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 periodic savings plan to do this. If he makes equal annual payments into savings account that pays 4% interest per year, how large must his payments be to enure that after retirement he will be able to draq $30,000 per year from this account until he is 80?

- (1/.04) (1-(1/1.04^15) = 11.118387

- 30,000 x 11.1183 = $333,551.62

- 333,551.62 / 29.778 = $11,201.27

To determine the size of Mr. Jones' annual payments, we need to calculate the present value (PV) of the retirement income stream he desires.

First, we need to calculate the total number of payments he will make. Since Mr. Jones intends to retire in 20 years at the age of 65 and wants to withdraw $30,000 per year until he turns 80, the total number of withdrawals will be 15 (from ages 65 to 79 inclusive).

Next, we calculate the present value of these future cash flows using the formula for the present value of an annuity:
PV = PMT * (1 - (1 + r)^(-n)) / r

Where:
PMT = payment per period (to be determined)
r = interest rate per period (4% = 0.04 in this case)
n = total number of periods (15 in this case)

Substituting in the values, we have:
PV = PMT * (1 - (1 + 0.04)^(-15)) / 0.04

Now, let's solve for PMT:
PMT = PV * r / (1 - (1 + r)^(-n))

Since the desired retirement income per year is $30,000 and we are assuming that the retirement account will provide this income perpetually until age 80, it means the present value of these future cash flows is $30,000 * 15 (total number of withdrawals) = $450,000.

Substituting this value along with the other known values, we have:
PMT = $450,000 * 0.04 / (1 - (1 + 0.04)^(-15))

Calculating this expression will give us the annual payments Mr. Jones needs to make in order to ensure he can withdraw $30,000 per year from the account until he is 80.