Question

You start a new job at age 39 with a beginning salary of $72,000 per year paid monthly. You expect a 2% increase annually in your salary. You invest 10% of your salary and your employer gives you an additional 3% of your salary each year paid into your retirement account monthly. At the end of year 6 you get a $10,000 raise instead of the 2% increase. At the end of year 10 you get another $10,000 raise instead of the 2% increase. The investment firm that you have your retirement account with is averaging 10% gains on your money over your lifetime. You expect to live until you are 95 years old and would like to retire by the time you are 65. At what age will you be able to retire to receive $4000 per month to live on until you die at age 95?

Answer 1

The amount required to receive $4000 every month after retirement is the present value of annuity at 10% annualised return. The formula for the PV of annuity is = Periodic Cash flow * [ 1 - (1+r)^-t]/r ; where r is the applicable monthly interest rate (in this case 10%/12) and t is the time period in months. We have two streams here - an investment stream from current time till retirement and annuity from retirement till 95. We have to calculate the time when the retirement corpus will be sufficient to provide annuity of $4000. We first as a benchmark calculate the PV of 4000 annuity at age 65 (for 30 years or t = 30*12 = 360, at age 60 and age 50.

PV (required at age 65)= 4000 * [1-(1+10%/12)³⁶⁰]/(10%/12) = 455803.3

PV (required at age 60)= 4000 * [1-(1+10%/12)⁴²⁰]/(10%/12) = 465293.5

PV (required at age 50)= 4000 * [1-(1+10%/12)⁵⁴⁰]/(10%/12) = 474567.3

Now let us model the savings in excel - given below:

We can compare the above savings stream and PV of annuity, and can see that in the year 17 from today that is age 55, the person can invest for annuity of $4000. We can do PV of annuity at age 55:

PV (required at age 55)= 4000 * [1-(1+10%/12)⁴⁸⁰]/(10%/12) = 471061.6

PV (required at age 54)= 4000 * [1-(1+10%/12)⁴⁹²]/(10%/12) = 471908.8

So in the 195 month i.e 16 years and 3 months from today.