Formula & Calculator
Present Value of a Single Sum
Calculates how much a future lump sum of money is worth today, given a discount rate and time period.
Interpretation
PV = FV / (1+r)^n. Value today of a future amount. Discounts future cash flows to account for time value of money. Used in investment appraisal and valuation.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| PV | Present value | currency |
| FV | Future value | currency |
| r | Discount rate per period | |
| n | Number of periods |
What it means
Present value (PV) is the current worth of a future sum of money or stream of cash flows, discounted at a given rate (r) over a period (n). It is the inverse of compounding. The formula PV = FV / (1+r)^n is fundamental in finance for valuing bonds, stocks, and capital projects. It also applies to personal finance (e.g., retirement planning). Understanding PV is essential for comparing cash flows occurring at different times, as it enables decisions based on economic equivalence. It is a core concept in discounted cash flow (DCF) analysis.
Worked example
Present Value of a Single Sum – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| FV | 10000 |
| r | 0.05 |
| n | 10 |
| Parameter | Value |
|---|---|
| FV | 50000 |
| r | 0.07 |
| n | 5 |
Common mistakes
- Discount rate r: In decimal form – must match the period of n.
- Number of periods n: The time until the future value is received.
- PV: The present value – always less than the future value if r > 0.
- Sign: PV is the amount needed today to achieve FV in the future.
- Continuous compounding: For continuous, use PV = FV·e^(−rt).
Applications
The present value of a single future sum, PV = FV/(1+r)^n, determines how much a future cash flow is worth today, given a discount rate. This is the core of the time value of money and is used in all areas of finance. Investors use it to value bonds, stocks, and real estate. Corporate finance uses it to evaluate projects and to set purchase prices. By discounting future cash flows, professionals can compare investments with different timing and risk profiles. The present value concept underlies net present value, internal rate of return, and many other financial metrics. Mastering this formula is essential for financial analysis, investment banking, and strategic planning.
- Valuation of fixed‑income securities and bonds
- Equity valuation – discounted cash flow (DCF) analysis
- Capital budgeting – project evaluation
- Real estate investment analysis
- Retirement and savings goal planning
Frequently Asked Questions
PV = FV / (1+r)^n. It calculates the current value of a lump sum to be received in the future, discounted at a given rate over a number of periods.
Because of the time value of money – a dollar today can be invested to earn interest, so it is worth more than a dollar received later. Discounting adjusts for this opportunity cost.
A higher discount rate reduces the present value (since the future cash flow is discounted more heavily). Conversely, a lower discount rate increases the present value. The choice of rate is crucial.
The discount rate reflects the risk and the opportunity cost of capital. For a risk‑free future cash flow, the risk‑free rate is used; for riskier cash flows, a higher rate is applied.
Adjust r and n to match the compounding period. For example, if compounding is monthly, use monthly rate and number of months: PV = FV / (1 + r/month)^(months).
PV is the value of a single future sum. NPV is the sum of PVs of multiple cash flows minus the initial investment, used for project evaluation.
The price of a bond is the present value of its future coupon payments and its face value at maturity, discounted at the market yield (required return).
Present value itself is a dollar amount. A positive PV means the future cash flow, when discounted, is worth a positive amount today. In project evaluation, a positive NPV (sum of PVs minus cost) indicates value creation.
The longer the time until the cash flow (larger n), the lower its present value. This is because there is more time for the discounting to reduce the value.
They are inverses: FV = PV × (1+r)^n and PV = FV / (1+r)^n. They represent the same amount at different points in time.