Correlation and Hedging

A mean–variance optimizer will hedge correlated assets. To see this, recall that the objective function of a basic mean–variance optimizer is

$\mathbf{w}^{\star} = \arg\!\max_{\mathbf{w}} \left\{ \mathbf{r}^{\top} \mathbf{w} - \gamma \mathbf{w}^{\top} \boldsymbol{\Sigma} \mathbf{w} \right\}, \quad \text{subject to }\sum_i w_i = 1, \quad \gamma \gt 0, \tag{1}$

where $\mathbf{w}$ is a vector of porfolio weights; $\mathbf{r}$ is a vector of expected returns; $\boldsymbol{\Sigma}$ is the covariance matrix of the asset returns; and $\gamma \gt 0$ is the risk-aversion parameter, so-named because it weights the risk term. Often, we might place additional constraints on $\mathbf{w}$, such as position limits, but that is not particularly interesting here. So in words, we want to find the optimal positions ($\mathbf{w}$) such that we maximize our portfolio’s expected return or “reward” ($\mathbf{r}^{\top} \mathbf{w}$) while minimize the volatility of that return or “risk” ($\mathbf{w}^{\top} \boldsymbol{\Sigma} \mathbf{w}$). See my post on mean–variance analysis for a deeper discussion of this framing.

The goal of this post is to understand the behavior of a mean–variance optimizer when dealing with correlated and anti-correlated assets. To simplify things, let’s stick to a portfolio with only two assets, which are correlated with coefficient $\rho$.

In the two-asset case, the covariance matrix is

$\boldsymbol{\Sigma} = \begin{bmatrix} \sigma_1^2 & \rho \sigma_{1,2} \\ \rho \sigma_{2,1} & \sigma_2^2 \end{bmatrix}, \tag{2}$

where $\rho$ is the aforementioned correlation coefficient; $\sigma_i^2$ is the variance of asset $i$; and $\sigma_{i,j}$ is the covariance between assets $i$ and $j$. The portfolio variance $\sigma_p^2$ is

$\begin{aligned} \sigma_p^2= \mathbf{w}^{\top} \boldsymbol{\Sigma} \mathbf{w}= w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \rho \sigma_{1,2}. \tag{3} \end{aligned}$

Now imagine that these two assets are perfectly correlated. Then $\rho = 1$, and the portfolio variance is

$\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \sigma_{1,2}. \tag{4}$

If we choose $w_1 = -w_2$, then we have

$\begin{aligned} \sigma_p^2 &= w_1^2 \sigma_1^2 + w_1^2 \sigma_2^2 - 2 w_1^2 \sigma_{1,2} \\ &= w_1^2 \left(\sigma_1^2 + \sigma_2^2 - 2 \sigma_{1,2} \right). \end{aligned} \tag{5}$

And our expected return is

$w_1 (r_1 - r_2), \tag{6}$

Clearly, this position may reduce our portfolio variance, and this works because we took an opposite position in the two positively correlated assets. This makes intuitive sense. If two assets are perfectly correlated, then they are the same asset in some sense, just scaled by their respective idiosyncratic variances. So we can hedge them against each other, with a net long position in the asset with the higher expected return.

Next, imagine these two assets are perfectly anti-correlated. Then $\rho = -1$, and we just pick $w_1 = w_2$. Then our portfolio variance is again Equation $6$, while our expected return is

$w_1 (r_1 + r_2). \tag{7}$

Notice that negatively correlated assets are even better than correlated assets! With correlated assets, we can hedge out our risk, but we drive down our expected return (Equation $6$). With negatively correlated assets, we can capture all the expected return while hedging out our risk.

Generalizing this to $n$ assets is easy. Simply observe that the portfolio variance can be written as a sum:

$\sigma_p^2 = \mathbf{w}^{\top} \boldsymbol{\Sigma} \mathbf{w}= \sum_{i=1}^n \sum_{j=1}^n w_i w_j \rho_{i,j}\sigma_{i,j}. \tag{8}$

When $i=j$, the term is simply $w_i^2 \sigma_i^2$. These idiosyncratic variances cannot be eliminated here. But when $i \neq j$, then the term is $w_i w_j \rho_{i,j} \sigma_{i,j}$, and our reasoning from the two-asset case applies. In other words, with $n$ assets, the portfolio’s variance dceomposes into a sum of idiosyncratic and cross terms, and we can simply apply our reasoning from the two-asset case to each term.

All that said, there’s a big caveat here, which is the risk-aversion parameter $\gamma$, as this changes the optimizer’s trade-off between risk and reward. So the reasoning above does not actually apply to all scenarios, but I think it’s a useful way to think about a common case.

We can visualize these trade-offs with a simple experiment. In Figure $1$, I have computed the optimal portfolio weights $w_1$ and $w_2$ over a range of correlations, assuming fixed $\sigma_1 = \sigma_2 = 1$ and expected returns $r_1=1$ and $r_2=0.1$.

Figure 1. A mean–variance optimizer's proposed portfolio weights $w_1$ and $w_2$ as a function of the correlation $\rho$ between two assets and as a function of risk-gamma $\gamma$. The first asset has an expected return of $1$, while the second asset has an expected return of $0.1$. Both assets of have a volatility of one.

We can see that when correlation $\rho$ is negative, we go long both assets. And when correlation $\rho$ is positive, we take a long position in the asset with the higher expected return, and we then hedge via a short position in the other asset. However, the risk-aversion parameter changes when this trade-off between capturing reward and hedging risk makes sense. When the risk-aversion parameter is higher, the optimizer demands more correlation before going long-short. My intuition here is that when the risk term is overweighted, the optimizer must put more emphasis on capturing the positive returns by going long-long for a broad range of the correlation spectrum.