Inelastic Markets ~ Flat SML

Stock returns are typically higher than bond returns. The average difference is somewhere in the neighborhood of $\mathbb{E}[\mathrm{Mkt}]{-}\mathrm{rf} \approx 4\%$ . Academic researchers call this the equity risk premium.

The CAPM is the most famous asset-pricing model in the literature. The theory predicts that each stock’s expected excess return will be proportional to the expected excess return on the market portfolio, $\mathbb{E}[\mathrm{Ret}_n {-} \mathrm{rf}] \;=\; \beta_n \times \mathbb{E}[\mathrm{Mkt} {-} \mathrm{rf}]$ . Suppose you plot each stock’s expected excess return, $\mathbb{E}[\mathrm{Ret}_n] {-} \mathrm{rf}$ (y-axis), against its market beta, $\beta_n$ (x-axis). The resulting line is called the “SML” (security market line)

(1) $\begin{equation*}\mathbb{E}[\mathrm{Ret}_n] {-} \mathrm{rf} \;=\; \beta_n \times \lambda\end{equation*}$

The CAPM predicts that you ought to get a line with slope $\lambda = \mathbb{E}[\mathrm{Mkt}] {-} \mathrm{rf}$ .

The same simple model predicts that aggregate demand elasticity should be roughly

(2) $\begin{equation*}\nu \;\approx\; \frac{1}{\mathbb{E}[\mathrm{Mkt}] {-} \mathrm{rf}}\end{equation*}$

If the level of the stock market goes up by $1\%$ for non-fundamental reasons, then it suddenly costs more money to buy the same future cash flows. Textbook theory predicts that investors ought to reduce their holdings. Demand elasticity tells you how much. If $\nu = 2$ , then a unilateral ${+}1\%$ increase in the current price of equities will cause a ${-}2\%$ reduction in investors’ stock holdings.

The security market line (SML) is much flatter than theory predicts. Instead of $\lambda = 4\%$ , the slope of the SML is basically zero. The aggregate stock market is much less elastic than theory predicts. Instead of $\nu = \frac{1}{4\%} = 25$ , Gabaix-Koijen puts the number in the neighborhood of $0.2$ . Investing an extra $\mathdollar 1$ in the stock market raises its aggregate value by about $\mathdollar 5$ .

This note shows both findings are related. They’re two perspectives on the same underlying problem.

CARA-Normal Model

Start with the simplest possible model: two periods (today and next year); one investor; one risky asset (the stock market); one riskless bond. Buying a share of the stock market costs $\mathrm{Price}_t$ today. If you own a share of the stock market today, then next year you are entitled to receive $\mathrm{Payout}_{t+1}$ . The riskless bond costs $\mathdollar 1$ today and will pay $(1 {+} \mathrm{rf})$ next year.

The representative investor has “constant absolute risk aversion” (CARA) preferences, $\mathrm{U}(\mathrm{C}) = {-}\tfrac{1}{\gamma} \cdot e^{-\gamma \cdot \mathrm{C}}$ . The stock market’s payout next year is normally distributed with variance $\sigma^2 > 0$ . The investor starts with wealth $\omega > \mathdollar 0$ . Today, he must choose how much to consume, $\mathrm{C}_t$ , and how many shares of the risky asset to purchase, $\mathrm{Q}_t$ . His goal is to maximize $\mathrm{U}(\mathrm{C}_t) + \mathbb{E}\big[ \, e^{-\rho} \cdot \mathrm{U}(\mathrm{C}_{t+1}) \, \big]$ where $\rho > 0$ is his rate of time preference. The investor parks any remaining wealth, $(\omega - [\mathrm{C}_t {+} \mathrm{Q}_t \cdot \mathrm{Price}_t])$ , in the riskfree bond. Next year, the investor eats the combined payout from his risky and safe investments

(3) $\begin{equation*}\mathrm{C}_{t+1} \;=\; (1 {+} \mathrm{rf}) \times \big( \, \omega - [\mathrm{C}_t {+} \mathrm{Q}_t \cdot \mathrm{Price}_t] \, \big) \,+\, \mathrm{Q}_t \cdot \mathrm{Payout}_{t+1}\end{equation*}$

Let $\psi > 0$ denote the supply of shares in circulation. The market clears when the investor’s demand for the risky asset equals the number of available shares, $\mathrm{Q}_t = \psi$ . An equilibrium is an allocation, $\{\mathrm{C}_t,\,\mathrm{Q}_t,\,\mathrm{C}_{t+1} \}$ , and a current price level for the risky asset, $\{ \mathrm{Price}_t \}$ , such that (i) the allocation solves the investor’s optimization problem given the price, and (ii) the price clear the market given the investor’s allocation.

