Question
What is the value today of a money machine that will pay $1,716.00 every six months for 26.00 years? Assume the first payment is made six months from today and the interest rate is 14.00%.
Answers
GPT-4o mini
To find the present value of a series of cash flows (an annuity) that pays $1,716.00 every six months for 26 years at an interest rate of 14.00% per annum, we first have to calculate the present value of an annuity formula.
### Given Data:
- Payment (PMT) = $1,716.00
- Number of years (n) = 26 years
- Interest rate (annual) = 14.00% (6.00% semi-annually)
- Number of payments per year = 2 (since payment is made every six months)
### Steps to Calculate Present Value:
1. **Calculate the Number of Payments:**
\[
\text{Total Payments} = n \times \text{Payments per Year} = 26 \times 2 = 52 \text{ payments}
\]
2. **Convert the Interest Rate for Each Period:**
\[
i = \frac{14.00\%}{2} = 7.00\% = 0.07 \text{ (per semi-annual period)}
\]
3. **Use the Present Value of Annuity Formula:**
The present value \( PV \) of an annuity can be calculated using the formula:
\[
PV = PMT \times \left(1 - (1 + i)^{-n}\right) / i
\]
where:
- \( PMT \) = payment per period
- \( i \) = interest rate per period
- \( n \) = total number of payments
Plugging in the values:
\[
PV = 1716 \times \left(1 - (1 + 0.07)^{-52}\right) / 0.07
\]
4. **Calculate \( (1 + i)^{-n} \):**
\[
(1 + 0.07)^{-52} \approx (1.07)^{-52} \approx 0.026304
\]
5. **Calculate \( 1 - (1 + i)^{-n} \):**
\[
1 - (1.07)^{-52} \approx 1 - 0.026304 \approx 0.973696
\]
6. **Calculate \( PV \):**
\[
PV = 1716 \times \frac{0.973696}{0.07}
\]
\[
PV = 1716 \times 13.19565714 \approx 22,648.65
\]
### Conclusion:
The present value of a money machine that will pay $1,716.00 every six months for 26 years at an interest rate of 14.00% is approximately **$22,648.65**.
### Given Data:
- Payment (PMT) = $1,716.00
- Number of years (n) = 26 years
- Interest rate (annual) = 14.00% (6.00% semi-annually)
- Number of payments per year = 2 (since payment is made every six months)
### Steps to Calculate Present Value:
1. **Calculate the Number of Payments:**
\[
\text{Total Payments} = n \times \text{Payments per Year} = 26 \times 2 = 52 \text{ payments}
\]
2. **Convert the Interest Rate for Each Period:**
\[
i = \frac{14.00\%}{2} = 7.00\% = 0.07 \text{ (per semi-annual period)}
\]
3. **Use the Present Value of Annuity Formula:**
The present value \( PV \) of an annuity can be calculated using the formula:
\[
PV = PMT \times \left(1 - (1 + i)^{-n}\right) / i
\]
where:
- \( PMT \) = payment per period
- \( i \) = interest rate per period
- \( n \) = total number of payments
Plugging in the values:
\[
PV = 1716 \times \left(1 - (1 + 0.07)^{-52}\right) / 0.07
\]
4. **Calculate \( (1 + i)^{-n} \):**
\[
(1 + 0.07)^{-52} \approx (1.07)^{-52} \approx 0.026304
\]
5. **Calculate \( 1 - (1 + i)^{-n} \):**
\[
1 - (1.07)^{-52} \approx 1 - 0.026304 \approx 0.973696
\]
6. **Calculate \( PV \):**
\[
PV = 1716 \times \frac{0.973696}{0.07}
\]
\[
PV = 1716 \times 13.19565714 \approx 22,648.65
\]
### Conclusion:
The present value of a money machine that will pay $1,716.00 every six months for 26 years at an interest rate of 14.00% is approximately **$22,648.65**.