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?

  1. 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?

  1. If we decide to also minimise the variance, how would we find the efficient frontier of the variance and neutralising wealth?

1 Answer 1


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.

  • $\begingroup$ When I meant neutralise, I meant dollar neutral, ie the expected PnL is $0$. Mainly for the situation let’s say a vendor, has a long position on AAPL, but there is no way to liquidate the position, so they try to find the most highly correlated asset, let’s say MSFT, what would their MSFT hedging amount be? I brought up variance because correlation doesn’t give the magnitude of the returns so I assumed you would need to incorporate variance to find the right hedging amount. $\endgroup$
    – Xerium
    Commented Apr 20 at 0:43
  • $\begingroup$ Ie if AAPL and MSF have 100% dependence, but AAPL has 2x deviation, then your MSFT position needs to be 2x of AAPL to be neutral. $\endgroup$
    – Xerium
    Commented Apr 20 at 0:51
  • $\begingroup$ @Xerium you make a confusion between the returns and the volatility. You can have AAPL that as an expected return that is twice the one of MSFT but the volatility of AAPL is one third of the one of MSFT. If you want to be dollar neutral, the only way is the one I cited. If you want to have an expected return of zero, the only way is to use the ratio of expected returns, and if you focus on volatility, the natural way is to minimise the variance of the obtained portfolio. $\endgroup$
    – lehalle
    Commented Apr 22 at 7:26
  • $\begingroup$ is it not just a beta-neutral portfolio? Take the covariance of AAPL and MSFT and then divide by the variance of MSFT. Then find the weight that gives a zero beta? And thank you for the help. Really appreciated it $\endgroup$
    – Xerium
    Commented Apr 23 at 11:51
  • $\begingroup$ @Xerium it is the case if you are in dimension one (one beta since you believe there is only one factor driving all market risk), in general you need a multi factor model, and then the "ratio approach" you propose does not work anymore: you need to solve the minimisation problem I explain in my answer. $\endgroup$
    – lehalle
    Commented Apr 23 at 14:03

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