4.Suppose John sells his house and earns a profit of $600,000. With the profit, he buys a 20 year annuity that earns 6.5% interest compounded monthly. What monthly payment will John get?
$3900
To calculate the monthly payment John will receive from the annuity, you can use the formula for the present value of an annuity. The present value (PV) represents the current value of a future stream of cash flows, discounted to reflect the time value of money.
The formula for the present value of an annuity is:
PV = PMT x [(1 - (1 + r)^(-n)) / r]
Where:
PV = Present value (the amount John invested, which is the profit of $600,000)
PMT = Monthly payment John will receive
r = Monthly interest rate (6.5% divided by 12, or 0.065 / 12)
n = Number of payments (20 years multiplied by 12 months, or 240)
By substituting the given values into the formula, we can calculate the monthly payment John will receive:
PV = PMT x [(1 - (1 + r)^(-n)) / r]
600,000 = PMT x [(1 - (1 + (0.065/12))^(-240)) / (0.065/12)]
To solve this equation for PMT, we need to isolate PMT on one side. Here are the steps to solve it:
Step 1: Simplify the equation:
600,000 = PMT x [(1 - (1 + (0.0054166667))^(-240)) / (0.0054166667)]
Step 2: Calculate the value inside the square brackets:
[(1 - (1 + (0.0054166667))^(-240)) / (0.0054166667)] = 125.0502
Step 3: Divide both sides of the equation by 125.0502:
600,000 / 125.0502 = PMT
Step 4: Calculate PMT:
PMT = 4799.4081
So, the monthly payment John will receive from the annuity is approximately $4,799.41.