You just turned 27 years old and plan to start a retirement account. You got a j

Question

You just turned 27 years old and plan to start a retirement account. You got a job that pays 68,000$ gross per year, and estimate you will bring home 75% of that number. Of the net take-home pay each month, you wanted to invest 10% in your retirement account each month. You can assume that you invest at the end of each month, starting at the end of the current month (i.e when you are 27 years and 1 month old). You expect your salary will increase by 3%/year for as long as you work.

To be conservative, you are going to estimate an APR of 6%, compounded monthly (despite the fact that the average market return is around 12%) on the account. Your goal is that you want your retirement years to be financially comfortable and you estimate that you will need 5,000$/month to do that – for 30 years. Given this, what age will you be when you retire? Show your work.

Explanation / Answer

First of all lets find the future value we will be needin in order to provide 5,000$/month for 30 years

PV(annuity) = A[1-(1/r)^n/r]

here A = 5000 , r = 6%/12 = 0.5%, n = 30*12 = 360

=5000[1-(1/1.005)^360 / 0.005]

=5000[1-0.16604 / 0.005]

=5000[0.83396/0.005]

=5000*166.792

=8,33,958$

Now at current we will save = 68000*75%*10% = 5100$ each month and the amount will grow by 3% yearly ie 0.25% monthly. thus we can use formula of growing annuity

FV of growing annuity = A[(1+r)^n - (1+g)^n/r-g]

833958 = 5100[(1.005)^n-(1.0025)^n/ 0.005-0.0025]

163.521 = [(1.005)^n-(1.0025)^n/ 0.0025]

0.4088 = (1.005)^n-(1.0025)^n

Assuming n = 108

(1.005)^108-(1.0025)^108 = 0.40418

Assumin n = 109

(1.005)^109-(1.0025)^109 = 0.40947

Assuming n = 108.8

(1.005)^108.8-(1.0025)^108.8 = 0.408

Hence no. of months = 108.8 ie 109 months

ie 108.8/12 = 9.0667 years

Thus asge of retirement = 27+9.0667 = 36.0667 years

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