How much mathematics do I need for investing?

Margin of safety, rules of thumb and simple arithmetics is all you need

The Rosetta Stone is right there at your fingertips.

When I started my first job at a broker firm in the spring of 1994, my mathematics proficiency was finally put to actual use. Throughout the school system, including business school, numbers and formulas had had no actual value, except for personal use in programming and writing exams. Well, that, and the money I got for being the top maths and physics student (directly from the hands of the King of Sweden on Sweden’s National Day, the year 1990)

Mathematics made the day at my first job as an equity analyst

During my first few months on the job I was happy to see that I surprisingly enough had use for exactly everything I knew. That included accounting, macroeconomics, microeconomics, economic history etc., all the way to the calculus used in Black & Scholes’ option valuation formula.*

* I was asked to put together an options arbitrage matrix in Excel and link it to our trading system to see if there was any easy money to be made. Our options broker thought so, but he wanted real-time proof. I said “Sure, I ‘ll get right to it”.

My boss and colleagues sceptically wondered if I shouldn’t get my books from school to check the formula (this was before the world wide web, web browsers and search engines), and I promptly answered that I knew the formula by heart.

Being able to right there and then recite both the formula and its derivation earned me a ton of respect (despite it being almost completely useless knowledge, except perhaps for the part where knowing it in my case meant understanding it, including its limitations — which is more than its inventors [Nobel Prize winners] did, since they almost crashed the entire global financial system four years later trying to use it).

That respect was my first sign that my studies had value.

Since that day, over 30 years ago, I’ve undulated between sometimes thinking nothing I learned at Stockholm School of Economics was valuable, and that I had had to slowly unlearn everything that academics had gotten wrong about financial markets; and sometimes swinging back to thinking that as an analyst and hedge fund manager with a global investment scope, I had used every little bit of knowledge more or less every single day at work.

The actual maths and B&S formulas turned out to be of no business value to us, which is something I thought I knew already when asked to design the arbitrage system. I had been taught that all financial arbitrage opportunities disappeared as soon as they materialized. The same reasoning stopped me from trying to create an automated trading system a few years later, since I concluded that it wouldn’t work in an efficient market; or for that matter a sentiment based trading system using internet discussion forum data that I thought about in the early 2000s. Talk about being naive!

So what maths, formulas, rules of thumb etc. do I think hold any value for ordinary private investors today?

That 15% annual growth leads to a doubling [2x = +100%] every 5 years (more precisely: +101,1%). Rule of 2x. Rule of 15.
That 10% annual growth leads to a doubling every 7 years (+95%; it actually takes 7.3 years at a 10% rate, but such details just ruin the whole system). Rule of 10, and of 2x.
That the S&P 500 stock market has returned 10% to investors per year in the long run, i.e., doubled their money every seven [7.3] years, quadrupling [4.2x] it every 15 years. Rule of 10.
That humans have ten fingers, ten digits; that we don’t really notice less change than 10% in a year’s time; that the church or state could demand a ‘tithe’, a tenth of your income in annual taxes without causing too much trouble; that saving a tenth of your income creates enough value for old age without really affecting your standard of living noticeably today; which all lead to an investor requiring at least a ten percent annual total return in order to take any idiosyncratic risk whatsoever, i.e., using a discount rate of at least 10% when valuing a stream of equity based cash flows. Rule of 10.

In particular, you should note that, because the stock market, when bought at its historical average valuation (measured as Price/Sales, P/GDP, cyclically adjusted P/E, or similar revenue based factor), returns 10% per year, you should demand significantly more than that to take risk on individual stocks, since the S&P is less risky, less exposed to bad luck etc. thanks to its diversification among 500 different companies. Rule of 10.

10% is the market’s return requirement for a broad stock market index like the S&P 500. The rule of 10.

15% is an intuitive level of required rate of return for a single stock with average risk characteristics; it’s enough extra MOS [margin of safety] to make a significant difference vs. just buying an index fund. The rule of 15 [10+5], leading to a doubling [Rule of 2x] every 5 years.

Note where this is leading… You don’t need calculus, integrals, knowing fancy formulas by heart, let alone implementing them.

You only need to remember a few rules of 2, 5 and 10. Analysing growth and value creation isn’t a precision sport, it’s about margin of safety and orders of magnitude.

If it were a precision sport, the total stock market index wouldn’t sometimes halve in value in a single year, despite all the economy, its people, tools, and resources still being present, or double in value almost as quickly.

