Defining Neutrality and Building a Factor-Neutral Stock Portfolio
Summary
The note explains that a stock hedge depends on what neutrality means. Dollar neutrality is one possible target; setting expected return to zero is another, though the answer describes it as uncommon. For market or factor neutrality, returns can be projected onto chosen factors, such as principal components or an externally defined risk model. A two-asset basket is then weighted to reduce its factor exposure, typically by minimizing the squared norm of its residual factor projection when exact cancellation is not possible.
It connects this target to variance minimization by splitting covariance risk into factor and orthogonal components. If the selected factors capture the main sources of ordinary return variation, minimizing variance may align with reducing factor exposure. The example supplies means, volatilities, and correlation, but the answer does not calculate a specific hedge amount or frontier. The relationship depends on the factor set and covariance structure; neutrality is not a unique objective, and factor-neutrality need not eliminate all portfolio risk.
Key ideas
- A hedge requires an explicit definition of neutrality, such as dollar, expected-return, market, or factor neutrality.
- Factor neutrality can be framed as minimizing the norm of a basket’s projected factor exposure.
- When exact factor cancellation is infeasible, the residual exposure can be minimized instead.
- Variance can be decomposed into factor and orthogonal components under the stated covariance assumptions.
- Variance minimization aligns with factor neutrality only when selected factors explain the main return risks.
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Full text
# How to construct a delta-neutral portfolio containing stocks using correlations?
# How to construct a delta-neutral portfolio containing stocks using correlations?
I’m aware of the mean-variance framework where we construct a portfolio such that we attempt to minimise the variance and maximise returns.
What if instead we’re in a scenario where the main goal is to neutralise our position, how would we construct a portfolio?
- In other words, we take the log-returns of stock A and stock B. Their statistics are
$$\mu_A=0.01 , \mu_B=0.02$$
$$\sigma_A = 0.2 , \sigma_B = 0.3$$
and correlation $\rho=0.8$. If we’re long stock A $\\\$200,000$, what would the optimal “hedge” amount be for stock B?
- If we decide to also minimise the variance, how would we find the efficient frontier of the variance and neutralising wealth?
## Answer by lehalle (score 3, accepted)
https://quant.stackexchange.com/a/79052
The answer depends on your definition of "neutralise".
For some people it is a matter of being dollar neutral. Then the answer is straightforward.
In your case it seems that you have also in mind a concept of zero expected returns. In this case the solution would be to invest 1 in $A$ and $-\mathbb{E}(r_B)/\mathbb{E}(r_A)$ in $B$. This is not a common choice.
For others, it means market neutral (or factor neutral with respect to a list of factors, that can be obtained via a PCA, or an expert-driven approach --like Axioma or BARRA factors--). In this case, you want to orthogonalize the basket made of a fraction $p$ of the instrument $A$ and $(1-p)$ of the asset $B$ (in dollars) with respect to the space spanned by the returns of our list of factors $F_1,\ldots,F_K$. You just have to project our basket on factors (say $\pi_A$, rest. $\pi_B$, is the projection of one dollar in risk of $A$, resp. $B$, projected on your set of factors), and to write that you search for $p$ such that $$p \pi_A + (1-p) \pi_B=0.$$ But since the projections are vectors, it is not achievable in general, hence you should target $$\min_p \|p \pi_A + (1-p) \pi_B\|^2.$$
The connect this with your question about the variance. Let say you want to minimise the variance, i.e. $$\min_p (p r_A + (1-p) r_B)^\top \Omega\; (p r_A + (1-p) r_B),$$ where $\Omega$ is your covariance matrix for all the assets, and assume you can diagonalise our matrix $\Omega$ by block:
- one block is made of your factors $F_1,\ldots,F_K$
- and the rest is the standard PCA of the orthogonal of these factors.
If you work on the formulation, you will see that this expression can be split in two parts: one that corresponds to $F_1,\ldots,F_K$, with a variance coefficient $\lambda_F$, and another that corresponds to the orthogonal: $$(p r_A + (1-p) r_B)^\top \Omega\; (p r_A + (1-p) r_B) = \lambda_F^2 \|p \pi_A + (1-p) \pi_B\|^2 + \lambda_0^2 \|p \pi'_A + (1-p) \pi'_B\|^2,$$ (where $\pi'$ is the projection in th orthogonal of $F_1,\ldots,F_K$) meaning that minimising the variance and being neutral to a set of factor is equivalent if your set of factors capture the essential of the variance of usual moves of returns.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.