The Multiplication Rule

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:

  1. 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
  2. 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
  3. 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.

– – – – – –

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.

– – – – – –

My own, intuitive application of the multiplication rule can also be understood by another example.

Some time ago, I read a story in The Economist which promoted me to quote it in a tweet

Screen Shot 2015-11-28 at 15.26.36

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.

– – – – – –

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.



22 thoughts on “The Multiplication Rule”

  1. Hi Professor, I would add that what you have explained above – i.e., have a sense of the base rate, and, adjusting the probability of an outcome as new information/evidence becomes available – can also be called the Bayesian way.

  2. Wonderful post Sir.

    I believe in most cases in investment this multiplication rule will have to be used with the Bayes rule(explained by you in an earlier post) in mind for best results.

    For example, if we consider the case you used where there were three types of risks and you assigned probabilities to each one of them, I believe you were using the prior odds and likelihood ratios to come out with posterior odds in each of the three risks independently before applying the multiplication rule on them.

    So if I learn correctly from your Bayes rule posts, I should be less prejudiced with the three probabilities you have assigned to the three risks. It may vary for different companies within the same industry, or rather for the same company it may vary over different periods of time in case of any change in facts.

    So, in my view multiplication rule and Bayes rue complement each other in investment decisions and both should be used together for best results.

    If I am wrong in my understanding of this, please correct me.


  3. Loved your article sir. An observation;

    Even in the aircraft example, ‘if shit did hit the ceiling’ – The multiplication rule would fail right. Like in case of external factors (accident, attack, pilot fault etc), the ‘triply redundant system’ fails. I am making the point of very low probability, possibly hard to imagine but high impact events. In case of investments, this would mean always having some allocation to almost guaranteed safety (cash, debt etc) at all times


  4. One of the most educative and reflective article that I have read in recent times .

    It again brings to point importance of cash / fixed income in asset allocation . Graham mentioned 20% rule that at no point debt or equity should be less than 20 % of portfolio .

    Second important thing is correlation of industries in equity pf .
    1) Manufacturer and User of commodities ,
    2) Low and high debt companies
    3) Market leaders and Market followers

    Third this rule also tells us important of right diversification & how many is enough .


  5. Thank you, sir.

    I wrote the below on a company sometime back, which never saw the light of the day (II). Thank you for reminding us again that a chain is no stronger than its weakest link.

    “Adding complexities. ABC continues to add more XX, the latest being DE and FG XX. Each additional XX adds multiple variables (sometime interacting with other XX) and associated convolutions and risks. While the strategy may work in the medium-term, foretelling it is difficult……………………………….”

  6. Thank you for the wonderful post, Sir. In my view, the multiplicative rule could also be applied to competitive strategy where the probability of success of a business is dependent on the performance of other competing firms, where the chances of failure of the subject business is determined by multiplying the individual probabilities of success of other competing firms.

  7. So an investor should aim for a shallow decision tree (less decision points or moving parts or variables) since assignment of probabilities to each decision path is in itself an estimate. Businesses that are simple with less variables and ideally independent of each other…reminds me of Sees candy investment logic where profits and future value was estimated based mainly on just one variable … change in price of candy [instead of focusing on volume.]
    Warren quoted … “When we looked at that business – basically, my partner, Charlie, and I, we needed to decide if there was some untapped pricing power there – whether that $1.95 box of candy could sell for $2 to $2.25. If it could sell for $2.25 or another $0.30 per pound that was $4.8 million on 16 million pounds. Which on a $25 million purchase price was fine. ” (Source:–warren-buffett–sees )

    1. I agree.

      On the subject of See’s Candy, here’s something I loved reading (from what transpired at BRK AGM in 1998)

      Munger: I’ve heard Warren say since very early in his life that the difference between a good business and a bad one is that a good business throws up one easy decision after another whereas a bad one gives you horrible choices – decisions that are extremely hard to make: “Can it work?” “Is it worth the money?”

      One way to determine which is the good business and which is the bad one is to see which one is throwing management bloopers – pleasant, no-brainer decisions – time aftertime after time. For example, it’s not hard for us to decide whether or not we want to open a See’s store in a new shopping center in California. It’s going to succeed. That’s a blooper. On then other hand, there are plenty of businesses where the decisions that come across your desk are awful. And those businesses, by and large, don’t work very well.

