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)/1.10][1.10]
$100[(1 − 1/1.1010)/0.10][1.10]
$1,000[(1 − 1/1.1010)/0.10][1.10]
$100[(1 − 1/1.1010)/0.10][.10]
Multiple choice question.
$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]
$100[(1 − 1/1.1010)/0.10][.10]
Answers
GPT-4o mini
To calculate the present value of an annuity where payments are made at the beginning of each period (known as an "annuity due"), we can use the formula for the present value of an ordinary annuity and then adjust it for an annuity due.
The formula for the present value of an ordinary annuity is:
\[
PV = PMT \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)
\]
Where:
- \( PV \) is the present value
- \( PMT \) is the payment per period
- \( r \) is the discount rate per period
- \( n \) is the total number of payments
For an annuity due, we multiply the result by \((1 + r)\) since we receive the first payment at time 0:
\[
PV = PMT \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \times (1 + r)
\]
Given:
- \( PMT = 100 \)
- \( r = 0.10 \) (10 percent)
- \( n = 10 \)
Substituting these values into the formula:
\[
PV = 100 \times \left( \frac{1 - (1 + 0.10)^{-10}}{0.10} \right) \times (1 + 0.10)
\]
Now, calculate \((1 + 0.10)^{-10}\):
\[
(1 + 0.10)^{-10} = (1.10)^{-10} \approx 0.3487
\]
Plugging this back into the annuity formula:
\[
PV = 100 \times \left( \frac{1 - 0.3487}{0.10} \right) \times 1.10
\]
Calculating \( \frac{1 - 0.3487}{0.10} \):
\[
\frac{0.6513}{0.10} = 6.513
\]
Continuing with the calculation:
\[
PV = 100 \times 6.513 \times 1.10 = 100 \times 7.1643 \approx 716.43
\]
Now, looking at the multiple-choice options:
1. \( 100\left[\frac{(1 - 1/(1.10^{10}))}{1.10}\right][1.10] \)
2. \( 100\left[\frac{(1 - 1/(1.10^{10}))}{0.10}\right][1.10] \)
3. \( 1,000\left[\frac{(1 - 1/(1.10^{10}))}{0.10}\right][1.10] \)
4. \( 100\left[\frac{(1 - 1/(1.10^{10}))}{0.10}\right][.10] \)
The correct option simplifies to the formula for the present value of the annuity due with the correct components for the given problem.
The correct choice is:
**\( 100\left[\frac{(1 - 1/1.10^{10})}{0.10}\right][1.10] \)**
So the answer is:
**\( 100\left[\frac{(1 - 1/(1.10^{10}))}{0.10}\right][1.10] \)**
The formula for the present value of an ordinary annuity is:
\[
PV = PMT \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)
\]
Where:
- \( PV \) is the present value
- \( PMT \) is the payment per period
- \( r \) is the discount rate per period
- \( n \) is the total number of payments
For an annuity due, we multiply the result by \((1 + r)\) since we receive the first payment at time 0:
\[
PV = PMT \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \times (1 + r)
\]
Given:
- \( PMT = 100 \)
- \( r = 0.10 \) (10 percent)
- \( n = 10 \)
Substituting these values into the formula:
\[
PV = 100 \times \left( \frac{1 - (1 + 0.10)^{-10}}{0.10} \right) \times (1 + 0.10)
\]
Now, calculate \((1 + 0.10)^{-10}\):
\[
(1 + 0.10)^{-10} = (1.10)^{-10} \approx 0.3487
\]
Plugging this back into the annuity formula:
\[
PV = 100 \times \left( \frac{1 - 0.3487}{0.10} \right) \times 1.10
\]
Calculating \( \frac{1 - 0.3487}{0.10} \):
\[
\frac{0.6513}{0.10} = 6.513
\]
Continuing with the calculation:
\[
PV = 100 \times 6.513 \times 1.10 = 100 \times 7.1643 \approx 716.43
\]
Now, looking at the multiple-choice options:
1. \( 100\left[\frac{(1 - 1/(1.10^{10}))}{1.10}\right][1.10] \)
2. \( 100\left[\frac{(1 - 1/(1.10^{10}))}{0.10}\right][1.10] \)
3. \( 1,000\left[\frac{(1 - 1/(1.10^{10}))}{0.10}\right][1.10] \)
4. \( 100\left[\frac{(1 - 1/(1.10^{10}))}{0.10}\right][.10] \)
The correct option simplifies to the formula for the present value of the annuity due with the correct components for the given problem.
The correct choice is:
**\( 100\left[\frac{(1 - 1/1.10^{10})}{0.10}\right][1.10] \)**
So the answer is:
**\( 100\left[\frac{(1 - 1/(1.10^{10}))}{0.10}\right][1.10] \)**