Section 8: Time Value of Money
1 lesson in this module. Work through each lesson top to bottom; download the templates and worksheets inside each lesson.
Present Value, Future Value, and Annuities matters because it’s a core skill every US bookkeeper and accountant relies on to keep the books accurate and decision-ready. This lesson covers the mechanics, a clear worked example, and the common mistakes that trip up beginners.
Apply the concepts of present value, future value, and annuities to a real US-market scenario, using the downloadable template attached.
Time Value of Money: Why a Dollar Today Is Worth More Than a Dollar 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 installments.
- 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.
Multi-sheet workbook · pre-built formulas · plug in your own numbers · yours to keep.
Disclaimer. This lesson is educational. The worked example uses representative numbers and does not constitute personalized financial, tax, legal, or investment advice. Trading and investing involve risk including loss of principal. US tax rates change annually — verify against the current IRS publications and your specific state’s tax code. See our Financial Disclaimer for the full statement.
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