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https://economics.stackexchange.com/questions/16115/what-is-the-equation-mathbbemr-1

The above post asks what $$\mathbb{E}[mR]=1$$ means and gets some great answers. From the first answer is seems that the physical rather than risk neutral probabilities are used to take the expectation.

I'm reading the paper "Characteristics are Covariances" on p2 they introduce the Euler equation as:

$$\mathbb{E}[mR]=0$$

I have not seen this formulation from the paper anywhere else. They say that the only assumption used was that of "no arbitrage". The paper does not state which measure was used to take the above expectation, but given the prices are arbitrage free I am assuming it is a risk neutral one.

  • A I correct to think that the expectation in the paper is taken under the risk-neutral measure?

  • Is it the switch from the physical to risk neutral measure that causes the difference on the rhs of the two expressions, if so can someone run me through the derivation?

  • Intuitively the first formulation seems to be a martingale but is calculated under the physical measure so I am assuming it is not arbitrage free can a price process be a martingale but still not be arbitrage free?

Thanks

Baz

Bob Jansen
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Bazman
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2 Answers2

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I don't have the paper, I think you mean this text.

On the Economics Stack Exchange $R$ denotes the asset return. In the linked paper, $r$ denotes the excess return, i.e. $R = 1 + r$. I believe all the rest is the same.

Bob Jansen
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$\mathbb{E}[mR]=0$ is the Euler equation under $\mathbb{P}$ for excess returns.

In general, the Euler equation is $$P_t=\mathbb{E}_t[M_{t,t+1}X_{t+1}],$$ where $P_t$ is today's price, $M_{t,t+1}$ the pricing kernel and $X_{t+1}$ tomorrow's payoff. The expectation is conditional on information available at time $t$ and under the real world probability measure $\mathbb{P}$.

The Euler equation is often stated for returns rather than prices. Then, with $R_{t+1}=\frac{X_{t+1}}{P_t}$, we have $$1=\mathbb{E}_t[M_{t,t+1}R_{t+1}].$$

For excess returns ($R^e_{t+1}$), the price is zero by construction ($P_t=0$). The Euler equation is thus $$0=\mathbb{E}_t[M_{t,t+1}R^e_{t+1}].$$

Here are more examples from John Cochrane's fantastic book (chapter 1). enter image description here

Kevin
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  • Kevin, I have some questions unrelated to this thread that I am struggling with. As a go-to expert on asset pricing, would you mind taking a look? Two of them have bounties on them. – Richard Hardy Dec 07 '23 at 12:17
  • @RichardHardy Thank you but I think phdstudent and Matt Gunn are much much better experts. I've also been recently less active here but I think I upvoted pretty much all of your questions! – Kevin Dec 07 '23 at 12:51
  • Thank you. I have contacted both of them. I have not seen Matt around for a while. I hope phdstudent will be able to contribute, though. (By the way: If I look at the top users for the asset-pricing tag, I find you right at the top.) – Richard Hardy Dec 07 '23 at 12:54
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    @RichardHardy Nice about the tag statistics. I didn't even know we can compare users this way. Still, in terms of answer quality and intellectual contributions, I bow deeply to both, phdstudent and Matt. I've learnt a tremendous amount from them and their brilliant answers. – Kevin Dec 07 '23 at 12:59