The Multiplication Rule
Remember, back in school, when you were learning elementary probability?
One of the topics covered was the “multiplication rule.” My favorite mathematician, John Paulos explains the rule.
If two events are independent in the sense that the outcome of one event has no influence on the outcome of the other, then the probability that they both occur is computed by multiplying the probabilities of the individual events.
For example, the probability of obtaining two heads in two flips of a coin is ½ x ½ = ¼ since of the four equally likely possibilities—tail,tail; tail,head; head,tail; head,head—one is a pair of heads.
Like you, I too encountered the multiplication rule in school. I learnt how to use it to solve questions of probability relating to flipping coins, drawing cards, rolling dice or picking marbles. And then I forgot all about it.
Many years later, when I became a student of business and value investing, I started appreciating the practical significance of the multiplication rule. I found that its utility lay beyond the textbook probability world of coins, cards, dice or marbles.
Multiplication Rule in Investing and Insurance
My first encounter with the multiplication rule outside the abstract world of coins, cards, dice and marbles occurred when I discovered Ben Graham.
Graham never really talked formally about the rule. But, he did recognize the need to “spread the risk” over somewhat uncorrelated (“independent”) risks.
For example, in these these passages in two of his books — Security Analysis and The Intelligent Investor — he wrote about diversification.
An investment might be justified in a group of issues, which would not be sufficiently safe if made in any one of them singly. In other words, diversification might be necessary to reduce the risk involved in the separate issues to the minimum consonant with the requirements of investment.
The instability of individual companies may conceivably be offset by means of thoroughgoing diversification.
There is a close logical connection between the concept of a safety margin and the principle of diversification. One is correlative with the other. Even with a margin in the investor’s favor, an individual security may work out badly. For the margin guarantees only that he has a better chance for profit than for loss—not that loss is impossible. But as the number of such commitments is increased the more certain does it become that the aggregate of the profits will exceed the aggregate of the losses. That is the simple basis of the insurance-underwriting business.
A margin of safety does not guarantee an investment against loss; it merely guarantees that the probabilities are against loss. The individual probabilities may be turned into a reasonable approximation of certainty by the well known practice of “spreading the risk.” This is the cornerstone of the insurance business, and it should be a cornerstone of sound investment.
Pure Grahamites believe in wide diversification because in their worldview, while bad things can happen to a handful of portfolio businesses at the same time, the probability of bad things happening to all portfolio businesses at the same time is remote, thanks to the multiplication rule.
Graham drew strong parallels between the worlds of value investing and insurance underwriting. Both the value investor and the insurance underwriter, according to Graham, should worry about aggregation of risks. Don’t put all your eggs (portfolio positions) in one basket (e.g. one industry). They may be different eggs, but they are one basket and if the basket falls, they will all break together.
Graham’s most famous student — Warren Buffett — has also written about the multiplication rule, although like Graham, he too didn’t mention it specifically. For example, in an essay in one of his letters, he described a principle followed by disciplined insurance underwriters.
They limit the business they accept in a manner that guarantees they will suffer no aggregation of losses from a single event or from related events that will threaten their solvency. They ceaselessly search for possible correlation among seemingly-unrelated risks.
The Deceptive Guise of Independence
Independence of events is a very important notion in probability but is also a deceptive one. This is especially true for some domains like financial markets which lie outside the world of coins, cards, dice or marbles.
Many global investors who practice wide diversification by spreading their money across “seemingly-unrelated risks” across geographies and asset classes found this in 2008 and 2009 when the global financial apocalypse of the time proved diversification to be a fair weather friend. He failed to come to their rescue just when they needed him the most just like an home insurance policy which expires moments before an earthquake strikes.
When shit hit the ceiling, their so-called diversified portfolios were slaughtered by the carnage that took place in asset prices across geographies and asset classes. The prices of almost every financial asset other than US treasuries crashed. Having one’s eggs in many baskets didn’t really help because what was thought to be “independent” and “uncorrelated” turned out to be anything but.
That experience, made Warren Buffett make an acute observation:
When there is trouble, everything co-relates.
So much for the multiplication rule!
My own thinking about diversification changed quite a bit post the 2008-09 experience but that’s not the subject matter of this post. Let’s stay focused on the multiplication rule instead.
Multiplication Rule in Aircraft Design and Engineering
A few years ago, while researching the idea of “margin of safety” for my class, I came across another idea related to the multiplication rule from the world of engineering. That idea is called “redundancy.” Here’s an example from the world of aircraft design, as illustrated by Wikipedia:
Duplication of critical components of a system with the intention of increasing reliability of the system, usually in the case of a backup or fail-safe… In many safety-critical systems, such as fly-by-wire aircraft, some parts of the control system may be triplicated. An error in one component may then be out-voted by the other two. In a triply redundant system, the system has three sub components, all three of which must fail before the system fails. Since each one rarely fails, and the sub components are expected to fail independently, the probability of all three failing is calculated to be extremely small. [Emphasis mine]
Obviously, this is a very powerful idea. The practical applications of the multiplication rule in engineering — of which aircraft design is just an illustration — have proven to be hugely beneficial for civilization by providing it with, amongst many other things, safer and more reliable planes, cars and nuclear power plants.
What I found interesting was that while “seemingly unrelated risks” in the world of financial markets proved to be not so unrelated after all, in the world of engineering, this wasn’t so. And so, my respect for the multiplication rule returned.
Multiplication Rule in Investment Thinking
Over the years, my appreciation of the multiplication rule has only increased. And even though the rule failed to protect widely-diversified investment portfolios (including mine) from collapse during the global financial meltdown of 2008-09, I continue to apply it to my investment process in other ways.
One of them involves the application of a related principle:
A chain is no stronger than its weakest link.
