Seeking Wisdom

by Peter Bevelin

  The effect of exponential growth

  john's son David made a proposal '1 take out the garbage every day for a month, and you only have to pay me a penny today and then every day afterward I want double as much as I got the day before. "

  The sequence of numbers 2, 4, 8, 16 grows exponentially. Day 2 his son would have 2 cents, day 3, 4 cents. After 27 days, he would have $1.3 million dollars. The individual growth is constant - 100% a day- but the sum gets higher faster and faster. This is the power of doubling.

  As we have seen there are limits to prolonged growth. Take bacteria as an example. Assume that a certain strain of bacteria divide in one minute. We put the bacteria in a bottle at 11 am and the bottle is full at noon. When was the bottle half full? - 11 :59 am. Just one minute earlier.

  Even a small number of steady growth leads eventually to doubling and redoubling. For example, a country whose population grows by 2% a year, double in size in 35 years and redoubles in 70 years. A simple formula for doubling time is found by dividing 70 by the percent growth per year.

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  Compounding refers to "interest on interest." If we invest $1,000 with a return of 6% a year, we receive $60 in the first year. If we reinvest that $60, next year we get another $60 from our original $1,000 investment, plus $3.6 from the $60 we reinvested. If we reinvest all our returns, the total value of our original $1,000 investment after 5 years is: $1,000 x 1.06 x 1.06 x 1.06 x 1.06 X 1.06 = $1,338.

  Time is the key to compound interest. Over short periods, compounding produces a little extra return. Over long periods, it has an enormous effect. Invest

  $2,500 each year for 40 years at 10% return and you will be a millionaire.

  The time value of money

  A bird in the hand is worth two in the bush.

  -Aesop

  Why must we reduce the value of money we receive in the future?

  Money paid in the future is worth less than money paid today. A dollar received today is worth more than a dollar received tomorrow. If we have a dollar today, we can invest it and earn interest making that dollar worth more than a dollar in the future. This means that money has a price and that price is interest.

  How much should we pay today for the right of receiving $1,000 a year from now? Or, how much do we need to invest today in order to have $1,000 a year from now? It's the same question. The answer depends on the interest rate. If the rate is 6%, then the answer is $943. If we invest $943 today at 6%, we have $1,000 one year from now. $943 is the present value of $1,000 a year from now. We have discounted or reduced $1,000 to its value today. The further out in time we receive the $1,000 or the higher the interest rate is, the less the present value is.

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  - FIVE -

  PROBABILITIES AND NUMBER OF POSSIBLE OUTCOMES

  Probability is the very guide of life.

  Marcus Tullius Cicero

  How likely do we believe it is that some event will occur? Probabilities are like guesses. But as Richard Feynman said in his Caltech Lectures on Physics: "There are good guesses and there are bad guesses. The theory of probability is a system for making better guesses."

  We can either estimate the probability based on its relative frequency (proportion of times the event happened in similar situations in the past) or we can make an educated guess using past experiences or whatever important and relevant information and evidence that is available.

  We can also count possible outcomes. The only time we can calculate the exact probability of an event in advance (over a large number of trials) is in cases where we know all possible outcomes and where all outcomes are equally likely. This is applicable for games of chance such as tossing a coin or rolling a die. However we use the notion of probability, we need to follow its basic rules.

  How like/,y is it that a hurricane strikes Texas?

  According to the National Hurricane Center, there have been 36 hurricane strikes in Texas from 1900 to 1996. Based on past experience and barring no change in conditions, we can estimate that there is about a 37% (36/97) chance that a hurricane will strike Texas in any given year. This figure - 36/97 - is also called the base rate frequency of outcomes (hurricanes in Texas).

  We must make sure that the conditions that produced the relative frequency can be expected to be pretty much the same before we can use it as a guide for the future. We must also look at variations of outcome and severity (how much damage an event may cause). Take tornadoes as an example. According to the National Climatic Data Center, between 1950 to 1999 there has been an average of 810 tornadoes yearly in the U.S. But in 1950 there was 201 tornadoes (causing 70 deaths), in 1975, 919 (causing 60 deaths), and in 1999, 1,342 tornadoes (causing

  94 deaths).

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  A doctor says, "This is the first time I've seen this disease. I estimate there is a 50-50 chance that the patient will survive. "

  This statement has only two possible outcomes. Either the patient dies or not. Does it really make any sense to say "a 50-50 chance" if there are no past data or other evidence to base the probability on? Does it really tell us something? If there are no historical, comparable or representative data or other evidence to base an estimate on, the probability figure only measures the doctor's belief in the outcome of the event.

  Another doctor says, "According to medical records of similar cases, under the same conditions, 50% of the patients survived five years or longer."

  The more representative background data or evidence we have the better our estimate of the probability.

  To narrow down the probability figure even more, we need a relevant comparison group, i.e., a group to which a frequency refers. In the hurricane example, we defined the probability for a specific comparison group, referring to the relative frequency with which hurricanes have occurred (37 times in Texas during the 97 years for which we have data).

