What is the equation $\mathbb{E}[mR]=1$?

Question

I have just joined this Stack Exchange site and I was reading the following question: What are the foundational equations of Economics?. I have seen a couple of times the following equation in that post:

$$\mathbb{E}[mR]=1$$

I have a certain idea of what it might mean, but could someone explain its meaning (or at least what do $m$ and $R$ stand for? It is difficult to Google it if I don't know what these 2 quantities are).

Also, is the expectation $\mathbb{E}[\cdot]$ taken w.r.t to any particular probability measure, or simply with respect to the real world measure?

PS: I apologize if the tags are not well-suited to the question, I have tried to guess what this equation might be related to.

Answer 1

This is an important result in financial economics (asset pricing) but not trivial to explain intuitively. I do my best to give you the big picture and get you started on your research.

R is the return on an asset or portfolio. Any asset or portfolio.

m is the stochastic discount factor or pricing kernel.

The expectation here is over possible states of the world given information at time t (m and R are as of t+1).

These are physical, not risk-neutral, probabilities. In fact, a (positive) stochastic discount factor is what is used to translate physical to risk-neutral probabilities. In a discrete case, the risk-neutral probability is the risk free rate times the physical probability times the value of the stochastic discount factor in that state.

The stochastic discount factor takes on a different value for each possible state of the world (that's the sense in which it is stochastic). But it doesn't change for each different asset. If it exists, a stochastic discount factor can be applied to every asset in the economy and price them all correctly.

Facts about the SDF:

• The existence of such a factor is equivalent to the law of one price holding.
• If it is always positive, there are no arbitrage opportunities in financial markets.
• If and only if markets are complete, then the stochastic discount factor is unique.
• The expectation of the SDF is the risk-free rate

There are a lot of theoretical uses of this relation and important related results---many fundamental results in finance can be interpreted in an SDF context, though many of those results came before development of SDF theory. For many common utility functions we can derive the form of the SDF for a representative agent theoretically.

I'm sorry if my answer isn't super accessible. The equation is simple, but stochastic discount factor theory can be a little opaque.

Resources to consider reading:

Cochrane and Culp's chapter in Modern Risk Management: A History

Asset Pricing and Portfolio Choice Theory, by Kerry Back

Answer 2

$Ε(mR) =1$ is a particular version of the general intertemporal long-run equilibrium/steady-state condition/relation

$$\text{One-period discount factor}\times \text{Gross Interest rate factor} = 1$$

$R$ is the Gross interest rate factor $R=1+r$ where $r$ is the real interest rate. The "discount factor" may take different forms depending on the exact formulation of each model. In the simplest case, it is a constant expressing the observed fact that people inherently prefer the present to the future, and so they "discount" future consumption (note: this is not a proxy for the existence of uncertainty, discounting of the future exists even in a deterministic setting).

The expected-value operator $E$ is used when the model is stochastic.

In a setting of intertemporal maximization of an additively separable utility function, this is the condition for a steady-state equilibrium imposed on the "Euler equation" which in turn emerges as a first-order necessary condition for the recursive optimization problem.

Answer 3

Thank you @farnsy for your explanation and references, particularly Cochrane & Culp's $-$ equations $(2)$ and $(3)$ on page 61.

Being more familiar with the technicalities of quantitative finance, I can give the following interpretation of the stochastic discount factor $m_t$ (SDF). Defining the SDF numéraire:

$$ \forall \, t \in [s,T], \ N_t = \frac{u'(C_s)}{\beta \, u'(C_t)}=\frac{1}{m_t}$$

The price $P_t$ of the claim with payoff $X_T$ at time $T$ can be interpreted as the expectation w.r.t. the real-world probability measure $\mathbb{P}$ of the payoff discounted by the SDF numéraire:

$$ P_t = \mathbb{E}^{\mathbb{P}}\left[\frac{X_T}{N_T}|\mathcal{F}_t\right]=\mathbb{E}^{\mathbb{P}}\left[m_TX_T|\mathcal{F}_t\right]=\mathbb{E}^{\mathbb{P}}\left[\beta \frac{u'(C_T)}{u'(C_t)}X_T|\mathcal{F}_t\right]$$

where $(\mathcal{F}_t)_{t \geq 0}$ is a filtration, i.e. represents the information available to the agent at time $t$.