Question

a bank offers a money market account paying 9.1% interest compounded annually. a competing bank offers a money market account paying 8.7% interest compounded daily (assume 365 days in a year.) which account is the better investment?
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choose the better investment.

a. 9.1% compounded annually
b. 8.7% compounded daily

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.
• $t$ is the time the money is invested for in years.

To compare the two accounts, we can assume $P = 1$ (since the principal will be the same for both accounts and will not affect the comparison of the growth rates) and $t = 1$ (we can compare the amount after 1 year).

Step2: Calculate for 9.1% compounded annually

For the first account:

• $r=0.091$, $n = 1$ (compounded annually), $t = 1$, $P = 1$.

Using the formula $A=P(1 +\frac{r}{n})^{nt}$, we substitute the values:
$A_1=1\times(1+\frac{0.091}{1})^{1\times1}=1 + 0.091=1.091$

Step3: Calculate for 8.7% compounded daily

For the second account:

• $r = 0.087$, $n=365$ (compounded daily), $t = 1$, $P = 1$.

Using the formula $A = P(1+\frac{r}{n})^{nt}$, we substitute the values:
$A_2=1\times(1+\frac{0.087}{365})^{365\times1}=(1+\frac{0.087}{365})^{365}$
First, calculate $\frac{0.087}{365}\approx0.000238356$
Then, $1 + 0.000238356=1.000238356$
Now, calculate $(1.000238356)^{365}$. Using the formula $a^b=e^{b\ln(a)}$, we have:
$\ln(1.000238356)\approx0.00023832$
$365\times0.00023832\approx0.087$
$e^{0.087}\approx1.0909$ (we can also calculate it directly: $(1.000238356)^{365}\approx1.0909$)

Step4: Compare the two amounts

We have $A_1 = 1.091$ and $A_2\approx1.0909$
Since $1.091>1.0909$, the account with 9.1% compounded annually has a higher amount after 1 year.

Answer:

A. 9.1% compounded annually