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Unfortunately, the needle on the instrument lags the movements of the stick, resulting in unintended consequences. Suppose you are trying to maintain 50 miles per hour, but the speed has decreased to 45 mph; the controls have become less effec- tive, and the wind noise has dropped. You push forward on the stick and monitor the airspeed indicator until it reaches the desired 50 mph. Proud of having avoided a stall; you look back out the wind- shield. Suddenly you are aware of the howling wind and look at the instruments again to see that you are now going You pull back hard on the stick and watch the airspeed indicator, waiting for it to drop back to By the time you reach 50 again, the nose is already pointed too high and you are destined to slow to 40, whereupon the cycle repeats with greater and greater deviations from the desired speed.

This results in the well-known phenomenon of pilot-induced oscillation PIO , which can be violent enough to destroy the air- craft. Ironically, the short-term fix is to let go of the stick altogether until the plane settles down. But a more proactive solution is to focus on the pitch nose-up or nose-down attitude instead of play- ing catch-up with the airspeed indicator.

This is done by using the seat of your intellect to create imaginary lines on the windshield corresponding to the position of the horizon at various speeds, as shown in Figure 5. Eventually this results in calibrating the seat of the pants, whereupon the speed is controlled by adjusting the position of the horizon in the windshield.

The airspeed indicator is barely needed anymore. A simple pitch control flight simulator is available at FlawOfAverages. So what is the most important instrument in the cockpit? The windshield! Correspondingly, what is the most important source of informa- tion in business, government, or the military?

Your customers, constituents, and adversaries are all out there waiting to be observed. The Most Important Instrument in the Cockpit 43 60 mph 50 mph 40 mph 40 0 60 9 1 8 2 80 7 3 6 4 14 5 1. Airspeed indicator 2. Attitude indicator 3. Turn coordinator 5. Heading indicator 6.

Vertical speed indicator Figure 5. And when it comes to analytical models, the most important ones, like the airspeed indicator, are for calibration. They change your thought process to the point that you may not need the models them- selves any more. Even birds avoid it. Under instrument flight rule IFR conditions, your intuition can misinterpret a death spiral for climbing flight.

Because you cannot trust your gut in these situations, you must rely much more on your intellect and use more sophis- ticated instruments. Fred Abrams, a seasoned flight instructor, describes an experience from his own instrument training. He had been flying on instruments in clouds for over an hour when he started experiencing serious vertigo. His inner ear had lost all confidence in the artificial horizon on his instrument panel, and he literally no longer knew which way was up.

The sharp but low-level G forces had the effect of reconnecting the seat of his intellect to the seat of his pants for the remainder of the flight. When you expect an analytical model to give you the right answer, as opposed to the right question, you are flying on instruments, and you better know exactly what you are doing. What is needed is some sort of interactive input that lets the user shake the controls the way Fred did when flying in the clouds.

Too bad the financial industry was not more skeptical of the VaR models on their instrument panels in and The best pilots do not fixate on their instruments during good visibility. However, no pilot would survive without them for more than a minute or two in the clouds. If managing a business is like flying a plane, then analytical models are analogous to the instruments. Use them to calibrate your intuition while visibility is good.

Then use them with caution if you are suddenly socked in by the fog of uncertainty. The subject is usually presented using classical theories, which like the steam locomotive are powerful and elegant, were developed around the same time, and are just as obsolete. This section attempts a different approach. In this section I will present five fundamental Mindles for grasping various aspects of uncertainty. If you achieve a seat of the pants understanding of this section, people will no longer be able to threaten you by blowing smoke with Red Words.

To assist you further in this regard, there is a sum- mary of the basic Mindles in Table P2. Risk Risk is in the eye of the Risk Attitude beholder. I have found that teaching probability and statistics is easy. The hard part is getting people to learn the stuff. Humans learn to read and write. Knowledge is preserved from generation to generation. Humans invent machines. This culminated in the Industrial Revolution, which involved harnessing the power of physics.

At the pinnacle of this era is the field of either physics or mathematics depending on whether you ask a physicist or mathematician , followed by engineering. Below engi- neering is the often ignored field of industrial design, which is devoted to the development of Handles that allow us to grasp the power of physics with our hands. Machines learn to read and write. During World War II machines began to both read and write information using electrical impulses, and the theoretical foundations of computer science were laid.

I would call this the start of the Informational Revolution, and by all accounts we are still just witnessing its dawn. We are not harnessing physics this time. You grasp it with your mind. In his famous book, The Selfish Gene, Richard Dawkins defined the meme rhymes with dream , as a societal analog of the gene.

