By
Sampson Iroabuchi Onwuka
Bernoulli and Risk Averse theories
Daniel Bernoulli
"value of an item must be not be based on its price" "but rather on the utility it yields. The price of the item is independent only on the thing itself and is the same for every one"
"Utility however is dependent on the particular circumstances of the person making the estimate"
"The determination of the value of an item must not be based on its price, but on the particular circumstances of the of the person making the estimate"
"It becomes evident that no valid measurement of the value of a risk can be obtained without consideration being given to its utility"
One man who theory is of great import in understanding Risk is Daniel Bernoulli who based his interpretation of risk and 'moral certainty' of the individual person making the decision and the utility. Of course, he is better known as a man who solved the puzzle of St. Petersburg Paradox, and did so by first citing the comment from his brother, Nicholas Bernoulli that "The utility resulting from any small increase in wealth will be inversely proportional to quantity of goods previously possessed" and then went on to describe that it was next to impossible for anyone to achieve a certain number for instance I, when the average expectations of the divisible half of the equal numbers does not give you 1. That is if you add 1/2 + 1/4 + 1/8 + 1/16 + 1/32 does not exactly give you 1. Although there is something of Pythagoras in the saying for instance, "if an odd number measures (+divides) an even number, then it also measures (+divides) half of it", which of course will prove that square root of 2 i irrational.
But subject raised by Daniel Bernoulli on the Wealth to Utility relationship has given reason to a whole lot of interpretation, and nearly every economist with interest in Risk management will not fail to rehearse some of the basic assumptions associated with the Bernoullis and why they or may not be relevant to the society such as the Welfare and Public housing, such as Shelter and prison. In the instance we have to measure that even the social services or Public Shelters houses has multiplied all of the place and is generating untold wealth for the 1% percent non-for profit, including women but yielding no results or helping few and fewer people. But this condition is not new that the more money you add the less result - for instance in Casino- it however appeals to the two 'EE' Efficiency and Effectiveness in business and not in terms of utility.
In terms of utility, some will argue that those who give the Residence less are forcing the equation of 'more' from them and not less, where as more is the case is dependency ratio. People are so to speak more dependent when they are denied of services or assistance or are not fed or not well fed and like Oliver Twist may demand that they want some more, than when they are well fed and begin to think. In the circumstance, we may also argue that most people do stupid things when they are drunk or commit crimes only when on drugs, so more in this case, should not lead to excess, perhaps a form of distribution along a mean, but greater tendency towards the positive. But this is not what the angle to the statement of Bernoulli that we shall delve into, this instance requires the comparing the reason why Bond rise and prices fall.
But this theory receives mixed reviews and sometimes can be applied different, and by different people. But it appeals to me that the instance raised by the Bernoulli may be quite fitting in describing why American Bond Yield may be inverse related to price.
William Poundstone 'Fortune Formula; The Untold of the Scientific Betting System, cited the early presumptions associated with Bernoulli statement, by citing the British Economist; William Stanley Jevons (1835-1882), who along with others described the statement as 'Logarithm utility applied to consumer wealth.'
And according to Poundstone, Jevon's commentary carries some weight that "As the quantity of any commodity, for instance, plain food, which a man has to consume, increases, so the utility or benefit derived from the last portion used decreases in degree" and Poundstone added 'that's how all you can eat restaurant stay in business'.
William Stanley Jevons 'Theory of Political Economy'; 1871, that "Value depends entirely upon utility" that in hindsight "We have only to trace out carefully the natural laws of the variations of utility, as depending upon the quantity of a commodity is our possession, in order to arrive at a satisfactory theory of exchange".>This last line appears elsewhere and one of the more popular quoted line of Jevons, yet the question raised about utility by one my favorite English man Jeremy Bentham concerning utility and the community, which must be considered in the light on individual interest, yet "...when the tendency it has to augment the happiness of the community is greater than any it has to diminish it" the greater emphasis should be placed on the community.
This was applied to many English projects including constructing a place for children who slept under the London bridge. For here, the comparison between the individual interest y was considered in terms of the x, such that as y draws to a 0, x moves to 1, which is Limits under normal restriction. Therefore x is risk averse to y when x closer to 0 than Y to 1.
