Question

Today is your 40th birthday (this is beginning of period, i.e., time 0). You expect to retire at age 65 and actuarial tables suggest that you will live to be 85. You want to move to Hawaii when you retire (on your 65th birthday). You estimate that it will cost you $50,000 to make the move on your 65th birthday. Starting on your 65th birthday and ending on your 84th birthday (all withdrawals are at the beginning of the year), you will withdraw $40,000 for annual living expenses. Assume the interest rate is 4%.

How much will you need to have saved by your retirement date?

What is your savings calculated in (1) worth today?

You start saving for this goal today till your 65th birthday. How much would you need to save each year?

Answer 1

(1): Amount needed to be saved by your retirement date = $50,000 + present value of annual withdrawals of $40,000 as on the date of your 65th birthday

Now withdrawals of $40,000 is made starting on your 65th birthday and ending on your 84th birthday. This is in the form of annuity due and period is 20 years (65th birthday to 84th birthday) and rate is 4%. Thus value of these withdrawals on 65th birthday = 40,000 + 40,000*[1-(1.04)^-(20-1)/0.04]

= 40,000 + (40,000*13.13394)

= $565,357.58

Thus you will need to have saved an amount of $50,000 + $565,357.58

= $615,357.58

(2): Your savings of $615,357.58 worths' today = 615,357.58/(1.04^20)

= $280,841.16

(3): Let the annual savings be "x". This is in the form of annuity due and time period involved is 20 years (40th birthday to 65th birthday). Thus 280,841.16 = x + x*[1-(1.04)^-(20-1)/0.04]

= x+ (13.13394x) = 280,841.16

or 14.13394x = 280,841.16

or x = $19,869.99