How to calculate Optimal Hedge Ratio when using more than one type of futures?

[Steve Jobs]

I had this confusion when I asked about the difference between Beta and Optimal Hedge Ration, but did't post it at that time because I thought I should not waste time on statistics and that it might be not included in the exam.

However, there is an example in the study materials explaining that since there is no futures for jet fuel, some companies might use a combination of crude oil futures and heating oil futures. I wonder, how to calculate the Optimal Hedge Ratio in case there was more than one type of commodity future? How the formula will be adjusted?
Optimal Hedge Ratio = p (s,f)* 6(s)/6(f)

I don't know if it's relevant but I remember that in portfolio allocation, the optimal weight of each asset class or and then each firm is calculated by a long formula using the standard deviation and correlation. Is the Optimal Hedge Ratio for more than one type of futures calculated in a similar fashion?

 

[chiyui]

It's just my guess.

If you want to optimally hedge jet fuel by using crude oil futures and heating oil futures, firstly you gotta determine which futures has a higher correlation with jet fuel spot.

Suppose you find the correlation of crude oil futures and jet fuel spot is 0.7, while the other one is 0.6. Then you just use crude oil futures to hedge, and just forget using heating oil futures then. Any combination of BOTH futures will reduce the correlation to less than 0.7, which is not what you want to see when hedging your jet fuel spot position. Thus in this case, you just use the ratio formula ρ(Δs/Δf) for crude oil futures only.

Portfolio allocation is another matter. In portfolio allocation, your primary objective is to find a minimum variance asset combination while keeping a desired level of portfolio expected return. But in hedging, your primary objective is to find a minimum variance asset combination and willing to reduce your desired level of portfolio expected return.

There is another point also. If a company want to hedge its jet fuel exposure by using crude oil futures, that's fine. But actually, the crude oil futures position also has some correlation with the company's other exposures as well, e.g. currency, electricity expenses, just name a few. Correlation is only a statistical measure, thus crude oil futures of course has a correlation with these things, no matter how small (or even ridiculous) it is. So in that case, does the company need to consider its other exposures in order to determine its "optimal" number of crude oil futures contracts?

If the company really wants to do this, then I think your guess is correct: just treat the crude oil futures position as if it is an asset class in the company's "portfolio" and use the "long formula" you mentioned. In this case, the company can minimize the "portfolio" variance while fixing the "portfolio" expected return to a given level. But in this case, the company's jet fuel position might lose money (while the other positions might gain money, giving an overall P&L in line with the mentioned level of portfolio expected return).

So I think it depends on what the company wants to achieve. Not every company wants to optimize the whole "portfolio" return/variance trade-off without considering individual exposure limits (like jet fuel exposure). This is because of many reasons we may or may not imagine (perhaps the major shareholders just hate jet fuel exposures because of religious reasons, which is not profit-maximizing!)

 

[chiyui]

BTW, actually you can use beta to calculate the optimal hedge ratio too (and this is the more convenient way to do so if hedging a stock portfolio using index futures).
β = cov(x,y)/var(x,y) = ρ(σx/σy)
So you may use β(σf/σs)(Δs/Δf) also.
FRM part 1 does contain this topic. So I think you have no loss studying it.

 

[Steve Jobs]

Hi Chiyui,

When I was making a presentation of a project for a statistics course in college, the prof. got me on that and said the same thing you mentioned for the correlation between crude oil futures and other variables such currency, electricity, etc.

Thanks and I got the points you explained.

I didn't see a single practice question in Kaplan materials for this matter, so I think it's not included in part 1. It could be in part 2.

 

[chiyui]

In my part 1 exam last November, I remember I have done some questions about optimal hedge ratio using stock index futures. The question provides the stock portfolio beta as an input. So I guess the topic will appear in part 1.

Actually, last time I found there were similar topic questions appearing in both part 1 and 2 (I took both parts in one time). So even though you didn't register for part 2 exam, you may consult part 2 topics in Kaplan for good.

