My uncle Samuel has $450,000 invested at 6.5%, and he plans to retire. He wants to withdraw $40,000 at the beginning of each year, starting immediately. How many years will it take to exhaust her funds, run the account down to zero?

450000=40000 + 40000(1 - 1.065^-n)/.065


I am using
PV = Paym(1 - (1+i)^-n)/n
but ....
that formula assumes that payments are made at the end of the period, which explains the single 40,000

.66625 = 1 - 1.065^-n
1.065^-n = .33375
log both sides
-n(log 1.065) = log .33375
-n = -17.425

n = appr 17.4 years

(not many years of retirement with money, I have already been retired for 17 years and hope to have lots left. )

To find out how many years it will take for your uncle Samuel's funds to be exhausted, we can use the concept of annuities. An annuity is a series of regular cash flows, in this case, $40,000 withdrawals at the beginning of each year.

The formula for calculating the number of years it takes for an annuity to be exhausted is:

N = log(A / (A - (P * r))), where:
N = the number of years
A = the initial amount
P = the periodic payment
r = the interest rate per period

In this case, the initial amount is $450,000, the periodic payment is $40,000, and the interest rate is 6.5% or 0.065.

Plugging these values into the formula, we have:

N = log(450,000 / (450,000 - (40,000 * 0.065)))

Now we can calculate the number of years it will take:

N ≈ log(450,000 / (450,000 - 2,600))
N ≈ log(450,000 / 447,400)
N ≈ log(1.00582)

Using a logarithm calculator or a calculator with a logarithm function, we find that N ≈ 199.47.

Therefore, it will take approximately 199.47 years to exhaust your uncle Samuel's funds, which can be rounded up to 200 years.