**This post is about how a single mathematical function can explain many businesses. Those that have done their math homework can skip straight to the “What the Heck Are You Talking About?” section.**

"Invert, always invert."

- Carl Jacobi, Berkshire's favorite mathematician

__Introduction__

The quote above is from a legendary mathematician and holds the key to how many famous mathematicians think. It also happens to be Berkshire Hathaway (NYSE:BRK.A)(NYSE:BRK.B)’s favorite way of thinking.

Numerical fluency can help investors beyond just teaching us new ways to think. Patterns already deciphered by mathematicians several centuries ago are commonplace in business.

Enough jibber-jabber, let’s use an example.

__One Mighty Useful Function__

Anyone who took mathematics at university would have come across logarithmic functions (perhaps you learned this in high school). To avoid evoking adverse reactions from insomnia sufferers, let’s skip the gory details (for those who simply can’t resist: Here is a refresher). All you need to know is that logarithmic (/exponential) functions are special equations that explain how stuff grows and, just as interestingly, how stuff *decays *(believe it or not, I'm not a professional mathematician).

Given a rate of growth/decay, a time period and a starting point – these functions can tell you what values growing or decaying variables will take in the future.

Maybe an example will clarify? The most common exponential function should be familiar to *all *investors: compound interest. Think about a bank account: it has a starting point (the principal that you put into your account), a growth/decay rate (the rate of interest you earn and reinvest) and a time period (how long you leave your money in the bank).

Although today's real interest rates may seem to suggest otherwise, bank accounts are a *growing *phenomenon. Logarithmic functions can also be used to model things that *decay *– i.e. things that are *declining* from their initial value at a constant rate. For example, this is the mathematical equation that dictates how objects cool (for those boffins that still can’t resist: __"Newton’s Law of Cooling"__).

Boiling water doesn’t go from 100°C to 25°C (apologies to our American readers) instantly. It *decays* in temperature over tiny intervals like this: 100°C -> 75°C -> 56.25°C -> 42.19°C and so on until it reaches 25°C – each time declining by 25% of its previous temperature.

Decaying functions also explain how radioactive materials work. Every 10 years, for example, a radioactive substance may be half as strong as it was previously (hence the term "half-life") and then 10 years later, it will be half as radioactive again. This is how scientists use "carbon dating" to estimate the age of really old stuff (believe it or not, I'm not a professional carbon dater).

__What the Heck Are You Talking About?__

“Ok,” you're thinking, “all very interesting, but what do logarithmic functions have to do with my stock portfolio?”

Decaying functions are more common in business than many analysts notice. Think of a hypothetical newspaper operating in a small regional town in the 1970s. Once this newspaper reaches a critical mass in its circulation, say 70% of this fictitious small town reads this dominant newspaper, its subscription rates (if left unattended by management) follow a somewhat decaying equation. Why? Each year, some small portion of that newspaper’s customers will stop subscribing to it: Some customers will move towns, some will perish, some will have a change of preferences, etc. Whatever the reason, each year there will be some slippage in its customer base.

In our example, let’s say 3% of the paper’s subscribers don’t renew their subscription the following year. This means that the newspaper in the following year owns 67% of the small town’s readership (70% - 3% = 67%). *To maintain their monopoly in percentage terms¸ *all things being equal, the newspaper now has to make up for this 3% deficit (by extolling the virtues of their ‘news hole’ to the rest of the unsubscribed population or by capturing new subscribers from those emigrating to the small town). To *grow *its subscriber base in percentage terms, it has not only to make up for this slippage, but the newspaper has to recruit even more, *new* customers on top of it. In our example, if the paper wanted a 75% market share, it would first have to make up the 3% natural decay in subscriber numbers and *then *win 5% *more *market share (okay, media analysts, this is a very simplified analogy).

This is more or less how all subscription-based businesses work, be it newsletters, satellite TV companies, database owners (Morningstar, Bloomberg, et. al.) or some other service providers.

__Banking (Very, Very Simplified)__

This principle also applies to mortgages. A bank’s loan book is also a decaying asset in a way. Loans are getting paid-off every day, thereby shrinking the size of the bank’s asset constantly. Banks, therefore, have to make up for this repayment rate through originating new loans or even through refinancing old ones.

The mortgage industry has a term for this; they call it the __Conditional Prepayment Rate__ or CPR for short. The CPR is the annualized percentage of the existing mortgage pool that is expected to be prepaid in a year. For example, if each year, 8% of a bank’s loan book prepays its loans earlier than scheduled, then the CPR is said to be 8%. The monthly equivalent to the CPR is the morbid-sounding Single Monthly Mortality rate.

Personally, I like the irony of the abbreviation CPR, as it can be thought of percentage of mortgages that the bank has to "resuscitate" back in to its balance sheet to keep its assets more or less unchanged.

__Predicting Profitability and Growth__

Those of you who have managed to battle through yawns and keep your eyes open thus far might have realized that this has implications for the profitability and growth of such businesses. Newspapers during their Golden Era were not really impacted by this slippage at all. Whatever they lost in market share percentage, they made up for in price hikes. Given their customer captivity and large fixed-cost base, a 5% increase newspaper prices (the impact of which was felt mostly by advertisers within the paper) dropped straight to the bottom line.

Even today's modern subscription-based businesses barely blink at the thought of customer-attrition. By definition, their customers are somewhat held captive in most cases. The services provided by some of these businesses (niche databases, mission-critical software, etc.) are often *essential* to the customer's livelihood. So "sticky" are these customers, they often offer to *pay in advance *for such services.

For banks*, however, the mathematics works differently. To achieve profit growth (all other variables, like interest rates, being equal), a bank pretty much *needs *credit growth – i.e. has to make more loans next year than this year. Credit growth, by extrapolation, requires more borrowers or larger loans. Thereby, banking *growth* can be seen as a "give out more to get back more" type of situation.

Now, fellow investors, you’re getting the picture: The worst time to buy a bank has often been when they look the most profitable, often coinciding with an analyst's mindless extrapolation of previous profits in their DCF models.

This too explains why, as pointed out by economists like Hyman Minsky, today’s credit growth can lead to tomorrow’s *credit decay.* Add to banking's 'lose-some-then-win-even-more' business model the aggressive leverage and poor reserving that occurs periodically in the industry and now you can see why *accelerated, industry-wide *decay becomes a mathematical inevitability. Some businesses literally are ‘radioactive’.

As blogged by: www.oddlotinvest.wordpress.com

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**Postscript: **A reader of our blog very astutely pointed out the connection between the CPR rate in banking industry and the 'Reserve Replacement Ratio' (RRR) in the oil industry. Oil fields are a decaying asset too. Hence, the RRR measures the percentage of decaying reserves that are replenished by a given oil company. It's a useful metric to gauge the industry as a whole, as well as evaluating projects/management teams on their ability to find prospects.