Congratulations on expecting your first child! Your parents would like to invest a sum of money on the day the child is born to provide for 5 annual payments of $50, 000 to cover the expense of tertiary education. The first payment is to be made on the child’s 19th birthday, and interest is 5% per year. How much will your parents need to invest?
Question
Congratulations on expecting your first child! Your parents would like to invest a sum of money on the day the child is born to provide for 5 annual payments of $50, 000 to cover the expense of tertiary education. The first payment is to be made on the child’s 19th birthday, and interest is 5% per year. How much will your parents need to invest?
Solution
To solve this problem, we need to use the formula for the present value of an annuity. An annuity is a series of equal payments made at regular intervals. In this case, the annuity is the $50,000 that will be paid annually for 5 years starting on the child's 19th birthday.
The formula for the present value of an annuity is:
PV = Pmt * [(1 - (1 + r)^-n) / r]
where:
- PV is the present value, or the amount that needs to be invested today.
- Pmt is the amount of each payment, which is $50,000.
- r is the interest rate per period, which is 5% or 0.05.
- n is the number of periods, which is 5 years.
However, since the first payment will be made on the child's 19th birthday, we need to discount the present value of the annuity back 19 years to today. This can be done using the formula for the present value of a single sum:
PV = FV / (1 + r)^n
where:
- FV is the future value, or the present value of the annuity we calculated earlier.
- r is the interest rate per period, which is 5% or 0.05.
- n is the number of periods, which is 19 years.
Let's calculate:
- First, calculate the present value of the annuity at the child's 19th birthday:
PV = 210,462.15
- Then, discount this amount back to today:
PV = 80,474.53
So, your parents would need to invest approximately $80,474.53 on the day the child is born.
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