Let me explain this with the help of an example. A few days ago, my colleagues and I were discussing the investment merits of a situation involving a company which had, a few quarters earlier, announced plans to manufacture a product related, but not identical to, it’s existing products. The new product required a new plant. Moreover the company would need to sell the new product to its existing customers. Also, before the company could start manufacturing the new product, it needed some environmental approvals which, as it happened, had already been delayed.
Furthermore, our analysis revealed that a very significant part of the total expected return from the proposed ownership of this business (acquired at prevailing market price) over the next few years was largely dependent on the success of this initiative. So, in a sense, the entire investment thesis rested on this project.
We saw three, independent risks on this project:
- a prolonged delay in receiving the environmental approvals. We figured the probability of this risk materialising was 50% which meant that there was a 50% chance of no further delays
- production related risks relating to product quality and cost, given that this was a new product which the company had never manufactured before. Considering the extensive experience of the company, however, we figured that there was only a 20% chance of this risk materialising, which meant that there was an 80% probability of no production hiccups
- the inability of the company to sell the new product to its customers. We figured the probability of this risk materialising to be only 10%, which meant that there was a 90% chance that it would be able to sell the product.
For the project to succeed, none of the risks should materialise and the probability of that was simply the product of the probabilities of each of these risks not materialising or
(1-0.5)*(1-0.2)*(1-0.1) = 0.36 or 36%
Therefore, there was only a 36% chance of success on all three parameters which, of course, meant that there was 64% chance of failure. As the consequences of failure were no return for us, we passed the opportunity.
To be sure, this type of thinking is deeply subjective but to paraphrase Keynes, we would rather be subjectively right than be objectively wrong.
Now, imagine that the company does indeed get the environmental clearances. So, risk # 1 is eliminated. What is the joint probability of success now? The multiplication rule tells us that the probability of success has now doubled to
(1-0.2)*(1-0.1) = 0.72 or 72%
Suppose, however the market ignores this development or under-reacts to it. Clearly then, there might be an excellent opportunity to make an investment in this situation, if it looks attractive in relation to other opportunities available at the time.
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A few years ago, Warren Buffett wrote on probability chains derived from the multiplication rule, which would serve as an even better example on how the rule should be used in one’s investment thinking.
Last year MidAmerican wrote off a major investment in a zinc recovery project that was initiated in 1998 and became operational in 2002. Large quantities of zinc are present in the brine produced by our California geothermal operations, and we believed we could profitably extract the metal. For many months, it appeared that commercially-viable recoveries were imminent. But in mining, just as in oil exploration, prospects have a way of “teasing” their developers, and every time one problem was solved, another popped up. In September, we threw in the towel.
Our failure here illustrates the importance of a guideline – stay with simple propositions – that we usually apply in investments as well as operations. If only one variable is key to a decision, and the variable has a 90% chance of going your way, the chance for a successful outcome is obviously 90%. But if ten independent variables need to break favorably for a successful result, and each has a 90% probability of success, the likelihood of having a winner is only 35%. In our zinc venture, we solved most of the problems. But one proved intractable, and that was one too many. Since a chain is no stronger than its weakest link, it makes sense to look for – if you’ll excuse an oxymoron – mono-linked chains. [Emphasis mine]
Clearly, Buffett learnt an important lesson there. The way I see it is that some business models, by their very nature are so complex (e.g. drug discovery) that one has to worry about lots of “moving parts” — independent risk factors. For the investment to be successful, all of those risks must be mitigated. And given the way the multiplication rule works, that’s a long shot. To be sure, long-shots can sometimes be offset by bonanza profits if success does occur, but that kind of investing is more in the nature of a venture capital operation than a value investing operation.
In contrast, other things remaining unchanged, simple, easy to understand businesses with fewer moving parts carry much lower risk of disappointment. As Buffett writes:
Our investments continue to be few in number and simple in concept: The truly big investment idea can usually be explained in a short paragraph. We like a business with enduring competitive advantages that is run by able and owner-oriented people. When these attributes exist, and when we can make purchases at sensible prices, it is hard to go wrong (a challenge we periodically manage to overcome).
Investors should remember that their scorecard is not computed using Olympic-diving methods: Degree-of-difficulty doesn’t count. If you are right about a business whose value is largely dependent on a single key factor that is both easy to understand and enduring, the payoff is the same as if you had correctly analyzed an investment alternative characterized by many constantly shifting and complex variables.
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My own, intuitive application of the multiplication rule can also be understood by another example.
I followed that tweet up with a blog post titled “Who will Bail Shale” in which I was asked to comment on the probability of oil prices remaining low for the next few years. While my original, tongue-in-cheek response was to estimate that probability to be “somewhere between zero and 1,” I subsequently wrote:
My head starts spinning when I think about the economics of shale, gas, regular good old crude oil, wind power, solar power and how they interact with geopolitical developments in Russia and USA and Syria and and Iran and Iraq and Saudi Arabia. I could go on and on but I hope you get the point. There are too many variables and too much variability. This one goes in my “too tough basket.”
Contrast the complexity involved in predicting the future price of oil or other commodities with the simplicity of investing in a business like Relaxo — India’s largest footwear manufacturer which despite volatility in input prices, does not experience volatility in its profit margins.
Because Relaxo follows the simple notion of buying commodities and selling brands. It has the ability to pass through cost inflation to customers without fear of loss of unit volume or market share. The business that manufactures EVA has lot more “moving parts” than the business that uses EVA to make and sell branded footwear.
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If you have used the multiplication rule in your investment process in a manner different from what I described above, I would love to know more about it.
Note: The use of Relaxo in the post was just an example to illustrate a point and must not be construed as a stock recommendation.