Perpetuities — Hard Practice Quiz

A Financial Math cheat sheet for Perpetuities — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Practice quiz

1. An investor wishes to receive a perpetuity with a present value of $PV_0$. If the annual interest rate is $r$ and compounding occurs $n$ times per year, what is the required periodic payment $PMT$?

   • $PMT = PV_0 \cdot \frac{r}{n}$
   • $PMT = PV_0 \cdot r \cdot n$
   • $PMT = \frac{PV_0 \cdot n}{r}$
   • $PMT = \frac{PV_0}{r \cdot n}$

   Answer: $PMT = PV_0 \cdot \frac{r}{n}$

2. If the annual interest rate $r$ for a perpetuity is halved, while the periodic payment $PMT$ and the number of compounding periods per year $n$ remain constant, how does the present value $PV$ change?

   • $PV$ is halved.
   • $PV$ is doubled.
   • $PV$ is quadrupled.
   • $PV$ remains unchanged.

   Answer: $PV$ is doubled.

3. Consider a perpetuity with a fixed annual interest rate $r$ and a constant periodic payment $PMT$. If the compounding frequency $n$ is doubled, how does the present value $PV$ of this perpetuity change?

   • $PV$ is halved.
   • $PV$ is doubled.
   • $PV$ is quadrupled.
   • $PV$ remains unchanged.

   Answer: $PV$ is doubled.

4. Perpetuity A offers a periodic payment of $PMT_A = \$100$ with an annual rate of $r_A = 5\%$ compounded annually ($n_A = 1$). Perpetuity B offers a periodic payment of $PMT_B = \$60$ with an annual rate of $r_B = 3\%$ compounded semi-annually ($n_B = 2$). Which perpetuity has a higher present value, and by how much?

   • Perpetuity A by $\$1000$.
   • Perpetuity B by $\$2000$.
   • Perpetuity A by $\$500$.
   • Perpetuity B by $\$1000$.

   Answer: Perpetuity B by $\$2000$.

5. An investment of $PV_0$ is made to generate a perpetuity with periodic payments of $PMT$. If compounding occurs $n$ times per year, what is the annual interest rate $r$?

   • $r = \frac{PMT \cdot n}{PV_0}$
   • $r = \frac{PMT}{PV_0 \cdot n}$
   • $r = \frac{PV_0 \cdot n}{PMT}$
   • $r = \frac{PV_0}{PMT \cdot n}$

   Answer: $r = \frac{PMT \cdot n}{PV_0}$

6. If the periodic interest rate $i$ for a perpetuity increases by $25\%$, how does the present value $PV$ change, assuming the periodic payment $PMT$ remains constant?

   • $PV$ decreases by $20\%$.
   • $PV$ increases by $25\%$.
   • $PV$ decreases by $25\%$.
   • $PV$ increases by $20\%$.

   Answer: $PV$ decreases by $20\%$.

7. An investor initially sets up a perpetuity to have a present value of $\$10,000$ with an annual interest rate of $4\%$ compounded quarterly. If the annual interest rate unexpectedly drops to $2\%$ (still compounded quarterly), what new periodic payment $PMT$ is required to maintain the original present value of $\$10,000$?

   • $\$50$
   • $\$100$
   • $\$200$
   • $\$400$

   Answer: $\$50$

8. If the annual payment $PMT$ of a perpetuity is $3\%$ of its present value $PV$, and compounding occurs annually ($n=1$), what is the annual interest rate $r$?

   • $0.03\%$
   • $3\%$
   • $30\%$
   • $0.3\%$

   Answer: $3\%$

9. For a given perpetuity, which of the following changes would result in the largest increase in its present value $PV$?

   • Doubling the periodic payment $PMT$.
   • Halving the annual interest rate $r$.
   • Increasing the number of compounding periods per year $n$ by $50\%$.
   • Doubling $PMT$ and halving $r$ simultaneously.

   Answer: Doubling $PMT$ and halving $r$ simultaneously.

10. A perpetuity currently has a present value of $PV_0$. If the periodic payment $PMT$ is increased by $20\%$ and the annual interest rate $r$ is decreased by $10\%$, while the compounding frequency $n$ remains constant, what is the new present value $PV'$ in terms of $PV_0$?

    • $PV' = 1.33 PV_0$
    • $PV' = 1.08 PV_0$
    • $PV' = 1.2 PV_0$
    • $PV' = 1.11 PV_0$

    Answer: $PV' = 1.33 PV_0$