Hyperbolic (or Simple) Discounting

I was critical of Doyne Farmer in my previous post, and in response I got a message from someone I respect

But I like Farmer's work... solar costs follow Moore's law; and we should value future more (hyperbolic discount rates).

What struck me is that there is an incongruity in associating Moore's Law and Hyperbolic discounting, which I shall explain, and when combined with the first statement points to something more peculiar: do ideas now derive their legitimacy from individuals rather than in relation to each other?

The Moore's Law statement is easier to explain and I assume it comes from this paper (a nice summary was published in Forbes) that investigates the theories

that cost [of production] decreases as a power law of cumulative production [Wright's Law]. An alternative hypothesis is Moore's law, which can be generalized to say that technologies improve exponentially with time.

and that

Our results show that technological progress is forecastable, with the square root of the logarithmic error growing linearly with the forecasting horizon at a typical rate of 2.5% per year. 

I think this final statement exemplifies the gulf between Farmer and myself: he presents technological growth as a matter of (statistical) fact, I see it as one of commercial values (the argument is presented by Deirdre McCloskey, particularly in Bourgeois Dignity).  But this post is not about that.

I suspect the reference to hyperbolic discounting is related to this paper.  Before commenting on Farmer's specific paper I would like to make some comment on the idea of hyperbolic discounting in general.  A hyperbolic discount factor is usually described as follows

where k is the rate of discounting and D is the maturity (I apologise for the notation I am lifting equation .png from elsewhere to save time).  They are contrasted with exponential discounting factors

.

The statement the hyperbolic discounting implies valuing the future more is a bit naive.  If we fix k then the decline in the exponential discount factor is faster than decline in the hyperbolic discount factor, but this assumes the k in both equations have equivalent meaning, which is not the case - and my undergraduate actuarial students get annoyed with me when suggest (in the context of discrete time derivative pricing) they are.  If we do not assume the rates are the same, we can see we can choose a rate for the exponential that gives the same discount factor as the hyperbolic:

Exponential discounting does not imply, a priori, that the future is valued less, the discount rate , k, determines how we value the future.

Farmer points out that hyperbolic functions decrease at a lower rate than exponential functions and so in the limit as time approaches infinity, hyperbolic functions will always value the future more than exponential discounting.  Farmer goes on to suggest that in certain cases of hyperbolic discounting "there is an infinite weight on the far future", this is problematic as if decision making placed infinite weight on the far future we would never consume now, but would have resources available at the end of time when the universe dies.  More technically, these discount functions would contravene the 'transversality condition', which originates in classical mechanics.

Classically in finance, a discount factor is given in terms of hyperbolic discount factor and one uses a D-specific value for k, obtained from the yield-curve.  This gets to the heart of the difference between exponential and hyperbolic discounting; in exponential discounting it is usually assumed that the rate k is constant, and the discounting is time-consistent, where as in hyperbolic discounting the rate k is time-dependent.  This is why the exponential discount function is deemed not to fit Thaler's empirical data, because it is assumed k is fixed, independent of D.

However, it should be noted that the whole field of interest rate models in mathematical finance is concerned with relating the exponential discount rate to the yield curve in order to identify the right discount factor, and in such models it is (generally) assumed that the exponential discount rate is time dependent, and usually stochastic.

To me the discussion of hyperbolic discounting is symptomatic of much of behavioural finance, in that it introduces a sophisticated term into finance for something that has been understood for some time.  My first reaction to hyperbolic discounting was "isn't this just simple (as distinct from compound) discounting", this is a naive but none the less not an unreasonable first approximation. I dislike the adoption of the difficult term "hyperbolic" to re-package the idea that the yield-curve observed in experiments is (peculiarly) decreasing (in itself not a trivial concept) as I see it as "mystifying" finance.  I often associate this process with claims that financial theory is novel.  For example Farmer asserts that exponential discounting was "originally posited by Samuelson (1937) and put on an axiomatic foundation by Koopmans (1960)."  It might come as both a surprise, and a disappointment, to Prof Farmer to learn that the number "e" was first identified by James Bernoulli in 1683,a few years before Newton's  Principia, by considering how a bank account grew as the time between interest payments became infinitesimally small, a year later Leibnitz tackled the problem in the abstract  and arrived at the same answer.  In fact, the problem of Brachistochrone curve, a key step in the development of dynamics, makes a related connection between Bernoulli, Leibnitz, Newton and the transversality condition mentioned above.

