Question

Use the following information regarding your retirement planning:

• You plan to work (and save) for 35 years, then retire (and spend money from your retirement account) for 25 years. After these 60 years, you expect that your retirement saving account will have $50,000 left to give to your family.
• You plan to save $4,000 in year 1, and you will increase this amount by 3% a year
• You want your retirement spending to increase by 2% per year
• You expect to earn a rate of return of 8% during your working years and 4.50% during retirement

To solve this problem, find the amount that you spend your first year of retirement.

Answer 1

Saving $4000 in the first year and increasing the savings at 3% is in the form of a Growing Annuity. The time period is here for 35 years for savings.

Future Value of Growing Annuity formula is

Future Value of Savings after 35 years of savings with r = 8%, g = 3%, P = $4000 (first payment)

Future Value = $957,718.5

Now from 36th year, this savings will earn an interest rate of 4.5%. Also every year there will be expenses which will grow every year at 2%.

To have $50,000 at the end of 60 years, it is equivalent to have a PV of $50,000 discounted at 4.5% per year at the end of 35 years.

PV of $50,000 at the end of 35th year = 50,000/(1.045)²⁵ = 16,636.53

So now we have two things in hand,

1) FV of all savings at the end of 35 years = $957,718.5

2) PV of $50,000(To be given to the family at the end of 60 years) at the end of 35 years = $16,636.53

The difference between these two would be the PV(at the end of 35years) of Expenses for the next 25 years = 957,718 - 16,636.53 = $941,081.47

The Expenses are also in the form of a growing annuity.

Present Value formula of a growing annuity

We already know Present Value = $941,081.47, r=4.5%, g=2%

P = $51,807.95

Amount to be spent in first year of retirement = $51,807.95

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