Actuarial Outpost Reconciling two "prepaid forward" formulas; understanding "cost of carry"
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 Financial Mathematics Old FM Forum

#1
01-11-2009, 03:36 PM
 Marid Audran Join Date: Nov 2008 Location: Minnesota Studying for MFE/3F Posts: 14 Blog Entries: 17
Reconciling two "prepaid forward" formulas; understanding "cost of carry"

Hi everyone --- I have two questions from Chapter 5 of McDonald, which seems like a prerequisite for the MFE material. I've searched the forums and not quite found these addressed.

The two "prepaid forward" formulas
Here are the two formulas for pricing prepaid forwards with dividends (McDonald, pp. 131-32):
With discrete dividends: $F^P_{0,T} = S_0 - \sum PV_{0,t_i}(D_{t_i})$

With continuous dividends: $F^P_{0,t} = S_0e^{-\delta t}$
I understand each of these formulas on its own, but I am having a hard time reconciling them. I did find a forum discussion about why the risk-free rate is omitted from the continuous-dividend formula, but that's not the reconciliation I'm looking for.

Here's as far as I got: I did some figuring using the approximation
$D_{t_i} \approx \delta S_t \Delta t$
(this seems to be the gist of the "daily dividend" formula on Page 132), which led me to
$\sum PV_{0,t_i}(D_{t_i}) \approx \delta\int_0^{T} e^{-rt}S_tdt$
and ultimately (after substituting and simplifying) to
$\delta\int_0^{T} e^{-rt}S_tdt = S_0(1-e^{-\delta t})$
. Is this accurate/meaningful? Is there some other way of understanding the two prepaid-forward price formulas simultaneously? Am I barking up the wrong tree?

Understanding "cost of carry"
I don't understand McDonald's discussion of this concept on Page 141:
Quote:
 Suppose you buy a unit of the index that costs S and fund the position by borrowing at the risk-free rate. You will pay rS on the borrowed amount, but the dividend yield will provide offsetting income of $\delta S$. You will have to pay the difference, $(r - \delta)S$, on an ongoing basis. This difference is the net cost of carying a long position in the asset; hence, it is called the "cost of carry."
This discussion seems to ignore the fact that the stock price changes over time. Also, further down, I'm a little confused about Formula (5.13):
Forward price = Spot price + Interest to carry the asset - Asset lease rate,
where the last two terms on the right-hand side are grouped together as "cost of carry." This formula seems to apply directly to the discrete-dividends case, but I don't see how to apply it directly to the continuous-dividends case. Is there a clearer interpretation?

I will munch on popcorn while you discuss TIA
#2
01-11-2009, 04:55 PM
 Nonpareil Note Contributor Join Date: Nov 2006 Location: Rocket City Studying for Exam C Posts: 833

Chapter 5 of McDonald is on the FM syllabus. You didn't study this material for that exam?
__________________
"One must do no violence to nature, nor model it in conformity to any blindly formed chimera." Janos Bolyai

"Theoria cum praxis." Gottfried von Leibniz
#3
01-11-2009, 05:07 PM
 Marid Audran Join Date: Nov 2008 Location: Minnesota Studying for MFE/3F Posts: 14 Blog Entries: 17

Actually I passed Exam FM in 2005 (after which I took a non-actuarial job...long story). I don't recall this being on it.
#4
01-11-2009, 05:10 PM
 Nonpareil Note Contributor Join Date: Nov 2006 Location: Rocket City Studying for Exam C Posts: 833

Keep reading, I think it would help for you to compare the pre-paid forwards to the forwards - see page 134 of McDonald.
__________________
"One must do no violence to nature, nor model it in conformity to any blindly formed chimera." Janos Bolyai

"Theoria cum praxis." Gottfried von Leibniz
#5
01-11-2009, 05:11 PM
 Nonpareil Note Contributor Join Date: Nov 2006 Location: Rocket City Studying for Exam C Posts: 833

Quote:
 Originally Posted by Marid Audran Actually I passed Exam FM in 2005 (after which I took a non-actuarial job...long story). I don't recall this being on it.
Yes, they added the derivatives material to FM in 2007.

You might get more responses if you post this question in the FM forum.
__________________
"One must do no violence to nature, nor model it in conformity to any blindly formed chimera." Janos Bolyai

"Theoria cum praxis." Gottfried von Leibniz
#6
01-11-2009, 05:16 PM
 Marid Audran Join Date: Nov 2008 Location: Minnesota Studying for MFE/3F Posts: 14 Blog Entries: 17

OK, I'll try the FM forum, thanks. As for comparing the prepaid forwards to the forwards: There is a very straightforward connection between discrete prepaid forward and discrete forward. Likewise for continuous prepaid forward and continuous forward. I am asking about discrete prepaid forward and continuous prepaid forward.
#7
01-11-2009, 05:52 PM
 Abraham Weishaus Member SOA AAA Join Date: Oct 2001 Posts: 7,257

Concerning your first question, have you considered that the two formulas are based on 2 different assumptions?

The continuous dividend formula assumes that the dividends are continuously reinvested in the stock

The discrete dividend formula does not assume reinvestment in the stock. The discrete dividends may be invested in a risk-free investment; their value is discounted at the risk-free rate.
#8
01-11-2009, 06:33 PM
 Final Fadeout Member Join Date: Aug 2004 Location: NYC Posts: 98

Intuitively I like to think of S0 - PV(dividends) is the discrete case, and S0e^(-st) is the same thing in the continuous case, and it works out because at time 0, you lose out the dividends because you don't get the stock until time t. If s=.05 and t=1, then S0e^(-st) is S0e^(-.05), and it's nearly equal to .95S0....but the discrete case would be S0 - .05S0 if the present value of the dividend was equal to .05*S0, and you would have exactly .95S0. Mathematically I'm not sure of a quick way to put them together, but I can't think of a reason to either answering or understanding an exam question.
#9
01-11-2009, 06:46 PM
 Marid Audran Join Date: Nov 2008 Location: Minnesota Studying for MFE/3F Posts: 14 Blog Entries: 17

Quote:
 Originally Posted by Abraham Weishaus Concerning your first question, have you considered that the two formulas are based on 2 different assumptions? The continuous dividend formula assumes that the dividends are continuously reinvested in the stock The discrete dividend formula does not assume reinvestment in the stock. The discrete dividends may be invested in a risk-free investment; their value is discounted at the risk-free rate.
Ah, OK. I thought about this more based on what you said. The two formulas are derived from positions that are in some sense equivalent, but different. They differ not only in the assumptions about whether to reinvest, but also in what the initial investments are to begin with (one share versus $e^{-\delta T}$ shares). It may not be meaningful to try and substitute one formula into the other. Thanks.
#10
01-12-2009, 04:47 PM
 Marid Audran Join Date: Nov 2008 Location: Minnesota Studying for MFE/3F Posts: 14 Blog Entries: 17

Addendum: in the first free online lecture at The Infinite Actuary -- the one where they review topics such as this one that have already been gone over -- the guy says that you can "calculate the present value by solving a stochastic differential equation" to get
$PV_{0,T}(Div) = S_0(1 - e^{-\delta T})$.
Maybe someday sooner or later I'll learn how those stochastic differential equations work...for now I'll accept that it's out of scope.

Last edited by Marid Audran; 01-12-2009 at 04:49 PM.. Reason: Correcting formula and formatting