Probability Management – A Cure for the Flaw of Averages

The success of both is measured by their ability to replicate widely, yet evolve to meet changing environments without losing their essential characteristics. Most memes have evolved naturally. In contrast a Mindle is the result of intentional informational design, that is, it is a designer meme. Did you ever take a statistics course? If so, was it the high point of your week? Probably not. But although people no longer teach steam locomotion, they do still teach Steam Era Statistics, whose precomputer Mindles look like bicycle equations to most of us and are better off forgotten.

Brad Efron, a Stanford University statistics professor who by his own admission is pretty good at Steam Era Statistics, is one of the founding fathers of its replacement, the modern school of Computational Statistics. Or perhaps, instead, you check your morning oatmeal for razor blades that slipped in at the factory? The tension introduced during the popular TV game show, Deal or No Deal, is based on the contrasts between the risk attitudes of members of the audience and that of the live contestant.

Imagine that you are the contestant. An idealized event on the show might go as follows. You are presented with two closed briefcases, each attended by an attractive model in a brief outfit. You must choose a briefcase to remove from the game, thereby keeping the contents of the other. Is it a deal or no deal? But suppose you were a penniless wino on skid row. If I had an extra million, my foundation could fund a science lab for a year and perhaps cure a new disease.

For example, a wino and Bill Gates might attach very different risks to the same uncertainty. I consider uncer- tainty to be an objective feature of the universe, whereas risk is in the eye of the beholder. I myself believe that uncertainty is an inevitable con- sequence of what a friend calls the Grand Overall Design, which I abbreviate as GOD. Regardless of where you believe uncertainty comes from, it is indeed an objective feature of the universe. Not so for risk. Not for me.

The flaw of averages : why we underestimate risk in the face of uncertainty in SearchWorks catalog

That is, I have engaged in a transaction that makes money only if XYZ loses value. I will suffer a loss only if it goes up. Risk is in the eye of the beholder. This is the first of the five basic Mindles of dealing with uncertainty. So if a tree with a coin sitting on a branch falls in the forest and no one is there to bet on the outcome, is there still risk? There is uncertainty as to whether the coin ends up heads or tails, but, because nobody knows or cares, there is no risk.

And as discussed earlier, risk is often associ- ated with uncertainty, but risk is subjective. If you own the stock, the risk is that it will go down, but if you have shorted the stock, the risk is that it will go up. If traffic on the way to the airport is worse than expected, you risk missing a flight to Europe for which you have a nonrefundable ticket. If traffic is lighter than expected, the airline risks not being able to keep your money, while reselling your seat to that standby passenger with the backpack.

This chapter presents a widely used Mindle for visualizing and communicating uncertain numbers. It is a shape with its own dis- tinctive patterns, easily viewed from the right side of the brain. Do you want to bet? The give-me-a-number mentality can hide the obvious fact that starting any new venture is a gamble. In his first career, however, he was a practicing psychologist helping indi- viduals, couples, and families make constructive and needed changes in their lives. Bob views the management of uncertainty as a discipline, requiring a commitment to trade short-term rewards for long-term gains.

He teaches that a new venture is a gamble, that a gamble has odds, and that odds must be at least estimated. What a concept. Statisticians call it the probability distribution, or just distribution for short, and so do I. This is the second basic Mindle for grasping uncertainty. The long-term economic value of the program an uncertain number if there ever was one might look something like Figure 8.

Note that the sum of the heights must be percent, because there is a percent chance that something will happen. The dis- tribution of economic value in Figure 8. Relative Likelihood The drug cures the lab The drug is a commercial rat, but after extensive success. The firm loses a money. Although statisticians view distributions in other ways as well, some of which will make your head spin, I like to start with this bar graph representation, which is called a histogram. The average of a bunch of numbers is the sum of the numbers divided by the size of the bunch. But averages also apply to uncertain numbers.

If you added up the economic outcomes of each one and divided by a million, you would get the average value of the project, which is a single number. Here is a Mindle that relates the average of an uncertain number to its shape. Imagine that the bar graph were sawed out of wood. Then it turns out that the average is where the thing would balance. But give me a break, is not equal to I call this the Weak Form of the Flaw of Averages, and here is an extreme example. But no one would characterize a hijacking as mak- ing a million dollars.

The only way to avoid these problems is to stop think- ing of uncertainties as single numbers and begin thinking of them as shapes, or distributions. The pointer will come to rest somewhere between 0 and 0. Thus a spin of 0. Oops, I forgot to tell you about your risk. The risk is that the value comes in at 0. You could answer all these questions by spinning a spinner thou- sands of times while tediously writing down each resulting number on a clipboard, but can you simulate it in your head?

