Minimum-Variance Hedging for a Leveraged Basket Liability

# Leveraged Porfolio Hedging


# Leveraged Porfolio Hedging












What is the right approach to hedge debt of 1 dollar who's value changes based on a basket composed of:

- 32 cents of short Asset A

- 26 cents long Asset B

- 43 cents long usd

The debt is leveraged by 2.6x, meaning if asset A's price goes down by 1%, ceteris-paribus, debt goes up by 2.6 * 32% * 1%. My crude attempt at it is to use the correlation between Asset A and Asset B, which historically is stable around 85%. And hedge by shorting 26 cents of Asset A and holding dollars with 74 cents, computed by 2.6*(32%-26%*0.85) and rebalance with this calculation. Or are there better approaches to this, since it seems to me, if the correlation were to breakdown to zero, this computation suggests not holding any of Asset B, which doesn't seem logical.

## Answer by mark leeds (score 0, accepted)

https://quant.stackexchange.com/a/70494

I'll just explain how I would do this without including the currency and making the assumption that the 32 cents of asset A represents one share and the 26 cents of asset B represents 1 share. (if not, you can figure out what amount of shares they actually are and modify below accordingly. ).

Let $r_{a}$ equal the return of stock a and $r_{b}$ = the return of stock b.

Then, the variance of the resulting portfolio is:

$var( w_{a} \times r_{a} + w_b \times r_{b}) = w_{a}^2 \times \sigma^2_{r_{a}} + w_{b}^2 \times \sigma^2_{r_{b}} + w_{a} w_{b} \times \rho_{ab} \sigma_{r_a} \sigma_{r_b} $

So, in order to minimize the portfolio variance, you would need estimates of

$\sigma^2_{r_{a}}$, $\sigma^2_{r_{b}}$ and $\rho(a,b)$.

Then, you would minimize that expression for the variance but you will still need another constraint involving $w_{a}$ and $w_{b}$ since you currently only have one expression and 2 unknowns.

This other constraint could represent how much you want to be totally long or short. So, assuming the correlation is positive, then you know that you will be short B and long A. So the second constraint might be that $abs(w_{a}) - abs(w_b) <= 0.5 $ meaning that the weight of a ( which is positive ) in the portfolio can't be more than 0.5 of the weight of b ( which will be negative ).

As far as the currency is concerned, once you figure out the $w_{a}$ and the $w_b$ and how much dollars in USD you need to invest to do the stock transaction of long a and short b, then you could just short that much USD in the currency market.

Of course, the result will depend heavily on your estimates of the variances and correlation of the returns of $a$ and $b$. So, you still have similar problems as those that a mean variance markowitz problem has. Someone else may want to comment or add insights because I've never actually done this in practice so there may be practical pitfalls - issues in addition to the instability of the estimates.

EDIT ========================================================

You should probably also add the two constraints that

$abs(w_{1}) <= 1$ and $abs(w_2) <= 1$ since the portfolio weights, regardless of their sign, should not be greater than 1.0.