Question

Suppose you wish to retire forty years from today. You determine that you need RM 50,000 per year once you retire, with the first retirement funds withdrawn one year from the day you retire. You estimate that you will earn 6% per year on your retirement funds and that you will need funds up to and including your 25th birthday after retirement

a) How much must you deposit in an account today so that you have enough funds for retirement?

b)How much must you deposit each year in an account, starting one year from today, so that you have enough funds for retirement.

Answer 1

Years remaining in retirement = 40 years
Funds needes after retirement for = 25 years
First post-retirement funds to be received after one year from retirement (it means that on first installment we will calculate interest for 40+1 = 41 years; on 2nd for 42 years and so on.)
Yearly funds needed after retirement = RM 50000 per year
Rate of return = 6% p.a.

a) When I am supposed to make a one-time investment in lumpsum now:

Investment Amount = [50000*(1/1.06)⁴¹] + [50000*(1/1.06)⁴²] + [50000*(1/1.06)⁴³] + ..... so on till..... [50000*(1/1.06)⁶⁵]

When we solve this, we get:

Lumpsum amount to be invested now = RM 62141.29

b) How much must I deposit each year in an account, starting one year from today, so that you have enough funds for retirement.

Let amount to be deposited each year = y

Now we know that:

The maturity amount of 'x' deposited every year till retirement must be equal to the present value (as on retirement) of RM 50000 to be received every year for 25 years after retirement. Let us bisect the preceding statement in to two, as follows:

Part 1 = The maturity amount of 'y' deposited every year till retirement
Part 2 = Present value (as on retirement) of RM 50000 to be received every year for 25 years after retirement

Let us formulize each part:

Part 1:
The maturity amount of 'y' deposited every year till retirement = [y*1.06³⁹] + [y*1.06³⁸] + [y*1.06³⁷] + ... so on till....+ [y*1.06⁰]

Part 2:
Present value (as on retirement) of RM 50000 to be received every year for 25 years after retirement = [50000*1/1.06¹] + [50000*1/1.06²] + [50000*1/1.06³] + .... so on till.... + [50000*1/1.06²⁵]

we know that if we want to find the value of y, then Part 1 must be equal to Part 2.

Let us compute Part 2 first, because we have no missing information here. After solving we get:
Present value (as on retirement) of RM 50000 to be received every year for 25 years after retirement = RM 639168

Now we can put the outcome of part 2 as equal to the whole formula of Part 1:

RM 639168 = [y*1.06³⁹] + [y*1.06³⁸] + [y*1.06³⁷] + ... so on till....+ [y*1.06⁰]

When we solve the above, we get y = RM 4130

(I took excel's help for calculations)