Where value comes from: people, prices, productivity, performance

The S&P returns 10% per year, because its companies grow profits by 8% per year and distributes dividends of 2% per year.
Their profits grow by 8% per year because they sell 6% more stuff per year and increase prices by 2% per year.
They can raise prices by 2% per year because people’s wages rise by at least the rate of inflation (2%), plus compensation for increased productivity.
Companies sell 6% more per year because their clients buy 6% more goods and services from them per year; and of course that they are able to produce 6% more stuff per year, using more employees, and better tools and input materials.
The companies can produce 6% more stuff every year because they employ 2-3% more people every year and every employee can produce 2-3% more per person thanks to better production processes, methods and tools (such as computers, e-mail, software and AI systems).
take note here that the annual productivity increase keeps falling, despite inventions like semiconductors, computers, internet, smartphones, AI…
S&P 500 companies can employ 2-3% more people every year because there are 2-3% more people available to employ, as well as 2-3% more people to sell stuff to available.
note that the total economy only grows by 6% per year, but S&P 500 companies grow by 8%, since they are better run, take market share from their smaller and less productive competitors and so on. S&P 500 companies are the pick of the litter, and they outperform the rest of the economy by 2% per year [Rule of 2] — that’s why they are in the index to begin with.
Think of it as a rule of five twos make a ten: 2+2+2+2+2=10. Population +2%, Productivity +2%, Prices +2%, Top 500 companies outperformance 2%, dividend yield 2% => 10% total annual return. Two hands, with five fingers each, a total of ten fingers. Twos, Fives, Tens. More or less all the maths you’ve ever needed has been with us since the Cambrian explosion 500 million years ago.

The only maths you need

The only maths I use for my own investments, private and professional, is no more advanced than what we all learned before puberty. Investing is a game of psychology, not of mathematics.

The value of a company is the sum of the cash flows it produces, that the owner can use to buy the consumables he or she actually wants. But the price of a stock is rarely the same as its value (or the best, rational estimate of that value). The price of a stock, or the entire stock market, typically tends to move away from its value, not toward it; not toward some kind of efficient market hypothesis [EMH] equilibrium. It’s only over very long periods of time, decades, that prices reflect the actual values. In the short run, prices move away from the supposed equilibrium, often increasingly fast.

Over periods of 10+ years [Rule of 10], the average price for that period is often also close to the average value for a stock or a stock market. Sometimes it takes 20 years. Or more. And when buying at a price that is about the same as the value, the buyer will be rewarded by a 10% annual total return for the subsequent long haul.

But in the short run, shorter than 10-20 years, you can sometimes buy the index at a price half or twice its value [Rule of two, Rule of 1/2], and get a 10 year annual return of 20% [2x the normal return], or just 0% per year for a decade.

Calculating cash flows and values

To estimate the value of a company you need to estimate its sales and profits, its investments and cash flows. To do that you need to estimate how much stuff it produces and sells, and at what prices, and how much it cost to produce and market & distribute. For that you need to know about their input resources (employees, tools etc.), their clients, and competitors.

The best guide for estimating all those variables is to identify trends and patterns in historical data — for that specific company, for similar companies (‘peers’) now and earlier, for the total economy, for situations such as product launches or geographic expansion etc., and extrapolate those trends cleverly. There is very little new under the business sun, just variations on the same theme of innovation and growth cycles.

If the total chart (past and future) of your select variables doesn’t look surprising, non-linear, or weird, then it isn’t. For example, historical sales growth levels/cycles/trends and so on should align nicely with your future projections.

The total economy is extremely predictable over long periods of time, and so is the stock market. But the long run doesn’t matter in the short run, which is where investors live. However, all companies and their stocks live inside the total economy and the total stock market, so there is a certain kind of predictability there to harvest if you’re diligent and patient.

Once you have an estimate of future ‘profits’, you have an estimate of the company’s value. That’s when the hard part begins: trying to estimate what price the market will put on the company, and not least when it will award it that price. That’s stuff that I teach in detail in my online courses The Investing Course (TIC) and Finanskursen.

Check them out, if you’re interested in stepping up your investment game. Just remember that all the mathematics you need is already right there at your fingertips: two hands of five digits each. The rest is all in your mind, and in the minds of the rest of the market, all reflexively trying to outguess each other.

Lack of patience leads to forced patience

Final note: Most of the time almost nobody cares about actual values, only about price trends. Very few have the patience to buy cheaply and simply wait. Most people think that a doubling every 5 or 7 years is too little, too late. They want doublings every two years (40% annual returns), rather than 10%, 15% or the 20% market panic troughs sometimes enable.

Remarkably, in order to get a chance at those 40%, people frequently buy at prices 2x the underlying value (or more), and thus risk losing half in short order, or suffer the fate of still being flat (0% per year) a full decade later.

The Investing Course