X$ per year, at Y% real interest, both growing at inflation as well...
Define K = 100%+Y%
X * (K^0 + K^1 + K^2 + ... + K^n)
if you do this for K years, =
X * [K^(n+1) - 1] / (K-1)
Say 3% real return, 10,000$ per year, 30 years:
10,000 * [1.5] / [.03] = 1/2 a million dollars in today's money.
Increase K to 1.04%: 600,000$ in today's money.
K to 1.05%: 700,000$ in today's money.
K to 1.06%: 850,000$ in today's money.
Every 1,000$ put in today -> 2500$/3300$/4500$/6000$ in 31 years. (3%/4%/5%/6% real return)
Increase savings from 10,000$ to 20,000$ per year: double the result.
Boost savings period to 40 years: (10000$ per year)
780,000$ @ 3%
1,000,000$ @ 4%
1,250,000$ @ 5%
1,650,000$ @ 6%
At 1,000,000$ (todays dollars -- in 30 years @ 3% inflation, that's 2.5 million then)... At 3% return with a 40 year "run to zero" target, you can manage 42,000$ per year (inflation adjusted). (roughly the same equations)
So aiming for 1 million dollars in liquid savings for a 40 year retirement-before-death doesn't leave you destitute.
Half half a million? ~21,000$ per year over 40 years.
1/4 a million? ~10,500$ per year over 40 years.
Want to hit that 1 million dollar savings target?
Let's be cautious and assume 4% real return (after inflation).
Aiming for a 1 million dollar nest egg:
Over 40 years: 10,000$ per year, -1000$/year every ~20k in savings you have invested
Over 30 years: 16,500$ per year, -1100$/year every ~20k in savings you have invested.
Over 20 years: 31,000$ per year, -1400$/year every ~20k in savings you have invested.
Over 10 years: 74,000$ per year, -2300$/year every ~20k in savings you have invested.
Note that there is the additional, hard problem of managing a consistent return of 4% over inflation when you need it.
As an example, investing all of your money in one country is a bad idea, both due to local economic problems and currency effects.
A simple version of some of the above math:
Let C(x,y) be the compound equation:
(x^(y+1) - 1) / (x-1)
Then we have:
R * C(1.04, 30) * 1.03^40 = I * C(1.03,40)
for a 30 year savings period @ 4% and a 40 year retirement period.
R * 193.5 = I * 78.66
R * 2.46 = I
If you already have savings "S", it becomes:
[R * C(1.04, 30) + S*1.04^30] * 1.03^40 = I * C(1.03,40)
You can change the real returns pre and post-retirement, the periods of each, etc with relative ease.
