# chapter 15: premium not a lump sum.

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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.

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.