To find the present value of a future amount that is compounded quarterly, we can use the formula for compound interest, which is:
[tex]\[ PV = \frac{FV}{(1 + \frac{r}{n})^{nt}} \][/tex]
Where:
- [tex]\( PV \)[/tex] is the present value
- [tex]\( FV \)[/tex] is the future value ([tex]$19,000 in this case)
- \( r \) is the annual interest rate (3% or 0.03)
- \( n \) is the number of times the interest is compounded per year (quarterly means \( n = 4 \))
- \( t \) is the number of years (4.5 years)
Plugging the values into the formula:
1. Future Value (\( FV \)): \$[/tex]19,000
2. Annual Interest Rate ([tex]\( r \)[/tex]): 0.03
3. Compounding Per Year ([tex]\( n \)[/tex]): 4
4. Time in Years ([tex]\( t \)[/tex]): 4.5
Let's take it step-by-step:
Step 1: Calculate [tex]\( \frac{r}{n} \)[/tex]:
[tex]\[ \frac{0.03}{4} = 0.0075 \][/tex]
Step 2: Calculate [tex]\( (1 + \frac{r}{n}) \)[/tex]:
[tex]\[ 1 + 0.0075 = 1.0075 \][/tex]
Step 3: Calculate [tex]\( nt \)[/tex]:
[tex]\[ 4 \times 4.5 = 18 \][/tex]
Step 4: Calculate [tex]\( (1 + \frac{r}{n})^{nt} \)[/tex]:
[tex]\[ 1.0075^{18} \approx 1.144474 \][/tex]
Step 5: Calculate [tex]\( \frac{FV}{(1 + \frac{r}{n})^{nt}} \)[/tex]:
[tex]\[ \frac{19000}{1.144474} \approx 16608.97 \][/tex]
Therefore, the present value is approximately \[tex]$16,608.97.
So, the present value for $[/tex]19,000 at 3% compounded quarterly for 4.5 years is \$16,608.97.