  Events may happen with great frequency or rarely. Some events are not repeatable and some events have never happened before. For certain events, past experience may not be representative. Others are characterized by low past frequency and high severity. Unforeseen events occur where our actual exposure (measures vulnerability and potential cost or loss) is unknown. Sometimes people react to an event by avoiding or preventing it in the future causing a change of the events future probability. Other times one bad event may increase the chance of another. For example, an earthquake may cause landslides, floods, or power blackouts. The more uncertainty there is, the harder it is to find a meaningful probability number. Instead our estimate must be constrained to a range of possible outcomes and their probabilities.

  Uncertainty increases the difficulty for insurers to appropriately pnce catastrophes, such as hurricanes or earthquakes. Warren Buffett says:

  Catastrophe insurers can't simply extrapolate past experience. If there is truly "global warming," for example, the odds would shift, since tiny changes in atmospheric conditions can produce momentous changes in weather patterns. Furthermore, in recent years there has been a mushrooming of population and insured values in U.S. coastal areas that are particularly vulnerable to hurricanes, the number one creator of super-cats. A hurricane that caused x dollars of damage 20 years ago could easily cost 1Ox now.

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  Occasionally, also, the unthinkable happens. Who would have guessed for example, that a major earthquake could occur in Charleston, S.C.? (It struck in 1886, registered an estimated 6.6 on the Richter scale, and caused 60 deaths.)

  But it may still be possible to price sensibly. Warren Buffett says:

  Even if perfection in assessing risks is unattainable, insurers can underwrite sensibly. After all, you need not know a man's precise age to know that he is old enough to vote nor know his exact weight to recognize his need to diet.

  Warren Buffett also considers a worst case scenario:

  Given the risks we accept, Ajit [Ajit Jain; manager of Berkshire's reinsurance operations] and I constantly focus on our "worst case," knowing, of course, that it is
difficult to judge what this is, since you could conceivably have a Long Island hurricane, a California earthquake, and Super Cat X all in the same year. Additionally, insurance losses could be accompanied by non-insurance troubles. For example, were we to have super-cat losses from a large Southern California earthquake, they might well be accompanied by a major drop in the value of our holdings in See's, Wells Fargo and Freddie Mac...

  We do, though, monitor our aggregate exposure in order to keep our "worst case" at a level that leaves us comfortable.

  How reliable is past experience for predicting the future? In Against the Gods, Peter Bernstein refers to a 1703 letter written by German mathematician Gottfried Wilhelm von Leibniz to the Swiss scientist and mathematician Jacob Bernoulli referring to mortality rates: "New illnesses flood the human race, so that no matter how many experiments you have done on corpses, you have not thereby imposed a limit on the nature of events so that in the future they could not vary." Even with the best empirical evidence, nobody knows precisely what will happen in the future.

  After the September 11, 2001, catastrophe, Warren Buffett wrote on the importance on focusing on actual exposure and how using past experience sometimes may be dangerous:

  In setting prices and also in evaluating aggregation risk, we had either overlooked or dismissed the possibility oflarge-scale terrorism losses... In pricing property coverages, for example, we had looked to the past and taken into account only costs we might expect to incur from windstorm, fire, explosion and earthquake. But what will be the largest insured

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  property loss in history (after adding related business interruption claims) originated from none of these forces. In short, all of us in the industry made a fundamental underwriting mistake by focusing on experience, rather than exposure, thereby assuming a huge terrorism risk for which we received no premium.

  Experience, of course, is a highly useful starting point in underwriting most coverages. For example, it's important for insurers writing California earthquake policies to know how many quakes in the state during the past century have registered 6.0 or greater on the Richter scale. This information will not tell you the exact probability of a big quake next year, or where in the state it might happen. But the statistic has utility, particularly if you are writing a huge statewide policy...

  At certain times, however, using experience as a guide to pricing is not only useless, but actually dangerous. Late in a bull market, for example, large losses from directors and officers liability insurance ("D&O") are likely to be relatively rare. When stocks are rising, there are a scarcity of targets to sue, and both questionable accounting and management chicanery often go undetected. At that juncture, experience on high-limit D&O may look great.

  But that's just when exposure is likely to be exploding, by way of ridiculous public offerings, earnings manipulation, chain-letter-like stock promotions and a potpourri of other unsavory activities. When stocks fall, these sins surface, hammering investors with losses that can run into the hundreds of billions.

  Even if we for some events can't estimate their probability, there may be some evidence telling us if their probabilities are increasing or decreasing. Ask: Do I understand the forces that can cause the event? What are the key factors? Are there more opportunities for the event to happen?

  Warren Buffett says on terrorism:

  No one knows the probability of a nuclear detonation in a major metropolis area this year... Nor can anyone, with assurance, assess the probability in this year, or another, of deadly biological or chemical agents being introduced simultaneously... into multiple office buildings and manufacturing plants...

