Increasing Annuity (Constant Growth) — Hard Practice Quiz

A Financial Math cheat sheet for Increasing Annuity (Constant Growth) — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Practice quiz

1. An investor aims to accumulate a future value of \$500,000 in 20 years. Payments are made annually, starting with $PMT_1$ at the end of the first year. The annual interest rate is $6\%$, and payments are expected to grow by $3\%$ annually. What should the first payment, $PMT_1$, be? Round to the nearest cent.

   • \$10,706.45
   • \$2,992.07
   • \$4,677.07
   • \$21,412.91

   Answer: \$10,706.45

2. How does the future value ($FV$) of a geometric gradient annuity change if the periodic payment growth rate $g$ increases, assuming the periodic interest rate $i$ and the number of periods $N$ remain constant, and $g < i$ remains true?

   • The future value increases.
   • The future value decreases.
   • The future value remains the same.
   • The change in future value cannot be determined without specific values.

   Answer: The future value increases.

3. A geometric gradient annuity has an initial payment $PMT_1$, a periodic interest rate $i$, and a periodic payment growth rate $g$. If $g=i$, which of the following expressions represents the future value ($FV$) of this annuity after $N$ periods?

   • $N \times PMT_1 \times (1+i)^{N-1}$
   • $PMT_1 \times \frac{(1+i)^N - (1+g)^N}{i-g}$
   • $N \times PMT_1$
   • $PMT_1 \times \frac{(1+i)^N - 1}{i}$

   Answer: $N \times PMT_1 \times (1+i)^{N-1}$

4. Given the formula for the future value of a geometric gradient annuity as $FV = PMT_{1} \times \frac{(1 + i)^{N} - (1 + g)^{N}}{i - g}$, where $g \ne i$, rearrange the formula to solve for the first payment, $PMT_1$.

   • $PMT_1 = FV \times \frac{i - g}{(1 + i)^{N} - (1 + g)^{N}}$
   • $PMT_1 = FV \times \frac{(1 + i)^{N} - (1 + g)^{N}}{i - g}$
   • $PMT_1 = FV \times \frac{i + g}{(1 + i)^{N} - (1 + g)^{N}}$
   • $PMT_1 = FV \times \frac{i - g}{(1 + i)^{N} + (1 + g)^{N}}$

   Answer: $PMT_1 = FV \times \frac{i - g}{(1 + i)^{N} - (1 + g)^{N}}$

5. An investor is considering two annuity options, both with an initial payment of \$1,000 and an annual interest rate of $5\%$ over 10 years. Option A's payments grow by $2\%$ annually, while Option B's payments remain level (no growth). How much more is the future value of Option A compared to Option B? Round to the nearest cent.

   • \$1,085.45
   • \$13,663.34
   • \$12,577.89
   • \$2,935.39

   Answer: \$1,085.45

6. Consider two geometric gradient annuities, Annuity X and Annuity Y, both with the same $PMT_1$, periodic interest rate $i$, and periodic payment growth rate $g$ (where $g < i$). If Annuity Y has twice the number of periods as Annuity X ($N_Y = 2 N_X$), which of the following statements is true regarding their future values ($FV_X$ and $FV_Y$)?

   • $FV_Y = 2 FV_X$
   • $FV_Y > 2 FV_X$
   • $FV_Y < 2 FV_X$
   • $FV_Y$ cannot be compared to $FV_X$ without specific values.

   Answer: $FV_Y > 2 FV_X$

7. An individual contributes to a retirement fund for 15 years. The first payment is \$2,000, made at the end of the first year. Payments grow by $4\%$ annually, and the fund earns $7\%$ annual interest. After 15 years, the individual stops making payments, but the accumulated fund continues to earn $7\%$ interest for another 10 years. What is the total future value at the end of 25 years? Round to the nearest cent.

   • \$125,646.95
   • \$63,872.54
   • \$94,546.96
   • \$184,106.42

   Answer: \$125,646.95

8. A geometric gradient annuity has $PMT_1 = \$500$, an annual interest rate $i = 8\%$, and $N = 12$ years. If the future value is \$12,649.34, what is the approximate annual growth rate $g$ of the payments? Round your answer to two decimal places.

   • $5.00\%$
   • $6.00\%$
   • $7.00\%$
   • $4.00\%$

   Answer: $6.00\%$

9. Under which condition is the given formula for the future value of a geometric gradient annuity ($FV = PMT_{1} \times \frac{(1 + i)^{N} - (1 + g)^{N}}{i - g}$) not directly applicable?

   • $g=0$
   • $i=0$
   • $N=1$
   • $g=i$

   Answer: $g=i$

10. If all other parameters ($i, g, N$) remain constant, and the first payment $PMT_1$ of a geometric gradient annuity is increased by $25\%$, how does the future value ($FV$) change?

    • The future value increases by $25\%$.
    • The future value increases by more than $25\%$.
    • The future value increases by less than $25\%$.
    • The future value decreases by $25\%$.

    Answer: The future value increases by $25\%$.