Which compound interest problem yields the most money?
Compounded quarterly
Compounded continuously
Compounded monthly
Compounded Annually

Question

Which compound interest problem yields the most money? Compounded quarterly Compounded continuously Compounded monthly Compounded Annually

UpStudy AI · Accepted Answer

We compare the following formulas for compound interest after a time period $$t$$ years and an initial principal $$P$$.

1. **Annually:**  
   $$
   A = P(1 + r)^t
   $$

2. **Quarterly:**  
   $$
   A = P\left(1 + \frac{r}{4}\right)^{4t}
   $$

3. **Monthly:**  
   $$
   A = P\left(1 + \frac{r}{12}\right)^{12t}
   $$

4. **Continuously:**  
   $$
   A = Pe^{rt}
   $$

For a fixed nominal interest rate $$r$$ and time period $$t$$, the effective annual rate increases as the number of compounding periods increases. The limit of the periodic compounding formula as the number of compounding periods $$n$$ tends to infinity is given by continuous compounding:
$$
\lim_{n \to \infty} P\left(1 + \frac{r}{n}\right)^{nt} = Pe^{rt}
$$

Thus, among the four options given, continuous compounding yields the highest accumulated amount.

Therefore, the compound interest problem that yields the most money is the one compounded continuously.
Continuous compounding yields the most money.

Which compound interest problem yields the most money?
Compounded quarterly
Compounded continuously
Compounded monthly
Compounded Annually

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