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25.3 Explain the Time Value of Money and Calculate Present and Future Values of Lump Sums and Annuities

Time Value of Money (TVM)​

Time value of money means a euro today is worth more than a euro in the future because it can be invested to earn a return. Capital budgeting requires discounting future cash flows to present value.

Future Value (FV)​

Formula: FV = PV × (1 + r)^n

  • PV = present value
  • r = interest rate per period
  • n = number of periods

Example:

  • Invest €10,000 at 5% for 3 years
  • FV = €10,000 × (1.05)^3 = €11,576.25

Present Value (PV)​

Formula: PV = FV ÷ (1 + r)^n

Example:

  • Receive €12,000 in 4 years at 6%
  • PV = €12,000 ÷ (1.06)^4 = €9,506.17

Present Value of Annuities​

Ordinary Annuity (payments at end of period): PV = Payment × [1 - (1 + r)^-n] / r

Example:

  • Receive €5,000 per year for 5 years at 7%
  • PV = €5,000 × [1 - (1.07)^-5] / 0.07 = €20,479.48

Annuity Due (payments at beginning):

  • Multiply PV of ordinary annuity by (1 + r)

Future Value of Annuities​

FV = Payment × [(1 + r)^n - 1] / r

Discount Factors​

Use present value tables or calculator functions (NPV, IRR) for efficiency.

Discount Rates​

Discount rate (cost of capital) reflects weighted average cost of debt and equity, adjusted for project risk.

Weighted Average Cost of Capital (WACC): WACC = (E/V × Re) + (D/V × Rd × (1 - Tc))

  • E = market value of equity
  • D = market value of debt
  • V = E + D
  • Re = cost of equity
  • Rd = cost of debt
  • Tc = corporate tax rate

Risk-Adjusted Discount Rates​

  • Base Case: WACC
  • Higher Risk Projects: Add risk premium
  • Lower Risk Projects: Use lower rate

Luxembourg Compliance Note​

Cost of capital in Luxembourg may consider bank lending rates, EU reference rates, and tax rate (~24-27%). Multinational SMEs must consider transfer pricing rules and cross-border financing.

Think It Through​

Why is the present value of future cash flows always less than the future value (assuming positive interest rates)? How does risk affect the choice of discount rate?