Question

Suppose that you are 22 today and that your salary for the year ending today is $61,250. Suppose that you are paid on an annual basis at the end of each year; so you are receiving $61,250 today.

You plan to retire at age 68 and will need 75% of your last year’s salary for your annual living expenses starting at age 69. You have saved $50,000 to date. All living expenses between now and retirement will be covered by your annual salary in each year. All living expenses after retirement will be covered by your retirement income.

Assume that your first withdrawal from your retirement account for post-retirement income occurs on your 69th birthday and that you next deposit into your retirement account will be one year from now. You have some older relatives that have always had an interest in you and have indicated that you are in their wills. Assume that you will inherit $100,000 when you turn 36.

Assume also that, given the state of health care costs and a family history of healthcare issues, that you will need to have an amount of $1.25 million available by age 57 to cover healthcare costs above and beyond what your health insurance plan is likely to cover. Assume that you will save 10% of your annual income while you work, and that you will not save any additional amount after you retire. You project that your salary will grow at a rate of 5% and that your retirement income needs will grow at 3% to reflect long-term inflation forecasts. The higher growth rate in your salary reflects your expectation that you will receive promotions and that absent promotions your salary will keep pace with inflation.

The appropriate interest rate for your working life is 9% and declines to 6% after you retire.

How much will you have to leave to your heirs if you live to age 90?

Answer 1

This involves 2 timelines. 1 - Pre-retirement and 2 - Post retirement.

Pre-retirement - Salary savings:

Age today is 22 years and retirement age is 68 years hence pre-retirement period = 68-22 = 46 yrs. This is the period to accumulate savings in order to form a retirement corpse.

This timeline further needs to be broken down to accommodate cash flows at age 36 and age 57 yrs.

Income and savings today are:

Current Savings = $50,000
Annual Income = $61,250 which is expected to grow at 5% p.a.
Interest rate = 9%
Savings rate = 10% of annual salary
Retire at = 68 yrs hence pre-retirement period N = 68- 22 = 46 yrs

Considering the first outflow at the age 57 for insurance, let us calculate FV from 22 to 57 years which is a period of 35 yrs.
N = 35
I = Interest to be earned is 9 % however salary is growing at 5%, therefore I = 14 %
Payment = 10% of annual salary which is $ -6,125 (Signage is negative as this is the amount to be invested)
FV = $ 4,248,132.80
Outflow due to insurance payment = $ 1,250,000
Hence balance at the end of 57 years = $ 4,248,132.80 - $ 1,250,000 = $ 2,998,132.80

This amount needs to be invested untill 68 yrs age (another period of 11 yrs) to arrive at for retirement amount.
N = 11
PV = $ -2,998,132.80 (signage is negative as this is the amount to be invested)
I = 9 % (as this is amount accumulated and not salary hence only 9%)
FV at 68 yrs = $ 7,736,461.04

Thus savings from salary, after deducting insurance at 57 years, will grow to $ 7,736,461 at the end of 68 yrs.

Pre-retirement - Current savings:
Current savings as given = $ 50,000
FV of the same at the end of 68 years is,
N = 46
PV = $ -50,000 (Signage is negative as this is the amount to be invested)
I = 9 % (as this is amount accumulated and not salary hence only 9%)
FV = $ 2,633,837.09

Pre-retirement - Inheritance:
At 36 years, expected inheritance amount is $100,000
FV of the same at the end of 68 years is, 68 - 36 = 32 yrs
N = 32
PV = $ -100,000
I = 9 % (as this is amount accumulated and not salary hence only 9%)
FV = $1,576,332.88

Considering all the 3 components above, total value of investment at the end of 68 yrs = FV of all =
$ 7,736,461 + $ 2,633,837 + $1,576,333 = $ 11,946,633

Post retirement:

Retirement expenses will be 75% of last drawn salary at 68 years.
Salary at the end of 68 yrs is:
N = 46
I = 5% (Rate at which salary is expected to grow)
PV = $ 61,250 (Salary today)
FV = $ 577,848
Retirement expenses = $ 577,818 * 75 /100 = $ 433,386
This is expected to growth at 3% p.a.
Return during retirement period is 6% p.a.
Life expectancy is 90yrs; hence retirement period is 22 yrs
Accumulated invvestment at the time of retirement is $ 11,946,633

Let us first calculate the PV of retirement expenditure;
N = 22
I = 3% (this is the rate at which expenses will grow)
PMT = $ 433,386 (annual amount required)
PV = $ -6,906,840

Total PV available for investment during retirement is $ 11,946,633 - $ 6,906,840 = $ 5,039,793
This is the amount to be invested at 6% until 90 yrs. FV value is,

PV = $ -5,039,793
N = 22
I = 6%
FV = $ 18,161,083

Therefore the amount available to leave to heirs at the end of 90 yrs after considering retirement expenses = $ 18,161,083