The payout to owning each share of the risky asset is positive on average. So, holding an extra share will lead to slightly higher consumption next year. At the optimum, this benefit will be exactly canceled out by the cost of the required reduction in consumption today, with each side weighted by its marginal utility

(4) $\begin{equation*}\mathrm{U}'(\mathrm{C}_t) \times \mathrm{Price}_t \;=\; \mathbb{E}\big[ \, e^{-\rho} \cdot \mathrm{U}'(\mathrm{C}_{t+1}) \times \mathrm{Payout}_{t+1} \, \big]\end{equation*}$

This is the Euler equation. An extra $\mathdollar 1$ that arrives in bad times (consumption is low; marginal utility is high) counts for more than a $\mathdollar 1$ that arrives in good times (high consumption; low marginal utility).

Here’s how to solve this model. First, note that the riskless asset costs $\mathdollar 1$ today and is guaranteed to deliver $(1{+}\mathrm{rf})$ next year, so its Euler equation is

(5) $\begin{equation*}\mathrm{U}'(\mathrm{C}_t) \;=\; \mathbb{E}\big[ \, e^{-\rho} \cdot \mathrm{U}'(\mathrm{C}_{t+1}) \times (1{+}\mathrm{rf}) \, \big]\end{equation*}$

If we replace the $\mathrm{U}'(\mathrm{C}_t)$ in Equation (4) with this expression, then the $e^{-\rho}$ cancels out, and the price becomes a marginal-utility-weighted average of the discounted payout. The definition of a covariance plus Stein’s lemma turn that weighted average into $\mathbb{E}[\mathrm{Payout}_{t+1}] - \gamma \times \mathbb{C}\mathrm{ov}[\mathrm{C}_{t+1}, \, \mathrm{Payout}_{t+1}]$ . What’s more, Equation (3) shows that next year’s consumption will be linear in the payout, so $\mathbb{C}\mathrm{ov}[\mathrm{C}_{t+1}, \mathrm{Payout}_{t+1}] = \mathrm{Q}_t \cdot \sigma^2$ . Given market clearing, $\mathrm{Q}_t = \psi$ , this leads to the following pricing rule

(6) $\begin{equation*}\mathrm{Price}_t = \frac{\mathbb{E}[\mathrm{Payout}_{t+1}] - \gamma \cdot \sigma^2 \cdot \psi}{1 + \mathrm{rf}}\end{equation*}$

Each extra share makes next year’s consumption covary more strongly with the payout, so the marginal buyer demands a larger discount. The numerator is the expected payout minus an adjustment for risk. The denominator adjusts for the time cost of money.

Security Market Line

Textbook asset-pricing theory puts every asset on a single line. Expected excess returns ought to be proportional to betas, and the constant of proportionality ought to be the equity risk premium. To see where this prediction comes from, define the stochastic discount factor as discounted marginal utility growth, $\mathrm{SDF}_{t+1} = e^{-\rho} \cdot \tfrac{\mathrm{U}'(\mathrm{C}_{t+1})}{\mathrm{U}'(\mathrm{C}_{t})}$ . Let $n = 1, \ldots, N$ index the cross-section of risky assets… i.e., each stock in the stock market. The same SDF should price every one of them. If you use the SDF to write stock $n$ ‘s Euler equation and divide by its current price, then you get a statement about its expected return

(7) $\begin{equation*}1 \;=\; \mathbb{E}\bigg[ \, \mathrm{SDF}_{t+1} \times \underbrace{\bigg(\frac{\mathrm{Payout}_{n,t+1}}{\mathrm{Price}_{n,t}}\bigg)}_{1+\mathrm{Ret}_{n,t+1}} \, \bigg]\end{equation*}$

Going forward, I’ll suppress time subscripts where it causes no confusion.

If you subtract the Euler equation for the riskless bond, $1 = \mathbb{E}[ \, \mathrm{SDF} \times (1{+}\mathrm{rf}) \, ]$ , then you get

(8) $\begin{equation*}0 \;=\; \mathbb{E}\big[ \, \mathrm{SDF} \times (\mathrm{Ret}_n{-}\mathrm{rf}) \, \big]\end{equation*}$

The difference being priced, $(\mathrm{Ret}_n{-}\mathrm{rf})$ , is stock $n$ ‘s excess return. It is the payout from a long/short portfolio that sells riskfree bonds and uses the proceeds to buy shares of the risky asset.

