Decreasing Annuities (Present Value) — Hard Practice Quiz

A Financial Math cheat sheet for Decreasing Annuities (Present Value) — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Practice quiz

1. If the periodic interest rate $i$ increases, how does the present value $PV$ required to support a fixed periodic withdrawal $PMT$ for a fixed number of periods $N$ change?

   • $PV$ increases.
   • $PV$ decreases.
   • $PV$ remains the same.
   • $PV$ change is indeterminate without specific values.

   Answer: $PV$ decreases.

2. Derive the expression for the periodic withdrawal $PMT$ in terms of $PV$, $i$, and $N$.

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

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

3. An annuity requires a present value of $PV_1$ to provide a periodic withdrawal of $PMT_1$ for $N$ periods at an interest rate $i$. If the desired periodic withdrawal is doubled to $2 \times PMT_1$, what is the new present value $PV_2$ required, assuming $i$ and $N$ remain constant?

   • $PV_2 = PV_1$
   • $PV_2 = 2 \times PV_1$
   • $PV_2 = 4 \times PV_1$
   • $PV_2 = \frac{1}{2} \times PV_1$

   Answer: $PV_2 = 2 \times PV_1$

4. Consider an annuity where $PMT$ and $i$ are fixed. If the total number of periods $N$ for withdrawals is increased, how does the required present value $PV$ change?

   • $PV$ increases.
   • $PV$ decreases.
   • $PV$ remains the same.
   • $PV$ change is indeterminate without specific values.

   Answer: $PV$ increases.

5. A retirement fund is set up to provide monthly withdrawals for $20$ years. The annual interest rate is $6\%$. If the compounding frequency changes from monthly to quarterly, while keeping the annual rate and total duration constant, how does this affect the periodic interest rate $i$ and the total number of periods $N$ used in the $PV$ formula?

   • $i$ increases, $N$ decreases.
   • $i$ decreases, $N$ increases.
   • $i$ increases, $N$ increases.
   • $i$ decreases, $N$ decreases.

   Answer: $i$ increases, $N$ decreases.

6. An individual plans to withdraw $PMT$ dollars at the end of each period for $N$ periods. The initial lump sum required is $PV$. Which of the following statements is true regarding the total amount withdrawn ($TotalWithdrawals = PMT \times N$) and the initial $PV$?

   • $TotalWithdrawals$ is always less than $PV$.
   • $TotalWithdrawals$ is always equal to $PV$.
   • $TotalWithdrawals$ is always greater than $PV$ if $i > 0$.
   • $TotalWithdrawals$ can be less than, equal to, or greater than $PV$ depending on $i$ and $N$.

   Answer: $TotalWithdrawals$ is always greater than $PV$ if $i > 0$.

7. Let $AF = \frac{1 - (1 + i)^{-N}}{i}$ be the annuity factor. If $PV = PMT \times AF$, express $AF$ in terms of $PV$, $PMT$, $r$, $n$, and $t$.

   • $AF = \frac{PV}{PMT}$
   • $AF = \frac{PV}{PMT} \times \frac{r}{n}$
   • $AF = \frac{PV}{PMT} \times (1 + \frac{r}{n})^{-nt}$
   • $AF = \frac{PMT}{PV}$

   Answer: $AF = \frac{PV}{PMT}$

8. Consider two annuities, A and B, both designed to provide the same periodic withdrawal $PMT$ for the same total number of periods $N$. Annuity A has a periodic interest rate $i_A$, and Annuity B has a periodic interest rate $i_B$. If $i_A > i_B > 0$, which annuity requires a larger initial present value $PV$?

   • Annuity A requires a larger $PV$.
   • Annuity B requires a larger $PV$.
   • Both require the same $PV$.
   • The required $PV$ depends on the specific values of $PMT$ and $N$.

   Answer: Annuity B requires a larger $PV$.

9. In the formula $PV = PMT \times \frac{1 - (1 + i)^{-N}}{i}$, what does the term $(1 + i)^{-N}$ conceptually represent in the context of the annuity's present value calculation?

   • The future value of a single dollar after $N$ periods.
   • The present value of a single dollar received $N$ periods from now.
   • The future value of an annuity of $N$ payments.
   • The present value of an annuity of $N$ payments.

   Answer: The present value of a single dollar received $N$ periods from now.

10. A fund provides a monthly withdrawal of $PMT$ for $t$ years with an annual interest rate $r$ compounded monthly. If the total duration of withdrawals is doubled to $2t$ years, while keeping $PMT$ and $r$ constant, how does the required present value $PV$ change?

    • $PV$ doubles.
    • $PV$ more than doubles.
    • $PV$ less than doubles.
    • $PV$ change is indeterminate without specific values.

    Answer: $PV$ less than doubles.