Financial and real decisions: history and method : Financial and real decisions: history and method
Jean-Sebastien Lenfant
Pierre-Charles Pradier
English language requirements : English language requirements You can interrupt me AT ANY MOMENT to ask
For repetition
For another way to tell the same idea
The meaning of a word
The spelling of a word
Whether I’m right or wrong from either a grammatical or substantial point of view
(note: most of the references are in French)
Organization : Organization Jean-Sebastien Lenfant:
Jean-Sebastien.Lenfant@univ-paris1.fr
Real decisions under certainty
Pierre-Charles Pradier
Pierre-Charles.Pradier@univ-paris1.fr
http://picha.univ-paris1.fr
Financial decisions under risk / uncertainty
Link between both parts (of the course) : Link between both parts (of the course) Concepts:
Uncertainty is a generalization of certainty
(or) certainty is a degenerated case of uncertainty
Every good is amenable to a money estimation
(or) finance is a projection of the real world
Method:
Historical approach
Methodological emphasis
Technical aspects : Technical aspects Every one of us (JSL, PCP) will teach one course every two weeks,
Formally, the two lectures are separated
Evaluation: a project following one of the proposed themes on the site.
Every project will then lead to a short academic defense in front of both JSL&PCP – who will mark it jointly.
Contents : Contents 0 – (Intro) What means risk – etymology, history
1 – Probabilizing risk : 17th and 18th centuries
2 – Risk in Political Economy : distribution, profit, entrepreneur and uncertainty
3 – Finance as an academic discipline : from statics to dynamics
4 – Applications of decision theory : statistics, operation research and macro
5 – Risk in financial markets : back to macro inquiries
0 – (Intro) : 0 – (Intro) Where does risk come from?
Believe it or not, both the etymology and origin of risk are uncertain
Proof: have a look at numerous etymological dictionaries (see e. g. Pradier [1998] ch. 1)
Origin: there is a bourgeois legend
The bourgeois legend : The bourgeois legend The Modern Era begins by the time of the great discoveries and religious reformation. The spirit of capitalism (Max Weber) sums up the development of commerce and new attitudes toward life, death and labour. It is associated with the rise of the bourgeoisie which quickly superseded the landed gentry. The risk is a modern bourgeois concept as opposed to old-fashioned notions of peril, venture, chance...
Etymological counterpart : Etymological counterpart There is an etymological support to this bourgeois legend, see e. g. Rey’s Dictionnaire historique de la langue française:
« There is a connection with the latin resecare “cut off” (réséquer), through popular latin °resecum “what cuts” hence, “reef”, then “risk run by a merchandise at see”. »
The commercial practice then seems the origin of the risk concept.
Unfortunately… : Unfortunately… The word risk itself emanates from medieval Italy
We cannot go beyond 12th century (arabic origin?)
However tempting, the bourgeois legend is erroneous
What we can learn from this story : What we can learn from this story There has been a cultural and economic revolution in medieval Italy
Financial innovations are well-known (double-entry bookkeeping, insurance, future and forward contracts, etc.)
The sociological background is more obscure
Emphasis shifted from “we know how Europe became modern” to “why did Italy lag then?”
Financial (r?)evolution in medieval Italy : Financial (r?)evolution in medieval Italy (Indo-)Arabic ciphers + double-entry bookkeeping (Fibonacci)
Insurance techniques: bottomry loan, share exchange, socitte diverse, fictive future sales, modern insurance contract
Betting?
or
Diversification?
Risk: probabilistic attempts : Risk: probabilistic attempts Pacioli, Summa de arithmetica… [1494]: 2 teams are playing football. Every set is 10 points. 60 points are needed to win the match. Overall bets is 10 ducats. When the match is stopped for whatever reason, on team scores 50 points while the other has 20 points. How must the bets be divided among the two teams?
Tartaglia, Prima parte del Generali Tratato di numeri e misure [1556]: “la risolutione di una tal question è piú presto giudiciale, che per ragione, tal che in qual si voglia modo la sarà risolta visi trovare da litigare” – no rational solution, every solution is arbitrary
Probabilistic approach: major breakthrough : Probabilistic approach: major breakthrough General scepticism toward any solution of the problem of points: too many arguments to decide which one is better
This pattern will be seen again further: D’Alembert will criticise Bernoulli on the same ground of arbitrariness, while Engel will offer one way out, etc.
Let us begin with the first probabilistic calculations ever.
Probabilizing risk: 17th and 18th centuries : Probabilizing risk: 17th and 18th centuries a – the problem of points
b – the mathematical expectation
c – applications: games of chance, insurance, annuities on lives
d – the Petersburg Problem (or Paradox)
e – the 1780’s: decision theoretic refinements, economic decision, statistical decision
The problem of points : The problem of points In 1654, the Chevalier de Méré asked Blaise Pascal the following question:
“Two players bet 32 pistoles each on a game of luck. They agreed that the first who won three games will take the wager. The game is interrupted after the first game. How to distribute the bets?”
(cf. supra Pacioli-Tartaglia)
Two words of math : Two words of math 4 games to be played:
(1) Either P1 wins 4 times and P2 0 time,
(2) or P1 wins 3 times and P2 1 time,
(3) or both players win 2 times,
(4) or P1 wins 1 time and P2 3 times,
(5) or P1 wins 0 time and P2 4 times.
P1 wins if (1) or (2) or (3) occurs, else P2 wins.
Probability of (1) = (1/2)4, P(2) = (14)(1/2)4 etc.
Thus probability given by triangle : 1+4+6 = 11 vs. 1+4 = 5 so P1 earns 64/16×11 = 44
P2 64/16×5 = 24
Pascal’s solution : Pascal’s solution Give every player his mathematical expectation
The expectation is computed using the Triangle
“a matter absolutely uncharted so far: the distribution of chance in games submitted to it; the uncertainty of fortune is so well mastered by calculation that we give every player exactly what he deserves according to justice.”
