Your aunt has $780,000 invested at 5.5%,and she now wants to retire.She wants to withdraw $45,000 at the beginning of each year,beginning immediately.She also wants to have $50,000 left to give you when she ceases to withdraw funds from the account.For how many years can she make the $45,000 withdrawals and still have $50,000 left in the end?

Question

Quizplus · Accepted Answer

Using the formula for the present value of an annuity:
PV = C[(1 - (1 + r)^-n) / r]

Where: 
PV = present value of annuity 
C = regular payment (withdrawal) 
r = interest rate per period (in this case, annual) 
n = number of periods

We can solve for n: 
780,000 = 45,000[(1 - (1 + 0.055)^-n) / 0.055] + 50,000(1 + 0.055)^-n

Simplifying: 
780,000 = 45,000[(1 - (1.055)^-n) / 0.055] + 50,000(1.055)^-n 
780,000 = 45,000[14.7304 - 1.055^-n] + 50,000(1.055)^-n 
780,000 = 658,868.04 - 45,000(1.055)^-n + 50,000(1.055)^-n 
780,000 = 658,868.04 + 5,000(1.055)^-n

121,131.96 = 5,000(1.055)^-n 
24.22639 = (1.055)^n

Taking the natural logarithm of both sides: 
ln(24.22639) = n*ln(1.055) 
n = ln(24.22639) / ln(1.055) 
n ≈ 42.58

Therefore, your aunt can make the $45,000 annual withdrawals for approximately 42.58 years and still have $50,000 left in the end.

Your Aunt Has $780,000 Invested at 5