I suggest that you take a shot at each of these questions now, before continuing. Figure 8. Are you ready for a bike ride? This is one of those times when a picture is worth a thousand words, and a simulation is worth a thousand pictures. The latest simulation software can perform 10, spins nearly instantaneously, but at Chapter 8 on the web site you will be able to run a slow-motion simulation of the spinner to see how a histogram is generated, and answer these questions.

If you ran the simulation on the web, you will have seen that each time the spinner is spun, the bar in the graph corresponding to that number is raised by one notch. That is, if the spin is between 0 and 0. Because the numbers are generated randomly, the exact results are different each time you run the simulation. But after 1 million spins, regardless of how things looked near the beginning, the histogram will have converged to that of Figure 8. The chance of falling into the ruinous first bin of the histogram is 20 percent, and the correct shape is Figure 8.

Although this flat shape is simple enough to work out from first principles, many graduates of statistics courses get it wrong. Of the thousands of subjects I have tested on these questions, roughly a third draw something like histogram c in Figure 8. In the words of Mark Twain, such students have let their schooling interfere with their education. Interactive displays of this type can help management visualize the risk implications of their decisions far better than single numbers.

Mindle 2: An Uncertain Number Is a Shape 63 The smooth line on this graph, known as the cumulative distribu- tion, displays a different representation of the same information. This curve indicates the chance of exceeding any given NPV. People typically use histograms to eyeball the relative odds of various outcomes, while the cumulative graph is for reading off more accu- rate probabilities of achieving different levels of success.

Ameo, now a private consultant, continues to model decisions made under uncertainty as gambles. Give Me a Distribution If we are ever to conquer the Flaw of Averages, bosses must begin to ask for distributions rather than numbers. How would you plug such a thing into your business plan?

This is where Probability Management comes in. In terms of the spinner, think of the DIST as consisting of the outcomes of one thousand spins, stuffed, like a genie in a bottle, into a single cell in your spreadsheet. This concept will be discussed in more detail in the last section of the book. Black Swans So once you have the histogram of an uncertain number, you know all there is to know about the distribution of outcomes.

Of course not. Take the spinner, for example. What if you spin the pointer so hard, that it flies off and hits you in the eye, or what if the friction of rotation sets the cardboard on fire, burning down your house? But they could happen. When a seventeenth-century explorer discovered a black swan in Australia, it made quite a stir. The best defense against Black Swans is the right half of your brain.

Psychologist Gary Klein has developed an exercise he calls the Pre Mortem 5 that helps in this regard. The idea is to vividly all imagine your plans in shambles, and then creatively explain how it all went bad. Mindle 2. The heights of the bars indicate the relative likelihood that the number takes on various values, and they must sum to percent. A Word from Your Author If this stuff is too technical for you, feel free to jump to the next chapter; otherwise, read on. If your left brain is obsessed with perfectly smooth curved distribu- tions, just make a histogram with a whole lot of really narrow bars.

If you think of an uncertain number as a bar graph, you will not be seriously misled. Median and Mode Two concepts related to the average are the median and the mode. That is, it is the point at which the bars add up to exactly 50 percent. If the histo- gram is symmetric in shape around its middle, the median is the same as the average, but this is not always the case. But on average, they are now all multimillionaires. The mode is also called the most likely, but from the figure you can see how misleading this descrip- tion is. In any event, the median and the mode are again single numbers, which fail to capture the shape.

The Law of Averages and Where It Fails The longer you run the simulation of the spinner, the closer the average will tend toward 0. This law is widely misinterpreted by people who think it justifies plugging in averages everywhere in place of uncertainties, and by now you know where that leads. But the Law of Averages sometimes fails all on its own. Consider the number 1 divided by the outcome of the spinner that is, take the reciprocal of the spinner. In the left-hand graph, by the time trials have been run, the average has reached 0.

In the right-hand graph, the average looks like it has converged to 8 by trial But shortly afterward, a single spin came so close to zero, that its reciprocal was huge, dragging the average from 8 to 16 in one shot. In theory, this graph would never converge, and the uncertain number would have no average, thereby kicking the law of averages in the teeth. One of his assignments at the bank was to establish a line of credit for the film company that had produced the hit Dirty Dancing and several less successful films.


From this and similar projects, Medress gained expertise in valuing theatrical and televi- sion property rights. After working as an entertainment banker for several years, he founded his own valuation firm, Cineval LLC, in the mids. In early , Medress attended a simulation seminar of mine in Palo Alto. During lunch he described how he was putting together an investment in film properties and wondered whether it made sense to simulate the uncertainty of the portfolio for the investors. I told him it would be dereliction of fiduciary duty if he did not run a simulation, and a few weeks later we had our meeting at the Georgian in Santa Monica near his office to explore the idea.

