Question

Assume the following scenario:
Bob plans to retire in 15 years from now and wants to have the following stream of CFs after retirement. Monthly payments of $6,000 for 10 years starting right after retirement (the first payment will be at the end of the first month of year 16). He then needs an extra 10000$ with the final payment (that is at the end of the final month of year 25). Starting from year 26 he wants the monthly payments to grow at 0.5% per month for another 15 years. (That is the first payment in year 26 will be 6000*(1+0.005). The APR is 8% with quarterly compounding.

What is the present value (at t = 0) of this plan? please give step by step solution!

Answer 1

Answer:

Step 1:

Present value of monthly payments during retirement starting from year 26 for 15 years:

Present of growing annuity = P / (r - g) * (1 - ((1+g)/ (1 + r))^n)

Where P = First payment

r = monthly interest rate

g = annuity growth rate

n = Number of months

The APR is 8% with quarterly compounding.

Effective annual rate = (1 + 8%/4)^4 - 1 =8.243216%

Monthly interest rate = (1 + 8.243216%)^(1/12) -1 = 0.662270956%

First payment = 6000 * (1 + 0.5%) = $6030

Number of months = 15 * 12 = 180

Present value as at start of year 26 = 6030/(0.662270956% - 0.5%) * (1 - ((1+ 0.5%) / (1 + 0.662270956%))^180)

=936566.7914

Present value (at t = 0) = 936566.7913 / (1 + 0.662270956%)^(25*12)

= $129,277.093

Step 2:

He then needs an extra 10000$ at the end of the final month of year 25.

Present value (at t = 0) = 10000 / (1 + 0.662270956%)^(25*12) =1380.3297

Step 3:

Present value of Monthly payments of $6,000 for 10 years starting right after retirement (the first payment will be at the end of the first month of year 16):

Present value at start of year 16 = PV(rate, nper, pmt, fv, type) = PV(0.662270956%,10*12, -6000,0,0)

= $495666.8383

Present value (at t = 0) = 495666.8383 / (1 + 0.662270956%)^(15*12) = $151,070.4624

Step 4:

Present value (at t = 0) of this plan =$129,277.093 + 1380.3297+ 151,070.4624

= 281727.89

Present value (at t = 0) of this plan = $281,727.89