Question Hull.05.11

[Hend Abuenein]

Hi David

Referring to the question mentioned, where did the conclusion about the 5% come from?
The question specifies the dividend yield of 2% for May, Aug, & Nov, but says nothing about the yields of the remaining two months.
Is there a convention that we're supposed to know of?

Thanks
Hend

Here's a copy of the question and answer

Category:Hull -> Chapter 05
Question:

Assume that the risk-free rate is 9% per annum with continuous compounding and that the dividend yield on a stock index varies throughout the year. In February, May, August, and November, dividends are paid at a rate of 2% per annum. Suppose that the value of the index on July 31, 2006 is 300. What is the futures for a contract deliverable on December 31, 2006?

Answer:

The futures contract lasts for five months. The dividend yield is 2% for three of the months and 5% for two of the months. The average dividend yield is therefore
1/5(3 x 2 + 2 x 5) = 3.2%

The futures price is therefore 300e^(0.09-0.032)x0.4167 = 307.34 or $307.34.

 

[David Harper CFA FRM]

Hi Hend,

We made a typo in that question (sorry!). It was previously noticed in the forum entry but hasn't found it's way back to a revision in the PDF (yet).

See link http://forum.bionicturtle.com/viewthread/4213/

Question should be:
Assume that the risk-free rate is 9% per annum with continuous compounding and that the dividend yield on a stock index varies throughout the year. In February, May, August, and November, dividends are paid at a rate of 5% per annum. In other months, dividends are paid at a rate of 2% per annum. Suppose that the value of the index on July 31, 2006 is 300. What is the futures for a contract deliverable on December 31, 2006?

Thank you, sorry for inconvenience, David

 

[phmpaiva]

Hi David

Referring to the question mentioned, where did the conclusion about the 5% come from?
The question specifies the dividend yield of 2% for May, Aug, & Nov, but says nothing about the yields of the remaining two months.
Is there a convention that we're supposed to know of?

Thanks
Hend

Here's a copy of the question and answer

Category:Hull -> Chapter 05
Question:

Assume that the risk-free rate is 9% per annum with continuous compounding and that the dividend yield on a stock index varies throughout the year. In February, May, August, and November, dividends are paid at a rate of 2% per annum. Suppose that the value of the index on July 31, 2006 is 300. What is the futures for a contract deliverable on December 31, 2006?

Answer:

The futures contract lasts for five months. The dividend yield is 2% for three of the months and 5% for two of the months. The average dividend yield is therefore
1/5(3 x 2 + 2 x 5) = 3.2%

The futures price is therefore 300e^(0.09-0.032)x0.4167 = 307.34 or $307.34.

Hend, why the book and you are using the 5% abd 2% dividend yield as constinuous rate?

It ok that the average is 3.2, if all the rates are continuous compounding, otherwise, its wrong.

In the same way, if the dividend rate arent continuous compound, shouldnt you do the convertion to use in the Forward Price formula?

 

[Nicole Seaman]

Hend, why the book and you are using the 5% abd 2% dividend yield as constinuous rate?

It ok that the average is 3.2, if all the rates are continuous compounding, otherwise, its wrong.

In the same way, if the dividend rate arent continuous compound, shouldnt you do the convertion to use in the Forward Price formula?

Hello @phmpaiva

This forum post has not been active since 2011, so Hend may not be an active user on our forum anymore. However, David did reply to her above, stating that there was a typo in our question.

Thank you,

Nicole

 

[phmpaiva]

Hi Hend,

We made a typo in that question (sorry!). It was previously noticed in the forum entry but hasn't found it's way back to a revision in the PDF (yet).

See link http://forum.bionicturtle.com/viewthread/4213/

Question should be:
Assume that the risk-free rate is 9% per annum with continuous compounding and that the dividend yield on a stock index varies throughout the year. In February, May, August, and November, dividends are paid at a rate of 5% per annum. In other months, dividends are paid at a rate of 2% per annum. Suppose that the value of the index on July 31, 2006 is 300. What is the futures for a contract deliverable on December 31, 2006?

Thank you, sorry for inconvenience, David

Thx Nicole, but the correction made by David just add the 5% dividend yield.

My dubt is that: they are assuming that just the risk free interest rate is continuous compounding. And the dividend yield is per annum (not specified the time compounding).

So, why in the way they solve the problem they consider all the rates as continuous compounding? Should i consider all rates when it is omitted to be as continuous compounding?

 

[David Harper CFA FRM]

Hi @phmpaiva You make a great point. Strictly speaking, if the question only says "dividends are paid at a rate of 2% per annum" it's not obvious how we should interpret that; is it 2% continuous; is it payable quarterly? It does not give frequency, so we don't know. As an investor, I am accustomed to quarterly or monthly dividends, so it's not clear why i would expect continuous. (that's the realistic problem here! we don't actually receive any dividends continuously!) Yet, the question does assume continuous just because it is consistent with the risk-free rate. So, your point is good in my opinion.

Technically, these is part of a theme (compound frequency, although this is dividend frequency) that GARP is aware of, due to years of feedback. As Nicole says, this is an old question such that in recent years, we've all been conditioned to be more precise about compound frequency. So, technically, the best questions do specify even the dividend frequency. As you apparently already understand, although many do not, the "per annum" does not convey frequency.

Okay, but realistically this is a grey area because it's also okay to use "common sense" and think: because the risk-free rate is explicitly given as continuous, and the frequency for the dividend is un-supplied, we can simply assume the same. I just quickly retreived three of Hull's end of chapter questions and notice how even he seems to be okay with giving the frequency assumptions once:

5.10. The risk-free rate of interest is 7% per annum with continuous compounding, and the dividend yield on a stock index is 3.2% per annum. The current value of the index is 150. What is the 6-month futures price?
5.11. Assume that the risk-free interest rate is 9% per annum with continuous compounding and that the dividend yield on a stock index varies throughout the year. In February, May, August, and November, dividends are paid at a rate of 5% per annum. In other months, dividends are paid at a rate of 2% per annum. Suppose that the value of the index on July 31 is 1,300. What is the futures price for a contract deliverable in December 31 of the same year?
5.12. Suppose that the risk-free interest rate is 10% per annum with continuous compounding and that the dividend yield on a stock index is 4% per annum. The index is standing at 400, and the futures price for a contract deliverable in four months is 405. What arbitrage opportunities does this create?

I hope that's helpful!

 

[phmpaiva]

Thanks @David Harper.

Its usefull. And this three question you quote was the same questions i was doubted. Now i can solve it easily due to you explanation.