What is the present value of an annuity of $100 payable at the end of each month for 1 year at 6% p.a interest
To find the present value of an annuity where payments are made at the end of each period (an ordinary annuity), you can use the following present value formula for an annuity:
PV = PMT * [(1 - (1 + r)^-n) / r]
Where:
- PV is the present value of the annuity.
- PMT is the payment amount per period ($100 in this case).
- r is the periodic interest rate (annual interest rate divided by the number of periods per year).
- n is the total number of periods (number of months in this case).
Given:
- PMT = $100
- n = 12 (since there are 12 months in a year)
- Annual interest rate = 6% or 0.06
First, we need to calculate the monthly interest rate by dividing the annual interest rate by 12:
r = 0.06 / 12 = 0.005 (monthly interest rate)
Now, we can apply the formula:
PV = $100 * [(1 - (1 + 0.005)^-12) / 0.005]
Next, calculate (1 + r)^-n:
(1 + 0.005)^-12 = (1.005)^-12 ≈ 0.940
Now compute the remaining part of the equation and the final present value:
PV = $100 * [(1 - 0.940) / 0.005]
PV = $100 * (0.060 / 0.005)
PV = $100 * 12
PV = $1200
It appears an error has been made in the calculation, as the present value of the annuity should be less than the total sum of the payments due to the time value of money. The correct calculation should take into account the exponent for the discount factor (1 + r)^-n accurately:
PV = $100 * [(1 - (1 + 0.005)^-12) / 0.005]
Calculating (1 + 0.005)^-12 accurately:
PV = $100 * [(1 - (1.005)^-12) / 0.005]
PV = $100 * [(1 - 0.94180) / 0.005]
PV = $100 * (0.05820 / 0.005)
PV = $100 * 11.64
PV ≈ $1164
So, the present value of an annuity of $100 payable at the end of each month for 1 year at 6% per annum is approximately $1164.