Annuities (Savings & Investments) — Practice Quiz

A Financial Math cheat sheet for Annuities (Savings & Investments) — every key formula with its symbols defined — plus a medium-level practice quiz to test recall.

Formulas & key concepts

The future value of an ordinary annuity, a series of equal payments made at the end of each period (typical savings plans). \(PMT\) is the periodic payment, \(i = r/n\) the periodic rate, and \(N = nt\) the total number of periods.

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

The regular payment required to accumulate a target future value \(FV\) with an ordinary annuity, where \(i = r/n\) is the periodic rate and \(N = nt\) the total number of periods.

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

The future value of an annuity due, where payments are made at the beginning of each period; it equals the ordinary-annuity value multiplied by \((1 + i)\). \(PMT\) is the payment, \(i = r/n\), and \(N = nt\).

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

Practice quiz

  1. Which of the following formulas correctly represents the future value of an ordinary annuity?

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

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

  2. A person deposits $200$ at the end of each month into an account that earns $6\%$ annual interest, compounded monthly. What will be the future value of this annuity after $5$ years?

    • $12,000.00$
    • $13,954.01$
    • $14,023.78$
    • $15,200.00$

    Answer: $13,954.01$

  3. How does the future value of an annuity due ($FV_{due}$) relate to the future value of an ordinary annuity ($FV$) with the same payment, interest rate, and number of periods?

    • $FV_{due} = FV \times (1 + i)$
    • $FV_{due} = FV / (1 + i)$
    • $FV_{due} = FV + PMT$
    • $FV_{due} = FV - PMT$

    Answer: $FV_{due} = FV \times (1 + i)$

  4. You want to accumulate $50,000$ in $10$ years by making equal monthly deposits into an account that pays $4.8\%$ annual interest, compounded monthly. What is the required monthly payment?

    • $300.00$
    • $326.67$
    • $416.67$
    • $350.12$

    Answer: $326.67$

  5. In the annuity formulas, what do $i$ and $N$ represent, respectively?

    • $i$ is the periodic interest rate, $N$ is the total number of periods.
    • $i$ is the annual interest rate, $N$ is the number of years.
    • $i$ is the periodic interest rate, $N$ is the number of years.
    • $i$ is the annual interest rate, $N$ is the total number of periods.

    Answer: $i$ is the periodic interest rate, $N$ is the total number of periods.

  6. If you deposit $100$ at the beginning of each month into an account earning $3.6\%$ annual interest, compounded monthly, what will be the future value after $3$ years?

    • $3,600.00$
    • $3,798.73$
    • $3,833.12$
    • $3,910.50$

    Answer: $3,833.12$

  7. All else being equal (same payment amount, annual interest rate, compounding frequency, and total time), which type of annuity will have a higher future value?

    • An ordinary annuity
    • An annuity due
    • They will always have the same future value.
    • It depends on whether the interest rate is above or below $5\%$.

    Answer: An annuity due

  8. If an ordinary annuity involves monthly payments for $15$ years, what is the total number of periods, $N$?

    • $15$
    • $12$
    • $180$
    • $30$

    Answer: $180$

  9. You are saving for a down payment on a house by depositing a fixed amount at the end of each quarter into a savings account. Which formula would you use to calculate the total amount saved at the end of your savings period?

    • Future value of an ordinary annuity
    • Future value of an annuity due
    • Payment required for a future value (PMT formula)
    • Present value of an ordinary annuity

    Answer: Future value of an ordinary annuity

  10. A student wants to save $10,000$ in $2$ years for a trip. They plan to make deposits at the *beginning* of each month into an account earning $3\%$ annual interest, compounded monthly. What is the required monthly payment?

    • $403.80$
    • $404.81$
    • $416.67$
    • $400.00$

    Answer: $403.80$

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