Average dividend yield for calculating futures price

[Liming]

Dear David,

I have encountered a phrase called "average dividend yield" when doing the practice question 05.11 from Hull. I couldn't understand what it means, can you kindly explain its definition as well the implication of this to calculation of price for a futures on stock index. For your reference, the original question is Hull.05.11 from one of your practice questions "FRM 2009 Practice Questions: Options, Futures & Derivatives, Hull Chapter 1-5".

Thank you for your enlightenment!

Cheers!
Liming

29/10/2009

 

[David Harper CFA FRM]

Hi Liming,

(I hope you are doing well!)

It is an elegance of the continous compounding assumption; where under the cost of carry, if the dividend is expressed as a constant % of spot, we are effectively assuming continuous compounding, along with the rate. In this way,

forward = spot * EXP(rate - dividend)

if the dividend is expressed this way, it is the negative of the rate (i.e., both are assuming continous compounding)

the elegance of this is, as you likely know, if we compound over two periods:
EXP(r1*T1) * EXP(r2*T2)

it is equal to EXP(r1*T1 + r2*T2); i.e., we can simply add the exponents

if the dividends are continous, they can be similarly treated, only they are negatives in regard to the forward (since the long forward will not receive them; they are benefits of ownership):

because exp(-div1*T1) * exp(-div2*T2) = exp(-div1*T1 - div2*T2)
...the average dividend is simply the weighted average
...in this case, exp(-2%*3/12) * exp(-5%*2/12) = exp(-2%*3/12 - 5%*2/12) = exp(-1.33%) over the 5 (3+2) months
... and 1.33% over 5 months = 1.33% *12/5 = 3.2% per "average" per annum

so i have taken the long way around to show (hopefully) why Hull is able to simply use (2%*3 = 5%*2)/5 = 3.2%; i.e., it's really due to the elegance of a continuous assumption, and the fact that under such the dividend is effectively a negative interest rates, and exponents add

Hope that helps, David