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The following sort of problem is increasingly fascinating to me, perhaps because it constitutes a profitable, prudent kind of thinking to which I'm not predisposed...

In Roger Lowenstein's excellent "Buffett --The Making of an American Capitalist", there is a passage which reads as follows:

"As Buffett liked to relate the [insurance] business to poker, it is illustrative to consider his response to an actual wagering proposition. Every other year, he got together with Tom Murphy, Charlie Munger, and some other pals for a golf and bridge weekend in Pebble Beach, California. The men did a lot of betting, and at one session, in the early eighties, Jack Byrne, the GEICO chairman, proposed a novel side bet. For a "premium" of $11, Byrne would agree to pay $10,000 to anyone who hit a hole-in-one over the weekend. Everyone reached for the cash --everyone, that is, except for Buffett, who coolly calculated that, given the odds, $11 was too high a premium. His pals could not believe that he --by then, almost a billionaire-- would be so tight and began to razz him for it. Buffett, grinning, noted that he measured an $11 wager exactly as he would $11 million. He kept his wallet zipped."

In another section of Lowenstein's book, he quotes National Indemnity's Jack Ringwalt, whom Buffett greatly admired and who profoundly observed, "There is no such thing as a bad risk. There are only bad rates."

Both of these quotes obviously allude to the same fascinating principle, but I wonder how this principle would best be mathematically represented. Any thoughts?
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Any thoughts?

I am most definitely a mathematical light-weight, but I am also fascinated by the concept of risk analysis.

When Buffett decided $11 was too great a price to pay for a $10,000 return he was evaluating his own ability at golf. He may have taken another look if Tiger Woods was the one at the tee.

So, perhaps the 'human input' aspect of the equation you are looking for is what makes it most difficult to evaluate risk. Buffett was also expected to risk his entire $11, win or lose, with no opportunity to recover a portion. If it had in fact been $11 million that he was investing, he would have done so with an expectation that a poor result would leave only a dent in his capital. I think the risk of total loss through wagering was what turned Buffett away from his early interest in horse-racing, as it is very difficult to protect capital and therefore stay in the game when any and possibly every wager may represent a total loss of the money-at-risk.

Professionals who bet on the outcome of races use their experience and knowledge to give each runner a value. If their best-value runner is offering an overlay, a higher return than they have essed it to be worth, then they bet a small portion of their total working capital against the market. If the market has taken to their selection and driven the offering down, then they walk away without taking a risk. If it were any easier than that the market would not exist, and if the essment of risk in insurance were any easier that market would falter also.

Your equation would need total capital times percentage at risk on one side and the probability of success or failure on the other, and the strange and unexpected things that people do make the essment of probability mind-boggling. The accuracy with which an insurer can manipulate such an equation will determine whether or not they attract business and remain liquid, both at the same time.
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How come the this Fool machine wrote 'essment' every time I wrote 'assessment'?
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I dunno. Ask on the "Censorship" board.

Me, all that happens is that I lose all my apostrophes.
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Very insightful! This very issue was one that was wrestled with for literally millenia before mathematicians found out the right way to deal with it. It's actually a simple concept, but like with many things, it requires that you "get it".

It's called "mathematical expectation". You define the "expectation" of something as the sum of products. The terms of each product are:
- the value of this occurrance x the probability of this occurrance.

So if I have a fair coin, and you win $5 if it's heads and win $1 if it's tails, what is the expectation?

Calculation: (0.5 x $5) + (0.5 * $1) = $3

A "risk neutral" player would be willing to pay $3 or less to get in to this game. [Very few people are risk neutral; I want to have some chance of getting ahead, so I'm risk-averse. Some people either like the thrill of gambling or need that $5 so bad that they are risk-seekers].

For a very thorough analysis, check out the book"
Against the gods: the remarkable story of risk by Peter Bernstein.

I'm sure Amazon has an excerpt online; I found another at:

If you read it, let me know what you think.
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jrr7 & fivepigs,

Thank you for your thoughtful responses. Actually, I picked up Peter Bernstein's "Against the Gods" last year with the hopes of better understanding probability and its application to these kinds of questions. The book gives an excellent historical perspective, but "Innumeracy", by John Paulos, is probably a better primer to practical application.

Perhaps I might frame the question of my previous post a bit differently: If Buffett believed his probability of shooting a hole-in-one was 1%, what premium would he be willing to pay for the chance to win $10,000? If he believed his probability was 25%, what premium would he be willing to pay?

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If he's completely risk neutral, he should be willing to enter into the deal when the premium equals the expectation.

For instance, if his chance of shooting a hole-in-one is really 1%, then the expectation of the bet is $100. A 25% chance implies an expectation of $2500. But he probably knows he can't forecast his chances of a hole-in-one that accurately.

Then there's the issue of how much someone needs the money (or feels they do). Emotional factors are viciously difficult to quantify. If they felt that the value of a potential $10,000 is huge and the cost of a loss of $10 is small, one could convince oneself to bet.

For instance, if one needs $10,000 for tuition for college, but one only has $100, the bet of $10 might make sense. Losing the $10 doesn't affect one too badly, but the $10,000 would be life-changing. This might make someone into a risk-seeker.
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i've read Lowenstein's book cover to cover several times. understanding WEB's mind is somewhat difficult but clearly you can see in Byrne's offer the way an insurer thinks. i would assume, using this anectdotal story, that byrne is thinking:

1. how much money can i blow without my wife killing me? (answer: around $30,000 bucks)

2. 30,000 bucks divided by three golf partners is 10,000 bucks.

3. a hole in one on a par three has about 1:1000 probability.

4. 10,000 bucks divided by 1000 is 10 bucks.

5. add my 10% profit for providing this service and the premium becomes 11 bucks.

Buffett, on the other hand is thinking to himself...i've played about 5000 or 6000 par threes in my lifetime, and have never had a hole in one. therefore my chance is probably less than that. so i might be willing to pay, based on my life's empirical evidence, 10,000 divided by 5000 or 6000, or around 2 dollars, hence the premium in his mind being "overvalued". WEB must have assessed his chances of a hole in one at significantly below the 1:1000.

But seriously, the emotional response to pull out the wallet simply because the potential payoff is high, is the reason that lotteries exist and why super-cat can be such a great business for Berkshire.

as far as the premium=insured value*probability of event, you are neglecting the fact that insurers also tend to tack on some service related fee, i.e., they have found that people will consistently pay slightly above that mathematical expectation for the intangible benefit of having the insurance. in the long run, WEB recognizes, as do all smart insurers, that profits in insurance derive primarily from the service premium they can get and from the benefits of float. any profit above that really is gambling in the true sense.


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