Annuities
Core study note
Current course guide + original explanation
Timing before formula
An annuity-immediate pays at the end of each period. An annuity-due pays at the beginning. That one-period shift changes the value by one accumulation factor (1+i).
For level payment R, rate i, and n payments, the immediate present-value factor is (1-(1+i)^(-n))/i; the future-value factor is ((1+i)^n-1)/i.
Useful habit
Draw the payment timeline before selecting a factor. Most annuity mistakes are timing mistakes disguised as algebra mistakes.