Mathematical expectation : Mathematical expectation Expression itself not in Pascal
Appears in Huyghens [1657], de ratiociniis in ludo aleae: (about) calculations in games of chance.
Theorized by Nicholas Bernoulli [1709], de usu artis conjectandi in iuri: (about) applications of probability theory to legal questions
Expectation is not seen from a religious (though… see Pascal’s wager) but juridical point of view
Applications : Applications Games of chance:
You already know all about this, remember your last high school year,
Social demand existed during Ancien Régime: passion for gambling (see e. g. Barry Lyndon),
Now this is of little interest.
But… Petersburg Paradox.
Applications - 2 : Applications - 2 Insurance: the premium could be set to the mathematical expectation of damage. Loading?
Maritime insurance is not challenging;
Fire insurance was impossible;
Life insurance: mathematically interesting, strong social demand.
Maritime insurance : Maritime insurance Premium calculation = too simple.
Mental habits of 18th century people made them think with proportions or fractions rather than percentages.
The mathematics of maritime insurance seems “natural”: 1 in 10 ships does not return → premium equals one tenth of the value.
Fire insurance : Fire insurance Fire: potentially uninsurable risk (Wikipedia)
The insurer must be able to charge a premium high enough to cover not only claims expenses, but also to cover the insurer’s expenses. In other words, the risk cannot be catastophic, or so large that no insurer could hope to pay for the loss.
(…)
The loss should be random in nature, else the insured may engage in adverse selection (antiselection).
Life insurance : Life insurance What means life insurance?
Every kind of contract connected with the duration of life:
Annuities, annuities on two or more lives, widow’s (or child) pension, etc. even wager on someone’s life… which were forbidden in continental Europe (and so life insurance too during 18th century therefore the Tontine spread).
The math beyond life insurance : The math beyond life insurance First you need a good mortality table. The first one was the “Graunt” Table of William Petty [1662]:
Slide27 : Life expectancy and annuities One can compute the residual life expectancy of an individual of age n from the mortality table:
(where N is the highest age of death)
The value of an annuity on life is then given by an actuarial calculus. Let r be the (constant) rate of interest:
Slide28 : More fun! The case of a widow’s rent is a bit more tricky. Let index w denotes the wife’s survival table and h the husband’s. Then the value of the widow rent paying 1 monetary unit every year is given by:
(the first probability is that of the husband being dead, the second one is of the wife being alive)
Slide29 : Mahematical expectation = rational decision criterion Mathematical expectation is the solution Pacioli, Tartaglia and thousands of others waited for. It is both:
Just (i. e. satisfying from a legal and moral point of view: insurance contracts sold at expectation will not bear undue profit)
Rational (insurance contracts sold at expectation will not cause the ruin of the insurer)
There is one problem, though…
A Counterexample to Mathematical Expectation : A Counterexample to Mathematical Expectation “A tosses a coin: B will give him one guinea if heads comes down, two guineas if heads comes down on the second throw, 4 guineas if on third throw, 8 if on fourth, etc.”
(Gabriel Cramer’s 1728 presentation of a game designed in 1713 by Nicolas Bernoulli)
The mathematical expectation of A is infinite, therefore A should bet an infinite amount of money to be granted the right to play.
Mathematical expectation of A : Mathematical expectation of A One chance in two to get 1 guinea,
One chance to carry on playing,
Then one chance in two to get 2 guineas, one in two to carry on, etc. The mathematical expectation is thus
Requiem : Requiem From the beginning of its 1731 paper, Daniel Bernoulli attacks the mathematical expectation as a rule of decision.
§ 1. “Ever since mathematicians first began to study the measurement of risk there as been a general agreement on the following proposition: Expected values are computed by [the mathematical expectation].”
§ 2. “Proper examination of the numerous demonstrations of this proposition that have come forth indicates that they all rest upon one hypothesis: since there is no reason to assume that of two persons encountering identical risks, either should expect to have his desires more closely fulfilled, the risk anticipated by each must be deemed equal in value.”
What is paradoxal there ? : What is paradoxal there ? A paradox is contrary to doxa = common opinion.
It seems unreasonable there to bet even a moderate amount on such game :
for a bet of 8 guineas, there is only a 1 in 8 chance to get the money back,
for any bet x, there is a 1 in (where […] stands for integer part), that you will get your money back !
Publication in the 1730-31 volume of the Commentarii Academiae Scientiarum Imperialis Petropolitanae made the paradox citizen of St-Petersburg.
Ways out : Ways out The main path of argument followed by authors was to challenge the potential infinity in the game on a legal basis.
If B has limited wealth, for instance1 billion ( 230) guineas, then the mathematical expectation is limited (to 30 x 1/2 = 15 in our example).
Unfortunately, 15 guineas seems too high a price to play. Moreover, allowing for higher gains would make the paradox more paradoxical !
Daniel Bernoulli’s solution : Daniel Bernoulli’s solution “The moral value of a given sum is inversely proportional to the amount of wealth already possessed.”
Hence, the value of any increment of wealth dW is worth , hence
a log utility f°
Eventually, a prospect which gives lot xi with probability pi is worth Σ pi . ln(W + xi) - ln W
Consequence: risk aversion : Consequence: risk aversion If the utility function is concave, then the expected utility is less than utility of mathematical expectation. wealth utility 2 possible outcomes with even probability Expectation = middle of segment Utility of expectation
Expected utility
Other utility functions by Cramer : Other utility functions by Cramer In a 1728 paper, Cramer proposed two other utility functions :
One is upward bound with upper limit = 224, hence game value = 12;
The other one is such that U(w) = w1/2 which gives fundamentally the same result as Bernoulli’s hypothesis.
