Question

You are planning for your retirement. You expect to earn a monthly salary of $7,000 starting on the 1st month after you retire, which will be able to provide comfortably for your daily expenses through your retirement years. You are currently 33 and plan on retiring when you become 64, and you expect to live 20 years after retirement. In addition to providing a salary for your retirement you would like to buy a house by the time you reach 55. The house you dream of would cost you $1,650,000. Now you have a down payment of $50,000 (ignore closing costs). In addition you would like to offer yourself a retirement gift, a Mercedes that you would buy brand new to serve you through your retirement years. The car is expected to cost you $76,000. It will be purchased when you reach 64 years of age. Assume you can earn 12% compounded monthly from now until you retire, and the rate will change to 6% monthly compounding after that.

a. How much should you save per month to be able to buy the car when you retire?

b. How much should you save per month to be able to buy the house at the age of 55?

c. Therefore, how much do you need to save in TOTAL per month if you can earn 12% compounded monthly from now till you buy your house taking into account your retirement expenses?

d. And how much do you need to save in TOTAL per month after you buy your house until you retire taking into account your retirement expenses?

Answer 1

(a) Car cost $76,000

Rate of interest per month = 12%/12 = 1% or 0.01

Period (N) = 64 yrs - 33yrs = 31*12 month = 372 months

Solving for annuity: FV = A ( ((1+r)^N -1) / r ). Therefore; A = FV / ( ((1+r)^N -1) / r )

A = 76000 / ( ((1+0.01)^372 -1) / 0.01 ) = 76000 / 3950.896 = $19.236 = $19.24 per month

In order to buy the car at age of 64, I need to save $19.24 per month starting 33rd year of my age.

(b) Car cost $1,600,000 ($50,000 already gone in down payment, hence pending amount is $1,600,000)

Rate of interest per month = 12%/12 = 1% or 0.01

Period (N) = 55 yrs - 33yrs = 22*12 month = 264 months

Solving for annuity: FV = A ( ((1+r)^N -1) / r ). Therefore; A = FV / ( ((1+r)^N -1) / r )

A= 1600000 /  ( ((1+0.01)^264 -1) / 0.01 ) = 1600000 / 1283.065 = 1247.014 = $1247.01 per month

In order to buy the house worth $1,650,000 by the age of 55, given I have paid the down payment of $50000, I need to save $1247.01 per month starting 33rd year of my age.

(c) For retirement,

A = $7000

Rate of interest per month = 6%/12 = 0.5% or 0.005

N = 20 * 12 month =240 months

Solving for annuity: PV = A * (1- (1/ (1+r)^N) / r)

= 7000 * (1- ( 1/ (1+0.005)^240) / 0.005)

PV = $ 977065.4

For earning $7000 every month from 64 yrs, I need to save total of $977065.4 uptill 64th year so that I can earn 6% interest on this amount compunded monthly to get my cheque of $7000 every month post that for 20 years.

We now need to calculate the amount need to save from 33 yrs so that I can accumulate $977065.4 till 64th year.

FV = 977065.4

N = 64 - 33 = 31yrs *12 month = 372 month

Rate of interest per month = 12%/12 = 1% or 0.01

Hence, solving for annuity:  FV = A ( ((1+r)^N -1) / r ). Therefore; A = FV / ( ((1+r)^N -1) / r )

A = 977065.4 / ( ((1+0.01)^372 -1) / 0.01 ) = 977065.4 / 3950.896 = $247.30 per month

Answer to (c) = Saving per month of (House + Car + Retirement)

= $1247.01 + $19.24 + $247.30

= $1513.55 per month

I need to save $1513.55 per month till i buy the house at 55th year of my age taking my retirement and car expense in to account.

(d) Saving per month of (Car + Retirement)

=  $19.24 + $247.30

= $266.54 per month

I need to save $266.54 per month after I buy house at 55th year uptill age of 64years in order to fullfill my retirement plan and car dream.