Compound Interest — Hard Practice Quiz

A Algebra cheat sheet for Compound Interest — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Practice quiz

1. An investment $P$ is made at an annual interest rate $r$. If it is compounded continuously for $t$ years, the final amount is $A_C$. If it is compounded annually for $t$ years, the final amount is $A_A$. Which of the following statements is always true for $r > 0$ and $t > 0$?

   • $A_C > A_A$
   • $A_C < A_A$
   • $A_C = A_A$
   • The relationship depends on $P$.

   Answer: $A_C > A_A$

2. An investment doubles in $t$ years under continuous compounding at an annual rate $r_C$. If the same investment were to double in the same $t$ years under annual compounding, what would be the required annual rate $r_A$ in terms of $r_C$?

   • $r_A = e^{r_C} - 1$
   • $r_A = \ln(1 + r_C)$
   • $r_A = r_C$
   • $r_A = 2r_C$

   Answer: $r_A = e^{r_C} - 1$

3. Consider two investments. Investment X: Principal $P$ is invested at rate $r$ for $t$ years, compounded quarterly. Investment Y: Principal $2P$ is invested at rate $r/2$ for $2t$ years, compounded semi-annually. What is the ratio of the final amount of Investment X to Investment Y, i.e., $A_X / A_Y$?

   • $1/2$
   • $1/4$
   • $1$
   • $2$

   Answer: $1/2$

4. The formula for compound interest is $A = P(1 + \frac{r}{n})^{nt}$. As the number of compounding periods per year, $n$, approaches infinity, this formula approaches which of the following forms?

   • $A = P(1 + r)^t$
   • $A = Pe^{rt}$
   • $A = P(1 + \frac{r}{t})^n$
   • $A = P \ln(rt)$

   Answer: $A = Pe^{rt}$

5. An investor aims to achieve a future value of $A$ in $t$ years at an annual interest rate $r$. What is the ratio of the principal required for continuous compounding ($P_C$) to the principal required for monthly compounding ($P_M$)?

   • $\frac{(1 + \frac{r}{12})^{12t}}{e^{rt}}$
   • $\frac{e^{rt}}{(1 + \frac{r}{12})^{12t}}$
   • $1$
   • $\frac{1}{12}$

   Answer: $\frac{(1 + \frac{r}{12})^{12t}}{e^{rt}}$

6. The effective annual rate (EAR) for an investment compounded $n$ times a year at a nominal rate $r$ is given by $EAR = (1 + \frac{r}{n})^n - 1$. If an investment with a nominal rate $r_1$ compounded quarterly has the same EAR as an investment with a nominal rate $r_2$ compounded continuously, what is the relationship between $r_1$ and $r_2$?

   • $r_1 = 4(e^{r_2/4} - 1)$
   • $r_1 = e^{r_2} - 1$
   • $r_1 = r_2$
   • $r_1 = \ln(1 + r_2)$

   Answer: $r_1 = 4(e^{r_2/4} - 1)$

7. An investment $P$ is made at an annual rate $r$. How much longer (in years) does it take for the investment to triple if compounded annually compared to being compounded continuously?

   • $\ln 3 \left( \frac{1}{\ln(1 + r)} - \frac{1}{r} \right)$
   • $\ln 3 \left( \frac{1}{r} - \frac{1}{\ln(1 + r)} \right)$
   • $\frac{\ln 3}{r} - \frac{\ln 3}{\ln(1 + r)}$
   • The time is the same for both.

   Answer: $\ln 3 \left( \frac{1}{\ln(1 + r)} - \frac{1}{r} \right)$

8. An investment $P$ grows to $A$ in $t$ years with continuous compounding at an annual rate $r_C$. If the same investment $P$ were to grow to the same amount $A$ in the same time $t$ with semi-annual compounding, what would be the required nominal rate $r_{SA}$ in terms of $r_C$?

   • $r_{SA} = 2(e^{r_C/2} - 1)$
   • $r_{SA} = e^{r_C} - 1$
   • $r_{SA} = r_C$
   • $r_{SA} = \ln(1 + r_C)$

   Answer: $r_{SA} = 2(e^{r_C/2} - 1)$

9. For a given principal $P$, annual rate $r$, and time $t$, how does the final amount $A$ change as the number of compounding periods per year, $n$, increases? What is the maximum possible value $A$ can approach?

   • $A$ increases and approaches $Pe^{rt}$.
   • $A$ decreases and approaches $Pe^{rt}$.
   • $A$ increases indefinitely.
   • $A$ remains constant regardless of $n$.

   Answer: $A$ increases and approaches $Pe^{rt}$.

10. An initial principal $P$ is invested for $t_1$ years with continuous compounding at an annual rate $r_1$. The accumulated amount is then reinvested for $t_2$ years with quarterly compounding at an annual rate $r_2$. What is the final amount $A_F$ after $t_1 + t_2$ years?

    • $A_F = Pe^{r_1 t_1} (1 + \frac{r_2}{4})^{4t_2}$
    • $A_F = P(1 + \frac{r_1}{4})^{4t_1} e^{r_2 t_2}$
    • $A_F = P e^{(r_1 + r_2)(t_1 + t_2)}$
    • $A_F = P (1 + \frac{r_1 + r_2}{4})^{4(t_1 + t_2)}$

    Answer: $A_F = Pe^{r_1 t_1} (1 + \frac{r_2}{4})^{4t_2}$