Finance question

Question

If I put $330,000 into an account at age 62 with 4 percent interest compounded annually, but I took $2300 out each month, at what age would my balance be $0?

Answer 1

You can set up a spreadsheet to do this with the numbers you have. But you can also solve the answer analytically using only pre-calculus algebra

Let N_0 be the balance to begin when you are 62, and N_t be the balance after t years. Let you withdraw amount d each month and the annualized interest rate is r paid at the year end.

Then, your monthly withdrawal reduce the balance in one year by

12*d

But you also earn interest at an annual rate r during the past year. The interest accrued one year is equal to

N_(t-1) * r/12 + (N_(t-1) - d) * r/12 + (N_(t-1) - 2*d) * r/12 + .... + (N_(t-1) - 11*d) * r/12

=(N_(t-1) + N_(t-1) - 11*d) * 12/2 * r/12

=N_(t-1) * r - 11/2 * d*r

Year-end balance is the sum of the above two terms so

N_t = (1+r) * N_(t-1) - (12 + 11/2*r) * d                                                                 (A)

Now we can solve N_t in terms of N_0

To solve this, observe that (A) is true for all t, so for t-1

N_(t-1) = (1+r) * N_(t-2) - (12 + 11/2*r) * d

hence

(1+r)* N_(t-1) = (1+r)^2* N_(t-2) - (12 + 11/2*r) * d * (1+r)                              (B)

Add (B) + (A) N_(t-1) will be cancelled out, you have solved N_t in terms of N_(t-2)

N_t = (1+r)^2* N_(t-2) - (12 + 11/2*r) * d * (1 + (1+r))

Repeat this until you reach N_0.

N_t = (1+r)^t* N_0 - (12 + 11/2*r) * d * (1 + (1+r) + (1+r)^2 + ... + (1+r)^(t-1))

Since 1 + (1+r) + (1+r)^2 + ... + (1+r)^t = ((1+r)^t -1)/ r

N_t = (1+r)^t* N_0 - (12 + 11/2*r) * d * ((1+r)^t -1)/ r

Combine the term (1+r)^t, you have

N_t = (1+r)^t * (N_0 - (12/r + 11/2)*d) + (12/r + 11/2) *d                      (C)

Now, to find out when your balance becomes 0, set N_t = 0, and solve for t in (C) which is given by

ANSWER

T = log_(1+r) { (12/r + 11d/2) / (12/r + 11d/2 – N_0) }            (D)

Note that if 12/r + 11d/2 < N_0, (D) has no answer, which means the balance will never run to zero.

Now use r = 4%, N_0 = 330,000, d = 2300, you have T = 16.17 year as your answer

Want to know how much monthly withdrawal you should make if you want the balance to run out in T-years?

Then solve (D) for d

d = N_0 * (1+r)^t / ((1+r)^t-1) * 2r / (24 + 11r)

Use r = 4%, N_0 = 330000, and T = 30  (i.e. run out balance in 30 years), the answer d = 1561 a month