM)

Substitute out consumption via \(c_t=g(z_t)\) or the value function \(V(W_{t+1},z_{t+1})\): \(m_{t+1}=\beta V_W(W_{t+1},z_{t+1})/V_W(W_t,z_t)\). Then linearize (Taylor / Stein / continuous time). In continuous time, defining the coefficient of relative risk aversion \(\mathrm{rra}\equiv-WV_{WW}/V_W\), substituting into the basic pricing equation gives the ICAPM:

One can substitute covariance with the wealth portfolio for covariance with wealth, and use factor-mimicking portfolios for the other factors. The essence of Merton's portfolio theory is proving that the value function depends on \(W\) and the state variables \(z\) for future investment opportunities, and that the optimal portfolio holds the market plus hedge portfolios against shifts in opportunities. Note: the log-utility CAPM holds even with time-varying investment opportunities — so the ICAPM only works if the utility curvature parameter is not equal to one (non-log).

9.3 Comments on the CAPM and ICAPM

Conditional or unconditional? The two-period quadratic-utility derivation gives a conditional CAPM (\(a_t,b_t\) time-varying); its investor holds a conditional-frontier portfolio, not necessarily on the unconditional frontier. The multiperiod quadratic-utility CAPM holds only if returns are i.i.d. (then conditional = unconditional). The log-utility CAPM in inverse form \(1=\mathbb E_t(R_{t+1}/R^W_{t+1})\) holds both conditionally and unconditionally (no free parameters) but makes additional, quickly-rejected predictions.

Should it price options? "The CAPM is not designed to price derivatives" depends on the derivation: the quadratic- and log-utility CAPMs should price all payoffs (Rubinstein 1976 derives Black-Scholes from the log CAPM); but if you assume normality to get a linear CAPM, you cannot price (non-normal) options.

Why linearize? The tricks arose when nonlinear models were hard to estimate; now GMM makes them easy, so linearization matters less — if the nonlinear model has important predictions, don't lose them for linearity.

The wealth portfolio? The log derivation shows how expansive "the wealth portfolio" is: to own a share of the consumption stream you must own all stocks, bonds, real estate, private/public capital, and human capital. So the value-weighted NYSE and similar proxies are a poor defense of the CAPM.

Identity of state variables. The ICAPM does not tell us what \(z_t\) is, and many use it as a "fishing license" (Fama 1991). But it is not so permissive: one could insist the factor-mimicking portfolios really are projections of identifiable state variables onto returns, and check that "investment-opportunity state variables actually do forecast something."

Portfolio intuition and recession state variables. The traditional portfolio view gives useful intuition: adding \(\varepsilon\) of \(R^i\) to a portfolio \(R^W\) raises portfolio variance by \(2\varepsilon\operatorname{cov}(R^W,R^i)\) — covariance (beta) measures how a marginal increase in \(R^i\) affects portfolio variance. At the optimum each asset's cost-benefit tradeoff is equal, so mean excess returns are proportional to covariance with the portfolio. The ICAPM adds long horizons and time-varying opportunities: a long-horizon investor (curvature above log) dislikes news that future returns are lower and prefers stocks that do well on such news (hedging reinvestment risk). Most current empirical work actually appeals to another source: investors have jobs, own houses, hold shares of small businesses — they prefer stocks that don't fall in recessions, bidding up prices and down expected returns. Such state variables need not forecast any traded return (so not strictly ICAPM). Crucially the extra factor must affect the average investor: if an event makes A worse and B better, they trade to transfer risk without affecting the price; only factors affecting the average investor change expected returns. So expect many common movements (e.g. industry portfolios) that carry no risk price.

9.4 Arbitrage Pricing Theory (APT)

The APT (Ross 1976) starts from a statistical characterization: returns have a big common component (when the market rises, most stocks rise), plus group co-movement (industry, size, value), plus a fully idiosyncratic part. The intuition: fully idiosyncratic movements should carry no risk price, since investors diversify them away in portfolios, so expected returns should relate only to covariance with the common components ("factors"). The job is twofold: (1) a factor decomposition modeling co-movement; (2) arguing the idiosyncratic part has zero risk price.

Factor structure. The decomposition \(x^i=E(x^i)+\beta_i'\tilde f+\varepsilon^i\), with \(E(\varepsilon^i)=E(\tilde f\varepsilon^i)=0\) (by regression construction). The real content (making it non-vacuous) is assuming residuals are mutually uncorrelated, \(E(\varepsilon^i\varepsilon^j)=0\). This is a restriction on the covariance matrix: \(\operatorname{cov}(x,x')=\beta\beta'\sigma^2(f)+\text{diagonal}\) (a singular matrix plus a diagonal one). When the factors' identities are unknown, estimate via factor analysis (eigenvalue decomposition of the covariance matrix, set small eigenvalues to zero).

Exact factor pricing (no residual). If \(\varepsilon^i=0\), then \(x^i=E(x^i)\mathbf 1+\beta_i'\tilde f\) says \(x^i\) can be synthesized from the factors and a risk-free payoff, so by the law of one price alone \(p(x^i)=E(x^i)p(1)+\beta_i'p(\tilde f)\), equivalent to

Approximate APT (small residual) — law of one price only. Actual returns do not satisfy the factor structure exactly, but residuals are often small (portfolio regressions have high \(R^2\)). Can we say "small residual ⟹ price not too far off"? Figure 9.1 shows the geometry: the factors span a payoff space, and \(x^i\) is not in it (\(\varepsilon^i\ne0\)). The \(f^*\) inside the factor space prices the factors and is orthogonal to the residual (zero price), but all discount factors pricing the factors form a line, along which \(m\)'s inner product with \(\varepsilon^i\) ranges over \((-\infty,\infty)\).