A payment of $1 is made at the start of each month for a period of 5 years. The annual effective interest is 10% p.a. Which of these annuity functions will give us the present value of the payments on the date of the first payment?
Question
A payment of $1 is made at the start of each month for a period of 5 years. The annual effective interest is 10% p.a. Which of these annuity functions will give us the present value of the payments on the date of the first payment?
Solution
The present value of an annuity can be calculated using the formula for the present value of an ordinary annuity. However, since the payments are made at the start of each period (in this case, each month), this is an annuity due, not an ordinary annuity.
The formula for the present value of an annuity due is:
PV = P * [1 - (1 + r)^-n] / r * (1 + r)
where: PV = present value of the annuity P = payment per period (in this case, $1 per month) r = interest rate per period (in this case, 10% per year, or 0.10/12 per month) n = number of periods (in this case, 5 years * 12 months/year = 60 months)
So, the calculation would be:
Step 1: Convert the annual interest rate to a monthly rate:
r = 0.10 / 12 = 0.008333
Step 2: Calculate the present value of the annuity due:
PV = $1 * [1 - (1 + 0.008333)^-60] / 0.008333 * (1 + 0.008333)
This will give us the present value of the payments on the date of the first payment.
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