The comedian Stephen Colbert coined the word “truthiness” to mean something that “feels right in the gut” but lacks empirical or theoretical support. Its counterpart in the realm of finance is “mathiness,” where academicians or marketers present seemingly rigorous mathematical proofs of assertions that, upon closer inspection, have little or no basis in reality. If the investment products you use rely on mathiness and not provable math, you and your clients will ultimately pay a price.

Mathematical sophistication in the investment industry is a sham. Most mathematical articles in peer-reviewed academic finance journals suffer from lack of model specificity, failing either to properly define terms or to specify exactly what they are trying to prove, or drawing conclusions in ordinary language that aren’t warranted by the mathematics. In these respects, they resemble more what economist Paul Romer has called mathiness than real mathematics. Even when the articles are mathematically sophisticated and do not have these failings, investment industry apostles tout, too strongly, conclusions based on them that cannot be found in the articles themselves.

I will illustrate this with a particular line of mathematical finance that has been pursued at great length in the literature (perhaps following up with others in future articles). To emphasize the positive, I will use two examples of sound mathematics applied to this topic, one by a physicist and the other by two mathematicians. However salutary – and rare – may be the contributions that these works make to the process of academic thought experimentation, nevertheless, like so many similar articles, they have, as the authors themselves reveal, no immediate practical implications for investors.

**The mean and the median**

It is well known that the topmost echelons of the wealth and income spectrum in the United States have experienced higher rates of real income growth in recent decades than the lower echelons. For example, the real after-tax incomes of the top 1% of earners rose by 138% since 1978, while those of the bottom 90% rose by only 15%.

Yet the only summary statistic typically cited for the economy as a whole is the growth of GDP per capita, the aggregate of all incomes divided by the number of recipients. The GDP per capita over the same period grew by 82%; this is the mean of the distribution of all income growth rates. Income growth at this high a rate was, however, experienced by only a small minority of the population. The median of the distribution was much lower.

But now suppose that each income recipient could experience the income growth rate of a different U.S. citizen every year. One year in a hundred, you’ll get a raise equal to the annualized equivalent of the 138% earned by the top 1% since 1978; 90 years in a hundred you’ll get a raise equal to the annualized equivalent of 15% over that period, or about 0.4%.

What kind of income growth would you experience over a long period of time? It would be a growth rate that is close to the median of all growth rates, or close to zero, not to the mean of 82%.

Which growth rate is more appropriate to use, the mean or the median – the one-year average of all income growth rates, or the long-term time-average?

The answer, of course, is that it depends on the context – the purpose for which you are using it. And, is it necessary to use only one of these averages when other information about the distribution of income growth rates is available too?

Yet this little question – that is, which one should you use, the mean growth rate or the median? – has been the underlying subject of scores and probably hundreds of finance articles. If this is not the underlying subject, it is difficult to discern what is. Yet most of these articles obscure this simple question under layers of obtuse and hard-to-interpret mathematics. Then, they draw conclusions that are unrelated to, or well beyond, what the mathematics implies.

**The simple math that subsumes all of these articles**

Growth rates can be expressed as being continuously compounded, periodically compounded or not compounded at all. For example, at a 6% annual interest rate, $1,000 will grow in two years to $1,126.50 if the 6% annual rate is continuously compounded; $1,126.15 if it is monthly compounded; $1,123.60 if it is compounded annually; and $1,120.00 if it is not compounded at all (that is, if the two-year uncompounded interest rate is 12%).

If the rate is continuously compounded, then rates add over time. For example if the continuously-compounded rates are 6% in the first year and 5% in the second, the two-year continuously-compounded rate is 11%, or 5.5% if annualized.

If each continuously-compounded annual growth rate is generated randomly from some probability distribution, then over a long period of time, the annual average rate will eventually converge with certainty, as time goes to infinity, to the mean of the distribution. If the continuously-compounded rate has a symmetric distribution, this number will also be its median, and the long-run time-averaged rate will converge to the median as well.