The issue I have with hyperbolic discounting is similar to the issue I have with"Prospect Theory"; a key insight of which is that utility functions are S-shaped, this was observed by Friedman and Savage in their 1948 paper The Utility Analysis of Choices Involving Risk (Kahneman and Tversky's paper of 1979, Prospect Theory: An Analysis of Decision Under Risk, has 10 times the citations as Friedman and Savage).  I wonder if this is a reflection of the higher status of the natural sciences over the social sciences: when an economist and a statistician make the observation, it is ignored (partially because it makes the optimisation problem hard), if psychologists make similar observations, admittedly armed with experimental data, it is taken seriously.  I have already discussed similar issues I have with "physics imperialism".  There is also the assumption that because when people are presented with the choices in the experiment choose to value future payoffs peculiarly highly, this is in some way "correct".  Mathematics has a long history of debunking "common sense" because it requires careful thinking.  It is peculiar to value distant payoffs more highly because there is the chance that you die waiting (the experiments usually involve cash payoffs to participants, not public goods).

What perplexes me is that, on the one hand Moore's Law, and the exponential growth of technology, is perceived as reasonable, where as on the other, exponential growth of the much simpler process of money in a bank account, is perceived as unreasonable.  Now some might argue that I miss the essential nature of hyperbolic discounting in associating it with interest accrual, but I would claim this is the essential point.

Financial "behaviour" is that interest is compounded, not simple, and so when discounting I suggest it is more coherent to employ exponential discounting with a non-constant discount rate than try and fit a constant hyperbolic discount rate.  I think this is a preferable approach because it emphasises the dynamic (and contingent) nature of discounting, rather than arguing about the "true" model and its seeking its "true" parameters.

Basically I see the role of academics as making things coherent, identifying the relationship between ideas.  I feel this places me outside the norm, where academic success seems to be measured by coming up with new ideas, even if not new and however problematic they are.  What I find worrying is that there appears to be an emergence of a "cult of personality" in science, where by people express an allegiance to an individual who personifies correct thinking, rather than autonomously challenging authority.  Maybe that's just because I have been too close to the Scottish independence referendum campaign recently and am a bit jaded by it all.

Economics, Ecology and Ethics

I spent the first part of the week at the eco**2 meeting, sponsored by the British Ecological Society and the LSE's Systemic Risk Centre and exploring the relationship between economics and ecology. I attended because I am in the early stages of a research programme to explore if distinctive financial network topologies emerge out of different commercial cultures, and whether the distinctive topologies are more/less efficient in distributing funds and resilient (robust) in responding to shocks; essentially how financial systems evolve and how the commercial environment influences the financial organisms- the ecology.  It was useful as I met for the first time Thomas Lux and Steffano Battiston.

I have a strange connection with ecologists; before becoming an academic I worked for a US oil company in London.  In the late 1990s, in response to over a decade of failed exploration, the firm invited John Krebs and Alex Kacelnik to review its approach to exploration based on their experience of optimal foraging strategies of animals. I was the internal project manager.  The research was cancelled before the project finished (part of a change in senior staff and a "re-focus") and the two tangible consequences are Oxford Risk and that I list Alex as one of my inspirations to leave industry for academia (along with a choice between a nice redundancy package and PhD in maths or a career in Houston).

One thing that Alex taught me is the sub-optimality of risk-aversion.  Some birds need to eat up to 40% of their body weight in a British winter to survive the night,  and so their very existence depends on making the right decisions about looking for food.  Let's say a bird has 6 hours to find 9 berries and it has two choices:

• [Play it Safe] The bird stays where it is, where it knows there are berries in the hope of finding a few.  The chance of finding one or two berries in the hour is 50:50.
• [Take a Risk] The bird flies off, in the hope of finding berry-bonanza but with a high chance of only finding enough to replace the energy lost in flying.  The energy cost of flying is one berry and the chance of only finding the one berry in the new field is 5 in 6, but there  is a 1 in 6 chance of finding 10 worms (and getting an excess of 9).

Notice that the expectation of both strategies is the same, one-and-a-half berries in an hour and so the bird can expect to get the nine berries in the six hours.  However the second approach is riskier (for any concave utility function the expected utility of the second strategy is less than the first).

Alex explained that if the bird has only found five berries after five hours, it is certain to die if it does not switch to the risky strategy, where it has a small chance of finding the 10 berries that will ensure survival.  A similar argument applies if the bird has found more than 8 berries after 5 hours - it can afford to take a risk.

The financial interpretation is that only the middle class should be risk averse the rich can afford to gamble, the poor have nothing (substantially) to lose by gambling but there is the small chance they become rich.  This is an explanation as to why the poor defy economic 'rationality' in buying lottery tickets, it is actually practically reasonable.

