Power Law

Most people think about money the way they think about arithmetic — linearly. Save $100 a month for 10 years and you have $12,000. But money does not follow arithmetic. Left to compound, it follows an exponential curve, and exponential curves are almost unimaginably steep once enough time passes.

The underlying principle is a power law: repeated multiplication by a constant factor. The concept appears everywhere in nature — earthquake magnitudes, city populations, the frequency of words in a language. In finance, it takes a form you can personally exploit: compound interest.

#1 % a Day, Every Day

Start with the simplest possible question: what happens if you improve by 1 % each day for a full year?

If growth were linear, 1 % a day over 365 days would give you a 365 % return — you would end up with $1 + 0.01 \times 365 = 4.65$ times what you started with. Decent, but nothing special.

Compounding works differently. Each day you multiply, not add:

Starting from $1$, you end with nearly $38 — almost ten times what the linear version promises. Now run the arithmetic in reverse: what if you slip back by 1 % every day?

You are left with less than 3 cents on the dollar. The gap between 1 % better and 1 % worse is not 2 percentage points — it is the difference between 37.78× and 0.026×, a ratio of nearly 1 500 to 1.

Self-improvement and portfolio returns are not literally 1 % a day. But the intuition transfers directly: small, consistent gains, reinvested, compound into a curve that logic alone cannot picture.

#The Formula

The engine behind all of this is one equation:

const A = P * Math.pow(1 + r / n, n * t)
// P = 10_000, r = 0.07, n = 12, t = 10
// A ≈ 20_097

The exponent $nt$ is what makes this equation violent. It grows with time — meaning the longer you wait, the faster the curve steepens. At year 1 the difference between investing and not investing is small. At year 30 it is a completely different life.

#Regular Contributions: The Multiplier

The formula above assumes a single lump sum. In reality, most people invest a fixed amount each month. A regular contribution is modelled as a future value of an annuity:

where $r_m = r/12$ is the monthly rate and $C$ is the monthly contribution.

const rm = r / 12
// future value of each monthly contribution
const fvContribs = C * ((Math.pow(1 + rm, n) - 1) / rm)

const total = P * Math.pow(1 + rm, n) + fvContribs

Two insights follow from this formula:

1. Start now, not when you earn more. Because of the exponent $n$, years at the beginning of your timeline are worth more than years at the end. Every year you delay is a year lost at the steepest part of someone else's curve.
2. Rate matters more than you think. Moving from 6 % to 8 % annual return does not grow your terminal balance by 33 %. Across 30 years it nearly doubles the gap between your final total and your contributions — the extra 2 % is compounded 360 times.

#Interest on Interest

The reason compounding feels counterintuitive is that you earn interest on your interest. In the first year, $10,000 at 7 % earns $700. In the second year, you earn 7 % on $10,700 — an extra $749. By year 20 you are earning almost $2,500 per year on a $10,000 original investment, without touching the principal.

#Time, Not Timing

A common mistake is waiting for the "right moment" to invest. But power law curves do not reward timing — they reward duration. Consider two investors:

• Early Ellie invests $5,000 a year from age 22 to 32 (10 years, $50,000 total) at 8 %, then stops.
• Late Louis invests $5,000 a year from age 32 to 62 (30 years, $150,000 total) at the same rate.

At 62, Ellie has roughly $602,000. Louis, who contributed three times as much for three times as long, has roughly $566,000 — still less. Ellie's decade of head start, compounded over 40 years, overpowers Louis's extra $100,000 and 20 extra years of contributions.

function ellieWealth() {
  let a = 0
  for (let y = 0; y < 10; y++) a = (a + 5000) * 1.08
  for (let y = 0; y < 30; y++) a = a * 1.08
  return a  // ≈ 602,000
}

function louisWealth() {
  let a = 0
  for (let y = 0; y < 30; y++) a = (a + 5000) * 1.08
  return a  // ≈ 566,000
}

The mathematics are unambiguous: the best time to invest was yesterday. The second-best time is today.

#The Three Levers

Every compounding model has exactly three inputs you can control:

Principal is the least powerful. A $100,000 windfall is wonderful, but it is a single multiplication. Rate matters more because it is applied repeatedly. Time dominates everything because it is the exponent's exponent — each additional year does not add a fixed increment; it multiplies everything that came before.

Compound interest is often described as the eighth wonder of the world. What it actually is, more precisely, is a power law applied to money — patient, indifferent, and brutally consistent. The curve does not care how smart you are or how hard you work in any given year. It cares only about one thing: how long you let it run.