      Buffett: I’ve been on the board of Coke for 10 years now. And during that time, we’ve had project after project come up to be reviewed by the board. And they always estimate the ROI – the return on investment. However, it doesn’t make much difference to me – because in the end, almost any decision you make that solidifies and extends Coke’s dominance around the world in a rapidly growing industry that enjoys great inherent profitability is going to be right. And you’ve got people there to execute ‘em well.

      Munger: You get blooper after blooper?

      Buffett: Yeah.

      Buffett: In contrast, Charlie and I sat on the board of USAir. And there, decisions would come along – and they’d be: “Do you buy the Eastern Shuttle?” And you’re running out of money. And yet, to play the game and keep traffic flows such that it will connect passengers, you just have to continually make these decisions where you spend $100 million more on some airport. You’re in agony – because you don’t have any real choice. And you also don’t have any great conviction that the expenditures are going to translate into real money later on.

      So one game is just forcing you to push more money onto the table with no idea of what kind of hand you hold. And in the other you get a chance to push more money in knowing that you’ve got a winning hand all the way.

  8. Dear Sir,

    Loved the article. Buffett is wonderful, no doubt, but i often wonder how to apply his theories in Indian market. Our market is quite new except TATA and Birlas who were there pre-independence. Unfortunately we don’t have 100+ year old solid businesses like COKE, WALLMART and so on. We have extremely narrow range of businesses which pass Buffett’s filter e.g. VST Ind, HDFC Twin etc. These are slow compounders and one has to wait for 9/11 or Lehman crisis kind of events to get them super cheap.

    We are evolutionary as a market so its good to refer to Buffett more as a theory but when it comes to choosing company in India than Peter Lynch looks more relevant. In my limited view, Buffett is more suitable for developed and mature market while Lynch is for growing and young economy like ours.


  9. Thank you for a thought provoking post. Some thoughts on the matter are listed below

    1. The way I understand Ben G and WEB’s thinking on diversification is that there is diversification on account of ignorance and diversification on account of independence of events. For eg. suppose there is a stock whose intrinsic value I am unable to determine with confidence though I know it is more determinable than my most confident assessment. In other words I am ignorant. I should diversify here so that I maximise the opportunities to nullify my overestimation of intrinsic value in some cases with underestimation with another, thus minimising downside. This is diversification on account of ignorance. In other cases I am able to assess the intrinsic value to the highest level of confidence determinable and there is a margin of safety. However I do know not when the price will rise up to intrinsic value, it may rise in 3 months or 3 years (or never of course). Such uncertainty reduces compounding opportunities if I am invested in only one stock. Suppose I diversify into 5 such similar opportunities, then chance that atleast one of them has price rising up to intrinsic value is improved by 5 times, thus increasing compounding opportunities on capital. This assumes that chances that one stock will rise is independent of another (unless it is very clear they will). So we may call it diversification on account of independence of events. This will be an additive rule though, not a multiplicative rule, and something I apply.
    2. Every event that needs to be handicapped will also carry with it a chance of error in estimate and our estimate is only that, an estimate. So an event like environment clearance estimated at a mean of 50% odds will have a wider range than a 50% odds of coin flipping, where the odds are deterministic. The best way to handicap odds for such situations is to know the base rate which in situations like the one above is not known. However it is known in many other situations better like say the mortality rate for an age cohort. We need to be cognizant of this fragility, if I may, in our estimate.
    3. If there are two events say where the odds have a wide range ( you know they do if a small poll shows similarly qualified people assign widely varying odds or the event has no history like say technology innovation) then their multiplication like above, will have a range that will have its error range multiplied or compounded. That may be fatal for returns or a lollapalooza. The best way I handle them is to just avoid (no called strikes).
    4. The magin of safety that Ben Graham conceptualized will have to be bigger if our intrinsic value estimates have a wider range. If hypothetically such value is dependent on 3 independent events with fragile odds, then such value will necessarily have a higher margin of safety.
    5. Lastly, often the best handicapper of events is not the investor and it may be a better idea to take the odds from the best, like management, in many cases. “If you don’t know jewellery, know the jeweller”.

    Thank you

  10. Thanks Prof for another enlightening article, Your ability to take simple concepts and weave them into beautiful core pillars of knowledge is outstanding.

    One of my key learning was

    – One should calculate odds on periodic basis, “When facts change, we should change our mind” , also this kind of thinking helps in avoiding self inflicted biases.

