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In the textbook Asset Pricing by John Cochrane, on p. 25, it says:

"This prediction holds even if the payoff $x$ is highly volatile and investors are highly risk averse. The reason is simple: if you buy a little bit more of such an asset, it has no first-order effect on the variance of your consumption stream."

What does "first-order effect" mean here? Why would buying more of such an asset have no first-order effect on the variance of consumption stream?

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    $\begingroup$ If you make a small change $\epsilon$ to a sytem and the result is a change proportional to $\epsilon$ that is a "first order effect" (i.e the impact is comparable to the change), if the change is proportional to $\epsilon^2$ that is a "second order effect" (small effect compared to the change), and so on. $\endgroup$ – Alex C Oct 29 '19 at 3:36
  • $\begingroup$ @AlexC Thank you for your comment. Am I correct that the reason why "it has no first-order effect on the variance of your consumption stream." is that only "a little bit more of such an asset" was bought? $\endgroup$ – Aqqqq Oct 29 '19 at 13:13
  • $\begingroup$ A slightly different and more general distinction between a "first-order" versus a "higher order" effect (the latter being exactly as above) is that there is no direct impact. dC/dX = 0. dC/d(X^y) can also be zero, which is the higher-order effect above (perfectly correctly stated). However dX/dZ and dC/dZ might not be zero, in which the case the chain rule kicks in. Buying the asset might not change my consumption; but it might change my preference/aversion for something else, which in turn might affect my consumption. Simple thought example to follow... $\endgroup$ – demully Jan 7 at 21:10
  • $\begingroup$ So do currency declines cause/correlate with higher real interest rates? No, because a strong economy tends to lead to stronger currency, and higher real rates. Both are first-order effects of a strong economy. However, if the currency tanks, that is inflationary, and inflation produces a policy response, in the form of tighter monetary policy. Higher real is a second-order effect from a weaker currency. Even if higher real and a stronger currency are first-orders of the economy. Different parts of the calculus can pull in different directions. $\endgroup$ – demully Jan 7 at 21:15
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Assume you have a consumption $c$ and an asset with the payoff $x$. Cochrane states that if you add "a little bit of this asset" in your portfolio first you care about the correlation between the payoff of the asset and consumption and ONLY then you care about variance. How you can see this?

Let's assume that you slighlty change your portfolio by $\xi$ (i.e. buy 0.0001 units of asset), then the variance of your consumption (which you care about) will be:

$$ \sigma^2(c + \xi x) = \sigma^2(c) + 2\xi cov(c, x) + \xi^2var(x). $$

Now, you clealry see that if $\xi$ is very small than second term will be always higher than the third term (if $var(x)$ finite). This means it has a higher impact - the first-order impact.

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