Now consider applying the definition of a covariance, $\mathbb{C}\mathrm{ov}[X, \, Y] = \mathbb{E}[X \cdot Y] - \mathbb{E}[X] \cdot \mathbb{E}[Y]$ , to this excess-return SDF formula

(9) $\begin{align*}0 &= \mathbb{E}[ \, \mathrm{SDF} \times (\mathrm{Ret}_{n}{-}\mathrm{rf}) \, ] \\ &= \mathbb{E}[\,\mathrm{SDF}\,] \times (\mathbb{E}[\mathrm{Ret}_{n}]{-}\mathrm{rf}) + \mathbb{C}\mathrm{ov}[ \, \mathrm{SDF}, \, \mathrm{Ret}_{n} \, ]\end{align*}$

By rearranging terms, we can arrive at the following expression

(10) $\begin{align*}\mathbb{E}[\mathrm{Ret}_{n}] {-} \mathrm{rf} &= \bigg( \frac{\mathbb{C}\mathrm{ov}[ -\mathrm{SDF}, \, \mathrm{Ret}_{n} ]}{\mathbb{E}[\mathrm{SDF}]} \bigg) \\ &= \underbrace{\bigg( \frac{\mathbb{C}\mathrm{ov}[ -\mathrm{SDF}, \, \mathrm{Ret}_{n} ]}{\mathbb{V}\mathrm{ar}[\mathrm{SDF}]} \bigg)}_{\beta_n} \times \underbrace{\bigg( \frac{\mathbb{V}\mathrm{ar}[ \mathrm{SDF}]}{\mathbb{E}[\mathrm{SDF}]} \bigg)}_{\lambda} \end{align*}$

The first $\beta_n$ term tells you how much asset $n$ ‘s return tends to comove with the SDF. The SDF captures growth in marginal utility. It is high when the economy enters into bad times. That’s when it becomes more valuable to have an extra dollar. Thus, stocks that tend to do well during booms and poorly during crashes will have large values of $\beta_n$ . The second $\lambda$ term is constant across stocks. It answers the following question: If a stock’s $\beta_n$ goes up by one unit, how much higher will its excess returns be on average?

In the CARA-normal model, the SDF is approximately linear in the change in aggregate consumption

(11) $\begin{equation*}\mathrm{SDF} \;=\; e^{-\rho} \cdot e^{-\gamma \cdot \Delta \mathrm{C}} \;\approx\; a - b \cdot \Delta \mathrm{C} \end{equation*}$

And what’s the main driver of the change in aggregate consumption in this model? The payout on the risky asset next year. Equation (3) shows that $\mathrm{C}_{t+1}$ is linear in $\mathrm{Payout}_{t+1}$ , and $\mathrm{Payout}_{t+1} = (1 {+} \mathrm{Mkt}_{t+1}) \cdot \mathrm{Price}_t$ by definition. So the SDF is approximately linear in the market’s return.

Under these assumptions, you can estimate stock $n$ ‘s $\beta_n$ by running a time-series regression of realized returns on the market return. Then, if you plot each stock’s average excess return against its estimated $\beta_n$ , the slope of the best-fit line will give you $\lambda$ . The stock market as a whole has $\beta_{\mathrm{Mkt}} = 1$ and an average excess return of $\mathbb{E}[\mathrm{Mkt}]{-}\mathrm{rf} \approx 4\%$ . The riskfree bond has $\beta_{\mathrm{rf}} = 0$ and an average excess return of $\mathrm{rf}{-}\mathrm{rf} \approx 0\%$ . Two points define the slope of a straight line. So textbook theory predicts that $\lambda = \mathbb{E}[\mathrm{Mkt}]{-}\mathrm{rf} \approx 4\%$ .

The estimated slope is far below the equity risk premium. Way back in 1972, Black-Jensen-Scholes ran the test on every NYSE stock from 1926 to 1966, grouped into 10 beta-sorted portfolios. Average excess returns do line up with betas. But the fitted line is too flat, $\hat{\lambda} \ll 4\%$ . Low-beta portfolios earn more than the model predicts, and high-beta portfolios earn less. The problem has only gotten worse. In 1992, Fama-French found basically no relation between average returns and betas from 1963 to 1990. Frazzini-Pedersen (2014) document the same issue in both US and global equities.