 

[Steve Jobs]

All the practice questions I found were only using 1 type of futures; there was no combination.

 

[chiyui]

Oh I see what you mean.....
Yeah becoz usually people won't consider hedging one exposure with two or more derivatives. People only hedge the exposure by using the futures contract with the highest correlation with that exposure. So usually you won't see any books or materials teaching people to use several types of derivatives to do the hedging.

-----------------------------Below off from the main topic-----------------------------

Just for discussion and not to be precise, I think there is only one situation people will use more than one type of derivative contracts to do the hedging - it is when those people don't want to fully hedge all kinds of risk embedded in that underlying asset.

For example, suppose you have a spot silver long position currently. You want to remove its directional risk (i.e. you want to be silver market neutral), but you don't want to remove its volatility risk. Then you may long a silver call plus a short position in silver futures. Another example is when you want to remove its directional risk just partially. Then you may short a risk reversal. This will hedge large directional movement but won't hedge small movement.

Some of the reasons why people intentionally engage in these complex strategy are, the hedging costs are lower, the hedging is more flexible, and the margin requirement of those derivatives transactions may be lower.

 

[Steve Jobs]

Thanks Chiyui,

It seems that the further we discussing this matter, we're getting into trading strategies of options and futures. However, I understand that there is only one formula for the Optimal Hedge Ratio (at least for the exam concerns) and there is no variations.

 

[chiyui]

The optimal hedge ratio is just derived by the same principle of simple regression, so don't worry there is only one answer. You're welcome and hope you get the FRM exam passed!

 

[Steve Jobs]

Thanks,

 

[Rocks]

Hi,

I am trying to work backwards. I have the optimal number of contracts (which I derived from Solver, Excel) and I want to find the Optimal Hedge ratio. I've rearranged the formula to do that, but what if I am using more than one type of futures contract?

So in my case, I have wheat yields and prices. I am using Wheat futures to hedge against wheat price, now I want to add Rainfall futures to hedge against wheat yields. How would that work? I'm not sure how the formula changes.

Thanks in advance.

 

[ShaktiRathore]

s=b*f+c is the regression of spot price s against its futures price f
let b be the number of futures contracts needed to hedge against any price movement.
taking co-variance on both sides,
Cov(s,f)=Cov(b*f+c,f)
=>Cov(s,f)=Cov(b*f,f)=b*Cov(f,f)
=>b=Cov(s,f)/Cov(f,f)=>b=Cov(s,f)/var(f) is the optimal hedge ratio or the number of futures contracts needed to hedge against any price movement.
if we now consider additional futures of rainfalll we run regression of spot price of wheat against wheat futures and rainfall futures and see the model,
s=b1*f(w)+b2*f(r)+c to perfectely hedge against spot prices we buy b1 futures of wheat and b2 futures of rainfall as is evident from our regression.
taking covariance w.r.t f(r) first on both sides,
Cov(s,f(r))=Cov(b1*f(w)+b2*f(r)+c,f(r))=>Cov(s,f(r))=Cov(b1*f(w),f(r))+Cov(b2*f(r)+c,f(r))=b1*Cov(f(w),f(r))+b2*Cov(f(r),f(r))
Cov(s,f(r))=b1*Cov(f(w),f(r))+b2*Var(f(r)) ...1
again taking covariance w.r.t f(w),
Cov(s,f(w))=b1*Cov(f(w),f(r))+b2*Var(f(w)) ..2
from eqs 1 and 2 solve for b1 and b2 which are the optimal hedge ratios for futures and wheat contacts for hedge against spot prices.
b2=[Cov(s,f(r))-Cov(s,f(w))]/[Var(f(r)) -Var(f(w))]
and b1=[Cov(s,f(r))*Var(f(w)-Cov(s,f(w))*Var(f(r))]/[Cov(f(w),f(r))*(Var(f(w))-Var(f(r)))]

thanks