How to solve Annuities — increasing, decreasing, increasing-perpetuity and geometric annuities questions on Exam FM?

Question

An annuity-immediate makes $n = 13$ annual payments. The first payment is $4800$, and each successive payment is larger by $4800$ (so $4800, 2\cdot4800, \ldots, 13\cdot4800$). Calculate its present value. The annual effective interest rate is $i = 4.4\%$.

Accepted Answer

Formula
$(Ia)_{\overline n|}=\sum_{t=1}^n t v^t=\dfrac{\ddot a_{\overline n|}-nv^n}{i}$, $(Da)_{\overline n|}=(n-a_{\overline n|})/i$, and $(Ia)_\infty=1/(id)$. For first payment $P$ growing at rate $g$, $PV=P[1-\{(1+g)/(1+i)\}^n]/(i-g)$ when $i\ne g$.

Why this formula applies
Represent each payment at its actual time and separate level and arithmetic or geometric growth when helpful. Standard increasing-annuity identities follow from summing those discounted payments.

Substitution and calculation
With $i=4.4\%$, With $v=0.957854$, $a_{\overline{13}|}=9.742306$, and $\ddot a_{\overline{13}|}=10.170967$, $PV=4800\dfrac{\ddot a_{\overline{13}|}-13v^{13}}{0.044000}=299298.1647$.

Answer
The answer is 299298.1647, option (B).

How to Solve Annuities — increasing, decreasing, increasing-perpetuity and geometric annuities Questions on Exam FM

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