Cointegration Does Not Guarantee CAPM Market Neutrality
Summary
The document asks whether a long-short portfolio formed from a cointegrated pair can be shown to have zero CAPM beta. It writes the portfolio’s market exposure as the beta of the long position minus the hedge ratio times the beta of the short position. Cointegration describes a long-run relationship between price levels, but the answer argues that this alone does not prove the portfolio is market neutral: market exposure depends on the assets’ return behavior and on the hedge weights.
The response suggests estimating a linear hedge weight from the relationship between the assets’ returns, using a regression-style covariance-to-variance ratio, while noting that a linear hedge may not be optimal if returns are non-normal or their relationship is nonlinear. This is a brief answer, not a complete derivation or a reliable universal prescription. In particular, a weight chosen to reduce pair-return variation does not necessarily make CAPM beta exactly zero; that objective requires checking covariance with the market or estimating portfolio beta directly.
Key ideas
- Cointegration between price levels does not by itself establish CAPM market neutrality.
- The long-short portfolio’s beta depends on the two assets’ betas and the hedge weight.
- A return-based regression weight may help hedge one asset against the other.
- A variance-minimizing hedge does not necessarily produce zero market beta.
- Nonlinear or non-normal return relationships can limit a simple linear hedge.
Tags
Full text
# Using cointegration to prove that a long-short strategy is market neutral (in CAPM sense)
# Using cointegration to prove that a long-short strategy is market neutral (in CAPM sense)
I am trying to prove that a long-short strategy invested according to the cointegrated relationship from Engle-Granger's. So essentially I'm trying to show that the return $r_{XY}$ of the portfolio (X long Y short) has zero $\beta$ (in other words it is market neutral in CAPM sense).
CAPM says that $r_Y = r_m\beta_Y + \alpha_Y$, $r_X = r_m\beta_X + \alpha_X$, hence $$r_{XY}=r_X - br_Y = \beta_Xr_m + \alpha_X - b\beta_Yr_m - b\alpha_Y = r_m(\beta_X-b\beta_Y) + \alpha_X - b\alpha_Y$$
In order to be market neutral we need that $\beta_X-b\beta_Y=0$ but I'm struggeling a bit to prove this.
Since $\beta$ is defined as $$\beta = \frac{Cov(r_p,r_m)}{Var(r_m)}$$ I firstly need to find $r_X$ and $r_Y$ to find the $\beta$'s.
This is were I get stuck. I have been trying to use that, assuming the stocks continues to be cointegrated, we need that $X_t(1+r_X) = bY_t(1+r_Y) + \mu + \epsilon_t$ $(\star)$. Then solve for $r_X$ or $r_Y$ and plug it into our equations, but it doesn't really work. For $\beta_X-b\beta_Y$ to be $0$ we need that $r_X = br_Y+constant$, but this doesn't work with $(\star)$$$$$Is it possible to do it like this, or is there some other way to prove that it is market neutral in CAPM sense? Or maybe it isn't market neutral according to CAPM?
## Answer by 4pie0 (score 1)
https://quant.stackexchange.com/a/7757
you get what should get. You can't prove that strategy long $X$ short $Y$ is market neutral: is strategy long EUR/USD short USD/CHF risk neutral? I wouldn't say that. It depends, on what? On relationship between these variables, so it is perfectly hedged only if `dX=dY` so your task is bad stated: it should be rather: what should be
$b$
to assure that
$r_{XY}=r_X - br_Y=0$
and answer is $b = \frac{Cov(r_X,r_Y)}{Var(r_X)}$ so regression coefficient, under assumption that errors are normally distributed, of course they might be not and then not linear combination is optimal but some else, nonlinear, it has just assure that if X moves dX then Y moves by dY so you just need to know what position to take. Since you want to determine constant $b$ size in asset then if they don't create linear combination you can't find linear market neutral combination and forget about CAPM and your $\beta$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.