Say I have these assumption:
- Start to work at 22. Retire at 65.
- Starting salary 50,000
- Yearly salary increase is 3%
- Return on my savings = 5%
- Die at 85, and want to have 80% of my last salary after I retire.
In order to get an idea, I started by making the assumption that I am saving 20% (and then I can see if this is enough or too little).
If I get 3% increases for 42 years, at 65 I should be making 173,035 starting at 50,000 --formula: $$A = 50,000*(1+0.03)^{42}$$
My savings should be 10,000 the first year, and then 34,607 the last year.
The geometric sum is $$\sum_0^{42} 1.03 = 1- \frac{(1.03)^{43}}{(1-1.03)} = 85.4838$$ So, at 65 I should have saved a total of 854,838.
Since I want 80% of my last salary, which is 138,428 for 20 years after I retire, I should really have saved 2,768,560. So, it looks like clearly 20% is not enough. However, I have not yet figured out how to calculate the 5% return on savings.
To do this I should calculate the yearly compounded interest, but each year I am depositing a different amount to add to that value. This is where I got stuck. Help?
Lets say $0,8*50000*(1+0,03)^{42}=x$
By the time you retire you need $x/(1+0,05)+x/(1+0,05)^2+...+x/(1+0,05)^{20}$ = $Y$
$x(1-(1/1,05)^{20})/0,05=Y$
Lets say each year from 22 till 65 you save $S$ .
Solution by user "N74"
$S(1+0,03)^{42}(1+0,05)+S(1+0,03)^{41}(1+0,05)^2+...+S(1+0,03)(1+0,05)^{42}=Y$
$(1+0,03)^{43}(S(1+0,03)^{-1}(1+0,05)+S(1+0,03)^{-2}(1+0,05)^2+...+S(1+0,03)^{-42}(1+0,05)^{42}=Y$
$(1,03)^{43}S\bigg({1,05\over 1,03}\bigg)\bigg({1,05\over 1,03}^{43}-1\bigg)/\bigg({1,05\over 1,03}-1\bigg)=Y$