You want to have Br. 10,000,000 when you retire in 40 years. You expect to earn 12% compounded semiannually over the entire 40-year period. How much extra money per month must you deposit if you choose to fund using an ordinary annuity technique rather than an annuity due technique?

You must deposit Br. 54.01 extra per month if you choose to fund using an ordinary annuity technique rather than an annuity due technique.

Explanation:

This can be determined using the following 3 steps:

Step 1: Calculation of monthly deposit under ordinary annuity

Theis can be calculated using the formula for calculating the Future Value (FV) of an Ordinary Annuity given as follows:

FV = Do * (((1 + r)^n - 1) / r) (1)

Where,

FV = Future value of the amount = Br. 10,000,000

Do = Semiannual deposit under ordinary annuity = ?

r = Semiannual interest rate = Annual interest rate / Number of semiannuals in a year = 12% / 2 = 6%, or 0.06

n = number of semiannuals = Number of years * Number of semiannuals in a year = 40 * 2 = 80

Substituting the values into equation (1) and solve for Do, wee have:

10,000,000 = Do * (((1 + 0.06)^80 - 1) / 0.06)

10,000,000 = Do * 1746.59989136857

Do = 10,000,000 / 1746.59989136857

Do = Br. 5,725.40972286698

Mo = Monthly deposit under ordinary annuity = Do / Number of months in a semiannual = Br. 5,725.40972286698 / 6 = Br. 954.23

Step 2: Calculation of monthly deposit under annuity due

Theis can be calculated using the formula for calculating the Future Value (FV) of an Annuity Due given as follows:

FV = Dd * (((1 + r)^n - 1) / r) * (1 + r) (2)

Where

Dd = Semiannual deposit under annuity due = ?

Other values are as already defined in Step 1 above.

Substituting the values into equation (2) and solve for Dd, wee have:

10,000,000 = Dd * (((1 + 0.06)^80 - 1) / 0.06) * (1 + 0.06)

10,000,000 = Dd * 1,851.39588485068

Dd = 10,000,000 / 1,851.39588485068

Dd = Br. 5,401.329927233

Md = Monthly deposit under annuity due = Dd / Number of months in a semiannual = Br. 5,401.329927233 / 6 = Br. 900.22

Step 3: Calculation of difference between monthly deposit under ordinary annuity and monthly deposit under annuity due

Difference = Mo - Md = Br. 954.23 - Br. 900.22 = Br. 54.01

Therefore, you must deposit Br. 54.01 extra per month if you choose to fund using an ordinary annuity technique rather than an annuity due technique.

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