# 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.

• Hi: based on what you explained, it's not clear ( atleast to me ) what exposure you are trying to hedge. you are overall slightly long equities and definitely long US currency. In order to figure out a transaction to makes you not exposed to something, you need to be clear on what the something is that you don't want to be exposed to. Apr 10, 2022 at 10:42
• sure, you can assume that Asset A is Ethereum and Asset B is Bitcoin, how do you hedge with the least amount of cost a leveraged position, taking advantage of the fact that Ethereum and Bitcoin are 85% correlated... I want to make this debt stable or indifferent to price changes, of course some assumptions will be need to be made here on the stability of the relationship between A and B Apr 11, 2022 at 7:33

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.

• Can you please confirm if my understanding on how to implement: (1) get an estimate of the variance of the components of the portfolio (2) get an estimate of the correlation between the components of the portflio (3) minimize the portfolio variance to compute w_a and w_b Few things here, I don't think we used the leveraged situation of debt I think. Also the objective is to try and achieve a hedge with the least amount of capital if possible. Apr 12, 2022 at 7:50
• HI: 1), 2) and 3) sound correct. Yes, I didn't use the leveraged situation of debt because I thought that the transaction described was financed by debt. I probably am understanding the problem perfectly but hopefully that gives you the idea-direction to go in. As far as least amount of capital, the amount of capital is proportional to the positions being taken so, as far as I can tell, it's not something that can be minimized because for every dollar you are in your positions, you need another dollar of debt. Apr 12, 2022 at 13:27
• I only really provided the statistical way of looking at it. The implementation details are somewhat unclear to me. Hopefully someone with practical experience in this can take my formula and apply it in a way that is more satisfactory and addresses your questions. Apr 12, 2022 at 13:29
• @Raginald Avto: In my comments above, I meant to write: "I am probably not understanding the problem perfectly ...". Apr 13, 2022 at 5:26