Targetting total duration of a portfoilio

[Liming]

Dear David,

There is a very basic formula that I'd like to double check with you, which concerns the targeting of total portfolio duration. The formula is shown on page 307 page of FRM 5th Handbook under the section ' Duration Hedging", where it says that in order for us to achieve a target duration Dv for an existing portfolio S (modified duration is Ds*) though futures hedging (modified duration of the future is Df*), the number of futures to short or long is:
N = (Dv*V - Ds*S) / (Df*F)
The point I'd like to double check is whether V = S+ NxF in this formula and S = existing portfolio value and F = contract size for one future contract. My point is that since future contracts will be entered into by the hedger, they should form a part of the new portfolio too, leading a change in the final portfolio value. However, I doubt my own thought since V = S+ NxF would mean that we have to transform this formula again to solve for N since N will appear on both sides of the equation.
Can you kindly correct my misunderstanding? And do you have any practice question that uses this formula, except those for targeting 0 duration?

Thanks for your enlightenment!

Cheers!
Liming
1/11/2009

 

[David Harper CFA FRM]

Hi Liming,

I *think* the assumption here is that the (initial) value of the futures contract = 0 (that is our typical assumption for the futures *value*, as opposed to delivery price, at inception) which may resolve?

I am not entirely sure I follow your question, however, it may be helpful to simplify what Jorion is doing here. The great thing about dollar durations is the can be added or subtracted. So, in regard to p 307, it seems to me here is using this formula:

DollarDuration(Cash Position) + N*DollarDuration(1 futures contract) = DollarDuration(Hedged Portfolio)
where
Hedged Portfolio = Cash position + futures, and
N = number of futures contract

i.e., he is just saying that you add the dollar duration of the total futures contracts (N*) to the cash position, and the sum is the dollar duratio of what I would call "the hedged porfolio" (cash + N futures contract)

to me, it is easier to view this as a simply sum, then it is a matter of solving for N, given the desired duration:
N = (DollarDuration(Hedged Portfolio) - DollarDuration(Cash Position)/DollarDuration(1 futures contract)

and the "optimal hedge" is just the special case where the hedged portfolio dollar duration = 0 such that
N = (0 - DollarDuration(Cash Position)/DollarDuration(1 futures contract)
= - DollarDuration(Cash Position)/DollarDuration(1 futures contract)

...please advise if i missed your point?
...in regard to questions, I definitely have hedging questions in the question bank i just current do not have time to pick thru to find them, sorry

Thanks, David