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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?

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    With excel I find a value between 14% and 15%. – 2017-01-16
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    @N74 Thank you, I am looking for the formula, or help. – 2017-01-17
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    BTW, how do you use Excel to solve this kind of problems? – 2017-01-17
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    Just make a simulation of what happens year after year. Put the savings percentage in a cell and use the value of this cell in all the calculations involved. Then change the value of this cell to achieve a value 0 in the last cell, or use the excel goal attainment to find the value that makes null the last cell. – 2017-01-17
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    Your last salary is USD 175.000? What job do you have?! And in many countries there is a mandatory deduction from your salary for your pension, which means the amount you have to save from your net salary may well be 0 (and you're probably saving too much, but there's nothing you can do about it). – 2017-01-17

1 Answers 1

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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"

$S(1+0,03)^{42}(1+0,05)+S(1+0,03)^{41}(1+0,05)^2+...+S(1+0,03)(1+0,05)^{42}=Y$

$(1+0,03)^{43}(S(1+0,03)^{-1}(1+0,05)+S(1+0,03)^{-2}(1+0,05)^2+...+S(1+0,03)^{-42}(1+0,05)^{42}=Y$

$(1,03)^{43}S\bigg({1,05\over 1,03}\bigg)\bigg({1,05\over 1,03}^{43}-1\bigg)/\bigg({1,05\over 1,03}-1\bigg)=Y$

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    You are assuming S is constant, but each year I save an amount S which is a percentage of my salary, and therefore changing. – 2017-01-16
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    Maybe you just need to change the last equations:$$ S(1+0,03)^{42}(1+0,05)+S(1+0,03)^{41}(1+0,05)^2+...+S(1+0,03)(1+0,05)^{42}=Y$$ Then take out of the sum a factor: :$$ (1+0,03)^{43}(S(1+0,03)^{-1}(1+0,05)+S(1+0,03)^{-2}(1+0,05)^2+...+S(1+0,03)^{-42}(1+0,05)^{42}=Y$$ And, finally, $(1,03)^{43}S({1,05\over 1,03})({1,05\over 1,03}^{43}-1)/({1,05\over 1,03}-1)=Y$ – 2017-01-17
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    I know that $$\sum_0^k x^n = \frac{1-x^{k+1}}{1-x}$$ what is $$\sum_0^k x^{k-n}y^{n} $$ – 2017-01-17
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    Did you see the edit? $x^k$ is constant and can be taken out of the sum, then $y\over x$ become your geometric ratio. – 2017-01-17
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    Oh, yes, thank you. Sorry, I was having trouble following your equation, but I was using your idea to write mine. Now it makes sense. Thank you! – 2017-01-17
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    It might benefit others if you edited your answer using what is in your comments. – 2017-01-17
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    LaTeX does not format decimal notation with commas properly. You should consider switching to dots instead. – 2017-01-17