• Global Frameworks: CFA Level 1 (Quantitative Methods), ACCA Financial Management (FM).

1. Future Value (FV) and Present Value (PV) of Single Cash Flows
The Time Value of Money (TVM) hinges on the premise that a unit of currency today is worth more than the same unit in the future due to its potential earning capacity.
The compounding formula determines how a current asset grows over time:
\(FV=PV\times (1+r)^{n}\)
Where:
  • FV = Future Value
  • PV = Present Value
  • r = Interest rate per period
  • n = Number of compounding periods

Discounting reverses compounding, pulling future sums back to the present to assess their current worth:

PV=FV(1+r)n

2. Ordinary Annuities vs. Annuities Due
An annuity is a series of equal, periodic cash flows.
  • Ordinary Annuities: Cash flows occur at the end of each period (e.g., standard corporate bonds, commercial mortgages).
  • Annuities Due: Cash flows occur at the beginning of each period (e.g., commercial lease payments, insurance premiums).

The present value of an ordinary annuity is calculated as:

PVord=PMT×[1−(1+r)−nr]

Because an annuity due receives payments one period earlier, each cash flow is compounded for an extra period. Its value is derived by multiplying the ordinary annuity value by (1 + r):

PVdue=PVord×(1+r)

3. Perpetuities (Constant and Growing)
A perpetuity is an annuity that continues indefinitely.
  • Constant Perpetuity: Pays a fixed amount (PMT) forever.
    PV = PMT/r
  • Growing Perpetuity: Pays a cash flow that grows at a constant rate (g) indefinitely. This model forms the foundation of the Gordon Growth Model for equity valuation.
    PV = (PMT_1)/(r − g)

4. Compounding Frequencies and Effective Annual Rate (EAR)
When interest compounds more than once a year (e.g., semi-annually, quarterly, monthly), the Stated Annual Interest Rate (or Nominal Rate) understates the actual return. The Effective Annual Rate (EAR) reflects the true annual interest earned or paid:

Formula: EAR = (1 + r/m)^m − 1

Where m is the number of compounding periods per year. For continuous compounding, the formula transitions to:

Formula: EAR = e^r − 1
 
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