Also remember that a spin of less than 0. Figure 9. In the prospectus for the new investment it states that two spinners will be spun. How does this second investment compare to the first? What is the shape of the average of two spinners? See Figure 9. Go ahead, make a guess before continuing. To understand what happens when you spin two spinners, I find it useful to think about what happens when you roll dice. If you roll a single die, you can get the numbers 1 through 6 with equal likeli- hood, as displayed in Figure 9.

When you roll two dice, you can get any of the numbers from 2 through There are more combinations for getting numbers at the center of the distribution than at the ends; so the shape goes up in the middle, as shown in Figure 9. Similarly, two spins of the spinner can average 0. Congratulations if you chose shape a in Figure 9. Only about half the graduates of statistics courses that I test get this one right. So what happens when you combine more than two spinners or dice? Figures 9. At FlawOfAverages.

Mindle 3: Combinations of Uncertain Numbers 71 Figure 9. So What? What does that have to do with risk? In fact, the chance of ruin with two spins is only 8 percent as compared to 20 percent with the single spin. This is proportionally a greater risk reduction than going from two bullets to one bullet in a game of Russian roulette!

It implies that, as you add up or average more and more independent uncer- tain numbers, the shape of the histogram approaches the famous bell-shaped, or NORMAL, distribution, as shown in Figure 9. Note that I have used smooth curves here instead of histograms, because five overlaid histograms on a single graph would cover each other up. It is far too red to mention in, say, a singles bar, but far too important to ignore. Luckily there is a Green Word that comes sufficiently close. Mindle 3: Combinations of Uncertain Numbers 73 What is a Green Word for why the distribution of two spins or two dice goes up in the middle?

Diversifying your investment across two dice or two spinners increases the chance of an average outcome and reduces the chance of extreme outcomes. As explained in the last chapter, spinners—and dice for that matter—are a great way to think about uncertainty and risk. Medress and I took just a few minutes to build a model in Excel and to simulate the outcomes of portfolios of various numbers of films on the computer. Conceptually, it worked like this. Imagine painting each of the 28 movie profits on ping-pong balls and throw- ing them into a lottery basket.

Now if you crank the lottery basket and pull out a ball, you will get an uncertain number. How is that number distributed? It is simply the distri- bution of the original numbers on the ping-pong balls. Technically, this approach is known as resampling, because all you do is resample the ping-pong balls over and over, throwing them back into the basket after each draw. It is a basic building block of computational statistics that will be covered in a later chap- ter.

Of course, a computer can simulate hundreds of thousands of cranks of a lottery basket per second, so it goes a lot faster than it did when the founding fathers of Steam Era Statistics compelled their graduate students to perform such drudgery with dice and real numbered balls to confirm their theories. The Power of Diversification If you invest in a single film chosen at random from this genre, Figure 9. Medress and I simulated what would happen if you diversified your investment over two or more films drawn independently from this distribution.

The simulation shows that the average profit does not change as you diversify. However, because the losers and block- busters tend to balance each other out, the shapes of resulting histo- grams and hence the risk of loss differ greatly, as shown in Figure 9. After viewing these results, I observed that only a fool would invest in a slate of less than four films. This clearly shows that, as you increase your diversification, you reduce both the upside potential and downside risk of your investment.

I hope this is of benefit to nonstatisticians who are nonetheless interested in statistician humor. Medress first used simula- tion while assisting a domestic television distributor who was raising funds for TV movies and miniseries. The distributor was able to pro- vide him with representative historical data to resample, as described earlier. But Medress warns that this is not always easy to come by.

Medress then excluded films that were not comparable to those planned for the fund. After draw- ing a flat histogram for the average of two spinners, a PhD student told me sheepishly that he thought it would take more than two spinners before the shape started going up in the middle. Actually the step from one spinner to two is the most dramatic in this regard. Hopefully, like Rick Medress, you will internalize the lessons of diversification and apply them often. So, long ago, even before the Steam Era, mathematicians came up with a yardstick for measuring the degree of uncertainty that has become the gold standard.

One Red Word is confusing enough, let alone three of them and a Greek letter that all mean pretty much the same thing. Basically, they all measure how wide the distribution of an uncertain number is. This is usually obvious from looking at the shape see Figure Although virtually all have heard the term, only a small fraction can actually define it. What kind of gold stan- dard is that? In the hands of technically competent managers, this can reduce the variability of a product or service, yielding more consistent quality. For comparison, I also ask people which pedal would they step on if a kid suddenly ran out in front of their car chasing a soccer ball.

For this question, the entire class responds immediately. And to their credit, even the few misan- thropes who claim they would step on the gas pedal instead of the brake respond quickly and decisively. Ah, some readers are thinking, the solution to this problem is a better way of teaching people what SIGMA really means. Imagine that learning the value of a particular uncer- tain number corresponded to apprehending a criminal. Then the average and SIGMA are comparable to the height and weight of the suspect, and they provide little detail. This may be important in a Sherlock Holmes mystery, but today we use security cameras, mug shots, and DNA samples to nail the bad guy.

