Question

If you make an initial contribution of $8000 to a retirement account 5 years from now and make annual contributions of $8000 thereafter, after how many contributions will your balance have reached at least $1000000 if your account earns 6% annual interest?

Answer 1

Here we can use the Future Value of Annuity (FVA) formula:

$\fn_jvn FVA = A\left [ \frac{(1+i)^n-1}{i} \right ]$

Where A = Annuity / Contribution

i = rate of interest

n = number of contributions

So let us put the values in the formula,

$\fn_jvn 1,000,000 = 8000\left [ \frac{(1+0.06)^n-1}{0.06} \right ]$

$\fn_jvn 125 = \frac{(1.06)^n-1}{0.06}$

$\fn_jvn 7.5 = {(1.06)^n-1}$

$\fn_jvn 8.5 = (1.06)^n$

$\fn_jvn \sqrt[n]{8.5} = 1.06$

$\fn_jvn n = 36$

It doesn't make difference whether you contribute from today or 5 years later, number of contributions will remain same.

This 36 contributions include the initial contribution.

Verification:

Year Opening Interest @ 6% on opening Contribution Balance 1,528 2,100 8,000 8,000 8,000 8,000 8,000 8,000 8,000 2,706 3,348 4,029 8,000 8,000 16,480 25,469 34,997 45,097 55,803 67,151 79,180 91,931 105,446 119,773 134,960 151,057 168,121 186,208 8,000 16,480 25,469 34,997 45,097 55,803 67,151 79,180 91,931 105,446 119,773 134,960 151,057 168,121 186,208 205,380 4,751 5,516 6,327 7,186 8,098 9,063 10,087 11,172 8,000 8,000 8,000 8,000 8,000 8,000 8,000 8,000

Year Opening Contribution Balance Interest @ 6% on opening 12,323 13,542 14,835 16,205 17,657 19,197 20,828 22,558 8,000 8,000 8,000 8,000 8,000 8,000 8,000 8,000 8,000 24,391 8,000 205,380 225,703 247,245 270,080 294,285 319,942 347,138 375,967 406,525 438,916 473,251 509,646 548,225 589,118 632,465 678,413 727,118 778,745 833,470 891,478 952,967 26,335 28,395 30,579 32,893 35,347 37,948 40,705 43,627 46,725 50,008 53,489 57,178 8,000 8,000 8,000 8,000 8,000 8,000 8,000 8,000 8,000 8,000 225,703 247,245 270,080 294,285 319,942 347,138 375,967 406,525 438,916 473,251 509,646 548,225 589,118 632,465 678,413 727,118 778,745 833,470 891,478 952,967 1,010,145

If you contribute $8000 from 5 years from now at the rate of 6% p.a, then your account will show balance of $1,010,145 at the end of 41st year. Total contributions including initial contribution is 36.