Question

1. ben is comparing savings accounts at different banks and finds that most are offering an interest rate of about 1%. how does this low interest rate impact the power of compounding?

Explanation:

Step1: Recall compound - interest formula

The compound - interest formula is $A = P(1+\frac{r}{n})^{nt}$, where $A$ is the amount of money accumulated after $n$ years, including interest, $P$ is the principal amount (the initial amount of money), $r$ is the annual interest rate (in decimal form), $n$ is the number of times that interest is compounded per year, and $t$ is the time the money is invested for in years.

Step2: Analyze the effect of low $r$

When $r$ (the interest rate) is low (in this case $r = 0.01$), the factor $(1+\frac{r}{n})$ is close to 1. As we compound over time, $(1+\frac{r}{n})^{nt}$ will grow more slowly compared to a higher interest rate. For example, if $n = 1$ (compounded annually) and $t = 1$, $A=P(1 + r)$. With $r = 0.01$, $A = 1.01P$. As $t$ increases, the growth of $A$ is less significant compared to a higher $r$.

Answer:

A low interest rate of 1% reduces the power of compounding. The growth of the investment over time is slower compared to a higher interest rate because the factor $(1+\frac{r}{n})$ is closer to 1, resulting in less exponential growth of the accumulated amount $A$ over time.