Outputs of Simulations In Figure The balance point of this graph is the average. The cumulative graph displays the chance that the uncertain number is less than any particular value. The same information is carried in the percentiles, shown on the right. For example, the 20th percentile is 1. One additional Mindle is required to understand groups of uncertainties: the interrelationships between them, as discussed shortly. This might correspond to the gang affiliation of our sus- pect. But when uncertain numbers travel alone, the graphs and percentiles of Figure In the special case that the uncertain number has a perfect bell- shaped distribution, the average and SIGMA, taken together, provide a shorthand notation for the same information.

But for all other dis- tributions, they provide less information. SIGMA did play an important historical role in the develop- ment of theoretical probability and statistics, and it is still used today in many areas of science and in quality control. But for most of us, it is irrelevant and becoming increasingly so with the spread of Computational Statistics. I would have liked to have included a bit more on this subject in the book, but did not want to risk making some of my readers lose their lunch. The beauty of the Internet is that I can put such material on the web for those who want it.

But the average state of the drunk is dead. The bank had difficulty in forecasting the total paid out in incentive bonuses, and Terri was looking for new perspectives on the problem. She explained her insight with a simplified example, as fol- lows. Suppose the number of checking accounts sold per year by employees varied widely but averaged So zero must be the average bonus.

Think again. Of course, the real problem is more complex, but it should be clear that designing an incentive plan around an average employee is futile. In represent- ing the profit of each film by its average, you are blind to the risk implications, but at least you get the average profit of the portfolio right. The bonus of the average employee is not the average bonus. This is the fourth Mindle for grasping uncertainty. Recall the following examples of the strong form from Chapter 1.

Like a successful first date, one is left with a sense of anticipation. But frankly, the most common result of such an encounter is deafening silence. But that was before I knew what to expect from Terri Dial.

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As it turned out, Wells Fargo was then in the process of acquiring another bank, which consumed their attention for the next couple of years. But with that behind them in , I got a call from Matthew Raphaelson, a former student and my primary contact at Wells. They wanted me to visit again. Wells was using Crystal Ball simulation software to model bonus incentive plans and several other activities involving uncertainty. When the light- bulb had gone on for Terri Dial two years earlier, it stayed on. Since then the bank has maintained a cul- ture of analytical thinking and Flaw-of-Averages awareness through continued executive education programs.

From the relaxed and interac- tive manner in which Pyles engaged my class, it was clear he had coached Little League baseball. In his day job, he manages and advises 45, bank employees and has to deal with complaints, questions, and sugges- tions on an individual level. With a group of that size, again like a baseball coach, he must also deal with the statistics. Mindle 4: Terri Dial and the Drunk in the Road 87 those at the bottom.

You need specialists for filling in holes in the product space. Suppose your firm is plan- ning to purchase a natural gas reservoir known to contain a million units of gas.

Unfortunately, if you decide to purchase the reser- voir, the transaction will take a month to complete, and the price of gas may have changed before you have the opportunity to pump it. Your boss asks you to estimate the price of gas in one month so that he can calculate the value of the property. Here is a Mindle for thinking about the average or expected value of the reservoir given the uncertain future price of gas. Imagine that instead of purchasing a single reservoir, you were pur- chasing one-thousandth of a share in 1, reservoirs, each on its own planet.

There is no future uncertainty in gas price. If the gas price goes up, the value goes up. You have the option not to pump, and this limits the downside to zero, as displayed in Figure See Figure Now what you choose to pay for such a property is your busi- ness, but if you do not recognize that its average value is way over the value based on average gas prices, then you will be outbid by people who recognize the option opportunity. The last example shows that the Flaw of Averages cuts both ways. Sometimes when you plug averages of uncertain values into your plans, it overestimates the average outcome, and sometimes it underestimates the average outcome.

Yes, a brash blond American woman running a stodgy British bank sounds like the basis of a Monty Python sketch, and it could have turned out that way. But instead in she was listed as one of the 50 most influential Americans in the United Kingdom and has earned the admiration of the British business press, who rather enjoy having a Human Cyclone in London. In , Stefan Scholtes, a colleague at the Judge Business School at Cambridge University, and I assisted in this effort by initiating a series of courses for Lloyds executives. Whereas top managers may pay lip service to executive education and will even fund a course for their middle managers, they rarely take the time to internalize the sobering lesson of the drunk in the road.

