Asked by Tsika
If $10,000 is used to purchase an annuity consisting of equal payments at the end of
each year for the next four years and the interest rate is 6% compounded annually, find
the amount of each payment?
Answers
Answered by
GPT 3.5
To find the amount of each payment, we can use the formula for the present value of an ordinary annuity:
\[PV = \frac{P \cdot (1 - (1 + r)^{-n})}{r}\]
Where:
- PV is the present value (initial investment) of the annuity (in this case, $10,000)
- P is the payment amount at the end of each year
- r is the interest rate per period (in this case, 6% or 0.06)
- n is the number of periods (in this case, 4)
Plugging in the values, we get:
\[10,000 = \frac{P \cdot (1 - (1 + 0.06)^{-4})}{0.06}\]
Now, we can solve for P:
\[10,000 = \frac{P \cdot (1 - 1.262476288)}{0.06}\]
\[10,000 = \frac{P \cdot (-0.262476288)}{0.06}\]
Multiplying both sides by 0.06:
\[600 = -0.262476288 \cdot P\]
Dividing both sides by -0.262476288:
\[P = \frac{600}{-0.262476288} \approx -2286.36\]
Since the payment amount cannot be negative, the amount of each payment is approximately $2,286.36.
\[PV = \frac{P \cdot (1 - (1 + r)^{-n})}{r}\]
Where:
- PV is the present value (initial investment) of the annuity (in this case, $10,000)
- P is the payment amount at the end of each year
- r is the interest rate per period (in this case, 6% or 0.06)
- n is the number of periods (in this case, 4)
Plugging in the values, we get:
\[10,000 = \frac{P \cdot (1 - (1 + 0.06)^{-4})}{0.06}\]
Now, we can solve for P:
\[10,000 = \frac{P \cdot (1 - 1.262476288)}{0.06}\]
\[10,000 = \frac{P \cdot (-0.262476288)}{0.06}\]
Multiplying both sides by 0.06:
\[600 = -0.262476288 \cdot P\]
Dividing both sides by -0.262476288:
\[P = \frac{600}{-0.262476288} \approx -2286.36\]
Since the payment amount cannot be negative, the amount of each payment is approximately $2,286.36.