Where did similar solutions come from?
Culture (common knowledge) : Culture (common knowledge) The upward-bound function was grounded in a moral-legal tradition: nobody can bet more than his wealth (law), hence nobody can experience more satisfaction than that given by the most important wealth (moral counterpart). It must be recalled that Pascal, Fermat, etc. were trained in law.
Where do the decreasing marginal utility reasoning of utility functions come from?
Culture - 2 : Culture - 2 “He sat down opposite the treasury and observed how the crowd put money into the treasury. Many rich people put in large sums.
A poor widow also came and put in two small coins worth a few cents.
Calling his disciples to himself, he said to them, "Amen, I say to you, this poor widow put in more than all the other contributors to the treasury.
For they have all contributed from their surplus wealth, but she, from her poverty, has contributed all she had, her whole livelihood."” (Mark 12, 41-44, from the New American Bible).
Diminishing marginal utility : Diminishing marginal utility Many instances of DMU of wealth in eighteenth century authors:
Buffon distinguishes necessities from superfluities, Condillac natural needs from fictitious ones.
Melon[3], Galiani[4] Casanova[5] : needs are divided in 3 categories.
Boisguilbert : one or two more categories [6], while Forbonnais writes about goods of fifth necessity.
Graslin is the author of a system where the number of needs (of decreasing necessity) is not limited.
[3] « ...si nous supposions ces îles abondamment pourvues de tout ce qui est de première nécessité, et ayant sous leur domination des îles de marchandises de nécessité secondaire, comme du vin, du sel, de la toile, etc. ; et de nécessité de luxe, comme de la soie, du sucre, du tabac, etc. » Melon [1734], p. 708.
[4] Galiani [1751] pp. 50-52. Pour qui les trois catégories sont les « victuailles », les titres de noblesse, les métaux et pierres précieux.
[5] Casanova ([1993a], p. 1178) distingue les produits de première, de seconde (« comme le vin, le sel, la toile ») et de troisième (« comme le café, le tabac ») nécessité.
[6] Boisguilbert [a] propose plusieurs typologies, par exemple (p. 362) : « ...dès qu’on a plus que le nécessaire, on se procure le commode ; qu’ensuite de cela, on passe au délicat, au superflu, au magnifique, et enfin, dans tous les excès que la vanité a inventé pour ruiner les riches... »
Breakthrough : Breakthrough The basic idea of DMU of wealth was common knowledge, only Daniel Bernoulli gave him a clear-cut expression. Hence his claim:
[T]hough a person who is fairly judicious by natural instinct might have realised and spontaneously applied much of what I have here explained, hardly anyone believed it possible to define these problems with the precision we have employed in our examples. Since all our propositions harmonize perfectly with experience it would be wrong to neglect them as abstractions resting upon precarious hypotheses. (Bernoulli [1738, §15)
Discussion : Discussion Nicolas to Daniel B.:
“Whatever clever your theory is, you must agree that it does not solve the heart of the problem. The point is not about measuring the usefulness or pleasure one can have from an earned sum, nor the lack of usefulness or sorrow one can experience from the loss of a sum; nor it is to find something equivalent to these things. The point is in finding how much a player must, according to justice or equity, give the other in exchange for the advantage given in this game of chance (…) in order for the game to be told equal…” (Bernoulli N. 1731 p. 566).
Understanding Daniel’s POV : Understanding Daniel’s POV § 2. “The relevant finding might then be made by the highest judges established by public authority. But really there is no need of judgment but of deliberation...”
§ 15. “The procedure customarily employed by merchants in the insurance of commodities transported by sea seems to merit special attention. This may again be explained by an example.”
How rich (resp. poor) one must be to reasonably supply (resp. demand) a given insurance contract?
§ 16. “Another rule which may prove useful can be derived from our theory. This is the rule that it is advisable to divide goods which are exposed to some danger in several portions rather than to risk them all together.”
Daniel Bernoulli shows that diversification of investments is profitable. The point is not (as with Markowitz) about covariances, it is only a matter of utility function concavity and independence of random variables.
Norms : Norms Nicolas’ norm: justice
Daniel’s (implicit) norm: economic rationality
On the 4th July 1731, Daniel wrote to Nicholas: “If only The Bernoullis, who lost so much when the Müllers got bankrupt, paid attention to the very principles that I establish actually, they would probably not have lost as much.” (Bernoulli N. 1731, pp. 566).
In April next year, Nicholas bitterly answered: “Man muβ nicht zu viel Eyer in ein Korb legen, say our fellows from Basel. But what would you do if you needed to make the most of your money by crediting it to merchants, without being allowed to divide it up?” Bernoulli N. 1731, pp. 566-7.
Note that the decision function is not specific of an a given person, it is given for mankind, hence it is normative.
Discussion - 2 : Discussion - 2 “because this principle is itself hypothetical, any amount of money is necessarily worth more to a less wealthy man; but it seems to us arbitrary to choose either Daniel Bernoulli’s law or any other one which could fulfil the same conditions” (“Notes sur la thèse de Nicolas Bernoulli”, in Condorcet 1994, p. 584).
The rejection of Bernoulli’s theory does not arise from decreasing marginal utility, but from an arbitrary specification of the utility function. Any specification (e. g. including those of Buffon or Cramer) would be arbitrary because there is no way to chose the right function.
This leads us to the research of late eighteenth century
The french way : The french way Around the 1770’s, 3 French mathematicians were developing probability theory: D’Alembert, then Condorcet and Laplace.
D’Alembert was an heretic, Condorcet set up the decision-theoretic foundations, Laplace gave fascinating applications thank to its analytical virtuosity (Laplace method then CLT).
