Seven Math Concepts with Real-World Application

During our school days, we often asked our teachers how the concepts that we were learning were going to help us in life. Most teachers didn’t have an answer to that question and would simply dismiss us.

Prof. Sanjay Bakshi challenges this conventional approach of monotonous rote learning and enlightens us with a fresh perspective on how even the simplest math ideas apply to the markets as well as our lives. These concepts were taught to us in school, but Prof. Bakshi’s insights provide a new way of thinking for us.

He is one of India’s best professors in the field of value investing and behavioural finance, and this article is a summary of the talk that Prof. Bakshi delivered to the students of Ashoka University in March 2019.

1. Proof by Contradiction 
   

   Falsification is a very powerful way of thinking, you can discredit a theory by assuming that it is true and then demonstrating that it ultimately leads to ridiculous conclusions.

   Warren Buffett used this technique to demonstrate the absurdity of the Dot-Com Boom.

   During the early 2000s, internet companies were soaring; the stock market was at its all-time high, and people were making huge amounts of money merely by investing in any company that had a dot com in at the end of its name.

   Buffet, however, refused to participate in this speculative mania and as a result, Berkshire Hathaway had significantly underperformed from the rest of the market. When questioned about this during a shareholder meeting, Buffett utilized the “proof by contradiction” method to demonstrate the market’s irrationality.

   He began by assuming that the current market valuation of internet companies was correct and assumed a 10% required rate of return. He then estimated the expected cash flows that these companies should generate to justify these valuations.

   He concluded that the internet companies would have to generate about $50 to $80 billion in annual cash flows (depending upon the timing of cash flows) to justify their current market valuation.

   Since no business around the time was generating such kind of cash flows, the only logical explanation was that the markets were in a bubble.

   By employing such a simple mathematical principle, Buffett solved the mystery of one of the biggest stock market bubbles. 

2. Inversion
   

   The second math idea is “Inversion.” It is a very simple yet powerful concept that is applicable in many areas.

   Here is an example to remind you of this concept: Suppose you flip a coin 10 times and want to know the probability of getting at least 1 head.

   You can solve the problem in two ways. You can calculate the probability of getting exactly one head and then add it to the probability of getting exactly two heads, and so on until you reach the probability of exactly ten heads.

   But that’s not how you remember solving these types of problems. You used to follow the principle of inversion to reduce the problem to a simpler form.

   To calculate the probability of at least 1 head, you invert the problem and instead, calculate the probability of no heads, and then simply deduct it from 1 since the total probability would always be equal to 1.

   You simplified the problem by inverting it and calculated the answer more efficiently.

   Similarly, in the business and investment world, you can apply the principle of inversion by asking yourselves, “How can I fail?” And then simply avoid doing those things.

   We see businesses repeating the same mistakes over and over again. They take too much debt when times are good and then get into trouble when times are bad.

   The principle of inversion teaches us that by studying the failures of other people, we can avoid making the same mistakes. Remember, it’s the least costly to learn from other people’s mistakes.

3. Law of Large Numbers
   

   The law of large numbers is another powerful math idea. It states that the results obtained from performing the same experiment over a large number of times will be closer to their expected values.

   For example, we know that the probability of getting a head or a tail in a coin toss is 50:50. However, when tossed 10 times in a row, it is very much possible to get 2 heads and 8 tails.

   When tossed 1000 times, however, the observed probabilities will be much closer to its expected.

   Casinos make their living following this principle. They know that the odds are in their favour, and thus they spread the size of the bets over a large number of trials by placing a cap on the bet. 

   They follow three principles that are as applicable in investing as they are in the casino business:

   Principle #1 – keep the odds in your favour.

   Principle #2 – place a cap on the maximum bet size.

   Principle #3 – follow the law of large numbers (diversification).

   Please note that for the law of large numbers to work in favor of the casinos, the expected value has to be positive. If the expected value was negative, the law of large numbers would actually make it more likely for the casino to lose money.

   Similarly, in investing, for diversification to work in your favor, it’s crucial to have a positive margin of safety. Without a positive margin of safety, the purpose of diversification would not be achieved. 

4. Compound Interest
   

   Compounding is all about delayed gratification. You sacrifice current consumption to have more in the future.

   It is as true in life as it is in investing. In 1972, professors from Stanford University conducted a study commonly referred to as the “marshmallow test.”

   In the study, kids of similar ages were offered a single marshmallow and were given the choice to either have it immediately or wait for 15 minutes to have another one. The researchers followed the lives of these children for nearly 40 years and found that those kids who displayed delayed gratification by waiting for the second marshmallow tended to be more successful in their lives than the kids that had the marshmallow immediately.

