Decreasing Annuities (Present Value) — Practice Quiz

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

Practice quiz

1. In the present value of an annuity formula, what does $PV$ primarily represent?

   • The total amount of all future payments.
   • The future value of a series of payments.
   • The current lump sum needed to fund future periodic withdrawals.
   • The interest earned over the life of the annuity.

   Answer: The current lump sum needed to fund future periodic withdrawals.

2. A person wants to withdraw $2,000$ at the end of each month for $10$ years from an account earning an annual interest rate of $6\%$ compounded monthly. In the formula $PV = PMT \times \frac{1 - (1 + i)^{-N}}{i}$, what are the correct values for $PMT$, $i$, and $N$?

   • $PMT = 2,000$, $i = 0.06$, $N = 10$
   • $PMT = 2,000$, $i = 0.005$, $N = 120$
   • $PMT = 2,000$, $i = 0.06/12$, $N = 10$
   • $PMT = 2,000$, $i = 0.005$, $N = 10$

   Answer: $PMT = 2,000$, $i = 0.005$, $N = 120$

3. Calculate the present value ($PV$) required to provide $PMT = 500$ per period for $N = 5$ periods, with a periodic interest rate $i = 0.02$. Round to two decimal places.

   • $2,358.00$
   • $2,500.00$
   • $2,450.00$
   • $2,200.00$

   Answer: $2,358.00$

4. A student wants to receive $1,500$ at the end of each month for $4$ years after graduation. If the account earns an annual interest rate of $3.6\%$ compounded monthly, what lump sum must be in the account at graduation to fund these withdrawals? Round to two decimal places.

   • $65,550.00$
   • $72,000.00$
   • $68,250.00$
   • $63,120.00$

   Answer: $65,550.00$

5. Holding all other variables constant, if the periodic withdrawal amount ($PMT$) increases, what happens to the present value ($PV$) required?

   • $PV$ decreases.
   • $PV$ increases.
   • $PV$ remains unchanged.
   • $PV$ becomes zero.

   Answer: $PV$ increases.

6. Holding all other variables constant, if the periodic interest rate ($i$) increases, what happens to the present value ($PV$) required to fund the same series of withdrawals?

   • $PV$ decreases.
   • $PV$ increases.
   • $PV$ remains unchanged.
   • $PV$ becomes infinite.

   Answer: $PV$ decreases.

7. Holding all other variables constant, if the total number of periods ($N$) for withdrawals increases, what happens to the present value ($PV$) required?

   • $PV$ decreases.
   • $PV$ increases.
   • $PV$ remains unchanged.
   • $PV$ becomes negative.

   Answer: $PV$ increases.

8. If you know the present value ($PV$), the periodic interest rate ($i$), and the total number of periods ($N$), which of the following expressions would correctly solve for the periodic withdrawal amount ($PMT$)?

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

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

9. A retiree wants to withdraw $4,000$ at the beginning of each month for $25$ years. The retirement account earns an annual interest rate of $4.8\%$ compounded monthly. What is the minimum lump sum needed in the account at retirement? (Note: The given formula assumes end-of-period payments. For beginning-of-period payments, multiply the result by $(1+i)$.) Round to two decimal places.

   • $699,000.00$
   • $701,796.00$
   • $1,200,000.00$
   • $685,500.00$

   Answer: $701,796.00$

10. In the formula $PV = PMT \times \frac{1 - (1 + i)^{-N}}{i}$, the term $\frac{1 - (1 + i)^{-N}}{i}$ is known as the:

    • Future Value Interest Factor of an Annuity.
    • Present Value Interest Factor of an Annuity.
    • Compound Interest Factor.
    • Discount Rate Factor.

    Answer: Present Value Interest Factor of an Annuity.