Present value
PV = sum_t CF_t DF(0,t)Each payment is translated to today's value with the discount factor for its own date.
Foundations
Discounting and Present Value
Present value converts future cash flows into today’s value using time, risk and a discount rate.
Mental model
A future cash flow is worth less today when time, risk, and the discount rate separate payment from valuation.
A cash flow farther away is exposed to more time and uncertainty, so the same nominal amount is usually worth less today.
For cash flows CFₜ and discount factors DFₜ, present value is PV = Σ CFₜ × DFₜ.
Explore the mechanism
How does the timing and discount rate of each cash flow change its value today?
A fixed stream of future cash flows can be viewed in two ways: their nominal amounts at payment dates, and their discounted contributions to present value at time 0. The discount rate compresses or stretches these contributions along the timeline.
Numbers
Sum of nominal CFs = 120
Present value at r ≈ 107.43
Interpretation
Increasing the discount rate shrinks the green bars more strongly at later times: far cash flows count less in today’s value. The present value is the sum of these discounted contributions, not the sum of nominal amounts.
In practice, each maturity has its own discount factor from the rate curve; here we use a single flat r for clarity. Mis-calibrated discount factors move the whole term structure of PVs.
UNDERSTAND
What to notice
A higher discount rate lowers the value of distant cash flows more than near-term cash flows.
Discounting helps compare a long commitment with the value of waiting, adapting or investing elsewhere.
The quantitative structure
Notation
CF_t- cash flow paid at time t
DF(0,t)- discount factor from time t to today
A small example
Two payments at different dates
A contract pays 100 in one year and 100 in two years. Use annual discount factors 0.95 and 0.90.
- Discount the first payment: 100 * 0.95 = 95.
- Discount the second payment: 100 * 0.90 = 90.
- Add the discounted values: 95 + 90 = 185.
ResultThe present value is 185, even though the nominal payments total 200.
Where it appears
- valuing bonds and projects
- discounting derivative payoffs
- comparing contracts with different payment dates
Where the intuition breaks
What this view leaves out
- A present value is only as defensible as the selected discount curve and its treatment of risk, liquidity, and optionality.
Common mistakes
“One discount rate is appropriate for every future cash flow.”
Not quite.Cash flows with different currencies, collateral, credit, or risk may require different discount curves or adjustments.
Where this concept sits
Follow the strongest prerequisites, applications, extensions, and contrasts.
Connections for Discounting and Present Value
Discounting and Present ValueYOU ARE HERE
Foundation ofDuration and Convexity (Intuition)Duration and convexity describe how discounted fixed-income values respond to changes in rates.
Foundation ofForwards and Futures Pricing (Cost of Carry)Forward values compare spot and future cash flows on a common present-value basis.
Used byBlack-76 Pricing for Forwards and FuturesBlack-76 discounts an option payoff written on a forward or futures price.
Supported byCurve Conventions and Day-CountDay-count and compounding conventions determine how quoted rates are converted into discount factors and present values.
Supported byYield Curve BootstrappingA bootstrapped discount curve supplies the discount factors used to turn future cash flows into present value.
Used byBlack–Scholes Option PricingBlack–Scholes uses discounting for the strike and risk-free growth in its valuation formula.
Foundation of · Duration and Convexity (Intuition)
Duration and convexity describe how discounted fixed-income values respond to changes in rates.
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