She came up to Cambridge from London her- self for two and a half days, bringing 16 of her managing directors with her. Stefan and I delivered a prototype version of the course to this group, which left its imprimatur on it for the many classes of Lloyds executives that followed. Terri had already left by the time Stefan and I taught our last course for Lloyds.

Because she was no longer there, I felt freer to gather opinions about her performance from the attendees. To a woman and man , they expressed admiration and amazement at what she had accom- plished in such a short time. I hope she comes out on top. Familiarity with the concept will help you avoid the former and seize the latter. In the last chapter we saw that the Flaw of Averages cuts both ways. When you plug averages of uncertain values into your plans, they sometimes overestimate the average outcome and sometimes underestimate the average outcome.

What follows are some general rules that characterize these two situations. Spreadsheets with Uncertain Inputs The prevalence of the Flaw of Averages is due to the fact that millions of spreadsheet users plug their best guesses of uncertain numbers into their models, naively believing that they are getting the best outputs. Of course, with a name like that, no wonder no one else has heard of it. On second thought, you would probably gag on the Red Words.

You had better go with my explanation. Johan Ludwig William Valdemar Jensen was a Danish telephone engineer in the late s, practically before there were telephones in Denmark. Jensen proved some important theorems, the best-known of which is his famous inequality. A Note from Your Author The paragraphs coming up may be challenging for some readers. They are also some of the most important for developing intuition into the Strong Form of the Flaw of Averages.

I suggest that you at least glance at the figures, and read the part about smiles and frowns. There are four cases. The average value of the formula equals the formula evaluated at the average input. The average value of the formula is greater than the formula evaluated at the average input.

The average value of the formula is less than the formula evaluated at the average input. The graph of the formula is none of the above. Now for a spreadsheet model to be LINEAR, it must pretty much use only addition, subtraction, and in some instances multiplication. In such cases, running a simulation is usually the only practical way to estimate the true average output of your model. English was his downfall, forcing him to be held back a year making us brothers in English teacher psycho drama.

Some 16 years later, he was a professor at Cambridge University in England, where we developed the courses for Lloyds TSB bank described in the last chapter. In teaching the Flaw of Averages to his MBA students, Scholtes describes two common characteristics of business plans that lead inexorably to the Strong Form of the Flaw of Averages: options and restrictions. The gas reservoir described earlier in this chapter demonstrates this principle. The average value of the property was more than the value associated with the average gas price because you had the option not to pump if the price fell see Figure Scholtes describes this graphically.

If it smiles at you, this is good news. Restrictions If some outcomes of an uncertainty lead to restrictions in your future actions, then the average outcome of your plan will be worse than the outcome of the plan based on the average value of the uncertainty. This problem is demonstrated by the microchip produc- tion case see Figure 1. A classic case of the Strong Form of the Flaw of Averages is related to the subprime mortgage fiasco. It involves the rela- tionship between mortgage default rates and housing prices. In times or locations where property values fall, defaults tend to go up with profits going down.

In times or locations where property values rise, defaults tend to go down with profits going up. It is tempting for analysts to base the profits of their mortgage portfolios on average property values, but this overestimates the average profit. Consider a diversified portfolio of mortgages spread across various housing markets. Suppose that property values are expected to rise in some of these markets but fall in others, remaining the same on average. What do you suppose the profit graph looks like with respect to property values? In locations where values increase, defaults drop slightly, increasing profit slightly.

But where values fall, defaults go up. In some cases values will fall to the extent that the equity in the houses drops below the amount owed. At that point defaults increase dramatically, with owners just dropping off the house keys at the bank and moving into Motel 6. This is reflected in Figure In this example, an 8 percent increase in value in one location improves profit by less than 5 percent, whereas an 8 percent devaluation in another location decreases profit by a whopping 40 percent.

Thus the profit of a mortgage portfolio based on what are expected to be average property values will overestimate average profit. I could cite tactical examples that we have used in the business, but this is confiden- tial stuff, right? Because he was math-phobic, he had put off taking any quantitative courses in college as long as possible. In his last term, with a perfect 4. The converted make the best proselytizers, and today Michael is president of Applied Quantitative Sciences, Inc.

The problem was to value the portfolio of exist- ing and new products over an uncertain future. Historically, each new product team was responsible for developing their own forecasts of future demand, whereas the marketing department created the demand forecast for all currently commercialized products. To their credit, some of these teams were not blindly churning out single average estimates of demand but were actually running simulations to produce distributions. This is one of those cases where each team was shaking its own ladder.

But the full portfolio of current and future products was like a bunch of ladders connected by planks, which meant that they could not be simply added up. Consider, for example, just one new product, as forecast by its development team, and one existing product, which might be replaced by the new one, as forecast by the marketing team. Up to this point everything is valid. However, if the corporation consolidated these results, it would come up with the four outcomes shown in Figure But the only way to keep high demand on the existing product is for the replacement product to fail, and the only way to get high demand on the new product is to cannibalize the existing product.

