Sizing a Cointegrated Three-Asset Mean-Reversion Basket
Summary
The document asks how to turn estimated coefficients from a three-asset cointegration relation into dollar allocations for a mean-reversion trade. Its example models the log price of one security as a linear combination of the other two log prices, then considers shorting the first security against a long basket of the other pair.
The answer cites a study and gives equations relating each asset's dollar weight to its price and the relevant cointegration coefficient. It also imposes a normalization on the sum of absolute weights. This indicates that coefficients alone are not the dollar allocations: prices and a scaling convention matter. The post does not explain how to derive or implement the equations, address estimation uncertainty, or answer whether log or raw prices are preferable. The cited relation is therefore a starting point rather than a complete portfolio-construction procedure.
Key ideas
- Cointegration coefficients describe a price relation but do not directly specify dollar weights.
- The proposed weight ratios depend on current asset prices and the fitted coefficients.
- A gross-exposure normalization is used to scale the three-asset position.
- The document does not resolve whether the regression should use log prices or raw prices.
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Full text
# How to determine ratios for mean-reverting basket
# How to determine ratios for mean-reverting basket
Suppose I have a basket of 3 securities A, B, and C. I believe that the basket is cointegrated and I want to create a mean-reverting trade. I fit the model: $\log(A)=\beta_b*\log(B)+\beta_c*\log(C)+\alpha$ where A, B, and C are the prices of the securities.
This gives me estimates of $\alpha$, $\beta_b$ and $\beta_c$.
Now suppose that I believe that the spread is out of line. I want to sell \$1 of A and buy \$1 of the B and C basket. How should I allocate that dollar to B and C? Is it simply $\beta_b*\$1$ units of B and $\beta_c*\$1$ units of C or is it more complex?
Related, is it more correct to regress log prices or raw prices when fitting the model?
(I know that this is related to How to build a mean reverting basket? but the answers there weren't very detailed and this is a more specific question).
## Answer by MGL (score 2)
https://quant.stackexchange.com/a/31175
In a recent paper - Cointegration and Relative Value Arbitrage by Binh Do and Robert Faff, the issue of relative value arbitrage with three stocks is addressed. On page 27 they formulate the cointegrating relation similarly to how you did: $$\ p_{1t} = \alpha + \gamma p_{2t} + \beta p_{3t} + \epsilon_t$$
They also say that the dollar weights of the asset should satisfy the following equations:
$$ w_1/w_2 = p_{1t}/-\gamma p_{2t} $$ $$ w_1/w_3 = p_{1t}/-\beta p_{3t} $$ $$ |w_1| + |w_2| + |w_3| = 2 $$
Hope this helps.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.