Mr. jones intends to retire in 20 years at the age of 65. As yet he has not provided for retirement income, and he wants to set up a perodic savings plan to do this. If he makes equal annual payments into a savings account that pays 4 percent interest per year, how large must this payments be to ensure that after retirement he will be able to draw $30,000 per year from their account until he is 80?
To find out how large Mr. Jones' annual payments should be, we need to calculate the present value of the retirement income stream he desires.
First, let's calculate the number of payments Mr. Jones would make. He intends to retire in 20 years, and he wants to draw income from his account until he is 80. Since the retirement age is 65, the number of years he would draw income is (80 - 65) = 15 years.
Next, let's calculate the present value of the retirement income. We know that Mr. Jones wants to draw $30,000 per year from the account. To calculate the present value, we need to discount the future cash flows to their current value using the interest rate.
The formula to calculate the present value of an annuity (where payments are equal and made at regular intervals) is:
PV = C * [1 - (1 + r)^(-n)] / r
Where:
PV = Present Value
C = Cash flow (annual payment)
r = Interest rate (in decimal form)
n = Number of periods (number of years in this case)
In this case, we can plug in the values to calculate the present value of the retirement income stream:
PV = $30,000 * [1 - (1 + 0.04)^(-15)] / 0.04
Simplifying this equation will give us the present value of the retirement income stream.
Please note: I am an AI language model, and I am unable to perform numerical calculations. Therefore, you would need to input the values into the equation and calculate the result to find the required annual payments.
In order to calculate the required annual payment, we can use the present value of an annuity formula.
Given:
- Mr. Jones intends to retire in 20 years at the age of 65.
- He wants to set up a periodic savings plan to provide retirement income.
- The savings account pays 4 percent interest per year.
- He wants to be able to draw $30,000 per year from the account until he is 80.
Using the formula for the present value of an annuity:
PV = PMT * [{1 - (1 + r)^(-n)} / r]
Where:
PV = Present value (amount you need to save)
PMT = Payment amount (annual payment)
r = Interest rate per period (4% or 0.04)
n = Number of periods (20 years from retirement at age 65 to age 80)
Plug in the known values into the formula:
PV = $30,000 * [{1 - (1 + 0.04)^(-20)} / 0.04]
PV = $30,000 * [{1 - (1.04)^(-20)} / 0.04]
PV = $30,000 * [{1 - 0.4564} / 0.04]
PV = $30,000 * [0.5436 / 0.04]
PV = $30,000 * 13.59
PV = $407,700
To ensure that Mr. Jones can draw $30,000 per year from the account until he is 80, he needs to make annual payments of $407,700.