Question

bit of a math problem. so if I retire at age 65 and withdraw 50,000 dollars from my 401k every year until I die at age 72, and there is an inflation rate of 2%, how much money do I need to initially be in my 401k to make it to my death without going broke?

Having trouble thinking of what formula to use because compound interest using the inflation rate is one thing but I'm not sure what to do about the 50,000 plus whatever interest that i would keep dishing out each year.

Answer 1

To answer your query, let us look at the problem in a different way.

Instead of withdrawing the the amount each year, assume you were to add $50,000 to the account starting at age 65 and each year after that you add 2% more than the previous year, for 7 years until you hit 72.
Let us see what such a formula would look like.

Let your initial amount to be P ($50,000),
The rate of inflation be r (at 2%),
The tenure by t (8 years)

So the formula comes out as:
Amount = P + [P+rP] + [P + r(P+rP)] + [P + r{P + r(P+rP)}] + ... upto t terms
which simplifies to
Amount = P + P + P + .. t terms
                    + rP + rP + .. (t-1) terms
                            + (r²)P + .. (t-2) terms
                                         + .... + (r^(t-1))P

Which can be written as:
Amount = tP + (t-1)(r¹P) + (t-2)(r²P) + (t-3)(r³P) + .... + (t-(t-1))r^(t-1)P
Taking the common P term out, and generalizing the terms, we get :
Amount = P [(t-0)r⁰ + (t-1)r¹ + (t-2)r² + (t-3)r³ + ... + (t-(t-1))r^(t-1)]
or

Substituting the values for P,t,r and we get:

which is the formula you need.