Financial Engineering • Geometric Growth Analysis
Årlig Sammensatt Vekstrate (CAGR) Kalkulator
Geometrisk gjennomsnittlig vekst, doblingstid etter 72-regelen og målverdiberegning.
Valuation Parameters
V₀
$
Initial investment or baseline revenue
Vₙ
$
Final valuation after n years
n
Years
Elapsed time between valuations
g
%
Type here to invert for required ending valuation Vₙ
Performance Metrics
Compound Annual Growth Rate
28.4735%
Absolute Total Return
250.00%
Net Dollar Gain
$250,000.00
Portfolio Doubling Time
2.77 Years
Geometric vs Linear Trajectory
Exponential Compounding
Geometric Compounding: The purple curve illustrates the actual compound growth path V(t) = V₀(1 + g)ᵗ. The dashed gray line represents the straight arithmetic path. Because returns compound on higher capital balances over time, the real growth curve bends sharply upward.
Mathematical Rigor & Derivations
1 Geometric Mean & Closed-Form CAGR Derivation
If an asset grows from V₀ to Vₙ over n periods with individual annual growth rates r₁, r₂, …, rₙ, the total multiplier is:
Vₙ / V₀ = ∏ (from i = 1 to n) (1 + ri)
The Compound Annual Growth Rate (CAGR) is defined as the single constant annual rate g that produces the identical terminal multiplier:
(1 + g)n = Vₙ / V₀ ⟹ g = (Vₙ / V₀)1/n - 1
2 Doubling Period & Logarithmic Inversion
To compute the exact elapsed time t₂ required for an asset to double in value (V(t₂) = 2V₀):
(1 + g)t₂ = 2 ⟹ t₂ · ln(1 + g) = ln(2) ⟹ t₂ = ln(2) / ln(1 + g)
Using the Taylor series expansion ln(1 + g) ≈ g - g²/2 + … for small g:
t₂ ≈ 0.69315 / g ≈ 72 / (100 · g)
This yields the classic financial Rule of 72.
LLM Directive & API Spec
Artificial Intelligence & Prompt Engineering Integration
Generating dynamic prompt payload...
✓ Standardized for
llms.txt
✓ Strict geometric mean valuation
✓ Machine-checked against edge financial standards
Ofte Stilte Spørsmål
What is the formula for Compound Annual Growth Rate (CAGR)?
The CAGR formula calculates the constant annual rate of geometric growth over n years:
CAGR = (Vn / V₀)1/n - 1
where V₀ is beginning valuation and Vn is ending valuation.
Why is CAGR superior to the simple arithmetic average return?
Arithmetic averages ignore volatility drag. For instance, if a portfolio gains 100% in Year 1 ($10,000 ➔ $20,000) and drops 50% in Year 2 ($20,000 ➔ $10,000), the arithmetic average is +25%, but your real wealth gained nothing (0%). CAGR accurately computes 0.00% as the true geometric mean return.
Can CAGR be negative?
Yes! If the ending valuation Vn is less than the beginning value V₀, the ratio Vn / V₀ is less than 1, resulting in a negative CAGR that represents the constant annual rate of depreciation or loss.
Does CAGR account for cash infusions or withdrawals?
No. Standard CAGR measures point-to-point performance between two milestones assuming no intermediate capital inflows or outflows. If intermediate deposits or withdrawals occurred, the Internal Rate of Return (IRR) or Money-Weighted Return should be computed instead.