Present Value
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Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| $PV$ | Present Value | currency | Current worth of a future sum of money. |
| $FV$ | Future Value | currency | Amount of money at a future point in time. |
| $r$ | Discount rate | decimal per period | The rate reflecting time value of money (opportunity cost). |
| $n$ | Number of periods | periods | Number of compounding periods between now and the future date. |
What Is Present Value?
Present Value (PV) answers the question: what is a future amount of money worth in today's terms? Money received in the future is worth less than the same amount received today, because today's money can be invested to grow.
$$PV = \frac{FV}{(1 + r)^n}$$
This is the inverse of the Future Value (compound interest) formula.
The Time Value of Money
The core principle: a dollar today is worth more than a dollar tomorrow, for three reasons: 1. Investment opportunity: Today's dollar can earn returns 2. Inflation: Future dollars have less purchasing power 3. Risk: Future payments may not materialise
The discount rate r captures all three factors — it represents the minimum return required to justify deferring receipt of money.
Discount Rate Selection
Choosing the right discount rate is the most important (and subjective) step:
| Context | Typical discount rate |
|---|---|
| Risk-free government bond | 3–5% |
| Corporate project (WACC) | 8–12% |
| Startup investment | 20–40% |
| Personal financial planning | Inflation rate (2–4%) |
Higher discount rates make future cash flows worth less today — a key reason why long-duration assets fall in value when interest rates rise.
PV in Investment Decisions
If an investment promises to pay $50,000 in 5 years, and your discount rate is 8%:
PV = 50,000 / (1.08)⁵ = 50,000 / 1.4693 = $34,029
You should be willing to pay at most $34,029 today for that future $50,000. If the investment costs more than $34,029, reject it; if less, accept it.
Net Present Value (NPV)
NPV extends PV to multi-period cash flows:
$$NPV = \sum_{t=1}^{n} \frac{CF_t}{(1+r)^t} - \text{Initial Investment}$$
NPV > 0 means the investment creates value; NPV < 0 means it destroys value.
Derivation & History
Present Value is simply the inverse of the compound interest formula. Since FV = PV × (1+r)^n, solving for PV gives PV = FV / (1+r)^n.
The concept of time value of money was formalised in the 16th century by Italian and Dutch merchant bankers who needed to compare cash flows occurring at different times. Leonardo Fibonacci's Liber Abaci (1202) contains early present-value-type calculations for trade finance.
In modern corporate finance, Irving Fisher's The Theory of Interest (1930) provided the rigorous theoretical foundation for discounted cash flow (DCF) analysis, the primary valuation framework used in investment banking today.
Worked Examples
Lump sum in 10 years
- PV = 100,000 / (1.05)^10
- PV = 100,000 / 1.6289
- PV = $61,391
Result: PV = $61,391 — you should pay at most $61,391 today for $100,000 in 10 years at 5% discount
Comparing payment options
- PV of Option B = 25,000 / (1.07)^3 = 25,000 / 1.2250 = $20,408
- Option B PV ($20,408) > Option A ($20,000)
Result: Option B is worth more in present-value terms — choose it
Edge Cases & Limitations
r = 0: PV equals FV — no time value of money (or zero-interest world).
Negative discount rates: Rare but possible in deflationary environments; FV is worth more than PV (future money is more valuable).
Very long time horizons: Even small discount rates make distant future cash flows nearly worthless; a payment in 100 years at 5% discount has PV ≈ 0.76% of its face value.
Inflation vs real rate: Use nominal cash flows with nominal discount rate, or real (inflation-adjusted) cash flows with real discount rate — not a mix.
Real-World Applications
Investment bankers use PV and NPV to value companies (discounted cash flow analysis). Insurance actuaries calculate present value of future liabilities to determine reserves. Pension funds use PV to assess whether assets cover future obligations. Courts use PV to calculate damages for future lost earnings. Real estate investors use PV to compare rental income streams to purchase prices. Central banks consider PV in quantitative easing decisions.