Annuities (Savings & Investments) — Hard Practice Quiz

A Financial Math cheat sheet for Annuities (Savings & Investments) — every key formula with its symbols defined — plus a hard-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. Consider an ordinary annuity and an annuity due, both designed to accumulate the exact same future value ($FV$) over the same number of periods ($N$) and at the same periodic interest rate ($i$). What is the ratio of the required periodic payment for the ordinary annuity ($PMT_{ordinary}$) to the required periodic payment for the annuity due ($PMT_{due}$)?

    • $\frac{1}{1 + i}$
    • $1 + i$
    • $(1 + i)^N$
    • $\frac{i}{(1 + i)^N - 1}$

    Answer: $1 + i$

  2. If the periodic payment ($PMT$) for an annuity due is doubled, how does its future value ($FV_{due}$) change, assuming the periodic rate ($i$) and total number of periods ($N$) remain constant?

    • The future value is halved.
    • The future value remains the same.
    • The future value is doubled.
    • The future value increases by a factor of $(1 + i)$.

    Answer: The future value is doubled.

  3. An individual wants to achieve the same future value as an annuity due that pays $PMT_A$ for $N_A$ periods at a periodic rate $i_A$. What periodic payment $PMT_B$ would be required for an ordinary annuity over the same $N_A$ periods at the same periodic rate $i_A$ to reach this identical future value?

    • $PMT_B = PMT_A \times (1 + i_A)$
    • $PMT_B = PMT_A \times \frac{1}{1 + i_A}$
    • $PMT_B = PMT_A \times (1 + i_A)^{N_A}$
    • $PMT_B = PMT_A$

    Answer: $PMT_B = PMT_A \times (1 + i_A)$

  4. Investor A makes payments of \$200 at the end of each month for 5 years into an account earning $6\%$ annual interest compounded monthly. Investor B makes payments of \$195 at the beginning of each month for 5 years into an account earning $6\%$ annual interest compounded monthly. Which investor accumulates more, and by approximately how much?

    • Investor A by \$280.82
    • Investor B by \$280.82
    • Investor A by \$139.54
    • Investor B by \$136.73

    Answer: Investor A by \$280.82

  5. An ordinary annuity with periodic payment $PMT$ and periodic rate $i$ accumulates to a future value $FV$ in $N$ periods. If the periodic rate increased to $2i$ (assuming $PMT$ remains constant), how would the number of periods $N'$ required to reach the *same* future value $FV$ qualitatively change?

    • The number of periods $N'$ would be exactly halved.
    • The number of periods $N'$ would decrease, but not necessarily by half.
    • The number of periods $N'$ would increase.
    • The number of periods $N'$ would remain the same.

    Answer: The number of periods $N'$ would decrease, but not necessarily by half.

  6. A student needs \$50,000 for college in 4 years. They plan to make monthly payments into an account earning $4.8\%$ annual interest compounded monthly. If they initially planned to make payments at the end of each month, but then decided to make payments at the beginning of each month instead, how much less would each monthly payment be, approximately?

    • \$3.81
    • \$4.57
    • \$952.25
    • \$948.43

    Answer: \$3.81

  7. Express the future value of an annuity due ($FV_{due}$) in terms of the future value of an ordinary annuity ($FV_{ordinary}$) with the same periodic payment $PMT$, periodic rate $i$, and total number of periods $N$.

    • $FV_{due} = FV_{ordinary} \times (1 + i)$
    • $FV_{due} = FV_{ordinary} \times \frac{1}{1 + i}$
    • $FV_{due} = FV_{ordinary} \times N$
    • $FV_{due} = FV_{ordinary} + PMT$

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

  8. An ordinary annuity with periodic payment $PMT$ and periodic rate $i$ accumulates to $FV$ in $N$ periods. If the number of periods is halved to $N/2$, and the periodic payment is kept the same, how would the new future value $FV'$ compare to the original $FV$?

    • $FV'$ would be exactly half of $FV$.
    • $FV'$ would be more than half of $FV$.
    • $FV'$ would be less than half of $FV$.
    • $FV'$ would be equal to $FV$ if $i$ is also halved.

    Answer: $FV'$ would be less than half of $FV$.

  9. A person wants to accumulate \$100,000 in 10 years. They can invest in an account earning $5\%$ annual interest. Compare the total annual contribution required if payments are made annually (ordinary annuity) versus if payments are made semi-annually (ordinary annuity).

    • Annual payments require a total annual contribution of \$7,950.50, while semi-annual payments require \$7,829.80 annually.
    • Annual payments require a total annual contribution of \$7,829.80, while semi-annual payments require \$7,950.50 annually.
    • Both strategies require the same total annual contribution.
    • Semi-annual payments require a total annual contribution of \$3,914.90, which is less than annual.

    Answer: Annual payments require a total annual contribution of \$7,950.50, while semi-annual payments require \$7,829.80 annually.

  10. An individual plans to save \$250 at the end of each month for 4 years into an account earning $3\%$ annual interest compounded monthly. At the end of 4 years, how much more would they have accumulated if they had made payments at the beginning of each month instead?

    • \$31.83
    • \$12,732.80
    • \$12,764.63
    • \$63.66

    Answer: \$31.83

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