7. Today you open a bank account and make a deposit of $6,000. Then you make the following deposits: in t=1 you deposit $6,500, in t=2 $45,000, in t=3 nothing, in t=4 nothing, in t=5 $7,000, in t=6 $8,700. You made your last deposit of $34,000 in t=7. Then, in t=10, you use all the money in your bank account to buy a financial security that guarantees 25 equal annual payments, with the first payment occurring in t=11. Assuming an annual interest rate of 10%, find the payment amount.
3245
To find the payment amount for the financial security, we will first calculate the future value of all the deposits made from t=1 to t=7 using the formula for future value of a series of deposits:
Future Value = Present Value (1 + r)^n - 1 / r
Where:
Present Value = $6,000
r = 10% (annual interest rate)
n = Number of periods (in this case, 7)
Applying the formula:
Future Value = $6,000 (1 + 0.10)^7 - 1 / 0.10
= $6,000 x 1.10^7 - 1 / 0.10
= $6,000 x 1.948717 - 1 / 0.10
= $11,692.30
Next, we need to calculate the payment amount for the financial security using the formula for the present value of an annuity:
Present Value = Payment Amount x (1 - (1 + r)^-n) / r
Where:
Present Value = $11,692.30
r = 10% (annual interest rate)
n = Number of payments (in this case, 25)
We need to solve for the payment amount:
$11,692.30 = Payment Amount x (1 - (1 + 0.10)^-25) / 0.10
Simplifying the equation:
Payment Amount x (1 - (1.10)^-25) = $11,692.30 x 0.10
Payment Amount x (1 - 0.142699) = $1,169.23
Payment Amount x 0.857301 = $1,169.23
Payment Amount = $1,169.23 / 0.857301
Therefore, the payment amount for the financial security is approximately $1,362.99.
To find the payment amount for the financial security, we need to calculate the present value of the future payments and set it equal to the amount in your bank account in t=10.
Here's how to calculate it step by step:
1. First, let's calculate the present value of the future payments. Since the payments are equal and occur annually, we can use the formula for the present value of an ordinary annuity.
PV = PMT * [1 - (1 + r)^(-n)] / r
Where:
PV = Present Value
PMT = Payment amount
r = Annual interest rate
n = Number of periods (number of payments)
2. In this case, we know that the annual interest rate is 10% (or 0.10 as a decimal). We also know that there will be 25 payments starting from t=11. Therefore, n = 25.
3. Now, let's calculate the present value using the given information in t=10. We need to set the present value equal to the amount in your bank account, which is the sum of all the deposits you made.
PV = 6,000 + 6,500 + 45,000 + 7,000 + 8,700 + 34,000
4. Now, we can substitute the values into the formula and solve for PMT:
PV = PMT * [1 - (1 + r)^(-n)] / r
PMT * [1 - (1 + 0.10)^(-25)] / 0.10 = 6,000 + 6,500 + 45,000 + 7,000 + 8,700 + 34,000
5. Next, solve for PMT by rearranging the equation:
PMT = [PV * r] / [1 - (1 + r)^(-n)]
PMT = [(6,000 + 6,500 + 45,000 + 7,000 + 8,700 + 34,000) * 0.10] / [1 - (1 + 0.10)^(-25)]
6. Finally, compute the value of PMT using a calculator:
PMT ≈ $3,353.42
Therefore, the payment amount for the financial security is approximately $3,353.42.