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.

• What exactly do you want to hedge? It sounds a little bit as if you were just talking about classical portfolio optimization which makes use of the covariance matrix. – vonjd Feb 24 '15 at 17:46
• @vonjd hedge to minimize risk? hedge my positions in instruments to minimize risk? – user793468 Feb 24 '15 at 18:06
• I am asking you! How do you define risk? – vonjd Feb 24 '15 at 18:14
• @vonjd Here we would be taking an offsetting position in related assets to profit from relative price movements. I wouldn't want to miss out on maximizing profits, but at the same time I would like to minimize my losses. Here I would say hedge risk(not buying optimal quantities of instruments) to minimize loss would be a risk for me. Hope I was able to explain clearly? – user793468 Feb 24 '15 at 19:26
• @vonjd will look into MPT – user793468 Feb 24 '15 at 20:47

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.

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.

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

• any way to hedge the risk when the correlated instruments directions are reversed? – user793468 Feb 24 '15 at 20:56