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?
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.