Geometric Increasing Annuity

Study guide: Exam FM-2

A geometric increasing (or decreasing) annuity has payments that grow (or shrink) at a constant geometric rate $g$ per period. For an $n$ -payment annuity-immediate with first payment $1$ and growth rate $g$ , payments are $1, (1+g), (1+g)^2, \ldots, (1+g)^{n-1}$ .

The present value (at effective rate $i \neq g$ ) is:

$\text{PV} = \frac{1 - \left(\frac{1+g}{1+i}\right)^n}{i-g}$

When $i = g$ , $\text{PV} = n \cdot v = n/(1+i)$ .
For a geometric Perpetuity with $i > g$ : $\text{PV} = 1/(i-g)$ .

Inflation-Adjusted Pension A retiree receives $20,000 at the end of year 1, with payments increasing 3% per year for 20 years. At $i = 7\%$ , find the present value.
Answer
 $\begin{align*} \left(\tfrac{1.03}{1.07}\right)^{20} &= 0.46673346 \\ \text{PV} &= 20{,}000 \cdot \frac{1-0.46673346}{0.07-0.03} \\ &= 20{,}000 \times \frac{0.53326654}{0.04} \\ &= 20{,}000 \times 13.3316635 \\ &= 266{,}633.27 \end{align*}$ 
The 20 payments total $537,407 undiscounted; growth at 3% offsets part of the 7% discounting, so the present value is 13.33 times the first payment rather than the level-annuity $a_{\overline{20}|0.07} = 10.59$ .