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. You start an investment with an initial payment of \$500 at the end of the first month. Subsequent payments increase by $3\%$ monthly. The account earns an annual interest rate of $6\%$ compounded monthly. What is the future value of this investment after $10$ years? Round to the nearest cent.

    • \$657,831.81
    • \$-657,831.81
    • \$81,939.67
    • \$7,448.86

    Answer: \$657,831.81

  2. In the formula for the future value of a geometric gradient annuity, what does the variable $g$ represent?

    • The periodic growth rate of payments.
    • The annual interest rate.
    • The total number of periods.
    • The first payment amount.

    Answer: The periodic growth rate of payments.

  3. An investor makes an initial payment of \$1,000 into an account at the end of the first year. Each subsequent payment increases by $5\%$ annually. The account earns an annual interest rate of $8\%$ compounded annually. Calculate the future value of this investment after $5$ years. Round to the nearest cent.

    • \$6,434.88
    • \$7,078.37
    • \$5,866.60
    • \$1,484.97

    Answer: \$6,434.88

  4. A company plans to deposit \$2,000 at the end of the first quarter, with each subsequent quarterly deposit increasing by $1.5\%$. The investment account offers an annual interest rate of $5\%$ compounded quarterly. The investment will last for $8$ years. In the geometric gradient annuity formula, what is the value of $PMT_1$?

    • \$2,000
    • \$2,030
    • \$64,000
    • $5\%$

    Answer: \$2,000

  5. What happens to the formula for the future value of a geometric gradient annuity if the periodic growth rate of payments ($g$) is equal to the periodic interest rate ($i$)?

    • The formula becomes undefined due to division by zero.
    • The future value is simply $PMT_1 \times N$.
    • The future value is calculated using a different, simpler formula.
    • The future value becomes infinite.

    Answer: The formula becomes undefined due to division by zero.

  6. An individual starts saving for retirement with a payment of \$1,200 at the end of the first quarter. Each subsequent quarterly payment increases by $4\%$ annually. The retirement account earns an annual interest rate of $7\%$ compounded quarterly. What will be the future value of this investment after $7$ years? Round to the nearest cent.

    • \$48,567.68
    • \$73,242.16
    • \$10,668.63
    • \$42,885.46

    Answer: \$48,567.68

  7. Consider an investment with an initial payment of \$500 at the end of the first month. Subsequent payments increase by $3\%$ *annually*. The account earns an annual interest rate of $6\%$ compounded monthly. What is the future value of this investment after $10$ years? Round to the nearest cent.

    • \$93,830.11
    • \$657,831.81
    • \$6,656.18
    • \$81,939.67

    Answer: \$93,830.11

  8. Which of the following best describes a geometric gradient annuity?

    • An annuity where payments increase or decrease by a constant percentage each period.
    • An annuity where payments increase or decrease by a constant dollar amount each period.
    • An annuity with equal payments made at regular intervals.
    • A single lump sum payment made at the beginning of an investment.

    Answer: An annuity where payments increase or decrease by a constant percentage each period.

  9. Compare two investment scenarios over $15$ years, with payments made at the end of each month. Both start with an initial payment of \$300. The account earns an annual interest rate of $5\%$ compounded monthly. Scenario A: Payments increase by $2\%$ monthly. Scenario B: Payments increase by $2.5\%$ monthly. What is the difference in future value between Scenario B and Scenario A (FV_B - FV_A)? Round to the nearest cent.

    • \$566,848.75
    • \$-566,848.75
    • \$629,187.68
    • \$1,196,036.43

    Answer: \$566,848.75

  10. In the formula $FV = PMT_{1} \times \dfrac{(1 + i)^{N} - (1 + g)^{N}}{i - g}$, which term represents the total number of payment periods?

    • $N$
    • $i$
    • $g$
    • $PMT_1$

    Answer: $N$

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