The Price of Waiting

Three short lessons for the parent of a teenager with a first account: why growth over time is bigger than it looks, the rule of 72, and what a small yearly fee costs over thirty years. Every rate here is an assumption, not a promise.

Spark & Whetstone · sparkwhetstone.org/learn/the-price-of-waiting · CC BY-NC-SA 4.0. Answers are at the end, so you can try each question first.

Lesson 1. Bigger than it looks

Suppose 1,000 sits untouched for 30 years and, as an assumption, grows 6% a year. Most people picture 60 a year added thirty times: 1,000 + 30 × 60 = 2,800. That is simple growth, where the growth never grows.

Compound growth adds 6% of whatever is there now, so each year’s step is a little bigger than the last: 1,000 × 1.06, thirty times over, is about 5,743, roughly twice the simple guess.

Economists call the habit of guessing low the exponential growth bias (Stango and Zinman, 2009), and in later studies a simple graph did not cure it. Working the numbers out, as here, is what helps. On Spark & Anvil, MarketQuest’s Compound Growth asks children to guess first, then shows the curve.

Question. A new example: at an assumed 7% a year, about what does 1,000 grow to in 20 years if the growth compounds?

About 14,000
About 3,900
About 2,400

Say it back. In your own words: why does the simple guess come out too low?

Lesson 2. The rule of 72

A quick way to feel compound growth: divide 72 by the yearly percentage to get the years it takes to double. At an assumed 6%, 72 ÷ 6 = 12 years; the exact answer is 11.9.

Doubling is why waiting costs. At the same assumed 6%, thirty years multiplies a sum by about 5.7, and forty years by about 10.3. The ten extra years at the start nearly double the end, because they come before every other year’s growth.

The rule is a rule of thumb: it is close between about 4% and 12% and drifts outside that. Teenagers meet it as The Rule of 72 on Spark & Anvil, and younger children meet doubling by folding paper.

Question. At an assumed 8% a year, about how many years does it take to double?

About 9 years
About 8 years
About 64 years

Say it back. In your own words: what does the rule of 72 tell you, and why does starting ten years earlier matter so much?

Lesson 3. A small fee, a long time

A yearly fee is taken as a share, so it compounds too. Take the same 1,000 and an assumed 6% a year, but a 1% yearly fee, so it grows 5% a year instead.

After 30 years that is 1,000 × 1.05³⁰ ≈ 4,322, against 5,743 with no fee. One per cent a year has taken about a quarter of the end amount (1 − 4,322 ÷ 5,743 ≈ 0.25). A half-per-cent fee would take about 13%.

This lesson does not say which account to choose; it shows how to read a fee. On Spark & Anvil, teenagers work the same sum in The Fee That Grows.

Question. A new example: at an assumed 6% a year, about what share of the 30-year amount would a 2% yearly fee take?

About 44%
About 60%, two per cent for each year
About 2%

Say it back. In your own words: why does a one per cent fee cost far more than one per cent over thirty years?

At the table: Doubling slips

Players: Two or more, ages 11 and up. You'll need: Paper slips, a pen, and one calculator to share.

Write growth rates on slips: 2%, 3%, 4%, 6%, 8%, 9% and 12%. Say out loud that each one is only an assumption.
Draw a slip. Everyone guesses how many years it takes to double, then works out 72 ÷ the rate.
Check on the calculator: start at 1 and multiply by 1.0 and the rate (1.06 for 6%) until it passes 2, counting the presses.
Talk about which guesses were low, and why. Nobody wins; the slips go back in.

Answers

About 3,900 — 1,000 × 1.07²⁰ ≈ 3,870. Simple growth gives 2,400, the low guess the bias leads to.
About 9 years — 72 ÷ 8 = 9 years. The exact figure is 9.0, so the rule is very close here.
About 44% — 1 − 1.04³⁰ ÷ 1.06³⁰ ≈ 0.44: the fee compounds, so two per cent a year takes almost half.