Question

a bank offers the following two investment options. find the value for each investment option if $20,000 is invested for 8 years. assume the full amount is withdrawn. long - term investment! 10 - year cd at 2.785% apy! apply online or at one of our convenient locations! note: cd means certificate of deposit. apy=(1 + \LXI0)¹² - 1 early withdrawal fee before 10 years is 2% of account balance. money maker savings! minimum balance: $10,000 earn 2.5% interest compounded monthly. loyalty program! every 4 years with us, your interest rate increases by 0.25%! the value of the long - term investment is $ _, and the value of the money maker savings is $ _. (round to the nearest dollar as needed.)

Explanation:

Step1: Calculate the future - value of the 10 - year CD

The formula for the future value of an investment with annual percentage yield (APY) is $A = P(1 + r)^{n}$, where $P$ is the principal amount, $r$ is the APY, and $n$ is the number of years.
Given $P=\$20000$, $r = 0.02785$ (2.785% APY), and $n = 10$.
$A_{CD}=20000\times(1 + 0.02785)^{10}$
$A_{CD}=20000\times1.02785^{10}$
Using a calculator, $1.02785^{10}\approx1.31777$.
$A_{CD}=20000\times1.31777=\$26355.4\approx\$26355$

Step2: Calculate the future - value of the Money Maker Savings

The formula for compound - interest is $A=P(1+\frac{r}{m})^{mn}$, where $P$ is the principal amount, $r$ is the annual interest rate, $m$ is the number of compounding periods per year, and $n$ is the number of years.
The initial interest rate $r_1 = 0.025$ (2.5%), $m = 12$ (compounded monthly), and $n = 8$.
Since there is a loyalty program that increases the interest rate every 4 years by 0.25%, for the first 4 years, $r_1 = 0.025$, and for the next 4 years, $r_2=0.025 + 0.0025=0.0275$.
For the first 4 years:
$A_1=20000\times(1+\frac{0.025}{12})^{12\times4}$
Let $x_1=(1+\frac{0.025}{12})^{48}$, using a calculator, $x_1\approx1.10516$.
$A_1 = 20000\times1.10516=\$22103.2$
For the next 4 years, with $P = A_1=\$22103.2$, $r_2 = 0.0275$, and $m = 12$, $n = 4$
$A_2=22103.2\times(1+\frac{0.0275}{12})^{12\times4}$
Let $x_2=(1+\frac{0.0275}{12})^{48}$, using a calculator, $x_2\approx1.11727$.
$A_2=22103.2\times1.11727=\$24704.5\approx\$24705$

Answer:

The value of the Long - Term Investment (10 - year CD) is $\$26355$, and the value of the Money Maker Savings is $\$24705$