- #1

issacnewton

- 1,024

- 35

## Homework Statement

Following is the statement of the problem.

Dear Financial Adviser,

My spouse and I are each 62 and hope to retire in three years. After retirement we will

receive $7,500 per month after taxes from our employers’ pension plans and $1,500 per

month after taxes from Social Security. Unfortunately our monthly living expenses are

$15,000. Our social obligations preclude further economies.

We have $1,000,000 invested in a high-grade, tax-free municipal-bond mutual fund.

The return on the fund is 3.5% per year. We plan to make annual withdrawals from the

mutual fund to cover the difference between our pension and Social Security income and

our living expenses. How many years before we run out of money?

Luxury Challenged

Marblehead, MA

You can assume that the withdrawals (one per year) will sit in a checking account (no

interest). The couple will use the account to cover the monthly shortfalls.

## Homework Equations

Compound interest formula, Annuity

## The Attempt at a Solution

Since the couple has $7500 from the pension plans and $1500 from social security, the monthly shortfall is

$15000 - (7500+1500) = $6000. So the total shortfall for the entire year is ##12 \times $6000 = $72000##. So this amount will be withdrawn each year from the invested fund. Now let ##r = 0.035## be the interest earned on the amount invested in the mutual fund. Let ##P = $1,000,000## be the amount invested. At the end of the first year, this will grow to ##P(1+r)##. They will withdraw $72000. Remaining amount, ##P(1+r) - 72000##, will grow to ##(P(1+r) - 72000)(1+r)## at the end of the second year. Again $72000 will be taken out. And this will go on till they run out of the money. So I am thinking of using a computer program to see how long does it take. Here is my little program in python 3

Python:

```
rate = 0.035
withdraw = 72000
amount = 1000000
counter = 1
while counter <= 20:
amount = (1+ rate) * amount - withdraw
print("The amount at the end of year %d is $%.2f " % (counter, amount) )
counter += 1
```

The amount at the end of year 1 is $963000.00

The amount at the end of year 2 is $924705.00

The amount at the end of year 3 is $885069.67

The amount at the end of year 4 is $844047.11

The amount at the end of year 5 is $801588.76

The amount at the end of year 6 is $757644.37

The amount at the end of year 7 is $712161.92

The amount at the end of year 8 is $665087.59

The amount at the end of year 9 is $616365.66

The amount at the end of year 10 is $565938.45

The amount at the end of year 11 is $513746.30

The amount at the end of year 12 is $459727.42

The amount at the end of year 13 is $403817.88

The amount at the end of year 14 is $345951.50

The amount at the end of year 15 is $286059.81

The amount at the end of year 16 is $224071.90

The amount at the end of year 17 is $159914.42

The amount at the end of year 18 is $93511.42

The amount at the end of year 19 is $24784.32

The amount at the end of year 20 is $-46348.23

Which tells us that in 20 years, the couple will run out of the money. Is this correct ? Now this problem is from

the Principles of Corporate Finance book and they don't mention anything about computer program to solve this problem. So is there another approach to solve this ?

Thanks