Question

What is the present value of $300 received at the beginning of each year for five years? Assume that the first payment is not received until the beginning of the third year (thus the last payment is received at the beginning of the seventh year). Use a 10% discount rate, and round your answer to the nearest $100.

soution given:
year 1: 1 x (1+0.1)^0 x 0
year 2: 1 x (1+0.1)^1 x 0
year 3: 1 x (1+0.1)^2 x 300
...

i would like to ask why for year 1,2,3 ..., the calculation is not as follow:
year 1: 1 x (1+0.1)^1 x 0
year 2: 1 x (1+0.1)^2 x 0
year 3: 1 x (1+0.1)^3 x 300
...

Answer 1

This is because the payment is being made at the beginning of the period, when the payment is made at the beginning of the year the discounting is not done for a particular period.

For Example : The payment is made at the beginning and the date of today is January - 1 - 2020 and if the payment is made at the end of the period then the payment will be made at 31-Dec-2020. So, to arrive at present value we will discount the payment of 31-Dec-2020 for a complete year. Thus, We deduct the number of periods by 1 when the payment is made at the beginning.

Therefore, The correct answer is

year 1: 1 x (1+0.1)^0 x 0
year 2: 1 x (1+0.1)^1 x 0
year 3: 1 x (1+0.1)^2 x 300