Time Value of Money: Why a Dollars Today Is Worth More Than a Dollars Tomorrow
The time value of money (TVM) is one of the most fundamental concepts in finance. It states that a sum of money available today is worth more than the same sum available in the future — because today’s money can be invested to earn a return.
Why Does Money Have Time Value?
- Opportunity cost — Money received today can be invested to earn interest or returns.
- Inflation — Prices generally rise over time, eroding the purchasing power of future money.
- Risk — Future payments are uncertain; present cash is certain.
Future Value of a Single Amount
How much will a sum grow to, given a rate of return over time?
FV = PV × (1 + r)^n
Where: PV = present value, r = interest rate per period, n = number of periods.
You invest $100,000 at 8% per annum for 5 years:
FV = $100,000 × (1.08)^5 = $100,000 × 1.4693 = $146,930
Present Value of a Single Amount
What is a future sum worth in today’s terms? This is the reverse of future value.
PV = FV ÷ (1 + r)^n
You will receive $200,000 in 4 years. Discount rate is 10%:
PV = $200,000 ÷ (1.10)^4 = $200,000 ÷ 1.4641 = $136,603
Receiving $200,000 in 4 years is equivalent to having $136,603 today at 10% discount rate.
Present Value of an Ordinary Annuity
An annuity is a series of equal payments at regular intervals. An ordinary annuity pays at the end of each period.
PV Annuity = PMT × [(1 − (1 + r)^−n) ÷ r]
$20,000 received at end of each year for 5 years; discount rate 8%:
PV = $20,000 × [(1 − (1.08)^−5) ÷ 0.08]
= $20,000 × [(1 − 0.6806) ÷ 0.08]
= $20,000 × 3.9927 = $79,854
Practical Applications
- Loan EMI calculation — Uses the annuity formula to determine equal monthly instalments.
- Bond pricing — A bond’s price is the PV of its coupon payments (annuity) plus PV of face value (lump sum).
- Investment appraisal — NPV discounts future cash flows to evaluate projects (covered in Section 7).
- Retirement planning — How much to save today to fund a desired future income stream.
The Rule of 72
A quick mental shortcut: divide 72 by the annual interest rate to estimate how many years it takes to double your money. At 8%, money doubles in approximately 72 ÷ 8 = 9 years.
Lesson Summary
- FV = PV × (1 + r)^n — compounding grows money forward in time.
- PV = FV ÷ (1 + r)^n — discounting brings future money back to today’s value.
- Ordinary annuity PV formula values a series of equal future payments.
- TVM is the foundation of bond pricing, loan calculations, and investment appraisal.
Time Value of Money: The Cornerstone of Finance
The core principle: a dollar today is worth more than a dollar tomorrow. Why? Because a dollar today can be invested to earn a return. This simple idea underlies all of corporate finance, investment analysis, insurance pricing, and pension planning.
Four variables govern every TVM problem:
- PV — Present Value (value today)
- FV — Future Value (value at a future date)
- r — Interest/discount rate per period
- n — Number of periods
Six Essential TVM Formulas
| Concept | Formula | Example |
|---|---|---|
| Future Value (lump sum) | FV = PV × (1+r)^n | $10,000 at 8% for 3 years = $12,597 |
| Present Value (lump sum) | PV = FV ÷ (1+r)^n | Receive $15,000 in 5 years at 7% → PV = $10,694 |
| Future Value (annuity) | FV = PMT × [(1+r)^n − 1] ÷ r | Save $2,000/yr at 6% for 10 yrs → $26,362 |
| Present Value (annuity) | PV = PMT × [1 − 1/(1+r)^n] ÷ r | $5,000/yr for 5 yrs at 8% → $19,964 |
| Effective Annual Rate | EAR = (1 + r/m)^m − 1 | 6% compounded monthly → EAR = 6.168% |
| Perpetuity PV | PV = PMT ÷ r | $1,000/yr forever at 5% → PV = $20,000 |
Practical Applications in Business
| Business Decision | TVM Used | Example |
|---|---|---|
| Should we buy this machine? | NPV — discount future cash flows to PV | Machine generates $50K/yr for 5 yrs; discount at WACC |
| What’s this bond worth? | PV of annuity (coupons) + PV of face value | 6% coupon bond, 10 yrs, 8% market rate |
| How much to save for retirement? | FV of annuity | Save $X/month for 30 years at 7% to reach $2M |
| What loan payment can we afford? | PV of annuity solved for PMT | $200K mortgage, 30 yrs, 5% → payment = $1,074/mo |
| What’s this startup worth? | PV of projected free cash flows | DCF valuation model |
Annuity vs Annuity Due: A Critical Distinction
An ordinary annuity pays at the end of each period (most common: mortgage, bond coupon). An annuity due pays at the beginning of each period (rent, lease). Annuity due is always worth more because payments arrive sooner.
Formula: PV Annuity Due = PV Ordinary Annuity × (1 + r)
Example: $3,000/year for 4 years at 8%:
PV ordinary annuity = $3,000 × 3.3121 = $9,936
PV annuity due = $9,936 × 1.08 = $10,731
Excel’s financial functions make TVM calculations instant: PV(), FV(), PMT(), RATE(), NPER(). These are essential skills for any finance professional. The logic you’re learning here is exactly what those functions compute.
Time Value of Money Practice Worksheet — Download, print, and complete to reinforce this lesson.