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. An ordinary annuity and an annuity due are established with identical periodic payments of $PMT$, periodic interest rates of $i$, and total number of periods $N$. Which of the following expressions correctly represents the ratio of the future value of the annuity due to the future value of the ordinary annuity, i.e., $\frac{FV_{due}}{FV_{ord}}$?

    • A) $1 + i$
    • B) $i$
    • C) $\frac{1}{1 + i}$
    • D) $N(1 + i)$

    Answer: A) $1 + i$

  2. An investor aims to accumulate a target future value $FV_{target}$ over $N$ periods. If they choose to make payments at the end of each period (ordinary annuity) with a periodic rate $i$, the required periodic payment is $PMT_{ord}$. If they instead decide to make payments at the beginning of each period (annuity due) to achieve the *same* $FV_{target}$ over the *same* $N$ periods with the *same* periodic rate $i$, what would be the required periodic payment, $PMT_{due}$, in terms of $PMT_{ord}$?

    • A) $PMT_{ord} \times (1 + i)$
    • B) $\frac{PMT_{ord}}{1 + i}$
    • C) $PMT_{ord} \times i$
    • D) $PMT_{ord} \times N$

    Answer: B) $\frac{PMT_{ord}}{1 + i}$

  3. An individual plans to save for $N$ periods to reach a specific future value $FV$. They calculate the required periodic payment $PMT$ using the ordinary annuity formula with a periodic interest rate $i$. If the periodic interest rate *doubles* to $2i$ (while $N$ and $FV$ remain constant), how will the new required periodic payment, $PMT'$, compare to the original $PMT$?

    • A) $PMT' = PMT \times \frac{1}{2}$
    • B) $PMT' < PMT \times \frac{1}{2}$
    • C) $PMT' > PMT \times \frac{1}{2}$
    • D) $PMT' = PMT$

    Answer: C) $PMT' > PMT \times \frac{1}{2}$

  4. An investor wants to accumulate a future value $FV_{target}$ over $N$ periods. They are considering two options: 1. An ordinary annuity with periodic payment $PMT_{ord}$ and periodic rate $i_1$. 2. An annuity due with periodic payment $PMT_{due}$ and periodic rate $i_2$. If $PMT_{ord} = PMT_{due} = PMT$ and $N$ is the same for both, but both options result in the *same* $FV_{target}$, what can be concluded about the relationship between $i_1$ and $i_2$?

    • A) $i_1 = i_2$
    • B) $i_1 < i_2$
    • C) $i_1 > i_2$
    • D) The relationship depends on $N$.

    Answer: C) $i_1 > i_2$

  5. An individual wants to accumulate $FV_{target}$ over $T$ years. They consider two savings plans, both with an annual interest rate $r$: 1. Plan A: Make annual payments of $PMT_A$ at the end of each year. 2. Plan B: Make semi-annual payments of $PMT_B$ at the end of each semi-annual period. If $PMT_A$ is the *total annual contribution* in Plan A, and $PMT_B$ is the *semi-annual contribution* in Plan B, and the total annual contribution in Plan B is also $PMT_A$ (meaning $2 \times PMT_B = PMT_A$), what is the ratio of the future value of Plan B to Plan A, i.e., $\frac{FV_B}{FV_A}$?

    • A) $\frac{(1 + r/2)^{2T} - 1}{(1 + r)^T - 1}$
    • B) $\frac{1}{2} \frac{(1 + r/2)^{2T} - 1}{(1 + r)^T - 1}$
    • C) $1$
    • D) $\frac{(1 + r)^T - 1}{(1 + r/2)^{2T} - 1}$

    Answer: A) $\frac{(1 + r/2)^{2T} - 1}{(1 + r)^T - 1}$

  6. A person makes monthly payments of $PMT$ at the end of each month for $T$ years into an account earning a monthly interest rate $i$. After $T$ years, they stop making payments but the accumulated amount continues to earn interest for another $T$ years. What is the total future value at the end of $2T$ years?