D’Alembert - doubts : D’Alembert - doubts From 1754 on, D’Alembert published his doubts on probability, “though he only brought himself
into disrepute by doing so” (Keynes 1921, p. 347)
Killing example: about the probability to bring in heads at least once in two consecutive tosses, D’Alembert wrote : “Heads, 1st toss. Tails, Heads, 1st & 2nd tosses. Tails, Tails, 1st & 2nd tosses. Hence odds are only 2:1”
D’Alembert: a genius? : D’Alembert: a genius? D’Alembert’s contributions to mathematics:
Traité de dynamique (1743),
Théorème fondamental de l’algèbre (1746),
partial derivatives equations (1747),
finite differences equations (1747),
D’Alembert’s rule (1754).
+ many probabilistic ideas
D’Alembert’s ideas : D’Alembert’s ideas Transformation of probabilities:
“instead of supposing the winning probability =(1/2)n, we could suppose it, for instance 1/2n(1+ß nn), where ß is whatever constant number…”
D’Alembert then gives a “simpler hypothesis” : 1/2n+αn; then a more complicated, not needed one. See Pradier [2003b] for more details.
D’Alembert’s ideas - 2 : D’Alembert’s ideas - 2 D’Alembert fought Daniel Bernoulli during the “controversy on inoculation”. By that time, there were no vaccines, and some physicians argued that it could be better to inoculate healthy children with small pox rather than wait for them to get the disease later, when they would be older (hence more heavily hit) and, with a non-negative probability, already subject to illness or starvation.
D’Alembert on inoculation : D’Alembert on inoculation
D’Alembert on inoculation - 2 : D’Alembert on inoculation - 2 « … Let there be two mortality curves AQCD, AOCD, with equal areas. One is but converging more quickly than the other one. Average life is equal in both cases. Shall we say that life expectancy is the same? Shall we say, as a consequence, that two people facing the two cases could indifferently exchange their destiny with one another? It seems to me on the contrary that the case with the AQCD mortality curve is less favourable, because there is more risk to die in the first years than when the mortality curve is AOCD. »
D’Alembert on inoculation - 3 : D’Alembert on inoculation - 3 D’Alembert then invented a concept of risk concentration that will become the standard expression of risk measurement in the 70’s…
He also showed that this argument had Bernoullian ground and that Bernoulli’s position toward inoculation was contradictory to its position toward the paradox.
D’Alembert: doubts : D’Alembert: doubts D’Alembert doubted:
The maths of mathematical expectation (2/3!),
The logical foundation of mathematical expectation (its superiority upon median, the need for taking risk/dispersion into account),
The specification of alternative theories (such as Bernoulli’s hypothesis),
The possibility to choose among various hypothesis.
Condorcet: foundations of decision : Condorcet: foundations of decision Condorcet principle :
“A reasonable man should enter into business only if he finds a fairly high probability to get his investment back, with common interest and the price of his labour.
Such man will doubtless require the probability not to lose his funds wholly (saving at least what is needed for the subsistence of his family) to be close to certainty; as well as the probability not to diminish his funds by more than a given amount to be very large too” (Condorcet 1784, p. 486).
the probability(ies) of the threshold(s) must be (experimentally) defined;
the theory is supposed to help decision-makers.
Condorcet’s epistemological stance : Condorcet’s epistemological stance Condorcet’s purpose is not to evaluate random prospects but to order them.
Theory is supposed to determine some boundary values: minimal expected profit needed for the merchant to undertake the investment, minimal insurance premium needed for the insurer to supply the insurance, etc. Condorcet does not design a price theory of “aleatory contracts” (as Pascal or Bernoulli did), but a theory that assigns rational boundaries to negotiation between contractors.
Cf. Turgot or Smith (determination of wage in the WoN) also presented theories of rational negotiation where rational boundaries are reservation prices of one side.
Risky decision may be more complex because it seems difficult to agree on what is rational.
Condorcet’s model : Condorcet’s model A decision-maker is involved in n identical trade operations. Every one can result either a failure with probability p or a success which will bring a£. The probability distribution of the insurer’s balance sheet is then a binomial. For instance, the probability to have m failure in n throws is Cmn pm(1-p)n-m. There is no explicit cumulative distribution function. One such function can be forged: if X is the binomial variable for number of failure in n throws, then P(X ≤ m) =
Problem with Condorcet : Problem with Condorcet The calculations involved are awful because m is the unknown: we want to get the greater m for a given P(X ≤ m). Summing up the terms of the binomial is so painful it destroys any practical application of the Condorcet principle.
For the principle / model to be useful, we need a fast approximation for the binomial.
The method of Laplace : The method of Laplace See Pradier [2003] for details & references;
Designed for studies about sampling (i. e. inferential statistics);
Looks like Central Limit Theorem but different: nothing conceptual (“variability of a sum of iid random variables”), the analogy is purely mathematical;
Applied to insurance management.
The result : The result
The probability P(X ≤ m) =
Can be approximated by
The integral can be tabulated (i. e. calculated using a Laplace Table Normal Table).
The application : The application The Laplace method can be used to determine the minimum loading rate such that the probability of failure of the insurance company is less than a given level (cf. Condorcet principle).
Remember: if insurance premiums are “pure” (unloaded) then the insurance company has a 50 % chance to end up the year with a net operating loss.
Mathematical analogy : Mathematical analogy An insurance company is formally identical to Condorcet’s model: n operations giving either a success or failure;
The Laplace method enabled the author to account for:
Binary variables with different parameters (probability or amount lost/earn);
Multinomial variables (modelling annuities).
This is impossible with CLT.
Anything else to declare? : Anything else to declare? Other authors involved in risky calculations:
Silvestre-François Lacroix: a condorcetian survivor popularized the calculation of loading rates;
Johannes Nikolaï Tetens: a Dane, member of the german combinatorial school, designed a Risico measure, wrote about sampling procedures in a laplacian way (though his confidence intervals had no associated probability), did not allow for loading of premiums…
One question, then: how evolved this field of research?
