# How to tail a hedge? (Question 3.26 from Hull, edition 10)

I am new to finance so I apologize if my question is really basic (which it probably is). If this is not the right "stackexchange" group for this, kindly refer me to the right one.

Let's say you own an asset and you want to cross-hedge using futures on a related asset. I'm going to establish a terminology here:

• $S,F$ is the spot price (per unit) of the owned asset, the futures price (per unit) respectively, and $\Delta S, \Delta F$ the corresponding changes during the life of the hedge.
• $Q_A, Q_F$ is the size (in units) of the position being hedged/one futures respectively.
• $V_A, V_F$ is the value of your hedged position, one futures contract respectively.
• $\sigma_S, \sigma_F$ is the standard deviation of $\Delta S, \Delta F$ respectively.
• $\hat{\sigma}_S, \hat{\sigma}_F$ is the standard deviation of the percent one-day change in $S,F$ respectively.
• $\rho$ is the correlation between $\Delta S, \Delta F$
• $\hat{\rho}$ is the correlation between percent one-day changes in $S,F$.
• $h^*, N^*$ is the minimum variance hedge ratio and optimal number of contracts (without tailing).
• $\hat h, \hat N$ are same as above but "with tailing".

According to the book and many other standard books, if we ignore daily settlements, (equivalently if we hedge using forward contracts) the minimum variance hedge ratio, that is the hedge ratio that minimizes the variance of the hedged portfolio, is: $$h^*=\rho \frac{\sigma_S}{\sigma_F}$$ and the corresponding optimum number of contracts is: $$N^*=h^*\frac{Q_A}{Q_F}$$ Here's where it gets confusing (for me at least): in page $62$ of the cited book, the author claims, that because of the daily settlements procedure, is we use futures instead of forwards and want to be accurate, we should actually use: $$\hat h=\hat{\rho}\frac{\hat{\sigma}_S}{\hat{\sigma}_F}$$ in place of $h^*$ and: $$\hat N= \hat{h}\frac{V_A}{V_F}$$ in place of $N^*$.

However, in other sources online, for instance here, $\hat N=h^*\frac{V_A}{V_F}=N^*\frac{S}{F}$. So my first question is:

• Which one is the right formula, and most importantly why?

A secondary question:

• If the formula in the book is to be believed, is it possible to find (estimate) $\hat h, \hat N$ from $\Delta S, \Delta F, \rho, \sigma_S, \sigma_F$, and if yes, how so?

For instance, consider part 4) of the following question (taken from Hull's book "Options, futures and other derivatives", edition 10):

A trader owns $55,000$ units of a particular asset and decides to hedge the value of her position with futures contracts on another related asset. Each futures contract is on $5,000$ units. The spot price of the asset owned is $28\$$and the standard deviation of the change in this price over the life of the hedge is 0.43\$$. The futures price of the related asset is$27\$$and the standard deviation of this over the life of the asset is 0.40\$$. The coefficient of correlation between the spot price change and the futures price change is $0.95$. 1. Find the minimum variance hedge ratio. 2. Should the hedger take a long or a short position? 3. What is the optimal number of contracts when adjustments for daily settlements are not considered? 4. How can the daily settlement of futures contracts be taken into account?

I have no idea how to even answer part $4$.

Thank you all for your time.

• What is this terminology "tail the hedge"? I for one have never heard that.
– dm63
Dec 18 '17 at 18:29
• Hi dm63, it is supposed to be an adjustment to the optimal number of contracts to account for the impact of daily settlements, specifically to account for possible cash inflows/outflows during the life of the hedge. Though, obviously I don't understand exactly how it works...
– Ted
Dec 18 '17 at 18:46
• Edited the question to hopefully make it more readable...
– Ted
Dec 18 '17 at 19:29
• Tailing the hedge: When interest rates are high and the expiration is far away you need fewer futures than the first formula suggests. Reason is today's P&L $\Delta F$ can be reinvested and will be worth $\Delta F \exp(r T)$ at maturity. However as expiration approaches you need to increase your future position gradually until you have the full position at expiry. This is what I learned in skool, in practice no one seems to do this, perhaps bcause i.r. are so low nowadays. Dec 20 '17 at 2:37
• "Tailing the hedge" was invented or publicized by Chicago futures trade Ira Kawaller in I think the 1980's, when interest rates were high. I haven't been able to find his original writeup, however. Anyway, my answer to Q4 would be: buy fewer futures at the beginning and build up the futures position gradually at the rate $r$ as time goes on. Dec 20 '17 at 2:45