[crazy canuck]

Lets assume your starting capital is X. Now assume that the stock market gives you a constant return of r, and assume you take out an amount A in each period pre tax.

So in period 0 you have X.

After 1 year you have (1+r)*X-A.

After 2 years you have (1+r)^2*X-A*(1+r)-A.

After 3 years you have (1+r)^3*X-A*(1+r)^2-A*(1+r)-A.

So after n years u have:

(1+r)^n*X - A*(1+r)^(n-1) - A*(1+r)^(n-2) .... - A.

Now u can use geometric sum formula for:

A*(1+r)^(n-1) + A*(1+r)^(n-2) .... + A =

= A*[1-(1+r)^n]/[1-(1+r)]=-A*[1-(1+r)^n]/r

So the final formula is:

(1+r)^n*X + A*[1-(1+r)^n]/r = (1+r)^n*(X-A/r)+A/r

So if after 50 years you want to have B, then set above to

B and you're starting capital is:

X=(B-A/R)/(1+r)^50 + A/r

So suppose you want to have 100K left (remember to readjust this amount for inflation) and stock market returns 4% above risk free interest rate and want to take out 30K each year then (in thousands):

X=(100-30/0.04)/(1.04)^50 + 30/0.04 = -91.5 + 750

=658.5 K

If you want to have 0 left then the starting amount is:

X=(0-30/0.04)/(1.04)^50 + 30/0.04 = -105.5 + 750 =

=644.5 K


Now there are several problems with this type of calculation.

First, of all it is hard to tell what the stock market return is going to be. Over the last 100 years in the US it was 6% above risk free interest rate. Many financial planners use this stupidly. However, economist studied other international markets and concluded it should be 3% (this is called equity premium puzzle), becasue the US had a phenomenal century.

And also the stock market return is not constant. This can ruin the above calculation. So the above calculation is kind of like planning based on the assumption that your poker earnings are the same each month. So if you start out with a disasterous year in which stock market drops 10% then you need way more initial capital than the above formula suggests to make up for this initial loss.