Choosing Hedge Weights for Negatively Correlated Assets

# hedging correlated instruments


# hedging correlated instruments












If two instruments have a significant negative correlation but the percent change in the price of the instrument moving in positive direction is always more by a fraction than the one moving in negative direction, in what ways can we leverage such correlations to minimize our risk? How would a hedging strategy look like in such situations? Simply buying both the instruments but a little more quantity of the instrument which you think will move in positive direction should work if I am using my common sense. But need some pro advise and/or tips to pointing me in the right direction to master the art of hedging in such situations.

## Answer by Richi Wa (score 2)

https://quant.stackexchange.com/a/16734

This sounds like quadratic hedging. If you have the return of the assets $r_X$ and $r_Y$ with negative correlation $\rho$ between the two (we could think of bonds and stocks) and more variance in one of them then the problem of weighting the two by $w$ is (assume zero expected returns for ease of presentation) $$ \text{risk} = E[(w r_X + (1-w) r_Y)^2] \rightarrow \text{Min} $$ Expanding the square we get $$ \text{risk} = w^2 E[r_X^2] + 2 w(1-w)E[r_X r_Y]+ (1-w)^2 E[r_y^2] =\\ w^2 \sigma_X^2 + 2w(1-w)\rho \sigma_X \sigma_Y + (1-w)^2 \sigma_Y^2. $$ Then we take the derivative w.r.t. $w$ and get $$ \frac{d}{dw} \text{risk} = 2 w\sigma_X^2 + (1-2w)\rho \sigma_X \sigma_y + 2 (-1+w) \sigma_Y^2. $$ Setting this (linear equation in $w$) to zero we get $$ w = \frac{\sigma_Y^2 - \sigma_X\sigma_Y \rho}{\sigma_X^2 + \sigma_Y^2 - 2\sigma_X\sigma_Y \rho}. $$

Why is this intuitive? First note that the numerator in $W$, the weight of $X$ increases if the risk of $Y$ increases and increases even more with negative $\rho$.

The approach above focuses on risk and will weight assets with less risk higher, if you want to introduce expected return also, then things get slightly more complicated.

Just looking at expexted return is easy. $$ \text{target} = E[w r_X + (1-w) r_Y] \rightarrow \text{Max} $$ has a trivial solution buy the asset with the bigger expeted return with $100\%$.

What you can do is combine the two by $$ \text{target} = E[w r_X + (1-w) r_Y] \rightarrow \text{Max} $$ constraint to $\text{risk} \le l^2 \%$ with some maximum level of risk $l$ (I put the square because above we were looking at variance).

Then you start from $100\%$ in the asset with the highest expected return and decrease the weight until you get below the desired level of risk.

## Answer by Jonas K (score 0)

https://quant.stackexchange.com/a/16722

One way to do this is to weight the assets in the portfolio to make it beta neutral against some benchmark. This would minimize risk in terms of the benchmark.

$W_i = \left | \frac { \beta_i } { \sum_{}{} \left | \beta \right | } \right | = \left | \frac{ Cov(R_i, R_m) / Var(R_m) } { \sum_{i}^{n} \left | Cov(R_i, R_m) / Var(R_m) \right | } \right |$