Forward price of a security with dividend (question 12.9)

Discussion in 'CM2' started by Amandeep Virdi, Jan 10, 2019.

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1. 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?

2. 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.

3. 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.