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in BUS 2016H - Financial Mathematics by (130 points)

Do we calculate the expected accumulation as E(10000 *annuity factor *Sn),  where the annuity factor is calculated using E(i)=j ?

Question : Calculate the expected accumulation in 15 years’ time if R10 000 is invested in the fund at the beginning of each of the next 15 years.

1 Answer

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by (280 points)
Hi Anokhanyo, thanks for the question. I see what you were trying to do but it isn't quite correct.

The easiest way to do questions on stochastic interest rate models is to approach them as a stats problem and derive everything from first principles using your stats and fin maths knowledge.

You are asked to find \( E[10000S_{15} + 10000S_{14} + 10000S_{13} + ... + 10000S_{1} ] \)

Now, factoring out the 10 000, we can write the above more neatly as a sum as follows:

\(10000E[\Sigma_{k=1}^{15}S_{k}] \)

and using properties of expectations we can write:

\(=10000\Sigma_{k=1}^{15}E[S_{k}] \)

Now consider each \(S_{k}:\)

It is a random variable and is equal to \(\Pi_{t=1}^k(1+i_t)\)

So \(E[S_{k}] = E[\Pi_{t=1}^k(1+i_t)] \) (*)

Of course you would need additional information about the distribution of each of the \(i_t\) in order to fully answer the question.

If for instance each of the \(i_t\) were independent we could then write:

\(E[S_{k}] = E[\Pi_{t=1}^k(1+i_t)] = \Pi_{t=1}^k E[(1+i_t)] \)

and if each of them were identically distributed i.e each \(i_t = i\) (where i is a random variable!) we could write:

\(E[S_{k}] = \Pi_{t=1}^k E[(1+i)] = \Pi_{t=1}^k (1 + E[i]) = (1+j)^k\) where \(E[i] = j\)

So that, what we are asked to find is:

\(10000\Sigma_{k=1}^{15}E[S_{k}] = 10000\Sigma_{k=1}^{15}(1+j)^k = 10000 \ddot{s}_{\bar{15}|j} \)

I hope the above helps. Please note that my solution only reconciled (sort of) with your approach because I assumed the \(i_t\) we iid. Had I not made the assumption, you would've had to proceed in a different way from step (*) above.

This general method of using first principles and your stats knowledge should seldom let you down for these sort of problems so I do advise you to adopt this approach especially for exams and stuff.