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

The 0% Floor Problem

A Fixed Indexed Annuity or IUL with a 0% floor produces a fundamentally different return distribution than an S&P 500 index fund. Consider this simulation:

Year 1: S&P +15% → IUL earns min(15%, cap=12%) = 12% | S&P fund earns 15%
Year 2: S&P -25% → IUL earns 0% | S&P fund loses 25%
Year 3: S&P +18% → IUL earns 12% | S&P fund earns 18%

After 3 years on $100,000:

• IUL: $100,000 × 1.12 × 1.00 × 1.12 = $125,440
• S&P fund: $100,000 × 1.15 × 0.75 × 1.18 = $101,903

The floor's protection in negative years outweighs the cap's limitation in positive years over long time horizons with typical volatility.

The Sequence of Returns Problem (Mathematically)

For an accumulation portfolio, order of returns doesn't matter — only average matters.
For a distribution portfolio (retirement), order matters critically.

Withdrawing $40K/year from a $1M portfolio:

• If returns are [-30%, +30%, +8%, +8%...]: portfolio may be depleted by year 15
• If returns are [+8%, +8%, +30%, -30%...]: portfolio may last 30+ years

Guaranteed income sources eliminate sequence risk by removing the distribution dependency on market performance.

Contact

Jessica Arnwine, CEO, CAER Financial Group
Jessica@caergroup.com | 502-677-0176 | caergroup.com
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