What is the relationship between the present value of an ordinary annuity and the present value of an annuity-due with the same number of payments and the same interest rate?

In this case, present value (PV) of an annuity due will be higher than the PV of an ordinary annuity.

Explanation:

An annuity is a financial product from which an investor receives a fixed stream of payments over period of time. The two types of an annuity are ordinary annuity and annuity due.

Under an ordinary annuity, investors receive payments at the end of each time period. The formula for calculating an ordinary annuity is given as follows:

PVo = P * [{1 - [1 : (1+r) ]^n} : r]

Where

PVo = Present value of an ordinary annuity

P = First payment

r = interest rate

n = number of years

But in an annuity due, investors get payments at the beginning of each time period. The formula for calculating an annuity due is not from the formula for calculating an ordinary annuity, but the only difference is that (1+r) is used to multiply the formula for calculating an ordinary annuity to obtain the formula for calculating an annuity due follows:

PVd = P * [{1 - [1 : (1+r) ]^n} : r] * (1+r)

Where

PVd = Present value of an annuity due.

P, r and n are as already explained above.

Consequently, the PV of an annuity due will be higher than the PV of an ordinary annuity, because the formula for calculating an ordinary annuity is multiplied by (1+r) to obtain the formula for calculating an annuity due which scales up the value.