Forward price of a security with dividend (question 12.9)

Hi
I've got my head around the basic forward pricing using the no arbitrage assumption and replicating portfolios, however I am missing some basic steps for this question in the CMP.

The question I am referring to is in chapter 12, question 12.9.

I am not quite following the summation required to give me (1 + D)^(T-t), hence why we need (1 + D)^-(T-t) units of S initially?

Thanks in advance.

 

The asset pays out dividends at a rate / yield of D per annum. Dividends are also reinvested into the asset when paid. Consider a simpler example of buying one unit of the asset at time \( t \) for the price of \( S_t \) pounds, then after exactly one year the dividend yield has grown our investment to the extent of giving us \( S_{t+1} * D \) pounds (because the dividend is paid at the end of the year and based on the share price at the time). This is immediately reinvested into the asset at price \( S_{t+1} \). Thus gives us \(S_{t+1} * D / S_{t+1} = D \) units of the asset (amount we invest dividing by the price we are investing at). Thus we now own \( 1 + D \) units of the asset.

If we repeat the above exercise for starting by buying \( x \) units of the asset then:

1. At the start we spend \( x * S_t \) to purchase the x units
2. At the end of the first year then our investment is now worth \( (x * S_{t+1})(1 + D) \)
3. Dividing the amount our investment is worth by the current price tells us how many units we own i.e. \( x * (1+ D) \)

We can see how this has generalised from the above case. Now consider a longer number of years than just one, extending the analysis, we can see that the number of units we hold of the asset after \( n \) years if we started with \( x \) units is \( x * (1+D)^n \). (For example consider the second year where we start with \( x * (1 + D) \) units which we know at the end of the year we times by \( (1 + D) \) as per point (3) above to then get \( x * (1+D)^2 \)).

Now thinking about the question in the notes. We want a portfolio that will grow to owning exactly one unit of the asset after an exact number of years given by \( T - t \). I.e. we want:

\( x * ( 1+D)^{T-t} = 1 \) Thus:

\( x = (1+D)^{-(T-t)} \) and this is the number of units we therefore need to purchase at the beginning.

 

Hi - that is really helpful, and you've explained it perfectly. It turns out I managed to get to the same solution after all. Not quite sure why I was I was summing each share proportion from t to T in the first place, but your comments have helped nail this bit down. Thanks.