1. An employee has contributed with her employer to a retirement plan for 20 years a
certain amount twice a year. The contribution earns an interest rate of 10%
compounded semiannually. At the date of her retirement the total retirement benefit
is Br. 300, 000. The retirement program provides for investment of this amount at an
interest rate of 10% compounded semiannually. Semiannual payments will be made
for 10 years to the employee of her family in the event of her death.
1. What semi annual payment should she made?
2. What semi annual payment should be made for her or her family?
3. How much interest will be earned on Br. 300, 000 over the 40 years?
1. To calculate the semi-annual payment the employee should make, we can use the formula for the future value of an ordinary annuity:
FV = Pmt * ((1 + r)^n - 1) / r
Where:
FV = Future Value (Br. 300,000)
Pmt = Payment
r = Interest rate per period (10% compounded semiannually = 5% per semiannual period)
n = Number of periods (20 years = 40 semiannual periods)
300,000 = Pmt * ((1 + 0.05)^40 - 1) / 0.05
300,000 = Pmt * (7.723665 - 1) / 0.05
300,000 = Pmt * 6.723665 / 0.05
300,000 = Pmt * 134.4733
Pmt = 300,000 / 134.4733
Pmt ≈ Br. 2,233.67 (semi-annual payment)
2. To calculate the semi-annual payment that should be made to the employee or her family, we can use the same formula as above but adjusted for the new number of periods (10 years = 20 semiannual periods):
Pmt = FV / ((1 + r)^n - 1) / r
Pmt = 300,000 / ((1 + 0.05)^20 - 1) / 0.05
Pmt = 300,000 / (3.86968 - 1) / 0.05
Pmt = 300,000 / 2.86968 / 0.05
Pmt = 300,000 / 57.3936
Pmt ≈ Br. 5,229.13 (semi-annual payment for the employee or her family)
3. To calculate the total interest earned on Br. 300,000 over the 40 years, we can use the formula for compound interest:
I = P * (1 + r)^n - P
Where:
I = Interest
P = Principal (Br. 300,000)
r = Interest rate per period (10% compounded semiannually = 5% per semiannual period)
n = Number of periods (40 years = 80 semiannual periods)
I = 300,000 * (1 + 0.05)^80 - 300,000
I = 300,000 * 416.11309 - 300,000
I = 124,833,926.7 - 300,000
I ≈ Br. 124,533,926.70