The end : The end Condorcet was the idéologue of the group. With his death, Laplace was left alone to his calculations. The mathematicians were not interested in social sciences (nor did the social scientists look for the use of math).
Only rigour was fashionable among the disciples of Gauss and Dirichlet: this leads to the proscription of both Laplace in France and the GCS in Germany + the abandonment of the mathematical social science.
Risk in Political Economy : : Risk in Political Economy : distribution, profit, entrepreneur and uncertainty
Risk in Political Economy : Risk in Political Economy
The English classical writers
The so-called “French tradition”
American idiots?
Risk and uncertainty
What then?
Adam : Adam Smith, WoN book 1 chapter X is about variable incomes:
If there is some annual variability in income, then the labourers will choose their income level to get insured against variability with a premium guaranteeing “certain” annual income;
If income is variable among individual as a result of personal success (e. g. singers, lawyers), then people are likely to overestimate their own luck a priori.
David : David Very few mentions of risk in the Principles of Political Economy and Taxation;
The idea of a risk premium is evident:
“A capitalist, in seeking profitable employment for his funds, will naturally take into consideration all the advantages which one occupation possesses over another. He may therefore be willing to forego a part of his money profit, in consideration of the security, cleanliness, ease, or any other real or fancied advantage which one employment may possess over another.” (PPET ch. IV)
Evidence : Evidence The classical British authors POV is about distibution, no longer decision;
They focus on what change risk might bring to any particular income;
Impact of risk only marginal: the bulk of distribution is performed according to functions (social classes = factors of production, labour, land, capital).
The so-called French tradition : The so-called French tradition Cantillon, Turgot, Say distinguished the entrepreneur from the capitalist;
The distinction was more theoretical than sociological, as genuine entrepreneurs were few;
The theoretical distinction had thus no real counterpart and was therefore useless for understanding of social phenomena;
Anyway it played an important ideological role.
Cantillon : Cantillon For Cantillon, the entrepreneur is the central agent in the economy: he is responsible for production and circulation of goods.
The entrepreneur can be distinguished from:
landlords because he depends on their consumption,
other workers because he his « on uncertain » or « on uncertain wages » ([Cantillon 1755] p. 30-31).
His function is « paying […] a given amount of money […] without being certain about the profit he will get back » ([Cantillon 1755] p. 28).
This theoretical lineage was carried on by Jean-Baptiste Say and Knight.
Cantillon – from a critical POV : Cantillon – from a critical POV Such an entrepreneur already existed in Vauban, see e. g. [Vérin 1982, 105-120]. Vauban built many strongholds for the king Louis XIV.
Cantillon’s work is aimed at vindicating his own social position. Merchants are infamous to the landed gentry, Cantillon invented the entrepreneur… One must remember that the first entrepreneur were the « entreprening knights » (chevaliers entreprenants) of the 12th century.
The risk concept appears only as an instrument in this sociodicy (fictitious theory of the society, cf. theodicy). It helps in identifying the best ones.
Say: riskless entrepreneur : Say: riskless entrepreneur “Everyone of us is the owner of some productive stock. We all have either some land, invested stock or some productive gift…” ([Say 1818], p. 128).
“The only way to make a really new fortune is either in undertaking industrially or in saving continuously for a long time, whatever the source of saved incomes.” ([Say 1821], p. 441)
Cf. Guizot: “get rich by working or saving” – bourgeois motto endorsed by Say. Entrepreneur: once again, part of a sociodicea.
Entrepreneurs in America: why? : Entrepreneurs in America: why? (Expanded form in my course in HET)
Because God bless America
Because American are different in substance
Because American have a different social organization:
No central money during the Free Banking Era (1837-1863),
Therefore people held bonds more easily than in Europe,
Difference between ownership and control of capital
Risk and entrepreneurship : Risk and entrepreneurship Some American authors thought profit was the reward for entrepreneurship / assumption of risks;
“It goes without saying that the hazard of business falls on the capitalist. The entrepreneur, as such, is empty-handed. No man can carry a risk who has nothing to lose.” ([Clark 1892] p. 46).
Entrepreneurs and America - II : Entrepreneurs and America - II Why so much concern about distribution by the end of the 19th century in America?
See Europe at the same time: 1891 = Fourmies + Rerum novarum.
No Pope in America, but
John Bates Clark
(son of a clergyman)
Slide77 : The welfare of the laboring classes depends on whether they get much or little; but their attitude toward other classes - and, therefore, the stability of the social state - depends chiefly on the question, whether the amount that they get, be it large or small, is what they produce. If they create a small amount of wealth and get the whole of it, they may not seek to revolutionize society; but if it were to appear that they produce an ample amount and get only a part of it, many of them would become revolutionists, and all would have the right to do so. The indictment that hangs over society is that of "exploiting labor." "Workmen" it is said, "are regularly robbed of what they produce. This is done within the forms of law, and by the natural working of competition." If this charge were proved, every right-minded man should become a socialist; and his zeal in transforming the industrial system would then measure and express his sense of justice. Distribution of wealth (1899)
Clark’s solution : Clark’s solution Statics:
Every factor get its marginal productivity,
Production functions are homogenous of degree 1,
Therefore under **perfect competition** everyone gets a fair reward for his contribution,
Entrepreneur supplies “coördination”.
Dynamics:
errors = either “profit” or “loss”
business risks are carried for their subjective actuarial value (1892, p. 42 = Daniel Bernoulli).
Replies to Clark : Replies to Clark “There is a magnificence in this generalization which recalls the youth of philosophy. Justice is a perfect cube, said the ancient sage; and rational conduct is a homogeneous function, adds the modern savant. A theory which points to conclusions so paradoxical ought surely to be enunciated with caution.”
