Question

a bank offers the following two investment options. find the value for each investment option if $20,000 is invested for 10 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 assume the full amount is withdrawn.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. (round to the nearest dollar as needed.) the value of the long - term investment is $, and the value of the money maker savings is $. 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%.

Explanation:

Step1: Calculate the future - value of the CD

The APY for the 10 - year CD is $r_{1}=2.785\%=0.02785$. The formula for compound - interest when compounded annually (since APY is annual) is $A = P(1 + r)^{t}$, where $P=\$20000$, $r = 0.02785$, and $t = 10$.
$A_{1}=20000\times(1 + 0.02785)^{10}$
$A_{1}=20000\times1.02785^{10}$
Using a calculator, $1.02785^{10}\approx1.31777$.
$A_{1}=20000\times1.31777=\$26355.4\approx\$26355$

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

The initial interest rate $r_{0}=2.5\% = 0.025$, compounded monthly. The compound - interest formula is $A=P(1+\frac{r}{n})^{nt}$, where $P = 20000$, $n = 12$, and $t = 10$.
First, for the first 4 years:
$A_{21}=20000\times(1+\frac{0.025}{12})^{12\times4}$
$A_{21}=20000\times(1+\frac{0.025}{12})^{48}$
$(1+\frac{0.025}{12})^{48}\approx1.105167$
$A_{21}=20000\times1.105167 = 22103.34$

Then, the interest rate increases by $0.25\%$ for the next 4 years. The new interest rate $r_{1}=0.025 + 0.0025=0.0275$
$A_{22}=A_{21}\times(1+\frac{0.0275}{12})^{12\times4}$
$A_{22}=22103.34\times(1+\frac{0.0275}{12})^{48}$
$(1+\frac{0.0275}{12})^{48}\approx1.11627$
$A_{22}=22103.34\times1.11627\approx24677.2$

Then, for the last 2 years, the interest rate increases by another $0.25\%$, so $r_{2}=0.0275+0.0025 = 0.03$
$A_{2}=A_{22}\times(1+\frac{0.03}{12})^{12\times2}$
$A_{2}=A_{22}\times(1 + 0.0025)^{24}$
$(1 + 0.0025)^{24}\approx1.061778$
$A_{2}=24677.2\times1.061778\approx26297.7\approx26298$

Answer:

The value of the Long - Term Investment is $\$26355$, and the value of the Money Maker Savings is $\$26298$