# Compounding with a monthly amount: time beats the rate > How a fixed monthly amount grows with compounding, why starting ten years earlier beats a higher rate, and what a three-year pause costs. - Course: [Long-Term Investing: Index Funds, Risk and Your Own Behavior](https://productovi.com/courses/long-term-investing.md) - Module: What can an investing app honestly promise? - Lesson 2/32, ~22 min - Source: https://productovi.com/courses/long-term-investing/compounding-monthly ## The goal screen **Grace:** Still February at Zapas, and today's design review is the "goal + monthly amount" screen. A user types a goal and a monthly sum, and we show how the balance could grow. **Ethan:** My idea: a big slider for the return. Users drag it to 10% and see a huge number. People love that. **Grace:** The slider moves the wrong lever. Users can't choose the market's return. They can choose when they start, how much they put in and whether they keep going. Let's see which of those matters most. **Grace:** **Compounding** means growth on growth: this year's gains earn their own gains next year. With a monthly amount, every payment starts its own small snowball, and the early ones roll the longest. $$ FV = P \times \frac{(1 + m)^{n} - 1}{m}, \quad m = (1 + a)^{1/12} - 1 $$ *Future value of P paid at the end of every month for n months at a yearly rate a* **Sofia:** So 100 dol. a month at 6% a year from 25 to 65 is 480 payments. **Grace:** Yes, and in real terms, so the result is in today's dollars. It ends at 190,768. Only 48,000 of that was paid in. **100 dol. a month at 6% real a year** | Age | Start at 25 | Start at 25, pause at 40–42 | Start at 35 | Paid in from 25 | | --- | --- | --- | --- | --- | | 25 | 0 | 0 | 0 | 0 | | 30 | 6949 | 6949 | 0 | 6000 | | 35 | 16247 | 16247 | 0 | 12000 | | 40 | 28691 | 28691 | 6949 | 18000 | | 45 | 45344 | 40935 | 16247 | 24000 | | 50 | 67629 | 61728 | 28691 | 30000 | | 55 | 97451 | 89555 | 45344 | 36000 | | 60 | 137360 | 126793 | 67629 | 42000 | | 65 | 190768 | 176627 | 97451 | 48000 | *An illustrative steady 6% real rate, not a forecast; real markets swing (next lesson).* > **Exercise:** 100 dol. a month at 6% real a year from 25 to 65 ends at 190,768, of which 48,000 was paid in. What share of the final balance is growth, in percent, to one decimal? ## Ten years later **Ethan:** Most of our beta users are 30-something. Does starting late really hurt that much? **Grace:** Look at the start-at-35 line. Same 100 dol., same 6%, 30 years instead of 40. It pays in 36,000, a quarter less, and ends at 97,451: about half. ![Starting at 25 pays 48,000 and ends at 190,768; starting at 35 pays 36,000 and ends at 97,451, about half.](https://productovi.com/courses/long-term-investing/diagrams/m1-l2-time-beats-rate.png) *A quarter less paid in, about half the result.* **Sofia:** Because the ten years we skip are the ones that would have compounded longest. > **Exercise:** At 100 dol. a month and 6% real, starting at 25 ends at 190,768 and starting at 35 ends at 97,451 by age 65. How many dollars does waiting 10 years cost, to the nearest dollar? ## Rate or years? **Ethan:** Still, a better rate must win. Double the rate and you double the money, right? **Grace:** Let's test it. Base: 100 dol. a month, 20 years at 4% gives 36,384. Doubling the rate to 8% gives 56,900. Doubling the years to 40 at 4% gives 116,106. **Sofia:** Doubling the years also doubles what's paid in, though. **Grace:** Fair: 48,000 instead of 24,000. Even so, 116,106 is more than twice 56,900. And unlike the rate, the years are something a user really controls. > **Exercise:** Base plan: 100 dol. a month for 20 years at 4% a year, 36,384 at the end. Which change gives the bigger balance? ## The pause **Grace:** One more line. A user starts at 25, then stops paying at 40–42, say after losing a job, but leaves the money invested. At 65: 176,627. **Ethan:** So skipping 3,600 of payments costs 14,141. That's a feature: "pause, don't sell". **Grace:** Exactly the right instinct. The pause costs something, but the balance keeps compounding. Selling everything to cover those three years would stop the whole snowball, and many people never come back. **Sofia:** And the emergency cushion is what makes the pause possible in the first place. > **Exercise:** Starting at 25 with 100 dol. a month at 6% real ends at 190,768. The same plan with no payments at ages 40–42 (money stays invested) ends at 176,627. How much does the pause cost by 65, in dollars? > **Exercise:** Order these 100-dol.-a-month plans by the balance at 65, from largest to smallest (6% real). > **Exercise:** Starting at 35 instead of 25 means paying a quarter less, so the balance at 65 is also about a quarter smaller. **Ethan:** Fine, the slider goes. But the screen still needs words that make people act, not a maths lecture. **Grace:** Agreed. One comparison users can feel, one action for bad months, and one honest line about risk. So what do we put on the screen instead of the slider? > **Exercise:** Write the 2–3 sentence helper text for the "goal + monthly amount" screen. It should nudge users to start early and keep going, use one number from this lesson, and not promise a return. > **Key ideas** > > - **Compounding**: growth earns growth, so early payments do most of the work. > - 100 dol. a month at 6% real: from 25 → 190,768 (48,000 paid, 74.8% growth); from 35 → 97,451. > - Doubling the years beats doubling the rate: 116,106 vs 56,900 from a 36,384 base. > - A 3-year pause costs 14,141 by 65, far less than a late start; selling is worse than pausing. > - The rate is the market's; start date, amount and persistence are the user's. > **Educational, not advice** > > This course explains how long-term investing works; it never tells you what to buy. Investments can fall, and you can get back less than you put in. Check the current rules where you live or ask a licensed professional. **Grace:** Next: real markets don't pay a steady 6%. We'll replace the single line with thirty years of possible paths. - Previous lesson: [Saving, investing, speculating and the real return](https://productovi.com/courses/long-term-investing/saving-investing-speculating.md) - Next lesson: [A range, not a number: twenty-five years of possible paths](https://productovi.com/courses/long-term-investing/range-not-a-number.md) --- Productovi Learning. Free, no sign-up needed to read.