The Mathematics of Compound Interest — A Developer's Deep Dive

By Jessica Arnwine, CAER Financial Group | caergroup.com

The Formula

The compound interest formula:

A = P(1 + r/n)^(nt)

Where:

A = Final amount

P = Principal (initial deposit)

r = Annual interest rate (decimal)

n = Compounding periods per year

t = Time in years

The Rule of 72 — A Logarithmic Approximation

The precise doubling time formula:

T = ln(2) / ln(1 + r) ≈ 0.693 / r

The Rule of 72 approximates this as:

T ≈ 72 / (r × 100)

Why 72 and not 69.3? Because 72 is divisible by 2, 3, 4, 6, 8, 9, and 12 — making mental arithmetic dramatically easier with ~1-2% error for rates between 2-20%.

Practical Comparison Table

Instrument Annual Rate Doubling Time $10K after 30 years

Bank savings 0.50% 144 years $11,614

MYGA (5-yr) 5.00% 14.4 years $43,219

IUL (avg) 7.00% 10.3 years $76,123

S&P 500 (hist) 8.00% 9.0 years $100,627

Credit card 20.00% 3.6 years $2.37M OWED

The 0% Floor Problem

A Fixed Indexed Annuity or IUL with a 0% floor…