Question

The nominal interest rate is 10​% compounded semiannually. What amount will need to be deposited every six months to be able to have enough money to pay three annuity payments of ​$5,000 for three years beginning at the end of year​ seven? The deposits begin now and continue every six months until six deposits have been made.   The amount to be deposited every six months is? 

Answer 1

Semiannual interest rate = 10/2 = 5% since interest compounded semiannually therefore effective interest rate = (1.05)^2 - 1 = 1.1025 - 1 = 0.1025 i.e 10.25%

First we need to calculate PV at the end of year 7, which is as below:

Year	Payments	Discounting factor @10.25%	PV
7	5000	1	5000.0000
8	5000	0.907029	4535.1474
9	5000	0.822702	4113.5124
Total			13648.6598

$13648.6598 is the PV at the end of year 7

Now deposit will start now and continue every six months until six deposits have been made therefore the deposits will continue till 2.5 years from now

Now we need to bring the PV of year 7 to the PV of year 2.5 considering PV at year 7 as FV

No of 6 months period from 2.5 years to 7 years = 9 periods

FV = PV (1+r) ^ n

13648.6598 = PV (1+0.05) ^ 9

13648.6598 = PV * 1.551328

PV at year end 2.5 = $8798.0478

Now no of deposit period in 2.5 years = 6

Now we need to calculate FV of annuity = [(1+r)^n - 1 / r]

FV of annuity = [(1+0.05)^6 - 1 / 0.05] = (0.340096 / 0.05) = 6.801913

Therefore amount to be deposited every six months =  $8798.0478 / 6.801913 = $1293.46671 i.e. $1293.47