A man stipulates in his will that $50000 from his estate is to be placed in a fund from which his three children are to each receive the same amount when they reach age 21. When the man dies (on January 2010), the children are aged 19, 15 and 13. The fund earns variable interest and the rates are as provided in the table below. How much will each child receive?
Duration
01/01/09 to 31/12/12
Variable interest rate
12% p.a compounded semi-annually
Duration
01/01/13 to 31/12/25
Variable interest rate
10% p.a compounded annually
28086.40
The answer is $28418.78
282451.78
To calculate how much each child will receive, we need to calculate the future value of the $50,000 contributed to the fund for each child individually.
Let's start with the oldest child who is 19 years old. We need to calculate the future value of $50,000 for 2 years (from 19 to 21) at an interest rate of 12% p.a. compounded semi-annually.
First, we need to find the number of compounding periods in 2 years, considering the semi-annual compounding. Since there are 2 compounding periods in a year, there would be a total of 4 compounding periods in 2 years.
Next, we can use the compound interest formula to calculate the future value:
FV = PV * (1 + r/n)^(n*t)
Where:
FV = Future Value
PV = Present Value (initial amount)
r = annual interest rate (as a decimal)
n = number of compounding periods per year
t = number of years
For the oldest child:
PV = $50,000
r = 12% = 0.12 (as a decimal)
n = 2 (since it's compounded semi-annually)
t = 2 (2 years)
FV = $50,000 * (1 + 0.12/2)^(2*2)
FV = $50,000 * (1 + 0.06)^4
FV = $50,000 * (1.06)^4
FV = $50,000 * 1.26247616
FV = $63,123.81
Therefore, the oldest child will receive $63,123.81 from the fund.
Next, let's calculate the future value for the second child who is 15 years old. We need to calculate the future value of $50,000 for 6 years (from 15 to 21) at an interest rate of 12% p.a. compounded semi-annually.
For the second child:
PV = $50,000
r = 12% = 0.12 (as a decimal)
n = 2 (since it's compounded semi-annually)
t = 6 (6 years)
FV = $50,000 * (1 + 0.12/2)^(2*6)
FV = $50,000 * (1 + 0.06)^12
FV = $50,000 * (1.06)^12
FV = $50,000 * 1.79084770
FV = $89,542.39
Therefore, the second child will receive $89,542.39 from the fund.
Finally, let's calculate the future value for the youngest child who is 13 years old. We need to calculate the future value of $50,000 for 8 years (from 13 to 21) at an interest rate of 12% p.a. compounded semi-annually.
For the youngest child:
PV = $50,000
r = 12% = 0.12 (as a decimal)
n = 2 (since it's compounded semi-annually)
t = 8 (8 years)
FV = $50,000 * (1 + 0.12/2)^(2*8)
FV = $50,000 * (1 + 0.06)^16
FV = $50,000 * (1.06)^16
FV = $50,000 * 2.01203635
FV = $100,601.82
Therefore, the youngest child will receive $100,601.82 from the fund.
In conclusion, each child will receive the following amounts from the fund:
- Oldest child (19 years old): $63,123.81
- Second child (15 years old): $89,542.39
- Youngest child (13 years old): $100,601.82