So outcomes 1 and 4 are both impossible, and the correct distribution of outcomes looks more like Figure By running the simulations separately, the product and market- ing teams had not taken into account the interrelationships between the demands of the existing and new products. The fifth Mindle for grasping uncertainty is the notion of interrelationships like this one.

So I will stick to interrelated uncertainties instead. It is not only more general but also does not require Red Words. As another example of interrelated uncertainties, consider the stock prices of two oil companies. But since oil price is not the only influence, it is also possible for the stocks to move in opposite directions, perhaps due to new taxation or environmental regulation that affect the firms differently. Interrelated uncertainties are at the heart of managing investments, as first described in the Nobel Prize—winning Portfolio Theory of Harry Markowitz in Here I will explain some Mindles underlying their Nobel Prize—winning work.

Three Idealized Investments For purposes of introducing interrelated uncertainties, I will present three hypothetical investment opportunities. Mindle 5: Interrelated Uncertainties 2. However, there is great uncertainty, as shown in Figure On the downside, for example, an experimental engine that runs on seawater might reduce petroleum demand. Airline stock: The second investment is airline stock. Imagine that the uncertainty here is exactly the same as that of petroleum. Licorice: The final investment is in licorice, the black rub- bery confection. For comparison, the three uncertainties are displayed in Figure Suppose you had to put every last penny in one of these three investments.

Where would you put it? An investment in any of these investments is equivalent to any of the others. Portfolios Of course, in the real world, no one ever has every last penny invested in a single asset. When you have money in more than one uncertain investment, it is called a portfolio, and two kinds of effects come into play. The first of these portfolio effects is the straight diver- sification that we observed in the film investments. The second effect arises from interrelations among the assets. Neither of these effects can change the average return of a portfolio, but they both can change the degree of uncertainty, which ties to risk.

Recall that people have different attitudes toward risk, ranging from those who go for broke to those who cover their butts. But from a risk perspective, they are not all the same due to the interrelationships among the constituent assets. What are the primary interrelationships? The cost of fuel is a large fraction of the overall operating expense of an airline.

So when petroleum prices go up, they cut into the bottom line, and you would expect airline stock to be driven down. Conversely, when petroleum goes down, you would expect airline stocks to rise due to lower fuel prices. The real world is more complicated than this, especially because the airlines already know what I am talking about, and they use derivative financial instruments to reduce their risk in this regard.

But for this chapter I will assume that the only significant interrela- tionship among the investments is between petroleum and airlines. A Mindle for Interrelated Uncertainties: The Scatter Plot Imagine that over the years, you had kept a record of the changes in both petroleum and airline stock prices and people have. If these changes were graphed in a scatter plot, they might look like the left side of Figure Each point corresponds to one year, with the x coordinate representing the change in petroleum price, and the y coordinate representing the change in airline stock.

Each uncertainty on its own is bell-shaped, but a high value of one tends to lead to a low value of the other, and vice versa. Now as a butt coverer, you are about to invest all your money in one of these three portfolios. To force yourself to make an intellectual commitment, scribble your choice on a piece of paper before going on. If I just gave you the answer, it would be too easy and you might not learn anything.

As they say, give a man a fish and you will feed him for a day; teach him to fish and he will sit in a boat all day and drink beer. So I will help you discover it on your own. We will start with licorice and petroleum because we are assuming they are not interrelated. I like to think of each investment as rolling a die. Then the licorice and petroleum portfolio is like rolling two dice, resulting in the shape in Figure What part of this shape do you hate as a butt coverer? If you answered the extremes then, again, your schooling has interfered with your education. Even butt coverers love to get rich.

The Flaw of Averages

What they hate is rolling a two on the far left. So suppose your broker comes to you with the bad news that the new Seawater Engine has made petroleum worthless. What does this do to the chances that licorice causes cancer? Nothing, they are independent. As a result, even with your petroleum investment in the toilet, there is still only one chance in six of the dreaded double dots known as snake eyes. Mindle 5: Interrelated Uncertainties Exactly the same behavior is exhibited by airlines and licorice because these two assets are also assumed to be independent.

But now suppose you owned petroleum and airlines. Again, news about the Seawater Engine will cause your petroleum investment to tank, but what does this do to the chance that your airline stock dives as well? The actual distribu- tions would look more like those shown in Figure When I present this example to my university and executive classes, a common knee-jerk reaction is to reject petroleum and airlines because the elements are interrelated.