In respect to the American Paul Samuelson, Poundstone adopted his position to Paul Samuelson that 'The nub of the issue is that Bernoulli's 'utility function' is psychologically unrealistic at the extremes of Wealth' but in the process of digesting the rest of the psychology implausibility of Bernoulli's, he resorted to the "bliss level".
It does appear to the best of us, that Paul Samuelson's position is based on a statement already made by the Gottfried Von Leibniz, who in a commentary on 'Jacob' Bernoulli's one of the Bernoulli citation's of Nicholas Bernoulli's comment, mentioned that "Nature has established patterns originating in the return of events, but only for the most part. New Illness 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"
It is has been said before and more than once, that the last line in the statement by Leibniz is a precursor to the invention of Large Numbers as propounded by Jacob Bernoulli, later day theories associated with 'Mean Regression', which in recent times was made popular by Harry Malkowitz and William Sharpe.
Whether or not nature or numbers have a way of making a return to the mean may or may not apply to the comment of Bernoulli, but for the fact that a part of it returns as opposed to the adoption of Pythagoras numbers, which divides
Peter L. Bernstein responding to this saying in his book 'Against the Gods' commentated on this saying by suggesting that "considering the nature of man, it seems to em that the foregoing hypothesis in apt to be valid for many people to whom this sport of comparison can be applied".
There is aloofness in this statement and there is no denying that the author had something in mind that may require a visit to early English society of Malthus to understand, but we are sent to a different direction by Bernstein, towards a clamping on the nearly poor in terms of the oddity of spending or social secularizing the poor or minority through benefit where as the more we spend on them seem to create the need to spend more. Yet we come to address that he was making a point based on what the Mathematician Bernoulli is saying. But History of Mathematics is not mathematics.
It is important to point out that theory that Peter L. Bernstein applies to Bernoulli does not immediately apply to Bernoulli and in fact Daniel Bernoulli's statement may not entirely apply to the original intent and design of Nicholas Bernoulli, that it is in fact quite wrong. There is nothing in the immediate commentary by Bernoulli of the inverse relationship between want and supply that mirrors the use of consumer judgement as William Jevon wanted to know.
For all intent purposes, we can summaries from guts of past interpretation of Bernoulli's statement as from Daniel Bernoulli St. Peterburg Paradox and as from Poundstone, that Wealthy English men/women or Wealthy Americans are not more apt to play lottery than the lesser fortunate American. That this is not so much a question of moral certainty, as for the quantitative reason that the needs of the poor may drive the poor into seeking money by any means necessary, for instance playing lottery for what they don't already have. To be fair, need in this case is replaced by utility and the human provisions which Bernoulli mentioned, has also been challenged by many experts.
What is missing from this argument is the theory of 'expectation' which Moivres argues to be different or changes with due respect to risk, what is the total expectation of the people or a people from what it considered...
It may be important to mention that the quantity as described by Bernoulli is borrowed from real life circumstances, and from that we can suggest that the fore going on Bernoulli's position that "The utility resulting from any small increase in wealth will be inversely proportional to quantity of goods previously possessed", can be represented graphically and arithmetically, and that his theory can better be understood through a Bond market, particularly U.S Bond yield.
Nearly all economist will agree that the U.S bond yield is inversely proportional to price and the reason for this sort inverse relationship is not exactly knowable.
The relationship between Interest Rate and Bond Prices is inversely co-related, bond prices fall as interest rates rise, for instance currency rate between the U.S dollars and commodity prices.
Scaling in physiology is generally defined as dividing a capacity by flow rate. That is, '...the scaling of average metabolic turnover times is estimated by dividing a capacity (volume) by flow rate for instance "urine production", or "food intake", which if given the constants in U.S IRS receipts and total American expenditure, we may arrive at a number that quite wide between the two measurements in other to understand the full measure of U.S Economy or U.S underground market.
