Actuarial Outpost Reconciling two "prepaid forward" formulas; understanding "cost of carry"
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#1
01-11-2009, 05:31 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.

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 sip coffee while you discuss TIA