Why is beta linear?

Question

Alpha and beta are commonly used to metaphorically describe decorrelated vs. correlated return, implicitly following a linear regression of returns relative to some benchmark. But isn't the choice of linear regression arbitrary? Aren't there ways of getting superlinear behavior, leading to a variable beta?

Why are returns expressed like this:

y = a + bx

And not like this:

y = a + Σ b_i x^c_i

Follow Up:

If there are practically no superlinear betas, is that caused by there being no zero-sum superlinear betas combined with a lack of counterparties willing to take on superlinear risk?

Answer 1

It really goes back to the Capital Asset Pricing Model (CAPM), which became a central pillar of modern finance and portfolio theory.

One can devise more complex models but that doesn't mean they'll be better. A lot of times, the simplest model that explains things "well enough" (parsimonious model) is preferred.

Answer 2

This linear assumption is done all over the place in science and engineering. Sometimes, it works. Not always.

For example, consider a spring. It has a force

F = k*x

...but does it? No. Actually it has a force:

F = k_1*x + k_2*x^2 + k_3*x^3 + ...

...and if x is small enough, the linear assumption is good.

Similar assumption is done in optics. It is there assumed that the material responds linearly to the electric field of electromagnetic waves. Usually it does. However, if the electric field is very strong, like in very intense lasers, it's possible to get nonlinear behavior in optics.

So, as long as market movements are small, linear assumption works. It's a model. It has a validity limit. When blood starts flowing in the streets, the validity breaks down. When Fed starts printing money like in Zimbabwe, the validity also breaks down.

This linear assumption is justified by the fact that every analytic function can be calculated by an infinite Taylor series. Assuming x is small, a good enough approximation is throwing all but the first term away, so in that case, the function is simply:

y = y_0 + k*(x - x_0)

...besides, I think this question might actually belong to Quantitative Finance instead of here.

Answer 3

Beta is intended to measure the risk of an asset (measured by variance in returns) versus some benchmark (typically defined vaguely as "the market"). A linear beta is simple - it does not vary depending on the level of return. A non-linear beta would mean that the risk of an asset is dependent on the return of the market which makes no fundamental sense to me. If the market makes a 10% return versus a 1% return, why does that change the risk of my investment? Is it just coincidence that a polynomial beta fits better, or is it due to other factors other than market return?

Modern finance has embraced the latter view, that the risk of an asset is not constant, but is dependent on multiple factors. So return is not a non-linear function of one variable, but a linear function of multiple variables:

r = a + b1*x1 + b2*x2 + b3*x3 + b4*x4 ...

where one of those factors might be market return. You may already be familiar with commonly used factors without knowing about them: Value, Growth, Momentum, Volatility, etc.

With a very simple one-factor model, beta is a quick indicator of how "risky" a stock is. With a non-linear risk profile, what would a large b1 mean? What would a small b2 mean? Those types of measurements would not be nearly as intuitive to most people as a simple linear model.