hull 05.11why averaging dividend yield is correct?

[Pflik]

in the question 05.11 the dividends over the months are averaged. However why is that correct?

In my calculation i assumed a constant yield of 2 percent and a excess yield of 3 percent for months (5-2 = 3) for months august and november.

i.e. my formula was (S0-I)e^(rf-i)

"I" was the return in august and november, discounted back to july 31 (respectively 5.41 and 5.41)

so it would have been (1300-5.41-5.41)e^(9-2) == 1327.2019 which is different from 1331.80

in the solution proposed in the solution is to average the yields, but that doesn't take into acount that the percentages should be discounted at different times.... or am i missing something?

 

[David Harper CFA FRM]

Hi Pflik,

Aren't you implicitly assuming discrete dividends, which is actually understandable and more realistic than the Q&A? The reason the solution works is due to the time-additive nature of continuous compounding/discounting, and the specification (in Hull 05.11) of dividends is actually meant to be continuous (if the question is not explicit, then the answer at least betrays this intention!).

So, we could break this down,

• Aug implied forward price = 1300*exp[(9%-5%)*1/12] = 1304.341,
• Sep implied forward price = 1304.341*exp[(9%-2%)*1/12] = 1311.971,
• Oct implied forward price = 1311.971*exp[(9%-2%)*1/12] = 1319.647 ...

so mathematically it is a produce sequence of exponents: 1300*exp(.)*exp(.)*exp(.) ...
... where exp(a)*exp(b) = exp(a+b) allows for the averaging

we could even play with it by computing a forward price with only the risk-free rate: F(dec) = 1300*exp(9%*5/12) = 1349.676
Then reducing by the dividends, one month at at time:

• 1349.676 *exp(-5%*/12) = 1344.064; i.e., the implied forward discounting Aug 5% dividend, then:
• 1344.064*exp(-2%*/12) = 1341.825 ; i.e., the implied forward discounting Sep 2% dividend, then:
• 1341.825*exp(-2%*/12) = 1339.591 ; i.e., the implied forward discounting Oct 2% dividend, then:
• 1339.591*exp(-5%*/12) = 1334.021 ; i.e., the implied forward discounting Nov 5% dividend, then:
• 1334.021*exp(-2%*/12) = 1331.799 ; i.e., the implied forward discounting Dec 2% dividend, then

But exp(-5%/12)*exp(-2%/12)*...*exp(-2%/12) = exp(average of dividends*5/12) = exp(3.2%*5/12). BTW, you can still assume discrete, discount and sum to an (I), per your method, that should work but (I think) should produce a nearby approximation. I hope that explains.