Using Peter Bernstein commentary in this case, we can also indicate De Moivre "On the Measurement of Lots" 1711 ;that "The Risk of losing any sum is the reverse of Expectation; and the true measure of it is the product of the sum adventured multiplied by the probability of the loss"
Bernstein believes that 'De Moivres advance in the resolution of these problems rank among the most important achievement in mathematics"
'De Moirvre's gift to mathematics was an instrument that made it possible to evaluate the probability that a given number of observations will fall within some specified bound around a true ratio'
That "De Moivres distribution is known today as a normal curve, or, because of its resemblance to a bell. as a bell curve, shows the largest numbers of observations clustered in the center, close to
'Drawing on both the calculus and on the underlying structure of Pascal's Triangle, Known as the binomial theorem, De Moivre demonstrated how a set of random drawings, as in Jacob Bernoulli jar experiment would contribute themselves around their average value'
One of the first things we learn from mathematics is that there is an 'inverse relationship between the length of a pendulum and its frequency' and that for instance exponential relationship of a 'square to its linear dimension'; that is surface is proportional to the square of linear dimensions and as we have also been told, this inverse relationship applies to 'volume to the cube'. The true relationship of the cubic and four side metric is best applied in Economics or in money, particular in determining the various topologies or changes or interface associated with long term bond and short term bond market. Or put it this way, why is that the square of the longest side of right angled triangle is always equal to the square root of both sides. Of course, the reasons include change in dynamics, fears of deflation, inflation, and so on that seem ultimately arbitrary but concerns are serious in the Short and interest and other economic incentives must be adjusted to accommodate it but overtime (long term) these prices tend to normalize.
Likewise we can use Riemann bi-axis or manifold to show that a right angle incurs an arbitrary beta angle which though apparent completes the three triangle of a right angle and which like Prism (a third level complex; multiplex) can switch to infinity planes but converge only the basic Euclidean R Complex (R) which some Physicist has suggested can be used to interpret almost anything.
The real demonstration of this movement is FROM LEFT TO RIGHT across the sine, cosine, tangent angles of an x and Y, the new shape cancels the old. The arbitrariness of this form is a googolplexus (googleplex) of time and space, as the future though part of the present is insubordinate to its present time. Time in respect to the future is therefore arbitrary an can only be reduced to fictional set of ordinals (limits) but in real time, time is expanding faster than the limits imposed on it, prove of things happen faster than we imagine and future breaks bad from the present, like futures and forwards, price has no history.
And in the words of Abraham De Moivre,
"Although Chance produces Irregularities, still the odds will be infinitely great, that in process of Time, those Irregularities will bear no proportion to recurrence of that Order which naturally results from ordinary designs'
From this statement we may also witness the hint that De Moivre is reaching the conclusion that new prices or irregularities if continued would in the long run create their of regularity.
Sampson Iroabuchi Onwuka
Bernoulli and Risk Averse theories
Daniel Bernoulli
"value of an item must be not be based on its price" "but rather on the utility it yields. The price of the item is independent only on the thing itself and is the same for every one"
"Utility however is dependent on the particular circumstances of the person making the estimate"
"The determination of the value of an item must not be based on its price, but on the particular circumstances of the of the person making the estimate"
"It becomes evident that no valid measurement of the value of a risk can be obtained without consideration being given to its utility"
One man who theory is of great import in understanding Risk is Daniel Bernoulli who based his interpretation of risk and 'moral certainty' of the individual person making the decision and the utility. Of course, he is better known as a man who solved the puzzle of St. Petersburg Paradox, and did so by first citing the comment from his brother, Nicholas Bernoulli that "The utility resulting from any small increase in wealth will be inversely proportional to quantity of goods previously possessed" and then went on to describe that it was next to impossible for anyone to achieve a certain number for instance I, when the average expectations of the divisible half of the equal numbers does not give you 1. That is if you add 1/2 + 1/4 + 1/8 + 1/16 + 1/32 does not exactly give you 1. Although there is something of Pythagoras in the saying for instance, "if an odd number measures (+divides) an even number, then it also measures (+divides) half of it", which of course will prove that square root of 2 i irrational.
But subject raised by Daniel Bernoulli on the Wealth to Utility relationship has given reason to a whole lot of interpretation, and nearly every economist with interest in Risk management will not fail to rehearse some of the basic assumptions associated with the Bernoullis and why they or may not be relevant to the society such as the Welfare and Public housing, such as Shelter and prison. In the instance we have to measure that even the social services or Public Shelters houses has multiplied all of the place and is generating untold wealth for the 1% percent non-for profit, including women but yielding no results or helping few and fewer people. But this condition is not new that the more money you add the less result - for instance in Casino- it however appeals to the two 'EE' Efficiency and Effectiveness in business and not in terms of utility.
