Authoritative Content & Methodology
Data Source and Methodology
Authoritative sources:
- Bodie, Z., Kane, A., Marcus, A. J. (2021). Investments, 12th ed. McGraw‑Hill. Chapter on Time Value of Money. Publisher link
- U.S. SEC – Investor.gov: Compound Interest and Annuities methodology. Investor.gov calculators
Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte.
The Formula Explained
Lump sum (discrete compounding):
PV = \( \dfrac{\text{FV}}{\left(1+\dfrac{r}{m}\right)^{m t}} \)
Lump sum (continuous compounding):
PV = \( \text{FV}\, e^{-r t} \)
Level annuity (ordinary):
PV = \( P \times \dfrac{1-(1+\frac{r}{m})^{-m t}}{\frac{r}{m}} \)
Annuity due:
PV = \( P \times \dfrac{1-(1+\frac{r}{m})^{-m t}}{\frac{r}{m}} \times (1+\frac{r}{m}) \)
Inflation adjustment (Fisher):
\( r_{\text{real}} = \dfrac{1+r}{1+i} - 1 \)
Where FV = future value, P = payment each period, r = annual nominal rate, m = compounding periods per year, t = years, i = inflation rate.
Glossary of Variables
- Present Value (PV): The amount today equivalent to a future cash flow or series.
- Future Value (FV): The amount received at a future date.
- P (Payment): Fixed cash flow per period in an annuity.
- r (Annual discount rate): Required return/interest rate per year (decimal form).
- m (Compounding frequency): Number of compounding periods per year (1, 2, 4, 12, 365 or continuous).
- t (Time): Number of years until payment(s).
- Discount factor: PV/FV for a lump sum, or the annuity present value factor for series.
- r_real: Real discount rate after accounting for inflation i.
How It Works: A Step-by-Step Example
Scenario: You expect $10,000 in 5 years. Your annual required return is 5% with monthly compounding.
- Inputs: FV = 10,000; r = 5% = 0.05; m = 12; t = 5.
- Compute periodic rate: \( r_p = r/m = 0.05/12 \).
- Compute total periods: \( n = m \times t = 12 \times 5 = 60 \).
- Compute PV: \( \text{PV} = \dfrac{10000}{(1 + 0.05/12)^{60}} \approx 7835.26 \).
Interpretation: Investing about $7,835.26 today at 5% compounded monthly should grow to $10,000 in 5 years.
Frequently Asked Questions (FAQ)
What discount rate should I use?
Use a rate reflecting your opportunity cost or the risk profile of the cash flow (e.g., your portfolio’s expected return, a WACC for projects, or a risk-free rate plus premium).
Does continuous compounding change the formula?
Yes. Lump-sum PV becomes \( \text{PV} = \text{FV} \, e^{-rt} \), where e is Euler’s number.
How is an annuity due handled?
An annuity due pays at the start of each period; multiply the ordinary annuity PV by \( (1 + r/m) \).
What if the rate is zero?
With r = 0, PV equals the undiscounted sum. Annuity PV is simply payment × number of periods.
Can I enter fractional years?
Yes. The calculator supports decimals for years to represent partial periods accurately.
Are taxes and fees included?
No. Results are pre-tax and exclude fees. Adjust your discount rate to reflect costs if needed.
Formula (LaTeX) + variables + units
Lump sum (discrete compounding): PV = \( \dfrac{\text{FV}}{\left(1+\dfrac{r}{m}\right)^{m t}} \) Lump sum (continuous compounding): PV = \( \text{FV}\, e^{-r t} \) Level annuity (ordinary): PV = \( P \times \dfrac{1-(1+\frac{r}{m})^{-m t}}{\frac{r}{m}} \) Annuity due: PV = \( P \times \dfrac{1-(1+\frac{r}{m})^{-m t}}{\frac{r}{m}} \times (1+\frac{r}{m}) \) Inflation adjustment (Fisher): \( r_{\text{real}} = \dfrac{1+r}{1+i} - 1 \)
- T = property tax (annual or monthly depending on input) (currency)
- Investor.gov calculators — investor.gov · Accessed 2026-01-19
https://www.investor.gov/financial-tools-calculators - Finance — calcdomain.com · Accessed 2026-01-19
https://calcdomain.com/finance - Investment — calcdomain.com · Accessed 2026-01-19
https://calcdomain.com/subcategories/finance-investment - Publisher link — mheducation.com · Accessed 2026-01-19
https://www.mheducation.com/highered/product/investments-bodie-kane/M9781260013832.html
Last code update: 2026-01-19
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