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. You deposit \$200 at the end of each month into an account that pays $4.8\%$ annual interest compounded monthly. What will be the future value of your account after $5$ years? Round to the nearest cent.

    • \$13,532.04
    • \$12,000.00
    • \$1,100.72
    • \$13,586.16

    Answer: \$13,532.04

  2. You want to accumulate \$50,000 in $10$ years by making equal payments at the end of each quarter into an account earning $6\%$ annual interest compounded quarterly. What is the required quarterly payment? Round to the nearest cent.

    • \$921.36
    • \$3,793.40
    • \$1,250.00
    • \$907.74

    Answer: \$921.36

  3. Suppose you deposit \$300 at the beginning of each month into a savings account that pays $3.6\%$ annual interest compounded monthly. What will be the future value of this annuity after $3$ years? Round to the nearest cent.

    • \$11,420.92
    • \$11,386.76
    • \$11,966.40
    • \$966.37

    Answer: \$11,420.92

  4. What is the primary difference between an ordinary annuity and an annuity due?

    • The timing of payments: ordinary annuities are paid at the end of each period, while annuities due are paid at the beginning.
    • The interest rate calculation method.
    • The total number of payments made over the annuity's term.
    • The frequency of compounding interest.

    Answer: The timing of payments: ordinary annuities are paid at the end of each period, while annuities due are paid at the beginning.

  5. You invest \$1,000 at the end of each year for $15$ years into an investment account that earns $7\%$ annual interest compounded annually. What is the future value of this investment? Round to the nearest cent.

    • \$25,129.02
    • \$26,888.05
    • \$30,750.00
    • \$15,000.00

    Answer: \$25,129.02

  6. A student wants to save \$25,000 for a down payment on a house in $5$ years. If they can earn $5\%$ annual interest compounded semiannually, how much must they deposit at the end of each semiannual period? Round to the nearest cent.

    • \$2,231.47
    • \$4,524.37
    • \$367.61
    • \$2,177.04

    Answer: \$2,231.47

  7. Which of the following formulas represents the future value ($FV$) of an ordinary annuity, where $PMT$ is the periodic payment, $i$ is the periodic interest rate, and $N$ is the total number of periods?

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

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

  8. You have two investment options. Option A: Deposit \$150 at the end of each month for $4$ years. Option B: Deposit \$150 at the beginning of each month for $4$ years. Both accounts earn $4.2\%$ annual interest compounded monthly. How much more will Option B be worth than Option A at the end of $4$ years? Round to the nearest cent.

    • \$27.39
    • \$25.20
    • \$150.53
    • \$7,825.27

    Answer: \$27.39

  9. A company makes quarterly contributions of \$5,000 to a retirement fund at the end of each quarter. If the fund earns $8\%$ annual interest compounded quarterly, what will be the value of the fund after $20$ years? Round to the nearest cent.

    • \$968,859.79
    • \$988,236.98
    • \$228,809.82
    • \$400,000.00

    Answer: \$968,859.79

  10. You wish to save \$100,000 for your child's college education in $18$ years. If you can invest in an account that pays $6.5\%$ annual interest compounded semiannually, how much must you deposit at the end of each semiannual period to reach your goal? Round to the nearest cent.

    • \$1,502.83
    • \$3,085.46
    • \$1,455.53
    • \$2,777.78

    Answer: \$1,502.83

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