Increasing Annuity (Constant Growth) — Practice Quiz

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

Formulas & key concepts

The future value of a geometric gradient annuity whose payments grow at a constant periodic rate \(g\) (with \(g \ne i\)). \(PMT_{1}\) is the very first payment, \(i = r/n\) the periodic rate, \(N = nt\) the total number of periods, and \(g\) the periodic growth rate.

$$FV = PMT_{1} \times \dfrac{(1 + i)^{N} - (1 + g)^{N}}{i - g} \quad\text{or}\quad FV = PMT_{1} \times \dfrac{\left(1 + \frac{r}{n}\right)^{nt} - (1 + g)^{nt}}{\frac{r}{n} - g}$$

Practice quiz

  1. What does $PMT_1$ represent in the future value of a geometric gradient annuity formula?

    • The last payment in the series.
    • The sum of all payments.
    • The very first payment in the series.
    • The average payment.

    Answer: The very first payment in the series.

  2. The formula $FV = PMT_{1} \times \frac{(1 + i)^{N} - (1 + g)^{N}}{i - g}$ is applicable under which specific condition regarding the periodic rate $i$ and the periodic growth rate $g$?

    • $i = g$
    • $i > g$
    • $i < g$
    • $i \ne g$

    Answer: $i \ne g$

  3. In the context of a geometric gradient annuity, what does the variable $g$ signify?

    • The annual nominal interest rate.
    • The total number of periods.
    • The periodic growth rate of payments.
    • The future value of the annuity.

    Answer: The periodic growth rate of payments.

  4. An annuity has a first payment $PMT_1 = $100$. The payments grow at a periodic rate $g = 2\%$. The periodic interest rate is $i = 5\%$. If there are $N = 10$ payments, what is the future value of this annuity?

    • $1366.33$
    • $1218.99$
    • $1628.89$
    • $1000.00$

    Answer: $1366.33$

  5. A series of payments starts with $PMT_1 = $500$. The annual nominal interest rate is $r = 6\%$ compounded semi-annually ($n=2$). Payments grow at an annual rate of $g = 3\%$ compounded semi-annually. The annuity lasts for $t = 5$ years. Calculate the future value.

    • $6112.50$
    • $5875.00$
    • $6350.00$
    • $5500.00$

    Answer: $6112.50$

  6. What is the implication if the periodic growth rate $g$ is greater than the periodic interest rate $i$ ($g > i$) in the geometric gradient annuity formula?

    • The future value will be negative.
    • The future value will be smaller than if $g < i$.
    • The future value will be larger than if $g < i$.
    • The formula becomes undefined.

    Answer: The future value will be larger than if $g < i$.

  7. If the periodic growth rate $g$ is equal to the periodic interest rate $i$ ($g = i$), how would you calculate the future value of a geometric gradient annuity?

    • The formula $FV = PMT_{1} \times \frac{(1 + i)^{N} - (1 + g)^{N}}{i - g}$ can still be used.
    • The future value is zero.
    • A different formula, often involving $N \times PMT_1 \times (1+i)^{N-1}$, must be used.
    • The future value is infinite.

    Answer: A different formula, often involving $N \times PMT_1 \times (1+i)^{N-1}$, must be used.

  8. An investor wants to accumulate $FV = $20,000$ in $N = 15$ periods. The periodic interest rate is $i = 4\%$, and payments are expected to grow at a periodic rate $g = 1\%$. What should the first payment $PMT_1$ be?

    • $937.52$
    • $1050.00$
    • $880.00$
    • $1120.00$

    Answer: $937.52$

  9. If an annual nominal interest rate is $r = 8\%$ compounded quarterly, what is the periodic rate $i$ to be used in the geometric gradient annuity formula?

    • $0.08$
    • $0.04$
    • $0.02$
    • $0.005$

    Answer: $0.02$

  10. An annuity has payments made monthly for $t = 7$ years. What is the total number of periods $N$ for this annuity?

    • $7$
    • $12$
    • $84$
    • $14$

    Answer: $84$

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