1. The Core Concept of TVM
A currency unit received today is worth more than the same unit received in the future. This foundational law exists because cash held today can be invested to earn a return (interest), alongside the impacts of inflation and purchasing power degradation.
 
2. Compounding and Discounting Formulations
Future Value (FV) of a Single Lump Sum
Moving a cash flow forward in time via compounding interest:

  • Future Value of a Lump Sum: FV_n = PV(1 + r)^n
  • Present Value of a Single Lump Sum:
    • FVₙ = PV × (1 + r)ⁿ
    • PV = FVₙ / (1 + r)ⁿ = FVₙ × (1 + r)⁻ⁿ
3. Annuities and Perpetuities
Ordinary Annuity vs. Annuity Due
An annuity is a stream of equal, periodic cash flows over a specified time horizon.
  • Ordinary Annuity: Cash flows occur at the end of each period (standard for corporate bonds).
  • Annuity Due: Cash flows occur at the beginning of each period (standard for lease payments). Annuity Due Value = Ordinary Annuity Value × (1 + r).

Present Value of an Ordinary Annuity
PV_annuity = PMT × [ (1 − (1 + r)⁻ⁿ) / r ]
Where: PMT = Periodic payment amount.
Perpetuities
An annuity stream that continues infinitely into the future with no terminal maturity date:

PV_perpetuity = PMT / r
If the perpetuity features a constant growth rate (g), it is evaluated using the Gordon Growth Model:

PV_growing_perpetuity = PMT₁ / (r − g), where r > g.
 
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