You\'ve decided to start saving this year for an early retirement. Your goal is

Question

You've decided to start saving this year for an early retirement. Your goal is to reture in 25 years with enough money to live comfortably for another 30 years after retirement (in other words, you have a 55 year investment horizon). You believe that it will take $200,000 per year to live well in retirement (you anticipate massive inflation). You have already saved $10,000 (an inheritance) that you have already invested. Additionally, you plan to make annual end-of-year investments for the next 25 years until you retire. You plan to increase your annual contribution 2.5% per year as your income grows. Assume a 9% interest rate applies for all of your contributions and the $10,000 you already have. How much will you need to invest at the end of this year in oprder to meet your retirement goals (don't forget to include the $10,000 inheritance in your calculations)

Explanation / Answer

This question has 3 parts :- 1. Retire after 25 yrs. Each withdrawl is $200,000 and will continue for 30 Yrs. So This withrwal will happen for Nr=30yrs. DUring this period, lump sum will reduce & will earn Interest Ir=9%. So here we have to find the PV of ANnuity of PMT = $200,000 for Ir=9% & Nr=30. 2. Inherited amt of $10,000 is inevsted at 9% for 25 Yrs. So this amount will be reduced from PVA in '1' above to calculat ethe FVA in '3' below. 3. SO at AGe T+25, he will have a Lump sum which is arrived at at '1' above which is Future Value of Annuity. This is a growing annuity & has PMT every year for N1=25Yrs at i=9% & g=2.5% to get the Retrtment Kitty. So we need to find PMT here So lets start :- See note 1 above. Present value of annuity due PVA = PMT(PVIFAi,n) So PVA = PMTr*(1/Ir - 1/(Ir(1+Ir)^Nr)) So we get PVA = 200,000*(1/9% -1/(9%*(1+9%)^30)) ie PVA =200,000*10.2737 So PVA = $ 2,054,730.81 So at Age T+25, He should have $ 2,054,730.81 ..............(A) See Note 2: FV of $10,000 at end of N=25 Yrs @I=9% will be given by FV = PV*(1+I)^N = 10000*(1+9%)^25 = $86,230.81 ..............(B) See note 3 above SO FVA is to be reduced by FV of $10,000 which is $1,968,500.00 We have to Find Fv of growing Annuity. Here we have FVA = $1,968,500, n=25, i=9%, g=2.5%, A = Initial Payment FVA = A*[(1+i)^n - (1+g)^n]/(i-g) ie FVA = A*[(1+9%)^25 - (1+2.5%)^25]/(9%-2.5%) ie FVA = A*104.1406 so A = FVA/104.1406 ie A = $1,968,50/104.1406 ie A = $18,902.34 .................Ans So I need to invest $18,902.34 each year for 25 Yrs to get my part Retirement kitty. Add to this my Inheritance Fv to get my Full kitty.

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