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Must dividend yield be continuously compounded

Jagan.Ganti

New Member
Hi David,

In reference to the basic Forward/Future formula for dividend yield (i.e. F0 = S0 * e^((r-q)*T)), must the dividend yield be continuously compounded (CC) given that the risk free rate is in the CC form.

This is in reference to the following questions from JC Hull:
The risk free rate of interest is 7% p.a. with continuous compounding and the dividend yield is 3.2% p.a. The current value of the index is 150. What is the six-month futures price?

Thank you.
Jagan
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
Hi @jag.aus.gvs It's a good question, and, the answer is: no, the dividend yield's compound frequency does not need to match the riskfree rate's compound frequency. We've found the cost of carry (COC) model to be almost infinitely flexible: not only is mixing compound frequencies fine, but we can mix/match lump sum costs/income (any future stream can be translated into its equivalent lump-sum, and vice-versa). In the case of your cited question ...
Hull 5.10. The risk free rate of interest is 7% p.a. with continuous compounding and the dividend yield is 3.2% p.a. The current value of the index is 150. What is the six-month futures price?

... for example, we can translate the continuous (C.C.) 3.20% dividend into, say, its quarterly equivalent: 4*[exp(3.20%/4)-1] = 3.21283% per annum with quarterly compounding. Then apply COC to retrieve the 6-month futures price but "mixing" the continuous riskfree rate with the discrete dividends (and realistically, dividends are paid discretely after all!):

$150.00*exp(7.0%*0.5)/(1+3.21283%)^(0.50*4) = $150.00*exp(7.0%*0.5)*(1+3.21283%)^(-0.50*4) = $152.8772473.
Exactly the same answer as the solution to the original problem.

I must give credit to @MilaBank as I was always doing the translation (into CC; or at least to match the compound frequencies) until I realized it's not really necessary, see https://www.bionicturtle.com/forum/...f-carry-model-hull-chapter-5.10601/post-85488 It may seem obvious to see it, but for many years I didn't realize you could just "tack" on each factor in the COC with its own frequency. I hope that's helpful,
 
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