    • A) $(PMT \times \frac{(1 + i)^{12T} - 1}{i}) \times (1 + i)^{12T}$
    • B) $(PMT \times \frac{(1 + i)^{12T} - 1}{i}) + (PMT \times \frac{(1 + i)^{12T} - 1}{i} (1 + i))$
    • C) $(PMT \times \frac{(1 + i)^{12T} - 1}{i}) \times (1 + i)^{24T}$
    • D) $(PMT \times \frac{(1 + i)^{12T} - 1}{i}) \times (1 + i)^{T}$

    Answer: A) $(PMT \times \frac{(1 + i)^{12T} - 1}{i}) \times (1 + i)^{12T}$

  7. An individual wants to accumulate a target future value $FV$ in $N$ periods using an ordinary annuity with a periodic interest rate $i$. If they decide to achieve the *same* $FV$ in *half the time* ($N/2$ periods), how would the new required periodic payment, $PMT'$, relate to the original $PMT$?

    • A) $PMT' = PMT \times ((1 + i)^{N/2} + 1)$
    • B) $PMT' = PMT \times (1 + i)^{N/2}$
    • C) $PMT' = PMT \times 2$
    • D) $PMT' = PMT \times \frac{1}{2}$

    Answer: A) $PMT' = PMT \times ((1 + i)^{N/2} + 1)$

  8. An investor makes monthly payments of $P_1$ at the end of each month for $T_1$ years. Immediately after the last payment, they decide to switch to making monthly payments of $P_2$ at the *beginning* of each month for an additional $T_2$ years. The monthly interest rate is $i$ throughout. Which expression represents the total accumulated future value at the end of $T_1 + T_2$ years?

    • A) $(P_1 \times \frac{(1 + i)^{12T_1} - 1}{i}) \times (1 + i)^{12T_2} + P_2 \times \frac{(1 + i)^{12T_2} - 1}{i} (1 + i)$
    • B) $(P_1 \times \frac{(1 + i)^{12T_1} - 1}{i}) + (P_2 \times \frac{(1 + i)^{12T_2} - 1}{i} (1 + i))$
    • C) $(P_1 + P_2) \times \frac{(1 + i)^{12(T_1 + T_2)} - 1}{i}$
    • D) $(P_1 \times \frac{(1 + i)^{12T_1} - 1}{i}) \times (1 + i)^{12(T_1 + T_2)} + P_2 \times \frac{(1 + i)^{12T_2} - 1}{i} (1 + i)$

    Answer: A) $(P_1 \times \frac{(1 + i)^{12T_1} - 1}{i}) \times (1 + i)^{12T_2} + P_2 \times \frac{(1 + i)^{12T_2} - 1}{i} (1 + i)$

  9. Consider two ordinary annuities, Annuity X and Annuity Y, both designed to accumulate a future value over $T$ years with the same stated annual interest rate $r$. Annuity X: Annual payments of $PMT_{annual}$ are made at the end of each year. Annuity Y: Semi-annual payments are made at the end of each semi-annual period, such that the *total annual contribution* is also $PMT_{annual}$ (i.e., each semi-annual payment is $\frac{PMT_{annual}}{2}$). Which annuity will result in a higher future value at the end of $T$ years, and why?

    • A) Annuity X, because payments are larger.
    • B) Annuity Y, because of more frequent compounding and earlier payments on average.
    • C) Annuity X, because the interest rate is applied fewer times.
    • D) Annuity Y, but only if $r$ is very high.

    Answer: B) Annuity Y, because of more frequent compounding and earlier payments on average.

  10. An investor wants to accumulate a future value $FV_A$ using an ordinary annuity with periodic payment $PMT_A$, periodic rate $i_A$, and $N_A$ periods. Another investor wants to accumulate $FV_B$ using an annuity due with periodic payment $PMT_B$, periodic rate $i_B$, and $N_B$ periods. If $PMT_A = PMT_B = PMT$, $N_A = N_B = N$, and $FV_A = FV_B = FV$, what is the relationship between $i_A$ and $i_B$?

    • A) $i_A = i_B$
    • B) $i_A < i_B$
    • C) $i_A > i_B$
    • D) The relationship cannot be determined without specific values for $N$.

    Answer: C) $i_A > i_B$

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