The correct knee-jerk reaction is to reject portfolios whose interrelations are direct or positive ; for example, a portfolio consisting only of two airline stocks might appear as shown in Figure The resulting value of 0. Figure However, an astute observer who created a scatter plot from the data might perceive a nonlinear relationship between the variables, as shown in Figure Scatter plots are my favorite way to grasp the interrelationships between uncertain numbers.

Mindle 5: Interrelated Uncertainties 1. While we are on the topic of interrelationships, we must not forget how uncertain numbers may be related to themselves over time. As an example, if a stock price goes up today, it has almost no bearing on whether it will go up or down tomorrow. On the other hand, historically, if interest rates get very high over some time period, they are more likely to go down in the future, whereas, if they are very low, they are more likely to go up.

There are also uncertainties that change dynamically over time, for which using a single number is even more misleading. Take housing prices, for example. What would happen if prices fell in some location to the point that a few homeowners now owed more to the bank than their houses were worth? They would either walk away or be foreclosed on. How would future housing prices relate to this? Well, the banks who now owned the houses would try to unload them quickly in an already depressed market, and the increased supply of houses would further drive down prices.

These new lower prices would force even more homeowners under- water on their mortgages, and guess what? There would now be even more abandoned houses to drive the prices even lower. In fact, it is called a bubble burst, and it can have very scary effects on the economy, as you may have noticed.

The problem is that even if you know a bubble is going to burst, it is very hard to predict when it will happen and how loud the bang will be. These situations are governed by what is known as chaos theory. So how exactly was the risk of the housing bubble monitored? According to Felix Salmon in a February article in Wired Magazine3, the huge market in mortgage backed securities that crashed and burned in was largely based on a formula called the Gaussian copula function, derived by mathematician David X.

As such, it must be worth some- thing. In fact, most people agree that we now have an information economy. Part 3 formalizes the connection between decisions and information. It was the winter of , when at age 6, I asked my father what he was working on. He had received a Guggenheim fellowship to write his book, The Foundations of Statistics,1 and was on sabbatical from the University of Chicago.

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A man in a restaurant is trying to decide between the fried chicken and the roast beef. The preceding restaurant decision would be deemed irrational, because whether the restaurant had duck was irrelevant to the choice between chicken and beef. But my father understood that life was so complicated that this could not easily be adhered to in practice. A few sentences later he writes: It is even utterly beyond our power to plan a picnic or to play a game of chess according to this principle. But he believed it was at least a good starting place for a theory of decision making. In point of fact, he has been proven wrong in the particular case of chess; today, computer programs are the reign- ing champions.

So score one for rationality. But when it comes to decision making under uncertainty by humans using their bare brains, experiments show that even sophisticated people often behave irrationally. Creating art, for example, requires at least tacit deci- sion making, yet art that springs from rationality instead of emotion is contrived. My position is that decisions are made using the seat of the intellect at one extreme and the seat of the pants at the other and that the best decisions are those upon which both extremities agree.

Because of his early work on decision making in the face of uncertainty, my father is often considered also to be one of the founding fathers of decision analysis. However, the field was not formally defined until the mids by Professor Ronald A. Howard of Stanford University. Just as you can actually see the minute hand move on a very large clock, you could sense yourself getting smarter in his class. I have heard many others describe their eye-opening experiences in this course on rational thinking; ironically, in quite emotional terms.

Decision Trees Decision Trees One of the great Mindles of decision analysis is the decision tree. I will introduce this concept with a simple example that can be done in the head, freeing the mind to more quickly absorb the mechanics of the process. Suppose you must choose between a Good Time and a Stick in the Eye. This situation can be displayed in a decision branch, as shown in Figure The basic idea is that, given multiple alternatives, you should pick the best one, which is displayed by the solid line. So far, this is hardly rocket science.

We cross the line into decision analysis, however, when we intro- duce uncertainty. This situation is displayed in Figure Following common convention: the uncertainty nodes are circular. Note: In true decision analysis, this formula would be modified to account for risk attitude, as discussed in Chapter 7.

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But where does that 10 percent chance of getting caught come from? At this stage, the naive decision analyst is in danger of getting laughed off the stage by trying to justify a precise probability. If you experimented with the model at FlawOfAverages. As John Tukey reminded us in Chapter 4, these answers are often to the wrong question.

On the other hand, simple models can sometimes quickly provide the right question. Savage finishes the book with a discussion of the emerging field of Probability Management, which cures this problem though a new technology that can pack thousands of numbers into a single spreadsheet cell. I highly recommend The Flaw of Averages. Perry , Former U.

Secretary of Defense. In this profound and entertaining book, Professor Savage shows how to make all this practical, practicable, and comprehensible. Request permission to reuse content from this site. Undetected location. NO YES. Selected type: Paperback. Added to Your Shopping Cart.

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