Present Value Calculator: Continuous Compounding Factor Tool
Present value with continuous compounding discounts a future dollar by $$e^{-rt}$$. At 7 %, $1 received in 10 years is worth only $0.496 ($1 × e^{-0.7})—a 50 % haircut (CFA Institute, 2020). Use our two-step tool: type the rate, type the years, press Calculate; multiply the displayed factor by any cash-flow amount.
Present Value - Continuous Compounding Factor Calculator
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How to use the tool
Enter the annual rate (%): type, for example, 3.75 or 8.20.
Enter time (years): type 2.5 or 0.75 to include partial years.
Click “Calculate”: the tool shows the discount factor to four decimals.
Find present value: multiply the factor by any future cash flow.
Formula used
The calculator applies the continuous-compounding present-value factor:
$$PV_{CCF}= rac{1}{e^{rt}}$$
r = annual rate (decimal)
t = years
Example A
r = 3.75 % ⇒ 0.0375; t = 2.5
Exponent: 0.0375 × 2.5 = 0.09375
Factor: 1/ e0.09375 = 0.9106
$5 000 × 0.9106 = $4 553
Example B
r = 8.20 % ⇒ 0.082; t = 0.75
Exponent: 0.0615
Factor: 1/ e0.0615 = 0.9403
$1 200 × 0.9403 = $1 128
Quick-Facts
e ≈ 2.71828, defined by $$\lim_{n→∞}(1+1/n)^n$$ (NIST, 2019).
Continuous compounding yields 0.012 % more than daily compounding at 5 % for one year (Investopedia, 2023).
Present value halves when $$t= rac{\ln2}{r}$$—≈ 9.9 years at 7 % (CFA Institute, 2020).
Federal Reserve prime rate ranged 3.25 %–8.5 % from 2010-2023 (FRB, 2023).
FAQ
What is continuous compounding?
Continuous compounding adds interest at every instant, giving a growth factor of $$e^{rt}$$ for principal $1 (Investopedia, 2023).
Why convert future cash flows to present value?
You compare investments on the same date and incorporate the opportunity cost of capital (CFA Institute, 2020).
How do I switch to a future-value factor?
Take the reciprocal: $$FV\;factor = e^{rt}$$, the mirror of the discount factor (Hull, 2022).
Which rate range is realistic for valuation?
Corporate analysts commonly model 2 %–12 % real rates depending on risk (Damodaran, 2023).
Does continuous compounding overstate returns?
No; it represents the limit as periods shorten and is only marginally above daily compounding—≈0.012 % at 5 % (Investopedia, 2023).
Where is the method used in practice?
Option pricing, swap valuation and risk-neutral discounting all embed continuous rates (Black & Scholes, 1973).
Can the rate be negative?
Yes; European government bonds traded near –0.5 % in 2020 (ECB, 2021). The formula still works.
How precise is the constant e?
“For engineering, 15 decimal digits of e ensure micro-unit accuracy” (NIST Handbook 100, 2019).
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