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Understanding the present value of annuities is crucial when comparing investments or making financial decisions involving cash flows that occur over time. The present value of an annuity is the total worth of a series of future annuity payments at the current point in time. We calculate it by discounting the future payments to the present using a specific interest rate, which accounts for the time value of money.

For an ordinary annuity, where payments occur at the end of each period, the formula used is
\[PV_{ordinary\_annuity} = PMT * \frac{1 - (1 + r)^{-n}}{r}\]
where PMT is the annuity payment, r is the periodic interest rate, and n is the number of periods. This calculation reflects the concept that money available today is worth more than the same amount in the future, due to its potential earning capacity.

The future value of annuities quantifies how much a series of annuity payments will be worth at a later date. It’s determined by applying compound interest over the term of the annuity. The future value shows the growth of an investment over time, given a specified rate of interest.

The formula for the future value of an ordinary annuity is
\[FV_{ordinary\_annuity} = PMT * \frac{(1 + r)^n - 1}{r}\]
while the future value of an annuity due can be calculated as
\[FV_{annuity\_due} = PMT * \frac{(1 + r)^n - 1}{r} * (1 + r)\]
The difference between them highlights the effect timing of payments has on the accumulation of interest, since annuity due payments contribute to the investment at the beginning of the period.

When comparing an ordinary annuity and an annuity due, the primary distinction lies in the timing of the payment. In an ordinary annuity, payments are made at the end of each period, while in an annuity due, payments are made at the beginning. This difference has a significant impact on the value of the annuity due to the time value of money.

Annuities due will always have a higher future value than ordinary annuities if they have the same terms and interest rate. This is because each payment is invested for an additional time period, allowing for more interest to be earned. Similarly, the present value of an annuity due is also higher than that of an ordinary annuity because the payments are considered more valuable when they occur earlier.

The time value of money is the principle that states that money available at the present time is worth more than the same amount in the future. This is due to the potential earning capacity of money; given time, money can earn interest or be invested, thus growing in value. It underscores the rationale behind discounting in present value calculations and compounding in future value assessments while showing why received or invested sums earlier are more beneficial.

To illustrate, having \(100 today is preferable to receiving \)100 in a year because you can invest that \(100 now and have more than \)100 in a year's time. This basic principle is pivotal in understanding the value of annuities, whether they are due or ordinary, as it affects the calculation and comparison of their worth.