Why an Asset’s Variance Has No First-Order Effect on Portfolio Risk

Summary

The document explains a first-order effect in the context of asset pricing and portfolio consumption risk. If consumption is initially c and an asset pays x, adding a small position of size ξ changes consumption to c + ξx. Expanding the variance gives the original variance, a term linear in ξ involving the covariance of consumption and the asset payoff, and a term quadratic in ξ involving the payoff variance.

For a sufficiently small position, the linear covariance contribution dominates the quadratic variance contribution, assuming the payoff variance is finite. Thus the initial marginal change in risk depends on how the payoff covaries with existing consumption, while the asset’s own variance affects risk at the next order. This helps explain why an asset can have high standalone volatility yet still have an attractive small-position effect if its covariance with consumption is favorable. The explanation is local: it concerns a small position and does not imply that payoff variance is irrelevant for larger holdings or that covariance alone determines an optimal portfolio.

Key ideas

• Adding a small position changes consumption by the asset payoff scaled by the position size.
• The variance expansion contains a linear covariance term and a quadratic payoff-variance term.
• For a sufficiently small position, covariance determines the leading change in consumption variance.
• Standalone payoff variance still matters as the position grows beyond the local, first-order comparison.

Tags

# What does "first-order effect" mean?


# What does "first-order effect" mean?












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?

## Answer by Koval  Boris (score 4)

https://quant.stackexchange.com/a/50581

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.