How to solve Annuities — present and accumulated value of level annuities-immediate and -due questions on Exam FM?

Question

An investor will receive $n = 13$ level payments of $P = 250$, one at the end of each period. The effective interest rate per period is $i = 4.4\%$. Calculate the accumulated value at the time of the last payment, using $s_{\overline{13}|}$.

Accepted Answer

Formula
$a_{\overline n|}=(1-v^n)/i$, $s_{\overline n|}=((1+i)^n-1)/i$, and due values multiply the corresponding immediate value by $(1+i)$.

Why this formula applies
Place the valuation date relative to the first payment, then apply the immediate or due annuity factor. A due value is an immediate value shifted one period earlier and therefore multiplied by 1+i.

Substitution and calculation
With $i = 4.4\%$, $v = \dfrac{1}{1+i} = 0.957854$, so $a_{\overline{13}|} = 9.742306$ and $s_{\overline{13}|} = 17.051722$. The requested value is $P\,s_{\overline{13}|} = P\cdot\dfrac{(1+i)^{13}-1}{i}\;= 4262.9306$.

Answer
The answer is 4262.9306, option (D).

How to Solve Annuities — present and accumulated value of level annuities-immediate and -due Questions on Exam FM

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