In terms of utility, some will argue that those who give the Residence less are forcing the equation of 'more' from them and not less, where as more is the case is dependency ratio. People are so to speak more dependent when they are denied of services or assistance or are not fed or not well fed and like Oliver Twist may demand that they want some more, than when they are well fed and begin to think. In the circumstance, we may also argue that most people do stupid things when they are drunk or commit crimes only when on drugs, so more in this case, should not lead to excess, perhaps a form of distribution along a mean, but greater tendency towards the positive. But this is not what the angle to the statement of Bernoulli that we shall delve into, this instance requires the comparing the reason why Bond rise and prices fall.
But this theory receives mixed reviews and sometimes can be applied different, and by different people. But it appeals to me that the instance raised by the Bernoulli may be quite fitting in describing why American Bond Yield may be inverse related to price.
William Poundstone 'Fortune Formula; The Untold of the Scientific Betting System, cited the early presumptions associated with Bernoulli statement, by citing the British Economist; William Stanley Jevons (1835-1882), who along with others described the statement as 'Logarithm utility applied to consumer wealth.'
And according to Poundstone, Jevon's commentary carries some weight that "As the quantity of any commodity, for instance, plain food, which a man has to consume, increases, so the utility or benefit derived from the last portion used decreases in degree" and Poundstone added 'that's how all you can eat restaurant stay in business'.
William Stanley Jevons 'Theory of Political Economy'; 1871, that "Value depends entirely upon utility" that in hindsight "We have only to trace out carefully the natural laws of the variations of utility, as depending upon the quantity of a commodity is our possession, in order to arrive at a satisfactory theory of exchange".>This last line appears elsewhere and one of the more popular quoted line of Jevons, yet the question raised about utility by one my favorite English man Jeremy Bentham concerning utility and the community, which must be considered in the light on individual interest, yet "...when the tendency it has to augment the happiness of the community is greater than any it has to diminish it" the greater emphasis should be placed on the community.
This was applied to many English projects including constructing a place for children who slept under the London bridge. For here, the comparison between the individual interest y was considered in terms of the x, such that as y draws to a 0, x moves to 1, which is Limits under normal restriction. Therefore x is risk averse to y when x closer to 0 than Y to 1.
In respect to the American Paul Samuelson, Poundstone adopted his position to Paul Samuelson that 'The nub of the issue is that Bernoulli's 'utility function' is psychologically unrealistic at the extremes of Wealth' but in the process of digesting the rest of the psychology implausibility of Bernoulli's, he resorted to the "bliss level".
It does appear to the best of us, that Paul Samuelson's position is based on a statement already made by the Gottfried Von Leibniz, who in a commentary on 'Jacob' Bernoulli's one of the Bernoulli citation's of Nicholas Bernoulli's comment, mentioned that "Nature has established patterns originating in the return of events, but only for the most part. New Illness 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"
It is has been said before and more than once, that the last line in the statement by Leibniz is a precursor to the invention of Large Numbers as propounded by Jacob Bernoulli, later day theories associated with 'Mean Regression', which in recent times was made popular by Harry Malkowitz and William Sharpe.
Whether or not nature or numbers have a way of making a return to the mean may or may not apply to the comment of Bernoulli, but for the fact that a part of it returns as opposed to the adoption of Pythagoras numbers, which divides
Peter L. Bernstein responding to this saying in his book 'Against the Gods' commentated on this saying by suggesting that "considering the nature of man, it seems to em that the foregoing hypothesis in apt to be valid for many people to whom this sport of comparison can be applied".
There is aloofness in this statement and there is no denying that the author had something in mind that may require a visit to early English society of Malthus to understand, but we are sent to a different direction by Bernstein, towards a clamping on the nearly poor in terms of the oddity of spending or social secularizing the poor or minority through benefit where as the more we spend on them seem to create the need to spend more. Yet we come to address that he was making a point based on what the Mathematician Bernoulli is saying. But History of Mathematics is not mathematics.
It is important to point out that theory that Peter L. Bernstein applies to Bernoulli does not immediately apply to Bernoulli and in fact Daniel Bernoulli's statement may not entirely apply to the original intent and design of Nicholas Bernoulli, that it is in fact quite wrong. There is nothing in the immediate commentary by Bernoulli of the inverse relationship between want and supply that mirrors the use of consumer judgement as William Jevon wanted to know.