  Here's what we do know: a. The probability of such mind-boggling disasters, though

  likely very low at present, is not zero. b. The probabilities are increasing, in an irregular and immeasurable manner, as knowledge and materials become available to those who wish us ill.

  The more opportunities (possible wanted or unwanted outcomes) an event has to happen in relation to what can happen (all possible outcomes), the more likely it is to occur.

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  Number of possible outcomes

  Toss a coin once. What can happen? There are 2 possible outcomes. Roll a die once. There are 6 possible outcomes. All equally likely. Roll a die twice. What can happen? There are 6 possible outcomes on each roll and therefore 36 possible combinations or outcomes when rolling a die twice. Roll a die 3 times. There are 216 possible outcomes.

  This is a simplified way of saying that the more possible outcomes an event has (in number or time), the less likely a specific outcome is (for example only one outcome satisfies the wanted event: "roll a die once and observe a six") and the more likely some outcome is (there are 6 possible outcomes to choose from). The more possible outcomes a specific event has, and the more they are unwanted, and the more independent events that are needed to achieve a scenario, the less likely it is that the wanted scenario happens. Some outcomes

  may be less likely than others (for example, due to constraints or limits).

  Treat rolling a die 3 times as 3 separate events where each event is "observe a six". What we see from the above is that the more events that must happen to achieve some scenario or wanted outcome ("3 sixes in a row"), the less likely the scenario is to happen. Observing "anything but 3 sixes in a row" is an unwanted event. There are 215 outcomes or ways for this unwanted event to happen out of

  216. This means that it is very likely that the unwanted event happens.

  We talk about what is likely to happen in the long run. We might be lucky and roll 3 sixes in a row. We must also consider the consequences of an unwanted outcome.

  What does this mean? If there are more ways of reaching a bad outcome than

  a good outcome, the probability of a bad outcome is higher. It is easier to destroy a system than to create one merely because there are more opportunities for destruction than creation.

  It means that surprises, coincidences, rare events and accidents happen, somewhere, sometime, and to someone if they have opportunities to happen.

  It also means that eliminating risk is preferable to finding out where the risk lies (since there are so many opportunities for an unwanted outcome). For example, we can reduce risk by increasing the number of wanted possible outcomes, reducing the number of unwanted possible outcomes, reducing the magnitude of consequences or avoiding certain situations.

  Ask some relevant questions:

  Event: Type of event? Frequent? Unique?

  Causes: What can initiate and cause the event? What factors contribute? What conditions and circumstances must be present? Have the causes changed over time?

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  Exposure: Known? Measurable? Possible consequences? Magnitude of consequences/loss? What's the worst that can happen?

  Probability: Distribution of possible outcomes over time? Stable? Relative frequency or relevant past experience? Number of observations? Relative likelihood of different size of losses? How is average frequency produced? Variability in outcome and severity? Dependence on human factors?

  Representative: Past data representative or change in conditions? Evidence of changes in causes or frequency of event? Temporary or permanent change? Small sample or too short observation time? Changing exposure as time proceeds?

  Backups: Backup failure rate?

  Let's observe some effects of what we've described in this chapter. More focus is put on the underlying ideas than the math. The theory of probability and its definitions, rules and calculations are found in Appendix Three.

  Low frequency events

  The chance of gain is by every man more or less overvalued, and the chance of loss is by most men undervalued

  - Adam Smith (Scottish philosopher and economist, 1723-1790)

  Supreme Court Justice Oliver Wendell Holmes, Jr. said: "Most people think dramatically, not quantitatively." We overestimate the frequency of deaths from publicized events like tornadoes, floods and homicides and u
nderestimate the frequency of deaths from less publicized ones like diabetes, stroke and stomach cancer. Why? As we learned in Part Two, we tend to overestimate how often rare but recent, vivid or highly publicized events happen. The media has an interest in translating the improbable to the believable. There is a difference between the real risk and the risk that sells papers. A catastrophe like a plane crash makes a compelling news story. Highly emotional events make headlines but are not an indicator of frequency. Consider instead all the times that nothing happens. Most flights are accident-free. Ask: How likely is the event? How serious are the consequences?

  john is boarding the day flight from Los Angeles to Washington and wonders, "How likely am I to die on this trip?"

  What is the risk of a disaster? First we need to know the available record of previous flights that can be compared to John's. Assume, we find that in 1 out of 10,000 flights there was an accident. The record also shows that when an accident happens, on average 8 out of 10 are killed, 1 injured and one safe. This means that

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  the chance that a passenger will be involved in an accident is 1 in 10,000; being killed, 1 in 12,500 (10,000/0.8); and being injured, 1 in 100,000 (10.000/0.1)

  According to the Federal Aviation Administration, Dr. Arnold Barnett of Massachusetts Institute of Technology (MIT), a widely recognized expert on air traffic safety, measured a passenger's odds of surviving the next flight. It related the probability of not being in a fatal air carrier accident and the probability of not surviving if a fatal accident happens. In the year 2000 the odds were 5.8 million to 1.