# No Magic in the Power of Compound Interest for Personal Finance, etc.

The U.S. stock market produces about a 7% inflation-adjusted annual return, on average. Consequently, the return on an investment in the whole market (or even an index fund of just the S&P 500) over a number of years can be approximated by the equation FinalValue = Principal × 1.07^Years. This fact is frequently used to show the impressive growth in an investment over time. For example, if a 25-year-old invests \$50,000 in Vanguard’s VTSAX index fund, s/he would have about \$50,000 × 1.07^40 after 40 years (Age 65) = \$50,000 × 14.97 = \$748,723. The same amount invested at Age 18 would yield a higher multiplication factor of 1.07^47 = 24.05, leading to a balance of about \$1,202,285 at Age 65. If, however, one invests the \$50,000 at Age 35, well… 1.07^30 is a measly 7.61, which is only \$380,613.

The stock market actually yields somewhere closer to 11.69% in nominal dollars on average, but the 7% figure is inflation-adjusted and somewhat conservative… Over 40 years, 7% returns yield 14.97× while 11.69% returns yield 83.29×. At first glance this might seem wrong, but only if you don’t notice the multiplicative element of 1.1169 × 1.1169 × 1.1169 × 1.1169 × 1.1169 … et cetera. Consequently, investing in the stock market at 7% annual average yield versus a CD at 1% annual yield is not just seven times better… It’s seven times better on average over a single year. Over multiple years, not only do the benefits of investing in stocks grow far faster than linear addition, but risk also declines. As a short-term investment, stocks are quite risky, but if you are investing for 40 years… there is much less risk. Our \$50,000 invested at Age 25 at a 7% annual yield grows by 15-fold; at a 1% annual yield… well, 1.01^40 = 1.49-fold, which is a pathetic \$74,443 versus a monstrous \$748,723.

So, 7% versus 1% annual yield yields a 10-fold difference over 40 years… but that’s with the principal included. If we deduct the principal of \$50,000, it’s \$24,443 versus \$698,723, which is actually 28.6× better. This is a legitimate mathematical maneuver, because obviously we don’t want the principal muddying the calculation… If it was one year, it would be \$50,500 vs. \$53,500 which is only a 1.06× difference with principal included, but a 7.00× difference if we just look at return on investment (ROI).

However, the power of compound interest too often becomes a demotivating counterfactual for people in middle age… or perhaps even people in their late twenties! The “magic” that happens here is not magical—it’s just exponential growth, or, literally multiplying the same number, the annual yield factor, over and over. That is to say, the compounding (exponential growth) curve remains the same regardless of entry point. At 7% annual yield, you could invest \$50,000 at Age 25 and have \$748,723 at Age 65… or, you could invest \$53,500 at Age 26 and have the same result. If you enter the curve at Age 35, you would need to invest \$50,000 × 1.07^10 = \$98.357.57 to have the same result at Age 65 as investing \$50,000 at Age 25. Of course, this oversimplifies risk, which is lower by investing in stocks for longer periods, but as long as you are over 15 years or so, it probably doesn’t make much difference… albeit, at Age 50 you would need \$271.371.63 to get to \$748,723 at Age 65… but that is still a tremendous return of 2.76× (or, 176%).

Because the exponential growth curve remains the same regardless of entry point, larger investments in middle age can make up for lack of investments in youth. As a corporate shill, one’s income tends to greatly increase in middle age, so it is not “too late” for you to achieve financial freedom in old age, provided you aren’t prevented from investing by the materialistic lifestyle that flushes every dollar you earn down the toilet. (Credit card interest compounds the same way, but against you, and often at 20% instead of 7%.)

Taxation aside, disconnecting exponential growth from your age is useful, because the two don’t actually have anything to do with each other, mathematically speaking. If Alfred Nobel puts \$50,000 in a VTSAX trust fund, in 100 years the account would be at \$50,000 × 1.07^100 = \$50,000 × 867.72 = \$43,385,816… and that’s an inflation-adjusted figure. But, recall that he could also put \$748,723 in for 60 years and still wind up with \$43,385,816. Not magic at all, but just a property of exponents.