“The theory of distribution”, Quarterly Journal of Economics, 1904.
Stop laughing please : Stop laughing please We’re getting closer to the light of science
Basically, Knight = JB Clark + a concept of “uncertainty”
i. e., no static profit, profit arises in a dynamic framework from uncertainty and rewards assumption of uncertainty by entrepreneurs.
Ideological shift: profit was purely fortuitous in JB Clark, it is vindicated by FH Knight’s RUP.
Another sexy guy : Another sexy guy
Frank Hyneman Knight 1885-1972
Weightlifter, Clown and Chippendale
More details in Pradier-Teira [2000]
What means uncertainty? : What means uncertainty? “To preserve the distinction which has been drawn in the last chapter between the measurable uncertainty and an unmeasurable one we may use the term "risk" to designate the former and the term "uncertainty" for the latter.” ([Knight, 1921], p. 234).
Knight is itself unmeasurable (and ununderstandable).
Trying to make things clear : Trying to make things clear 3 kinds of probabilities:
A priori probabilities (e. g. games of chance with given probabilities),
A posteriori probabilites (e. g. statistical frequencies told equal to probabilities by the “law of great numbers” = Jacob Bernoulli’s Theorem),
“estimates” = subjective probabilities when no relevant information exists (unique or near-unique events).
Economic life = uncertainty : Economic life = uncertainty It is impossible to insure economic activity because phenomena are unique (hence the need for entrepreneur)
Hence there can be no (rational) connection between anticipations and what really happens
Profit depends upon luck
(Even past is uncertain! – see e. g. Usher)
What implies uncertainty : What implies uncertainty Give up studies about distribution:
“The only conclusion as to social policy which we shall insert here is the insistence that "society" must get rid of the idea that because income is "earned" it is "deserved" and not otherwise. We are already far from this view in practice, as is shown by the indiscriminate taxation of large "service" incomes and assistance of the unfortunate and incapable. If we are to have organized society and maintain human standards of life, we must either radically eliminate weakness or impose upon strength the burdens which weakness cannot bear. (And even then there are limits to the possible toleration of weakness, and the luck element would still remain!)”
What is economics, then? : What is economics, then? “Economic analysis may be truly said to deal with ‘conduct,’ in the Spencerian sense, of acts adapted to ends, or of the adaptation of acts to ends, in contrast with the broader category of ‘behavior’ in general. It assumes that men's acts are ruled by conscious motives; that, as it is more ordinarily expressed, they are directed toward the ‘satisfaction of wants.’”
Cf. Robbins…
Knight, Robbins &c. : Knight, Robbins &c. Robbins (An Essay on the Nature and Significance of Economic Science, 1932): “Economics is the science which studies human behavior as a relationship between given ends and scarce means which have alternative uses.”
Economics = decision theory!
Groenwegen: “Robbins’ definition destroyed political economy”
Political economy? : Political economy? Something dirty (has to do with politics)
“Political Economy you think is an enquiry into the nature and causes of wealth—I think it should be called an enquiry into the laws which determine the division of the produce of industry amongst the classes who Concur in its formation. No law can be laid down respecting quantity, but a tolerably correct one can be laid down respecting proportions. Every day I am more satisfied that the former enquiry is vain and delusive, and the latter only the true objects of the science.”, Ricardo, “Letter to TR Malthus”, October 9, 1820.
The new economics : The new economics No poor
No classes
No politics
Nothing at stake
Plenty of meaningless equations
Only experts can benefit from the holy science
Risk and Uncertainty - II : Risk and Uncertainty - II What about him?
He’s got a moustache too
He’s prettier… or more handsome?
He’s smarter
His name is John Maynard Keynes!
A Treatise on Probability : A Treatise on Probability Keynes PhD Dissertation (1909)
Re-written and published in 1921 (same year as Frank H. Knight’s Risk, Uncertainty and Profit)
Hailed by many mathematicians: Whitehead, Wittgenstein, Borel…
What is a probability : What is a probability 18th century: probability can be either
Ontic (1 chance in 6 to get a “6” with one dice)
= related to essence of things
Epistemic (the dice has already fallen I’m wondering which side is up)
= related to knowledge
19th century: probability can be either
Objective (i. e. a priori or a posteriori)
Subjective (pointless) subjectivism…
What is a probability after Keynes : What is a probability after Keynes “Let our premisses consist of any set of propositions h, and our conclusion consist of any set of propositions a, then, if a knowledge of h justifies a rational belief in a of degree α, we say that there is a probability-relation of degree α between a and h.”
Treatise on Probability p. 4
Probability is thus related to an information set.
Weight of arguments : Weight of arguments “As the relevant evidence at our disposal increases, the magnitude of the probability may either decrease or increase, according as the new knowledge strengthens the unfavourable or favourable evidence; but something seems to have increased in either case, — we have a more substantial basis upon which to rest our conclusion. I express this by saying that an accession of new evidence increases the weight of an argument.” (p. 77)
Do probability (always) exist? : Do probability (always) exist? “There appear to be four alternatives. Either in some cases there is no probability at all; or probability do not all belong to a single set of magnitudes measurable in terms of a common unit; or these measures always exist, but in many case are, and must remain, unknown; or probabilities do belong to such a set and their measures are capable of being determined by us, although we are not always able so to determine them in practice.” (p. 33)
4 alternatives : 4 alternatives “in some cases there is no probability at all” = Keynes (when he’s pessimistic)
“probability do not all belong to a single set of magnitudes measurable in terms of a common unit” = Keynes? (when he’s optimistic)
“these measures always exist, but in many case are, and must remain, unknown” = Knight (objective probabilities do exist).
