Time Value of Money (TVM)

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Time Value of Money (TVM)

What is TVM?

The time value of money (TVM) is a fundamental financial concept: a dollar today is worth more than the same dollar in the future. That difference arises because money you have now can be invested to earn returns, and because inflation erodes future purchasing power.

Key ideas

  • Money invested today grows through interest or returns; money received later sacrifices that earning opportunity (opportunity cost).
  • Compound interest—interest earned on prior interest—magnifies growth over time.
  • TVM calculations assume positive returns; they do not account for capital losses or negative interest rates unless explicitly modeled.

Core formulas

Future value (FV) with periodic compounding:
FV = PV × (1 + i/n)^(n×t)

Present value (PV) from a future amount:
PV = FV / (1 + i/n)^(n×t)

Where:
* PV = present value (amount today)
* FV = future value (amount at time t)
* i = nominal annual interest rate (decimal)
* n = number of compounding periods per year
* t = number of years

For continuously compounded interest:
FV = PV × e^(i×t)

Examples

  1. Simple future value

  2. Invest $10,000 for 1 year at 10% compounded annually:
    FV = $10,000 × (1 + 0.10/1)^(1×1) = $11,000

  3. Present value

  4. What amount today at 7% annually will become $5,000 in one year?
    PV = $5,000 / (1 + 0.07) = $4,673 (rounded)

  5. Effect of compounding frequency (10% annual rate, $10,000 for 1 year)

  6. Annual: $11,000
  7. Quarterly: $10,000 × (1 + 0.10/4)^(4) = $11,038
  8. Monthly: $10,000 × (1 + 0.10/12)^(12) = $11,047
  9. Daily: $10,000 × (1 + 0.10/365)^(365) = $11,052

More frequent compounding increases FV because interest is reinvested more often.

Explain Like I’m 5

If someone gives you $100 today or $100 a year from now, take it today. You can put it in a savings account or invest it so it grows. Waiting loses both the chance to earn more and some buying power because prices tend to rise (inflation).

Why TVM matters

  • Investment decisions: Compare cash flows occurring at different times (e.g., a payout now vs. later).
  • Valuation: Discounted cash flow (DCF) uses TVM to value businesses, projects, and investments.
  • Personal finance: Retirement planning, mortgages, savings goals, and loan comparisons all rely on TVM.
  • Risk management: Ensures future obligations are met by accounting for how money must grow to reach future targets.

Opportunity cost

Receiving money later carries an opportunity cost: what you could have earned if you had the money now. Even a guaranteed future payment loses value when discounted to present terms.

Practical notes

  • TVM models should include realistic rates and consider inflation, taxes, fees, and the possibility of negative returns.
  • For annuities, perpetuities, or uneven cash flows, use specialized PV and FV formulas or a DCF approach to aggregate multiple periods.

Bottom line

The time value of money helps translate money across time. Use FV to estimate how current sums will grow and PV to determine what future sums are worth today. Applying TVM correctly leads to better investment, borrowing, and financial planning decisions.

Note: The idea of valuing money over time has historical roots going back centuries and underpins modern finance.