Question

Suppose that you are planning for retirement. You want to have exactly $4 million dollars when you retire. You just turned 26 and plan to retire on your 65^th birthday. For the next 15 years, you can save $10,000 per year (with the first deposit being made one year from now), and at the end of that time (t=15) you plan to buy a car for your child as a gift that will cost $50,000.   How much will you have to save each year in years 16 until retirement so that you meet your retirement goal? Assume the discount rate is 10 percent per year.

Answer 1

You want to have exactly $4 million dollars when you retire, therefore future value of your deposits at the time of retirement FV =$4 million

Discount rate or Interest rate = 10% per year

Time period for retirement n = 39 years (65 years – 26 years)

For the next 15 years, you can save $10,000 per year; we can use FV of an Annuity formula to calculate the future value of after 15 years savings by you

FV = PMT *{(1+i) ^n−1} / i

Where,

Future value of annual savings of $10,000 FV =?

PMT = Annual savings = $10,000

n = N = number of payments = 15 years

i = I/Y = interest rate per year = 10%

Therefore,

FV = $10,000 *{(1+10%) ^15−1} /10%

FV = $317,724.82

Therefore you will save $317,724.82 till year 15 to achieve your retirement goal.

But you plan to buy a car for your child as a gift that will cost $50,000

Therefore remaining saving after buying car = $317,724.82 - $50,000 =$267,724.82

Now at the time of retirement this amount will grow at 10%; therefore its future value at that time

FV = PV * (1+i %) ^n

Where,

Future value of deposit FV =?

Present value of deposit PV = =$267,724.82

Interest rate i = 10% per year

Time period n = 24 years (39 years – 15 years)

Therefore

FV = =$267,724.82* (1 + 10%) ^24

FV = $2,637,017.88

But to achieve retirement goal; amount required at the time of retirement is $4 million or $4,000,000

Your retirement account will have $2,637,017.88 at the time of retirement, therefore remaining amount for annual savings

Remaining amount = $4,000,000.00 - $2,637,017.88

= $1,362,982.12 (now this amount will be future value of your annual saving for 24 years)

We can use FV of an Annuity formula to calculate the annual savings by you

FV = PMT *{(1+i) ^n−1} / i

Where,

Future value of annual savings FV = $1,362,982.12

PMT = Annual savings =?

n = N = number of payments = 24 years

i = I/Y = interest rate per year = 10%

Therefore,

$1,362,982.12 = Annual savings *{(1+10%) ^24−1} /10%

Annual savings = $15,401.39

Therefore you have to save $15,401.39 per year for remaining 24 years to achieve your retirement goal.