Question
What is formula to calculate the present value of an annuity that makes payments of $100 per year for 10 years if the first payment is made immediately and the discount rate is 10 percent per year?
Multiple choice question.
$100[(1 − 1/1.1010)/0.10][.10]
$100[(1 − 1/1.1010)/1.10][1.10]
$100[(1 − 1/1.1010)/0.10][1.10]
$1,000[(1 − 1/1.1010)/0.10][1.10]
Multiple choice question.
$100[(1 − 1/1.1010)/0.10][.10]
$100[(1 − 1/1.1010)/1.10][1.10]
$100[(1 − 1/1.1010)/0.10][1.10]
$1,000[(1 − 1/1.1010)/0.10][1.10]
Answers
GPT-4o mini
To calculate the present value of an annuity where the first payment is made immediately, you can use the present value of an annuity formula adjusted for an immediate payment (also known as an annuity due).
The formula for the present value of an annuity due is:
\[
PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \times (1 + r)
\]
Where:
- \(P\) is the payment per period ($100 in this case),
- \(r\) is the discount rate (10% or 0.10),
- \(n\) is the number of periods (10 years).
Plugging in the values:
\[
PV = 100 \times \left( \frac{1 - (1 + 0.10)^{-10}}{0.10} \right) \times (1 + 0.10)
\]
Now, simplifying further:
1. Calculate \(1 + r = 1 + 0.10 = 1.10\).
2. Calculate \((1 + r)^{-n} = (1.10)^{-10}\).
3. Calculate \((1 - (1.10)^{-10})/0.10\).
4. Finally multiply by \(1.10\) to adjust for the annuity due.
Now matching it with the options given in the question:
Let's analyze the choices:
1. **First Option**: \(100\left[(1 - 1/1.10^{10})/0.10\right][0.10]\) - Incorrect. The last factor should multiply by 1.10 instead of 0.10.
2. **Second Option**: \(100\left[(1 - 1/1.10^{10})/1.10\right][1.10]\) - Incorrect. The divisor should be \(0.10\) not \(1.10\).
3. **Third Option**: \(100\left[(1 - 1/1.10^{10})/0.10\right][1.10]\) - This matches the annuity due formula.
4. **Fourth Option**: \(1,000[(1 - 1/1.10^{10})/0.10][1.10]\) - Incorrect. The payment was $100, not $1,000.
Thus, the correct choice is **$100\left[(1 - 1/1.10^{10})/0.10\right][1.10]**.
The formula for the present value of an annuity due is:
\[
PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \times (1 + r)
\]
Where:
- \(P\) is the payment per period ($100 in this case),
- \(r\) is the discount rate (10% or 0.10),
- \(n\) is the number of periods (10 years).
Plugging in the values:
\[
PV = 100 \times \left( \frac{1 - (1 + 0.10)^{-10}}{0.10} \right) \times (1 + 0.10)
\]
Now, simplifying further:
1. Calculate \(1 + r = 1 + 0.10 = 1.10\).
2. Calculate \((1 + r)^{-n} = (1.10)^{-10}\).
3. Calculate \((1 - (1.10)^{-10})/0.10\).
4. Finally multiply by \(1.10\) to adjust for the annuity due.
Now matching it with the options given in the question:
Let's analyze the choices:
1. **First Option**: \(100\left[(1 - 1/1.10^{10})/0.10\right][0.10]\) - Incorrect. The last factor should multiply by 1.10 instead of 0.10.
2. **Second Option**: \(100\left[(1 - 1/1.10^{10})/1.10\right][1.10]\) - Incorrect. The divisor should be \(0.10\) not \(1.10\).
3. **Third Option**: \(100\left[(1 - 1/1.10^{10})/0.10\right][1.10]\) - This matches the annuity due formula.
4. **Fourth Option**: \(1,000[(1 - 1/1.10^{10})/0.10][1.10]\) - Incorrect. The payment was $100, not $1,000.
Thus, the correct choice is **$100\left[(1 - 1/1.10^{10})/0.10\right][1.10]**.