Futures # of contracts - Risk Free rate or not?

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pollfre

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Hi guys,
I get confused when calculating the # of futures contracts to hedge an equity position. When do you account for the risk free rate?
I would usually use this formula:
(equity position to hedge/value of futures) x (beta of equity position/beta of futures)
but apparently this is incorrect as you should be multiplying the equity position by (1+Rf)^t over the life of the futures.
Can somebody clarify when to use Rf? And why we account for Rf?
Thanks!
 
You incorporate Rf into these formulas when creating syntheic positions (sythetic cash from equity, synthetic cash from fixed income, synthetic equity from cash…can’t recall if we creat synthetic fixed income from cash). The key is to remember that cash grows at the risk free rate. When adjusting the fixed income or equity allocations, betas, durations or any combination of the four, you use your target dollar beta or target dollar duration, ie we are not creating a synthetic cash position - no use of the Rf growth of cash.
 
So if synthetic position (equity=>cash; cash=>equity; bonds=>cash; cash=>bonds), use Rf. Else not. Correct?
 
I believe that is correct. Although I don’t want to overgeneralize and say that questions that ask you to adjust allocations or beta / duration will not also include creating an allocation of syntetic cash. For me, the easiest approach is just to remember that cash grows at the risk free rate. Other than that, the formuals for adjusting allocation or market sensitivity of fixed income and equity are structured identically, its only a matter of using beta for equity and modified duration and yield beta for fixed income (and obviously (CTD) futures contract price and any applicable mulitplier to get the $Beta/$Duration of the contract).
 
i guess this is the key from S2000:
“The upshot is that if you start or end with cash, you use the future value; otherwise, you use the present value.”
 
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