   Similar examples can be found in the business world. Companies that delay instant gratification and focus more on their long-term plan tend to be more successful than those that focus on short-term profits.

   Valeant Pharmaceuticals, a Canadian pharma company, had a very short-sighted approach. They acquired the patents for essential drugs and drastically increased their prices to earn more profits. They prioritized greed (instant gratification) over the long-term well-being of their customers.

   Over time, the company saw pushback from society and regulators and ultimately perished.

   Amazon, on the other hand, always adopted a far-sighted approach and concentrated on creating value for its customers. Over time, it became one of the largest companies in the world.

   The concept of compound interest is omnipresent. Like Naval Ravikant said, “All the returns in life, whether in wealth, relationships, or knowledge, come from compound interest.”

5. Multiplication Rule
   

   We learned multiplication rules in school while studying probability. If you roll a pair of dice and want a hard 3, the probability would be 1/6 * 1/6 = 1/36.

   The idea is that if two events are independent, meaning that the outcome of one event does not influence the outcome of the other, the probability of both events occurring is calculated by multiplying the probabilities of each individual event.

   Many smart people fail to apply this simple concept in real life. Suppose you are considering a project where there is a 90% chance that a bad thing will not occur, and hence a 10% chance that it will occur.

   If there are five such independent bad things with identical probabilities, the actual probability of a bad thing happening jumps to 41% (1 – 0.9^5).

   This simple math rule is often overlooked by people when making investment decisions. Warren Buffett made this mistake with his investment in a company called Mid-American Energy.

   The company had a major investment in a zinc recovery project and was facing some problems but was expected to overcome them. However, every time the company solved one problem, another one appeared, and it turned out that Buffet had significantly underestimated the risk involved with the project. There were simply too many ways in which the project could go wrong, even though the likelihood of each bad thing was small.

   Buffett discussed this experience in one of his shareholder letters and emphasized the importance of the multiplication rule and how he routinely incorporates it into his thinking about the world of business and investing. 

6. Small Probabilities
   

   In school, we always studied probabilities in terms of likelihoods but never in terms of consequences.

   Prof. Bakshi emphasizes the importance of thinking of probabilities in terms of consequences because when the consequences of failure are significant, small probabilities can become big.

   He gives the example of a fund called Long-Term Capital Management (LTCM), founded by Myron Scholes and Robert Merton, Nobel laureates who developed the BSOP equation for option pricing. 

   The fund’s strategy involved arbitraging between 30-year treasury bonds and 29-year & 3-quarters treasury bonds. Since these bonds were essentially identical, they had a very narrow spread. Therefore, to achieve a good return on capital, they had to operate at very high leverage.

   They had $4.7 billion in equity and a whopping $135 billion in debt. A debt-to-equity ratio of 29 times.

   They did thorough back-testing and concluded that there were no logical reasons for the spread between the two bonds to get wide enough to cause them trouble. However, they ignored the fact that they were dealing with an asset class that was affected by human emotions, and as John Maynard Keynes famously said, “Markets can remain irrational longer than you can remain solvent.”

   And that’s exactly what happened. When the markets behaved irrationally, the fund was not able to hold itself together and went belly up. 

   Scholes and Merton’s strategy had a small chance of failure, but the consequences of the failure were substantial. And given a long enough time, the unlikely event becomes likely, i.e. it will definitely occur.

7. Bayesian Reasoning
   

   We all hated the Bayes rule when studying in school. It was because we were taught the Bayesian principles mathematically and not philosophically.

   The Bayes’ theorem, is a fundamental principle in probability theory that allows us to update probabilities based on new evidence.

   In the book “Thinking, Fast, and Slow,” Daniel Kahneman introduced the concept of “Bayesian Thinking” – how one should change his mind in light of new information.

   For instance, when Ramalinga Raju confessed to manipulating the financials of Satyam, people should have immediately changed their minds and disregarded all the past information since it no longer remained relevant.

   However, many people went into denial and failed to change their minds even when the evidence suggested. These people obviously ended up paying a hefty price for their mistake.

   People often get attached to their priors and apply a selective filter in their minds that only allows the information that concurs with their priors to pass. 

   Bayesian reasoning is about how one should change his mind objectively in light of new evidence.

Conclusion

In conclusion, the practical utilities of the math ideas that Professor Sanjay Bakshi discussed, provide valuable insights for decision-making in our day-to-day life.

We all learned these basic math ideas in our schools but never really thought about it much. However, these principles are omnipresent and can help us make more educated decisions both in the investment world and in life.