Let's calculate the present value of an ordinary annuity step-by-step.
### Step-by-Step Solution:
1. Identify the given values:
- Annuity amount (PMT): [tex]\( 15,100 \)[/tex] USD
- Payment frequency: Quarterly (4 payments per year)
- Time: 5 years
- Annual interest rate: 5%
2. Calculate the number of periods (n):
Since the payments are quarterly, there are 4 payment periods in one year.
[tex]\[
n = 5 \text{ years} \times 4 \text{ quarters per year} = 20 \text{ periods}
\][/tex]
3. Calculate the rate per period (r):
The annual interest rate is 5%, which needs to be converted to a quarterly rate.
[tex]\[
r = \frac{5\%}{4} = \frac{5}{100 \times 4} = 0.0125 \text{ (or 1.25% per quarter)}
\][/tex]
4. Use the present value of an ordinary annuity formula:
[tex]\[
PV = PMT \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)
\][/tex]
5. Substitute the given values into the formula:
[tex]\[
PV = 15,100 \times \left( \frac{1 - (1 + 0.0125)^{-20}}{0.0125} \right)
\][/tex]
6. Calculate the factor inside the parentheses first:
[tex]\[
1 - (1 + 0.0125)^{-20} \approx 1 - 0.75941 \approx 0.24059
\][/tex]
7. Divide by the rate per period:
[tex]\[
\frac{0.24059}{0.0125} \approx 19.2472
\][/tex]
8. Multiply by the annuity amount:
[tex]\[
PV = 15,100 \times 19.2472 \approx 290,832.52
\][/tex]
### Final Answer:
The present value (amount needed now to invest to receive the annuity) is:
[tex]\[
\boxed{265,749.67 \text{ USD}}
\][/tex]
This is the amount you would need to invest now at a 5% annual interest rate, compounded quarterly, to receive quarterly payments of $15,100 for 5 years.
