Investing Math Matters
by Charles Cheng, CFA
The math of investing returns is something easily overlooked or misunderstood for the average investor. But even simple differences in calculation or understanding can make all the difference when making investment plans.
One of the fundamental concepts in finance is compound interest. There is an apocryphal quote attributed to Albert Einstein where he jokes that compound interest is the most powerful force in the universe. The significance is that over long periods of time, compounding returns at different rates or times leads to large differences in wealth. It’s hard to get a sense of the magnitude of the differences unless you actually do the calculations.
As seen in the chart below, starting to save 10 years earlier can mean almost double the wealth after 20-30 years despite only putting in 33% more money in total.
Source: Business Insider. Saving $200 per month at 6%.
Source: Business Insider
This effect also dramatically changes the amount needed to be saved each in order to reach a return target. One would need to save almost four times more per month at age 40 vs age 20. Also, even small differences in return can make big differences when compounded over many years. A single lump sum of $10000 will grow to almost $150,000 after 40 years at 7% compounded returns vs around $70,000 at 6%.
The math for calculating return itself is actually not very straightforward. One 6% average return can be worse than another 6% average return. How is this possible? Because average returns can be either arithmetic or geometric averages. For example, if an annual return series is 6% in year one, and -4%, 2%, and 12% in year two to four, the arithmetic mean is (16%-15%+22%-3%)/4 = 5%. The geometric mean is (1.16)(0.85)(1.22)(.97)^(1/4) = 3.9%, a lower return, but one which more accurately represents the annual compounded rate that results in the actual total return of 16.7%.
計算回報本身的數學實際上並不是非常直觀。一個6%的平均回報率有時表現會劣於另一個6%的平均回報率。這是怎麼回事？因為平均回報可以是算術平均值或幾何平均值。例如，如果第一年的年回報率為6％，接下來三年的回報率為-4％，2％和12％，則算術平均值為(16%-15%+22%-3%)/4 = 5%，幾何平均值為(1.16)(0.85)(1.22)(.97)^(1/4) = 3.9%。後者回報較低，但卻是更準確地顯示得到4年總回報16.7%的複利率。
An even more extreme example is as follows. Above are two hypothetical return series. Investor A receives 30% for four years followed by losing -60% every 5th year. Investor B receives +12% every four years followed by losing -10% every 5th year. Which return would give you more money over time? Many people would choose A, as it appears that the total percentage would be higher. Indeed, the arithmetic average return is higher at 12% vs 8% for investor B. However, the reverse is true.
Investor B (Grey) vs Investor A (Blue)
After a period of 30-40 years, the returns to Investor B far outpace those to Investor A. It is also much less volatile, making it even more attractive as an investment. Indeed, it is because of its high volatility that investment A does not have as high a geometric return and ending value as investment B. Investing is often described as being a tradeoff between risk and return. In this case, and in some real-life investing decisions, understanding the math and nature of investment returns can help you have both higher returns and lower risk.
Mr. Cheng is a managing partner at a Hong Kong based independent private investment office. This article reflects his personal views and not his firm’s and should not be viewed as an investment recommendation.