For all intent purposes, we can summaries from guts of past interpretation of Bernoulli's statement as from Daniel Bernoulli St. Peterburg Paradox and as from Poundstone, that Wealthy English men/women or Wealthy Americans are not more apt to play lottery than the lesser fortunate American. That this is not so much a question of moral certainty, as for the quantitative reason that the needs of the poor may drive the poor into seeking money by any means necessary, for instance playing lottery for what they don't already have. To be fair, need in this case is replaced by utility and the human provisions which Bernoulli mentioned, has also been challenged by many experts.
What is missing from this argument is the theory of 'expectation' which Moivres argues to be different or changes with due respect to risk, what is the total expectation of the people or a people from what it considered...
It may be important to mention that the quantity as described by Bernoulli is borrowed from real life circumstances, and from that we can suggest that the fore going on Bernoulli's position that "The utility resulting from any small increase in wealth will be inversely proportional to quantity of goods previously possessed", can be represented graphically and arithmetically, and that his theory can better be understood through a Bond market, particularly U.S Bond yield.
Nearly all economist will agree that the U.S bond yield is inversely proportional to price and the reason for this sort inverse relationship is not exactly knowable.
The relationship between Interest Rate and Bond Prices is inversely co-related, bond prices fall as interest rates rise, for instance currency rate between the U.S dollars and commodity prices.
Scaling in physiology is generally defined as dividing a capacity by flow rate. That is, '...the scaling of average metabolic turnover times is estimated by dividing a capacity (volume) by flow rate for instance "urine production", or "food intake", which if given the constants in U.S IRS receipts and total American expenditure, we may arrive at a number that quite wide between the two measurements in other to understand the full measure of U.S Economy or U.S underground market.
Using Peter Bernstein commentary in this case, we can also indicate De Moivre "On the Measurement of Lots" 1711 ;that "The Risk of losing any sum is the reverse of Expectation; and the true measure of it is the product of the sum adventured multiplied by the probability of the loss"
Bernstein believes that 'De Moivres advance in the resolution of these problems rank among the most important achievement in mathematics"
'De Moirvre's gift to mathematics was an instrument that made it possible to evaluate the probability that a given number of observations will fall within some specified bound around a true ratio'
That "De Moivres distribution is known today as a normal curve, or, because of its resemblance to a bell. as a bell curve, shows the largest numbers of observations clustered in the center, close to
'Drawing on both the calculus and on the underlying structure of Pascal's Triangle, Known as the binomial theorem, De Moivre demonstrated how a set of random drawings, as in Jacob Bernoulli jar experiment would contribute themselves around their average value'
One of the first things we learn from mathematics is that there is an 'inverse relationship between the length of a pendulum and its frequency' and that for instance exponential relationship of a 'square to its linear dimension'; that is surface is proportional to the square of linear dimensions and as we have also been told, this inverse relationship applies to 'volume to the cube'. The true relationship of the cubic and four side metric is best applied in Economics or in money, particular in determining the various topologies or changes or interface associated with long term bond and short term bond market. Or put it this way, why is that the square of the longest side of right angled triangle is always equal to the square root of both sides. Of course, the reasons include change in dynamics, fears of deflation, inflation, and so on that seem ultimately arbitrary but concerns are serious in the Short and interest and other economic incentives must be adjusted to accommodate it but overtime (long term) these prices tend to normalize.
Likewise we can use Riemann bi-axis or manifold to show that a right angle incurs an arbitrary beta angle which though apparent completes the three triangle of a right angle and which like Prism (a third level complex; multiplex) can switch to infinity planes but converge only the basic Euclidean R Complex (R) which some Physicist has suggested can be used to interpret almost anything.
The real demonstration of this movement is FROM LEFT TO RIGHT across the sine, cosine, tangent angles of an x and Y, the new shape cancels the old. The arbitrariness of this form is a googolplexus (googleplex) of time and space, as the future though part of the present is insubordinate to its present time. Time in respect to the future is therefore arbitrary an can only be reduced to fictional set of ordinals (limits) but in real time, time is expanding faster than the limits imposed on it, prove of things happen faster than we imagine and future breaks bad from the present, like futures and forwards, price has no history.
And in the words of Abraham De Moivre,
"Although Chance produces Irregularities, still the odds will be infinitely great, that in process of Time, those Irregularities will bear no proportion to recurrence of that Order which naturally results from ordinary designs'
From this statement we may also witness the hint that De Moivre is reaching the conclusion that new prices or irregularities if continued would in the long run create their of regularity.
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