Question

Hi I am doing an intro to finance the the present value and future values are a little bit confusing so my question is:

How is the difference between an ordinary annuity and an annuity due affect the Present Value and Future Value of two otherwise identical annuities?

Answer 1

Present value:

Annuity is a series of fixed cash flow for a certain number of periods. An ordinary annuity states that cash flow is paid at the END of each year. An annuity due states that cash flow if paid st the BEGINNING at the each year. The present value of annuity due will always be greater than present value of ordinary annuity. This is because annuity due has a period less to discount the cash flows. This is can be best illustrated with a example:

Ordinary annuity:

You are paid a cash flow of $1,000 for 10 years at an interest rate of 6%. The cash flows are paid at the end.

Present value = Annuity * [1 - 1 / (1 + r)ⁿ] / r

Present value = 1,000 * [1 - 1 / (1 + 0.06)¹⁰] / 0.06

Present value = 1,000 * 7.360087

Present value = $7,360

annuity due:

You are paid a cash flow of $1,000 for 10 years at an interest rate of 6%. The cash flows are paid at the end.

Present value = (1 + r) * Annuity * [1 - 1 / (1 + r)ⁿ] / r

Present value = (1 + 0.06) * 1,000 * [1 - 1 / (1 + 0.06)¹⁰] / 0.06

Present value = 1.06 * 1,000 * 7.360087

Present value = $7,802

As you can see, present value of annuity due is greater.

Future value:

Annuity is a series of fixed cash flow for a certain number of periods. An ordinary annuity states that cash flow is paid at the END of each year. An annuity due states that cash flow if paid st the BEGINNING at the each year. The future value of annuity due will always be greater than future value of ordinary annuity. This is because annuity due has a period less to discount the cash flows. This is can be best illustrated with a example:

Ordinary annuity:

You are paid a cash flow of $1,000 for 10 years at an interest rate of 6%. The cash flows are paid at the end. What is the future value

Future value = Annuity * [(1 + r)ⁿ - 1] / r

Future value = 1,000 * [(1 + 0.06)¹⁰ - 1] / 0.06

Future value = 1,000 * 13.180795

Future value = $13,181

annuity due:

You are paid a cash flow of $1,000 for 10 years at an interest rate of 6%. The cash flows are paid at the beginning. What is the future value

Future value = (1 + r) * Annuity * [(1 + r)ⁿ - 1] / r

Future value = (1 + 0.06) * 1,000 * [(1 + 0.06)¹⁰ - 1] / 0.06

Future value = 1.06 * 1,000 * 13.180795

Future value = $13,971.6