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

Question

A level annuity pays $P = 250$ for $n = 13$ periods, but the stream is deferred: the first payment period does not begin until $k = 5$ periods from now. The annual effective interest rate is $i = 4.4\%$, with $v = 1/(1+i)$. Assume the payments are made at the END of each period. Calculate the present value today (a deferred annuity-immediate).

Accepted Answer

Formula
${}_k|a_{\overline n}=v^k a_{\overline n}$, so $PV=P v^k a_{\overline n}$; use $\ddot a_{\overline n}=(1+i)a_{\overline n}$ for payments due.

Why this formula applies
Place the first payment on a timeline. An annuity-immediate factor is valued one period before its first payment, while an annuity-due factor is valued at its first payment; discount that correctly dated value back to time zero.

Substitution and calculation
With $i = 4.4\%$: $a_{\overline{13}|} = 9.742306$, $v^{5} = 0.806302$, so $\text{PV} = P\,a_{\overline{13}|}\,v^{5}\;= 1963.8092$.

Answer
The answer is 1963.8092, option (A).

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

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