I You are retired and have at t0 3600000 invested in an acco

I. You are retired and have at t-0 $3,600,000 invested in an account that is expected to earn 0.225% per month in the future. You g Security Administration (SSA) every month but estimate your monthly expenditure to be $18000. Assuming that the first withdrawal and income from SSA occur at t-1, how long would it be before you run out of money? Suppose your answer for the number of month your savings will last is m, where m is most likely not an integer. Let n be the portion of m that ignores the values after the decimal point. (For instance, if m = 216.45, then n- 216). Confirm that the present value of n withdrawals is less than your savings, but the present value of (n +1) withdrawals is not et an income of $2070 from the Social

Solution

Monthly estimated expenses = $ 18,000

Income from SSA = $ 2,070

Requirement of monthly withdrawal = $ 18,000 - $ 2,070 = $ 15,983

Formula for PV of annuity can be used to compute the no. of withdrawals as:

PV = P x [1-(1+r)^-n/r]      

PV = Present value of annuity = $ 3,600,000

P = Periodic cash flow = $ 15,983

r = Rate per period = 0.225 % or 0.00225 p.m.

n = Numbers of periods

Substituting the values in above formula, we get n as:

$ 3,600,000 = $ 15,983 x [1-(1+0.00225)^-n/0.00225]

$ 3,600,000 = $ 15,983 x [1 - (1.00225)^-n/0.00225]

$ 3,600,000/$ 15,983 = [1 - (1.00225)^-n/0.00225]

225.2393168 x 0.00225 = 1 – (1.00225)^-n

0.506788463 = 1 – (1.00225)^-n

(1.00225)^-n = 1 -   0.506788463

(1.00225)^-n = 0.493211537

Taking log of both sides, we get:

- n x log 1.00225 = log 0.493211537

- n x 0.00097606492 = -0.30696677289

n = 0.30696677289/0.00097606492

= 314.4942171 or 314 months

The money will last for 314 monthly withdrawals.