Present value, P=1000000
annual interest rate, i=3%
annual withdrawal, A=60000
Future value, F=0
n=number of years (periods).
We set up and equate the future value of the present value, as well as the 60000 withdrawals, and solve for n.
Future value F1 of the $1000000 amount after n years:
[tex]F1=P*(1+i)^n[/tex]
Future value F2 of the withdrawal after n years:
[tex]F2=\frac{A((1+i)^n-1)}{i}[/tex]
Equate the F1 and F2, solve for n:
[tex]P(1+i)^n=\frac{A((1+i)^n-1)}{i}[/tex]
[tex]1000000(1+.03)^n=\frac{60000((1+.03)^n-1)}{.03}[/tex]
To solve for n, we first set u=1.03^n, and solve for u.
[tex]1000000u=\frac{60000(u-1)}{.03}[/tex], where
u=2=1.03^n
Now solve for
n=log(2)/log(1.03)=23.45 years.
Answer: funds will be exhausted after about 23.5 years.
Check:
Present value after withdrawal of 23 years = 986616.50
Present value after withdrawal of 24 years = 1016132.53
Therefore funds are exhausted after about 23.5 years (PV=1000000)
