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Optimal Hedge Ratio and Beta, are they the same?

Steve Jobs

Active Member
There is an illustration by formulas in the books showing how the Optimal Hedge Ratio(OHR) is equal to Beta. I found the idea very confusing because OHR is used for hedging and Beta is a symbol of systematic risk and risk should be reduced always unless the return is increasing.

Also, the formulas for the number of contracts to buy/sale for OHR, tailing, Beta and (even Duration Based hedging) are all almost the same. The total Asset value is divided by futures price and multiplied by x(which could be OHR , tailing, Beta and Duration Based hedging)

The Beta here is the same one we studied in econometrics/finance principles course which can be calculated manually or generated in excel by regressing one company stock prices against the market index. Right?

But in FRM book 3, or in banking course, instead of giving us the whole series of data (S and F prices for a specific period), the correlation and standard deviation are provided(which are calculated again by the author on excel based on data series) to students to calculate the OHR which is actually the same as Beta. So the OHR formula is the shortcut to Calculate Beta? Unless for historical reasons, why to introduce the Beta again in a different way?

Also the following observations:

Tailing: Since the used correlation and standard deviation are not updated to include the new prices, we somehow are updating them by multiplying them by the new prices. Right?

Duration based hedge: This is not the same as Beta, since it’s not being calculated based on historical data but rather on PV of future cash flow, so for sure it should be updated by time as cash flows are received/paid.

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
It's a whole layer cake, but just to bite off the easiest, most exam-relevant piece, I think beta is the symbol for a generic partial slope coefficient (multi-variate regression). The rest are generally special cases. Including a univariate regression, Y = bx + a, which illustrates that (like correlation) beta is a linear relationship between any variable and another, specifically cov(xy)/var(x) = correlation(xy)*vol(y)/vol(x). This can be any two variables; for example, in analytical portfolio VaR, beta(position, portfolio including the same position) informs marginal VaR. When we encounter this marginal VaR-beta, I think some get confused because they give beta too much respect beyond it's simple partial slope status: it's just a rescaled correlation, so it's between anything even semi-self referencing variables.

In another, albeit famous "use case," the regression (relationship) is between a security and a portfolio of all risky assets; this beta happens to be the special case of CAPM beta. Which, being hard to measure, is further proxied with a market index like S&P 500 (which is not at all the same thing as the theory). But i think Hull Chapter 3 keeps us grounded in the realization that beta is "merely" the slope of a regression line, which consequently is the minimum variance hedge ratio (and this keeps us mindful of it's limitation as a linear artifact). I hope that's helpful w.r.t the first layer of a fascinating cake .... Thanks,

Steve Jobs

Active Member
Thanks David, so I'll think about it as:
The number of Contracts = (Total Assets Value/Security Price) x Relation
The first part of the above formula is intuitive and the Relation is Beta, Optimal Hedge Ratio, Tailing, Duration Based Ratio, etc.

Another point is the selection of futures for hedging. The books explain about basis risk which is the result of difference in price volatility, delivery and closing dates, etc, but no statistical method is provided for selection. Focusing only on the price volatility of the basis risk, can we say that the best securities to purchase/sell for hedging are the one with high correlation?(either positive or negative)