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Write the payment dates explicitly
| Pattern | Payment dates | First payment |
|---|---|---|
| Ordinary | 1, 2, 3, 4, 5 | End of the first period |
| Due | 0, 1, 2, 3, 4 | Immediately at time zero |
Compare values at a common valuation date
Shifting every payment one period earlier multiplies its time-zero value by 1 + r under the same periodic-rate convention. The total annuity value changes by that same factor. At a positive rate, due PV is higher; at zero, it is equal; at a negative rate, it is lower.
Define the future-value date as well
The calculator compares both patterns at time N. The due pattern’s final payment is then one period before that date and compounds for one period. Comparing one pattern at N and another at N − 1 would mix valuation dates.
Separate the model from an insurance product
This model values specified equal deterministic payments. An actual insurance annuity can involve mortality, guarantees, charges and other contract terms that this cash-flow pattern does not represent.