    However as an investor using this approach, I should take caution on two factors

    – Negative black swan. Low probability high impact events can derail any calculation.
    – % of success of an event is a value assumed by me and nobody will compensate me for my irrationality


  11. Fantastic post Sir. The principle applies in another form. The probability that we assign to an event may itself be product of several other probabilities. I mean that the variables that we consider to be independent are seldom independent. They are semi-independent. In your example above, the probability of the company to sell the new product will itself will be dependent on the product quality which you had covered as a separate point. That makes multiplication rule all the more interesting. Imagine that product quality is not to the mark vis-a-vis the competition. Then despite the strong sales and distribution machinery, the probability of company’s ability to sell the product will be relatively weak. Is that why Coca Cola was not able to get into readymade fruit juice market? Would love to hear your thoughts.

    Few months back, when Maggi was banned and price of Nestle tanked, you wrote a note about some important questions an investor must consider before deciding to take an investment call. While the variables in these questions are not exactly independent, multiplication rule provides to the investor one good framework for business analysis.

  12. In theory, the multiplication rule sounds great but in practicality how do you assign probabilities to events? Sounds a bit arbitrary to me. Garbage in, garbage out! However, I do understand the larger point that if multiple things need to fall in place for a successful “event”, the probability of that happening is pretty low…

  13. Awesome.. Good to know the multiplication rule and that can be applied in calculating the risk. As usual, your all post are very thoughtful. Thank You for sharing.

  14. On Independence of Events, here is something I read and wanted to share here:

    Source: Charlie munger – Caltech 2008 DuBridge Distinguished Lecture in Beckman Auditorium

    Transcript from:

    Professor Tom Tombrello: Charlie, You’re in the insurance business so I’m going to be a professor and tell a story about derivatives.

    Twenty years ago my wife and I decided to build a house on a tropical island and yet they had hurricanes. Being a Cal Tech professor, thats a risk I didn’t want so I constructed a financial instrument which was to build a house and buy an insurance policy. A hurricane did come and damaged the house severely, and now, more out of blind, dumb luck than skill, I cashed the check from the insurance company about two weeks before the insurance company went belly-up because the insurance company had only insured houses on this island and therefore…


    I hadn’t constructed a hedge at all. I had constructed a common mode failure. The real question in society right now, whether we talk about bundled mortgages or any other hedge funds is whether the risks are really independent and you can determine them or whether somehow there’s these common mode failures that take down both systems at once…

    1. At some level all events are non-independent, for instance if earth were hit by a large meteor destroying humanity like it possibly did dinosaurs. So if we were to plot two events seemingly independent over long periods of time we are likely to find some variable that will affect them simultaneously; say once in 50, 100, 500 or billion years. But its impractical to factor that because we do not live our lives based on the occurrence of an event which is mostly unlikely in the near term but sure in the future. This was pointed out by Borel (Borel’s Law of Large Numbers). Borel said “that events with a probability on the “cosmic” scale of 1 in 10 to the power 50 simply will not happen.”

      Prof. Tom above may have used the illustration to suggest non-independence of events but what if he had been insured by Allianz or National Indemnity for instance. His insurer collapsed because the underwriting was not independent but was dangerously dependent – like having a bond portfolio with all types of fixed income securities belonging to the same conglomerate rated B-; and wise insurer would have spread it out among many ‘independent’ events.

      Of course the fractional reserve system of banking works with the presumption of Borel’s law – that not all depositors will come calling at the same time, in other words are independent – but that came to be severely tested in late 2008.


  15. Brilliant Post, Professor. Thank you!

    Key learning:
    Invest in simple easy to understand businesses with few variables.
    And, while assigning probability value, ensure that is analysed with adequate depth.

  16. you may enjoy considering independance as a function of 4 macro quadrants relative to the sources of value. the matrix is growth +/- on one dimension and inflation +/- on the other dimension. BRK opco’s map to the 4 quadrants as does one of the world’s largest investment managers. I discuss it in my book. here is a graphic.

  17. Sir, As a mathematics teacher to science students I had not given much importance to chapters like ‘probability’ till I found your lectures on Baye’s theorem. Now I got new dimensions on multiplication rule too. It is interesting that my questions on ‘vast diversification’ is also explained !! Thanking you a lot…and loving to study more..

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