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. The formula for the future value of a geometric gradient annuity is given as $FV = PMT_{1} \times \frac{(1 + i)^{N} - (1 + g)^{N}}{i - g}$. What happens to this formula when the periodic interest rate $i$ is exactly equal to the periodic growth rate $g$?

   • The future value becomes infinitely large.
   • The formula becomes undefined, and a different formula must be used.
   • The future value simplifies to $FV = PMT_{1} \times N \times (1 + i)^{N-1}$.
   • The future value is always zero.

   Answer: The formula becomes undefined, and a different formula must be used.

2. An investor aims to accumulate a future value of $FV$ in $N$ periods. The periodic interest rate is $i$, and the payments are expected to grow at a periodic rate $g$ (where $g \ne i$). Which of the following expressions correctly represents the initial payment $PMT_{1}$ required to achieve this goal?

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

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

3. A series of payments begins with $PMT_{1} = \$1000$. These payments grow at a periodic rate of $g = 3\%$. The investment earns an annual nominal interest rate of $r = 8\%$, compounded semi-annually. If the payments are made semi-annually for $10$ years, what is the future value of this annuity? Assume $g \ne i$.

   • $\$38,501.20$
   • $\$35,070.00$
   • $\$40,000.00$
   • $\$32,187.50$

   Answer: $\$38,501.20$

4. Consider a geometric gradient annuity where $i > g$. If the periodic growth rate $g$ increases, while $PMT_{1}$, $i$, and $N$ remain constant, how does the future value ($FV$) of the annuity change?

   • $FV$ increases.
   • $FV$ decreases.
   • $FV$ remains unchanged.
   • $FV$ increases if $N$ is large, but decreases if $N$ is small.

   Answer: $FV$ increases.

5. Consider a geometric gradient annuity where $i > g$. If the periodic interest rate $i$ decreases, while $PMT_{1}$, $g$, and $N$ remain constant, how does the future value ($FV$) of the annuity change?

   • $FV$ increases.
   • $FV$ decreases.
   • $FV$ remains unchanged.
   • $FV$ decreases only if $i$ remains greater than $g$.

   Answer: $FV$ decreases.

6. An investor has two options for an annuity, both with $PMT_{1} = \$500$ and $N = 10$ periods.\nOption A: A geometric gradient annuity with a periodic growth rate $g = 2\%$ and a periodic interest rate $i = 5\%$.\nOption B: An ordinary annuity (level payments) with $PMT = \$500$ and a periodic interest rate $i = 5\%$.\nWhich option will result in a higher future value, and by approximately how much?

   • Option A by approximately $\$542.73$.
   • Option B by approximately $\$542.73$.
   • Option A by approximately $\$683.17$.
   • Option B by approximately $\$628.90$.

   Answer: Option A by approximately $\$542.73$.

7. For a given geometric gradient annuity with fixed periodic interest rate $i$, periodic growth rate $g$ ($g \ne i$), and total number of periods $N$, if the initial payment $PMT_{1}$ is doubled, how does the future value ($FV$) of the annuity change?

   • $FV$ doubles.
   • $FV$ quadruples.
   • $FV$ increases by a factor of $(1+g)$.
   • $FV$ increases by a factor of $(1+i)$.

   Answer: $FV$ doubles.

8. Under what condition, assuming all other parameters ($PMT_{1}$, $i$, $N$) are identical, would the future value of a geometric gradient annuity be *less* than the future value of an ordinary annuity (where payments are constant, i.e., $g=0$)?

   • When the periodic growth rate $g$ is negative.
   • When the periodic growth rate $g$ is greater than the periodic interest rate $i$.
   • When the total number of periods $N$ is very small.
   • When the periodic interest rate $i$ is very high.

   Answer: When the periodic growth rate $g$ is negative.

9. An investment offers a geometric gradient annuity with $PMT_{1} = \$200$, $N = 15$ periods, and a periodic interest rate $i = 4\%$. If the observed future value is $\$5,000$, which of the following statements about the periodic growth rate $g$ is most likely true?

   • $g$ is negative.
   • $g$ is positive and less than $i$.
   • $g$ is positive and greater than $i$.
   • $g$ is exactly equal to $i$.

   Answer: $g$ is positive and less than $i$.

10. An investor wants to determine the number of periods $N$ required for a geometric gradient annuity to reach a specific future value $FV$, given $PMT_{1}$, $i$, and $g$ ($g \ne i$). Which of the following statements best describes the process to solve for $N$?

    • $N$ can be directly isolated using basic algebraic operations.
    • $N$ can be solved for using logarithms, but it requires advanced algebraic manipulation due to the presence of two exponential terms.
    • $N$ can only be determined through numerical methods or iterative calculations.
    • $N$ is indeterminate unless $g=0$.

    Answer: $N$ can only be determined through numerical methods or iterative calculations.