Question

a bank offers the following two investment options. find the value for each investment option if $25,000 is invested for 20 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 assume the full amount is withdrawn.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 value of the Long - Term Investment (CD)

The formula for APY (Annual Percentage Yield) is $APY=(1 + \frac{r}{n})^{n}-1$, where $r$ is the annual interest rate and $n$ is the number of compounding periods per year. For a 10 - year CD with an APY of $2.785\%$, and assuming the principal $P = 25000$. Since it's not clear if it's compounded monthly, quarterly etc. and APY is given, we can directly use the compound - interest formula $A=P(1 + APY)^{t}$, where $t = 10$ years.
$A_{CD}=25000\times(1 + 0.02785)^{10}$
$A_{CD}=25000\times1.02785^{10}$
$1.02785^{10}\approx1.3177$
$A_{CD}=25000\times1.3177 = 32942.5\approx32943$

Step2: Calculate the value of the Money Maker Savings

The initial interest rate $r_0=2.5\%=0.025$, compounded monthly ($n = 12$). The compound - interest formula is $A = P(1+\frac{r}{n})^{nt}$.
For the first 4 years:
$A_1=25000\times(1+\frac{0.025}{12})^{12\times4}$
$A_1=25000\times(1+\frac{0.025}{12})^{48}$
$(1+\frac{0.025}{12})^{48}\approx1.10516$
$A_1=25000\times1.10516 = 27629$

For the next 4 years, the interest rate increases by $0.25\%$, so $r_1=0.025 + 0.0025=0.0275$
$A_2=A_1\times(1+\frac{0.0275}{12})^{12\times4}$
$A_2=27629\times(1+\frac{0.0275}{12})^{48}$
$(1+\frac{0.0275}{12})^{48}\approx1.1167$
$A_2=27629\times1.1167\approx30853.3$

For the next 4 years, the interest rate increases by another $0.25\%$, so $r_2=0.0275+ 0.0025 = 0.03$
$A_3=A_2\times(1+\frac{0.03}{12})^{12\times4}$
$A_3=30853.3\times(1+\frac{0.03}{12})^{48}$
$(1+\frac{0.03}{12})^{48}\approx1.12727$
$A_3=30853.3\times1.12727\approx34704.4$

For the last 8 years, the interest rate increases by another $0.25\%$, so $r_3=0.03 + 0.0025=0.0325$
$A_4=A_3\times(1+\frac{0.0325}{12})^{12\times8}$
$A_4=34704.4\times(1+\frac{0.0325}{12})^{96}$
$(1+\frac{0.0325}{12})^{96}\approx1.2919$
$A_4=34704.4\times1.2919\approx44834$

Answer:

The value of the Long - Term Investment is $\$32943$ and the value of the Money Maker Savings is $\$44834$