Demand Elasticity

The demand-system approach to asset pricing takes the same CARA-normal model but solves each investor’s problem before imposing market clearing. Let $i = 1, \ldots, I$ index individual investors, each with his own risk-aversion coefficient, $\gamma_i$ . Repeat the steps that led to the pricing rule in Equation (6), but stop short of market clearing. Isolating investor $i$ ‘s demand on the left-hand side, you get the formula below

(12) $\begin{equation*}\mathrm{Q}_i \;=\; \frac{\mathbb{E}[\mathrm{Payout}] - (1 {+} \mathrm{rf}) \cdot \mathrm{Price}}{\gamma_i \cdot \sigma^2}\end{equation*}$

If you hold investor $i$ ‘s curve fixed and move the price, then the investor’s demand elasticity is given by

(13) $\begin{equation*}\nu_i = - \frac{\partial \log \mathrm{Q}_i}{\partial \log \mathrm{Price}} = \frac{(1 + \mathrm{rf}) \cdot \mathrm{Price}}{\gamma_i \cdot \sigma^2 \cdot \mathrm{Q}_i}\end{equation*}$

Define the aggregate risk-aversion parameter, $\gamma$ , as the harmonic average of the individual coefficients, $\tfrac{1}{\gamma} = \sum_i \tfrac{1}{\gamma_i}$ . In equilibrium, each investor’s position in the CARA-normal model will be inversely proportional to his risk aversion, $\gamma_i \cdot \mathrm{Q}_i = \gamma \cdot \psi$ . Thus, the denominator in the elasticity formula is the same for everyone, $\nu_i = \nu$ . A single elasticity describes every investor. More risk-tolerant investors will hold bigger positions, but their percentage response is identical. This is analogous to the common $\lambda$ across assets.

Notice that the denominator in the elasticity formula is just the risk discount in the CARA-normal model, $\gamma \cdot \sigma^2 \cdot \psi = \mathbb{E}[\mathrm{Payout}] - (1 {+} \mathrm{rf}) \cdot \mathrm{Price}$ . Replace the denominator in Equation (15) with this expression and divide through by the current price. If the riskfree rate isn’t too large, then you get

(14) $\begin{equation*}\nu \;=\; \frac{(1 + \mathrm{rf}) \cdot \mathrm{Price}}{\mathbb{E}[\mathrm{Payout}] - (1 + \mathrm{rf}) \cdot \mathrm{Price}} \;\approx\; \frac{1}{\mathbb{E}[\mathrm{Ret}] {-} \mathrm{rf}} \end{equation*}$

The reward for bearing a unit of stock-market risk is the equity risk premium, $\mathbb{E}[\mathrm{Mkt}]{-}\mathrm{rf} \approx 4\%$ . A $1\%$ rise in the price will increase the cost of financing a share by $1\%$ , consuming roughly a quarter of the $4\%$ margin and causing demand to fall by $\frac{1\%}{4\%} = 25\%$ . In other words, theory predicts that $\nu = 25$ .

Deeper Connection

The slope of the SML, $\lambda$ , is the exchange rate between risk and expected returns. How much higher must a stock’s expected excess return be in order to compensate investors for holding one more unit of exposure to market risk? One number common to every asset. The demand elasticity, $\nu$ , is the exchange rate between flows and prices. How much do investors have to adjust their holding in response to a $1\%$ change in the price? One number common to every investor. Every assumption about preferences and beliefs reaches returns data only through $\lambda$ , and reaches price-impact data only through $\nu$ .

In one sense, these two parameters are two sides of the same coin. Neither is consistent with the observed equity risk premium, $\mathbb{E}[\mathrm{Mkt}]{-}\mathrm{rf} \approx 4\%$ . This point estimate is enormous compared to the observed slope of the SML, which is basically zero. However, the same $4\%$ number implies a demand elasticity of $25$ , far above the value near $0.2$ in the data. What’s more, the two predictions pull in opposite directions. Any effort that pushes $\mathbb{E}[\mathrm{Mkt}]{-}\mathrm{rf}$ down to fit the slope of the SML makes the elasticity error worse and vice versa. The too-flat SML and the too-steep demand curve are both manifestations of the same underlying problem.

But there’s also a deeper connection. A flat SML is a trading opportunity. Buy levered positions in low-beta stocks and short high-beta stocks. Frazzini-Pedersen calls this trade “betting against beta”, and it has been profitable for decades. That sort of thing shouldn’t survive. Investors ought to pour capital into the trade, bidding up the prices of low-beta stocks and pushing down the prices of high-beta stocks until the SML steepened back to $4\%$ . That correction is a demand response to price, which is exactly what $\nu$ measures. When demand barely responds to price, mispricings do not get traded away. They just sit there. So the too-steep demand curve is not merely a second manifestation of the same problem. It offers a reason why the first one never went away. The betting-against-beta alpha is what inelasticity looks like in returns data.