# Short thoughts on Efficiency

Eugene Fama, Nobel laurate in 2013 for his empirical analysis of asset prices Source: http://www.nobelprize.org

The Efficient Market Hypothesis (EMH): The holy grail of the theory of finance, establishes that asset prices should reflect all information about future cash flows. But why should they? Who are we to decide how prices must behave? It reminds me of the Einstein-Bohr debate on the nature of quantum mechanics:

Einstein: God does not play with dice

Bohr: Stop telling God what to do

Should we be like Einstein, and impose a normative explanation on asset prices? Or should we be like Bohr, and just accept that if theories do not fit reality, we need a better explanation?

A not so formal Introduction to the EMH

Let me analyse how we think assets behave in a frictionless environment using a simple model .  Consider a representative agent that lives for two periods (present and future) $t=0,1$. The agent has some endowment (wages, dividends, etc) in each period that he/she uses to consume or invest. To invest, there is an asset that pays a random amount $X$ in the future. The budget constraint in period $0$ is:

$c_0=e_0-\eta p$

Where $c_0$ is present consumption, $e_0$ is present endowment, $p$ is the unitary price of the investment, and $\eta$ is the units of the asset bought (if $\eta<0$ the investor shorts the asset). Also, the constraint in period $t=1$ is:

$c_1=e_1+\eta X$

How much units to buy (sell)? The representative agent solves the following two period maximization problem:

$\max_\eta u(c_0)+\beta \mathbb{E}(u(c_1))$

Subject to the above constraints on $c_0$ and $c_1$. We impose the standard assumptions of risk aversion and non satiation on the utility function $u()$. The factor $\beta$ represents the impatience of the consumer and the way he/she values future consumption. A high $\beta$ means that the investor values present and future consumption almost equally, while a low $\beta$ means that the investor values more present consumption. Replacing the constraints in the objective function leads us to an unconstrained optimization problem with first order condition equal to:

$u'(c_0)p=\mathbb{E}(\beta u'(c_1) X)$ The marginal utility of consuming one extra unit in the future equals the marginal desutility of buying one marginal unit in the present. Rearranging terms give us the following expression for prices:

$p=\mathbb{E}(\frac{\beta u'(c_1)}{u'(c_0)}X)$

Defining $M=\beta \frac{u'(c_1)}{u'(c_0)}$ as the Stochastic Discount Factor (for short SDF which is a random variable as well, since consumption in the future is random)  give us one of the most important equations in finance

$p=\mathbb{E}(MX)$

According to this model, the price of any asset is the expected value of its cash flows times the stochastic discount factor. We can rearrange the equation to define a formula not for prices but for returns. Defining $R=\frac{X}{p}$ as the gross return, and using the formula for covariance we get the following expression:

$1=\mathbb{E}(MR)$

$1=Cov(M,R)+\mathbb{E}(M) \mathbb{E}(R)$

We can define the return on a risk-free asset (An asset thay pays a return $R_f$ independent on the state of nature) as

$1=\mathbb{E}(M)R_F \rightarrow R_F=\frac{1}{\mathbb{E}(M)}$

Using the above definition:

$R_f=\frac{Cov(M,R)}{\mathbb{E}(M)}+\mathbb{E}(R)$

Finally:

$\mathbb{E}(R)-R_f=\frac{Cov(M,R)}{Var(M)}(\frac{-Var(M)}{\mathbb{E}(M)})$

Defining $\beta=\frac{Cov(M,R)}{Var(M)}$ as the “sensitivity” to the stochastic discount factor, and $\lambda=\frac{-Var(M)}{\mathbb{E}(M)}$ as the market price of ris we get the following expression:

$\mathbb{E}(R)-R_f=\beta \lambda$

Which implies that the excess return on any asset (the return of borrowing at the risk free rate and investing in the asset) is equal to the product of the sensitivity of the asset to the stochastic discount factor and the so called market price of risk. If the asset pays more, when the stochastic discount factor is high (Low consumption) the asset will have a lower return. Why? Well for a risk averse agent having an extra income when things are bad, is more valuable than having it when things are good.

Testing Market Efficiency

What are the implications if markets are efficient? Well the above equation give us the answer. If markets are efficient, running a regression of the form

$R-R_f=\alpha+\beta \lambda$ must lead to an $\alpha$ that is undistinguishable from zero. Technically, the distribution of all alphas must be centred near zero. However there is one problem with this methodology. We have no idea about what really is the Stochastic discount factor (Measure of bad times). Every market hypothesis test is a joint test of the EMH and a specific asset pricing model.

An asset pricing model tries to model the stochastic discount factor using a measure of bad times. The CAPM, Fama and French 3-5 factor model, Momentum strategy, etc are ways to model the stochastic discount factor to explain the cross section of expected returns.

Off course just running an OLS regression just wont do the job. Why? well there are several reasons, there is a lot of cross sectional variation between assets, and we are using as an independent variable something that we can not even observe (the $\beta$ in the regression). There are some ways to deal with this, the most common ways are the so called Fama-Macbeth regression, and the GRS test (Gibbons, Ross, Shanken, 1989) that provides a joint hypothesis test on those $\alpha$‘s.

My thoughts

In my opinion, this methodology does not work properly. As explained in Garleanu and Pedersen (2011) and Imbett (2016) (A working paper I promise to upload shortly :p), in the presence of market frictions the methodology fails. In the presence of capital or liquidity requirements (requirements on the composition of assets or liabilities) assets exhibit $\alpha$‘s different from zero in equilibrium even if the market is efficient.

Suppose there is a liquidity requirement for investing in the asset. For every euro borrowed at the risk free rate investors must borrow an extra amount $\epsilon$ just to satisfy the liquidity requirement. What do you need to invest in the risky asset? Borrow $\latex (1+\epsilon)$ to invest. What is the payoff of this strategy?

$R-R_f(1+\epsilon)$

In equilibrium ( I skip the derivation)

$R-R_f=\epsilon R_f+\beta \lambda$

Which clearly give us an alpha different from zero. This is also the case when investors have margin requirements, or default risk. In the presence of margin requirements, investors are not able to leverage their positions to achieve higher expected returns. This increases the demand for risky assets (lowering their alphas below their natural levels) and decreases the demand for safer assets.

Conclusions

Well, we do not have yet a robust methodology to test if markets are efficient. Our tests are based on frictionless models. However as we discussed before, including real frictions to our models (something that ironically was not done until some years ago) has important implications on the distribution of the $\alpha$‘s.

References

N. Garleanu, L.H. Pedersen (2011) Margin-based asset pricing and deviations from the law of one price . Rev. Financial Stud., 24 (6), pp. 1980–2022

Imbett, J.F. (2016) Asset pricing and reserve requirements, Working Paper