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Advanced Compound Interest: Mathematical Origins & Research

Before reading this analysis, it is advised to read the Regular Investing analysis, as it develops the basis behind many of our foundational formulas represented here.


We know that any polynomial can be represented as a finite, descending sum via Polynomial Reduction Theory:


f (x) = L * Σ { xᵃ(a!) / xⁿ(a-n)! }, where the summation is bounded by {n =0, a}


As an example, let's display f (x) = 3x³ + 12x² + 14 as a finite sum:


f (x) = L * Σ { x³(3!) / xⁿ(3-n)! }, where the summation is bounded by {n =0, 3}


= L * Σ { 6x³ / xⁿ(3-n)! }


= 6x³ L * Σ { 1 / xⁿ(3-n)! }


= 6x³ L * (⅙ + 1/(2x) + 1/x² + 1/x³)


= x³ L + 3x²L + 6xL + 6L


(3x³ + 12x² + 14) / (x³ + x² + 6x + 6) = L


Essentially, we're drawing out the constants of the equation so that only the base values remain, so that when we take a derivative, the base values (the variables) are the only things that change. This makes it extremely easy to do otherwise-impossible calculations in the span of minutes.


Let's simplify L down.


(3x³ + 12x² + 14) / (x³ + x² + 6x + 6) = L


L = 5x + (5/2)x² + ½x³ + 12/x + 14/x³ + 14/x² + 7/(3x) + 52/3


From the regular investing analysis, we also learned that investment interest could be modeled (via compound interest) as


I = Prᵗ + Mr ᵗ ( r ᵗ-1 ) / ( r -1 )


T will always end up as a constant, no matter how the formula is manipulated. Therefore, the compound interest itself can be viewed as a polynomial: all we need to do is apply PRT to it, so we can find the optimal strategy for our situation.


f(x) = Prᵗ + Mr ᵗ ( r ᵗ-1 ) / ( r -1 )


Let a = t


L = [ Pr ᵗ + Mr ᵗ ( r ᵗ-1 ) / ( r -1 ) ] / [ r ᵗ + tr ᵗ⁻¹ + (t) (t-1) r ᵗ⁻² + ... + (r! - (r-n)! ) r ᵗ/rⁿ