Laplace in 4th place : Laplace in 4th place “If an intelligence, for one given instant, recognizes all the forces which animate Nature, and the respective positions of the things which compose it, and if that intelligence is sufficiently vast to subject these data to analysis, it will comprehend in one formula the movements of the largest bodies of the universe as well as those of the minutest atom: nothing will be uncertain to it, and the future as well as the past will be present to its vision” (Essai philosophique sur les probabilités).
Together with Condorcet : Together with Condorcet Condorcet was dead when Laplace wrote his Essay
But he shared one common view with Laplace:
Perfect science is accessible to a “sufficiently vast” intelligence,
When perfect science is not accessible, a probabilistic knowledge is possible,
There exist something as THE (objective) probability of a phenomenon.
What is uncertainty? : What is uncertainty? By uncertain knowledge “...not only mean merely to distinguish what is known for certain from what is only probable. The game of roulette is not subject, in this sense to uncertainty; nor is the prospect of a Victory bond being drawn. Or, again, the expectation of life is only slightly uncertain. Even the weather is only moderately uncertain. The sense in which I am using the term is that in which the prospect of a European war is uncertain, or the price of copper and the rate of interest twenty years hence, or the obsolescence of a new invention, or the position of private wealth owners in the social system in 1970. About these matters there is no scientific basis on which to form any calculable probability whatever. We simply do not know.”
The application of uncertainty to conduct : The application of uncertainty to conduct Again from Keynes 1937: “[Under uncertainty] there is no scientific basis on which to form any calculable probability whatever. We simply do not know. Nevertheless, the necessity for action and for decision compels us as practical men to do our best to overlook this awkward fact and to behave exactly as we should if we had behind us a good Benthamite calculation of a series of prospective advantages and disadvantages, each multiplied by its appropriate probability waiting to be summed.”
The application of uncertainty to conduct - II : The application of uncertainty to conduct - II “Most, probably, of our decisions to do something positive, the full consequences of which will be drawn out over many days to come, can only be taken as a result of animal spirits - of a spontaneous urge to action rather than inaction, and not as the outcome of a weighted average of quantitative benefits multiplied by quantitative probabilities. […] Only a little more than an expedition to the South Pole, is [enterprise] based on an exact calculation of benefits to come. Thus if the animal spirits are dimmed and the spontaneous optimism falters, leaving us to depend on nothing but a mathematical expectation, enterprise will fade and die; - though fears of loss may have a basis no more reasonable than hopes of profit had before.”
Keynes: assault on micro? : Keynes: assault on micro? 1921 p. 344 (“application of probability to conduct”) : “The first assumption [of economic calculus], that quantities of goodness are duly subject to the laws of arithemtic, appears to me open to a certain amount of doubt. […] The second assumption, however, that degrees of probability are wholly subject to the laws of arithmetic, runs directly counter the view which has been exposed in Part I of this Treatise…”
Saving micro (and decision theory)? : Saving micro (and decision theory)? Friedman’s instrumentalism:
“A meaningful scientific hypothesis or theory typically asserts that certain forces are, and other forces are not, important in understanding a particular class of phenomena. It is frequently convenient to present such a hypothesis by stating that the phenomena it is desired to predict behave in the world of observation as if they occurred in a hypothetical and highly simplified world containing only the forces that the hypothesis asserts to be important.”
Thinking of rational applications.
Last words about Keynes : Last words about Keynes Keynes is often referred to as a man of two projects:
A radical one: sort of gay cultural revolution against poverty, victorianism, utilitarianism, stupidity etc.
A more pragmatic one: helping the country out of the mess.
“Economics is a science of thinking in terms of models joined to the art of choosing models which are relevant to the contemporary world.” To Harrod, July 4, 1938.
See the excellent paper by Favereau: “L’incertain dans la révolution keynésienne : l’hypothèse Wittgenstein”, Economie et Société — Œconomia, 1985. See also:
http://www.psychanalyste-paris.com/Keynes-et-Freud.html
Risk, uncertainty and profit after Keynes : Risk, uncertainty and profit after Keynes Real-world business continued
Economic delirium continued, too:
High Theory: in Debreu’s Theory of value (1959), the number of “states of the world” is known, no reference is made to decision theory whatsoever.
In handbooks etc. the entrepreneur still plays an ideological role… the same as a century ago: he is useful to make profit acceptable.
(See e. g. Pradier 1998 chapter 3)
Let’s get rich! : Let’s get rich! 3 – Finance as an academic discipline : from statics to dynamics
Finance as an academic discipline : from statics to dynamics : Finance as an academic discipline : from statics to dynamics Static models
Before Markowitz
Markowitz’s E/V approach
Risk after Markowitz
Dynamic models
Bachelier
Modern Market Finance
Methodology?
Before Markowitz : Before Markowitz Reference: Pradier 2000.
Risk / return correlation:
“qui ne risque rien n’a rien” (1798),
“Chi non risica non guadagna” (Botero 1589)
“un bourgeois qui ne risque point son argent n’en peut retirer qu’au taux du Roy” (Furetière 1680)
Condorcet, Laplace…
Risk/return in the Condorcet-Laplace Framework : Risk/return in the Condorcet-Laplace Framework
Intuitively we can say:
If dispersion rises,
there will be more weight
in tails,
then the profit rate
(or loading rate) must be
increased to compensate.
Edgeworth (1888) : Edgeworth (1888) “A mathematical theory of banking” = application of the Condorcet-Laplace framework to banking activity.
“Probability is the foundation of banking. The solvency and profits of the banker depend upon the probability that he will not be called upon to meet at once more than a certain amount of its liabilities.”
More on banking : More on banking Banking business:
collect “free” deposits (short-term or overnight);
lend for interest rate (medium or long-term);
leverage effect: the more you lend, the higher the profit.
Potential problem:
illiquidity (i. e. cash retrievals exceed cash balances);
cash balances = security.
