Annuity due problem

Question

Kindly help me. This is under Annuity Due 

Mrs. Semion had accumulated 2.3 million in her retirement plan, she wanted to raise it to 6 million with equal payments every 6 months for eight years, if the money is worth 10% compounded semi annually, how much will be the further contribution?

Answer 1

The first thing we need is the formula for the future value of an annuity due

 

P = (PMT [((1 + r)n - 1) / r])(1 + r)
Where:
P = The future value of the annuity stream to be paid in the future= = 6M
PMT = The amount of each annuity payment
r = The interest rate = 10%/2 since compounding semi-annually
n = The number of periods over which payments are to be made = 16 (8x2)

 

We need to amend this formula for the $2.3 she has already accumulated.

 

P = 2.3M(1+r)ⁿ + (PMT [((1 + r)ⁿ - 1) / r])(1 + r)

 

Let's fill in the info we have

6M = 2.3M(1.05)¹⁶ + (PMT[(1.05)¹⁶ - 1)/.05])(1.05)

6M = 5,020611.55 + (PMT[(2.183-1)/.05])(1.05)

979388.45 = PMT(24.843)

PMT = $39,423.12

 

Mrs Semion will need to make semi-annual payments of $39,423.12 for the next 8 years in order to accumulate $6M given she has already accumulated $2.3M and will invest that $2.3 million at the same rate of 10% compounded semi-annually.