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# Optimal Hedge Ratio Correlation Understanding

#### materio

##### New Member
Hi,

I have a doubt about the meaning of the hedge ratio.

Hedge ratio = ρ * σ_spot / σ_fut

Number of contracts = HedgeRatio * PortfolioValue / ValueFuturesContract

Therefore, the lower the correlation, the lower the number of contracts.

So, let's say that I have a portfolio of $1.000.000 of Crude Oil (σ_spot = 100) that I want to hedge. I have two products that I can use to hedge it: - 1. Crude Oil futures: ρ = 0.99 σ_fut = 100 Value Futures Contract =$ 1.000​
- 2. Weather futures:
ρ = 0.10
σ_fut = 100
Value Futures Contract = $1.000​ If I calculate the number of contracts to hedge the position for each product I have: 1. N=0.99*100/100*1.000.000/1.000=990 2. N=0.10*100/100*1.000.000/1.000 = 100 How is it possible that with less correlation the number of contracts will be less? Would it cost less to hedge the position with a less correlated product? It is counterintuitive for me. What am I missing? Thank you in advance. #### bpdulog ##### Active Member Hi, I have a doubt about the meaning of the hedge ratio. Hedge ratio = ρ * σ_spot / σ_fut Number of contracts = HedgeRatio * PortfolioValue / ValueFuturesContract Therefore, the lower the correlation, the lower the number of contracts. So, let's say that I have a portfolio of$ 1.000.000 of Crude Oil (σ_spot = 100) that I want to hedge. I have two products that I can use to hedge it:
- 1. Crude Oil futures:
ρ = 0.99
σ_fut = 100
Value Futures Contract = $1.000​ - 2. Weather futures: ρ = 0.10 σ_fut = 100 Value Futures Contract =$ 1.000​

If I calculate the number of contracts to hedge the position for each product I have:

1. N=0.99*100/100*1.000.000/1.000=990

2. N=0.10*100/100*1.000.000/1.000 = 100

How is it possible that with less correlation the number of contracts will be less? Would it cost less to hedge the position with a less correlated product? It is counterintuitive for me. What am I missing?

This is driven by the modern portfolio concepts in the Foundation of Risk Management section. If you have a portfolio of assets that have low correlation with each other, the overall standard deviation (and variance) of the portfolio will be lower because the securities "naturally" hedge each other. Just think of your hedge as another asset in your portfolio, with the intention of reducing beta, duration, variance, etc. For example, if I have a portfolio of oil stocks and I hedge with crude futures, correlation will be high because the portfolio and the crude futures will move in the same direction. If I choose to hedge with S&P500 futures, I will need less futures contracts because impacts from the oil sector will not affect the index as much and will have lower correlation. However, the basis risk will be higher. In addition, this ignores the movement of all correlations toward 1 in a financial crisis.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @materio This is a provocative scenario. To add to @bpdulog 's answer, this classic approach (i.e., solving for the the number of contracts based on the minimum variance hedge ratio) is solving for the minimum variance of a portfolio with two positions: the underlying exposure (in your example, $1.0 million in spot crude oil) plus a futures position. In the case where your rho is nearly 1.0, you can can almost achieve portfolio variance of zero; so, your futures contract position will be large with exposure to almost as many barrels and your underlying long. For example, if the oil price is$40.00 per barrel, then the $1.0 million position is 25,000 barrels. If the ρ is 0.99, then the minimum variance portfolio (i.e., where portfolio = exposure + hedge) is achieved by a short position in futures contracts on almost 25,000 barrels (if ρ=1.0, your futures would short 25,000 barrels and you'd have a perfect hedge) with a portfolio variance that will be closer to zero than to the variance of the underlying exposure. However, consistent with @bpdulog 's point, if the rho is only 0.1, you won't be able to reduce your portfolio variance very much below the variance of the original exposure. This isn't exactly the same as investing$1.0 million and reducing the variance of the invested $1.0 million by diversifying; here, you have a$1.0 exposure and you are adding a (hedge) position, so a hedge instrument with low correlation will quickly increase your position's variance! So, here where correlation is rho, there are fewer contracts to utilize in the hedge because the net variance can only be reduced a little, compared to the superior hedge instrument. Because it uses (and can use) more contracts, the net variance of our first hedged position is significantly less than the net variance of your second position. It's very interesting, I think!

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#### materio

##### New Member
Thank you @bpdulog and @David Harper CFA FRM for you answers. For what I understand, it is all about minimizing the variance of the portfolio, and I was focusing on the hedge concept.

So, given the following formula for the variance of the portfolio: Of course, the best way of reducing the variance is to minimize the correlation.

In the second example I stated, trying to hedge the spot crude oil with weather futures could barely be called a hedge, but the variance of the portfolio would be minimum.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @materio Your formula should subtract the final term as we will short the futures contract; otherwise this is the correct idea. When the final term--that is, 2*x(1)*x(2)*ρ(1,2)*σ(1)*σ(2)--subtracts, we can see that ceteris paribus any higher correlation will reduce the total portfolio variance. Below illustrates the difference. The almost-zero correlated contracts are still volatile so adding more of them tends to increase the portfolio variance (y-axis; my intervals are rough so you can't really see that the righthand variance is local minimum at 100 contracts with variance of \$994,987). I hope that's interesting! 