Banking business = risk/return arbitrage.
Generalizations of Edgeworth : Generalizations of Edgeworth Wicksell: every business is facing the same arbitrage;
Pigou: why only normal random variables?
Keynes, Knight: uncertainty prevalent in economic life.
Bottom line: we need a theory to account for randomness in economic decisions. Bernoulli’s theory make gambling and insurance behaviors incompatible. Why not try something else?
No insurance & gambling : No insurance & gambling wealth utility 2 possible outcomes with even probability Expectation = middle of segment Utility of expectation
Expected utility wealth utility 2 possible outcomes with even probability Expectation = middle of segment Expected utility
Utility of expectation Insurance: concave utility function Gambling: convex utility function
Describing curves : Describing curves “…the shape of any frequency-curve may be studied by its moments – in the statistical sense. Every curve could be rigidly defined by a sufficiently large number of moments, and an approximation to the situation obtained by taking a small number” (Hicks [1934], p. 195).
Moments (order k): mk =
Indifference curves : Indifference curves
Graph from Chambers [1934] Preferences are shown in a variance/expectation diagram.
« an individual being given 2 percent without risk will be indifferent to these 2 percent riskless and two and a half percent with a unit standard error, etc. for all values of the indifference curve (iii). »
More on moments : More on moments Centered moments : µk =
Moments for continuous distributions
mk = , where f(.) is the density function of the random variable. Centered moment of order k:
Partial moments are computed up to a limit:
or
Is that true that curves can be defined by their moments? : Is that true that curves can be defined by their moments? Borch counterexample (1969) shows that indifference curves in a (E,V) plane intersect :-/
Let (E1,S1) and (E2,S2) be 2 points on the same indifference curve with (E1, E1>E2 and S1>S2. Then let:
Borch counterexample, continued : Borch counterexample, continued Now, if Ni = (x, p, yi, 1-p) with i = 1 or 2, then:
E(N1) = E1, V(S1) = S12 and E(N2) = E2,
V(S2) = S22 hence they must be indifferent BUT
If E1 > E2, then N1 is preferrable to N2 for any increasing utility function (it is said to “stochastically dominate”):
With probability p, both N1 and N2 give outcome x ;
With probability 1-p, N1 gives y1 superior to y2 (N2);
Hence N1 always better!
Conversely, if E1 < E2…
Sometimes a curve can be defined by its moments : Sometimes a curve can be defined by its moments Normal distributions can be defined by their two first moments (hence the N (m,s) notation);
More general: curves satisfying the “location and scale condition” can be defined by their two first moments.
Let F and G be two cumulative distribution functions, the “location and scale condition” is said to hold between them if there exist (a,b) such that for any x,
F(x) = G (ax+b)
Applications : Applications 1936: Keynes asserted that interest rate is determined on the money market (liquidity preference).
Hence, interest for “financial macro” : Hicks (“a suggestion for simplifying the theory of assets”), Marschak (“Money and the of assets”);
And, why not, general equilibrium under “uncertainty”: Marschak (“Assets, prices, and monetary theory”).
Then : Then WAR
Monetary theory seems a bit futile, isn’t it?
There are but many application of probability theory: sampling inspection (of ammo), gunfire setup, routing of cargo fleet, etc. This led, among other operations research developments, to linear programming.
Linear Programming : Linear Programming Ever seen this?
This look very much like a minimization program of a vector function. If V(.) is linear then any mathematically able economist of the 40’s could find the solution.
Quadratic programming : Quadratic programming Now, let us assume that V(.) is quadratic, for instance:
Let W be the variance-covariance matrix of a set of assets and w be the weights of every asset of a portfolio, the program states minimization of variance for a given r(eturn).
Harry Marowitz’s E/V approach : Harry Marowitz’s E/V approach The minimisation of variance program results in an efficient frontier. Any point of the frontier is (generally) on a higher indifference curve than any single asset.
And now : And now This is the basis of Markowitz’s “portfolio theory” (=Marschak =Hicks).
No conceptual breakthrough involved.
Computational breakthrough:
Markowitz H. M. [1952], « Portfolio selection », Journal of Finance.
Markowitz H. M. [1956], « The optimization of a quadratic function subjet to linear constraints », Naval research logistics quarterly, III, pp. 111-133.
Markowitz H. M. [1959], Portfolio selection : efficient diversification of investment, New Haven, Yale University Press.
E/V approach - continued : E/V approach - continued The E/V approach then leads to diversification: investing does no longer mean pick the best stock but choose a portfolio of the frontier that satisfies the investor’s preference for a given risk/return balance.
E/V approach – continued II : E/V approach – continued II The idea that there existed an increasing relation between risk and return is ancient;
Markowitz’s idea is that the relation is not given, it must be constructed through theory;
Moreover Markowitz gives the method and proves it is possible (with a little help from the US Army computers) to compute the efficient frontier.
Refinements of the E/V approach : Refinements of the E/V approach Sharpe’s Capital Assets Pricing Model (1964):
if the capital market is “perfect” (=lend and borrow infinitely at the same rate), then the only rational portfolio to invest in is the “market portfolio”.
The CAPM : The CAPM Return of an asset:
Where rj = return on asset j (random), rm = return on market portfolio (random), rf = risk free rate.
Alternative formulation:
Leads to thinking of b as a measure of riskiness of the asset.
b?
What about risk? : What about risk? Markowitz: pure variability;
Sharpe:
Specific risk (sj) can be eliminated through diversification,
Systematic risk (sm) linked to market portfolio cannot be diversified away.
Market price of risk:
(exchange rate between risk and return)
= Same as Markowitz (with details).
Refinements : Refinements Using more complex risk measures (semi-variance) leads to more computations for a similar results;
Problem: specification of risk still ov