Question

You are currently 24. Based on your calculations, you will need $120,000 of income each year once you retire. You plan on retiring at age 65, and want to have enough money in your account so that you can draw payments out for 25 years (with the first payment taking place immediately upon your 65th birthday). Assume that all your funds earn an 11% annual return. You will begin making payments into your retirement account immediately and will make 41 payments into the account. How much must you invest each year in order to make your retirement goal? Now, assume that all other facts stay the same, but that your start with a balance of $10,000 in your account. Now what should your annual payments be?

Answer 1

Fund required facilitating cash out flow of $ 120,000 per year at the beginning of year for 25 years can be calculated using formula for present value of annuity due as:

PV (at the time of retirement) = P + P x [1-(1+r)^-(n-1)/r]

P = Periodic payment = $ 120,000

r = rate per period = 11 % or 0.11 p.a.

n = number of periods = 25

PV = $ 120,000 + $ 120,000 x [1-(1+0.11)^-(25-1)/0.11]

    = $ 120,000 + $ 120,000 x [1-(1.11)^-(24)/0.11]

     = $ 120,000 + $ 120,000 x [(1-0.081704976)/0.11]

    = $ 120,000 + $ 120,000 x (0.918295024/0.11)

   = $ 120,000 + $ 120,000 x 8.348136578

   = $ 120,000 + $ 1,001,776.3893 = $ 1,121,776.39

Future value of 41 number of immediate payment is $ 1,121,776.39.

Periodic payment can be calculated using formula for future value of annuity due as:

FV = (1 + r) x P x [(1+r) ⁿ – 1/r]

P = Periodic payment

r = rate per period = 11 % or 0.11 p.a.

n = number of periods = 41

$ 1,121,776.39 = (1 + 0.11) x P x [(1+0.11)⁴¹ – 1/0.11]

$ 1,121,776.39 = 1.11 x P x [(1.11)⁴¹ – 1/0.11]

$ 1,121,776.39 = 1.11 x P x [(72.15096271 – 1)/0.11]

$ 1,121,776.39 = 1.11 x P x (71.15096271/0.11)

$ 1,121,776.39 = 1.11 x P x 646.8269337

P = $ 1,121,776.39/ (1.11 x 646.8269337)

P = $ 1,121,776.39/717.9778964

P = $ 1,562.410758 or $ 1,562.41

$ 1,562.41 must be invested each year to achieve retirement goal.

Suppose you have $ 10,000 in your account.

So Future value of $ 10,000 after 41 years @ 11% compounding yearly is:

FV = $ 10,000 x (1+0.11)⁴¹

     = $ 10,000 x (1.11)⁴¹

     = $ 10,000 x 72.15096271 = $ 721,509.63

Required fund at the time of retirement = $ 1,121,776.39 - $ 721,509.63 = $ 400,266.76

Again applying the formula for future value of annuity due, we get cash flow as:

FV = (1 + r) x P x [(1+r) ⁿ – 1/r]

$ 400,266.76 = (1 + 0.11) x P x [(1+0.11)⁴¹ – 1/0.11]

$ 400,266.76 = 1.11 x P x [(1.11)⁴¹ – 1/0.11]

$ 400,266.76 = 1.11 x P x [(72.15096271 – 1)/0.11]

$ 400,266.76 = 1.11 x P x (71.15096271/0.11)

$ 400,266.76 = 1.11 x P x 646.8269337

P = $ 400,266.76/ (1.11 x 646.8269337)

P = $ 400,266.76/717.9778964 = $ 557.49176 or $ 557.49

$ 557.49 need to invest each year to get retirement goal if there is balance of $ 10,000 in the account.