Apart from presenting my own work, my first contribution came in the second plenary when Thomas Lux and Michel Loreau discussed "Important Models in Economics/Ecology".  Michel discussed Malthusian catastrophes (possibly in response to a question) and positively contrasted Malthus with Condorcet.  This becomes interesting to me, given Condorcet is associated with maths and finance.  The moderator asked the audience of mixed "economists" and ecologists if any economist took Malthus seriously, I don't think anyone raised their hand.  I felt this left the ecologists with the impression that economists were a bit short sighted.  My contribution was the following: Malthus was a Tory cleric worried about the effects of the political changes -the collapse in social hierarchy- in France, where as Condorcet was committed to French Revolutionary principles of equality, brotherhood and freedom.  I suggested the reticence of economists to follow Malthus was that his ideas legitimated the liberal policies of the British government during the Irish famine.  Restricting growth on the basis of Malthus requires we address the problem of (global) inequality; I question the morality of people with stuff telling people without stuff that they can't have any more stuff. Lord May commented that Malthus was right but the time scale was wrong, I think this is a peculiar view of an empirical scientist: I can point to the problems of the Malthusian catastrophe theory, the theory has yet to be shown to be true. I think the consensus is that famine and pestilence is not an issue of availability of resources but the distribution of resources, what I think economists call the co-ordination problem.

At the end of the day Doyne Farmer captivated his audience by describing his success as a hedge fund quant in the 1990s.  I though it a bit odd that the previous evening Lord May had been critical of the type of activities Doyne had been involved with, yet 24 hours later the audience seemed to see the sense of it all.  Doyne's story was essentially that he did not believe in the Efficient Markets Hypothesis, and as a skilled mathematician he could go below the economic theory of "first order efficiency" and mine riskless profits in identifying "second order inefficiency".  He produced a series of convincing plots, which all seemed to end around 1998.  Let's be straight- a priori I am sceptical about Farmer's claims given he claims he built the first wearable computer in the 1970s, where as I believe it was Claude Shannon and Ed Thorpe in the early 1960s.  At the end of his presentation a young ecologist asked the question "If you were making money out of these inefficiencies, who was losing money".  Farmer skirted round the question, first stating that economics is not a zero-sum game (that is true, but finance IS a zero-sum game) and that the activities of arbitrageurs made markets more efficient (again that is true).  What he did not answer was who was losing money.

I then asked Prof Farmer  "Is making riskless  profits ethical?" My reasons were three-fold.

1. I felt he had not been sincere in answering the ecologist's question.
2. I have argued that the Efficient Markets Hypothesis can bee seen not a statement of fact but one of values: it is a contemporary version of the Scholastic injunction on making riskless profits.  Farmer's criticism of the EMH, in my mind, is a criticism of this moral injunction.
3. Both the US and UK legislators (in the Financial Crisis Inquiry Commission report  of 2011 and the Changing Banking for Good report of 2013) stress the degradation of commercial ethics as a cause of financial crises since 2007.

I do not think I am alone in thinking Doyne did not really understand the question when it was asked. I  do not think he really understood what riskless profits means, as opposed to high probability profits, but I also feel he had never considered the ethics of his mathematical modelling (the Professional Standards Board of the Chartered Institute of Banking are interested in my work because they have found it virtually impossible to get the ethics agenda into the "quant" world).  I believe the moral vacuum in finance is not endogenous to commerce but has entered the discipline from academia, that is dominated by a strict fact/values dichotomy.  Doyne's response was a suggestion that I was evoking some sort of Protestant shaming.  I pointed out usury prohibitions were Catholic, the Protestants removed them. But in a sense I was; the previous night Lord May had called for a "professionalisation" of banking, so that those who bought the profession into disrepute could be expelled.  The legal, accounting, actuarial, and banking professions all emerged in Presbyterian Scotland, with a tradition of shaming, before Episcopal England.  One of the features of contemporary finance is that it is impenetrable for public scrutiny, every thing happens behind closed doors, and so it is difficult to "shame" bankers into probity.  If no one asks you if you are behaving ethically, because they cannot see what you are doing, you do not have to worry aebout behaving ethically.  Mauss made the point that the difference between science and magic is science is a public activity (like religion) where as magic is purposefully secret.

After the session closed I went to the pub with a couple of ecologists and explained that I asked the question because before the nineteenth century commerce was seen as a civilising (as in supporting civil society) force, not the destructive force we think of today.  Shakespeare personified the virtue of charity in Antonio, The Merchant of  Venice, something that is inconceivable today.  The origins of actuarial science are in the Scottish Ministers' Widows' Fund, a synthesis of Faith, Hope and Charity.  Later I was asked "What changed", I took a breath and answered "Darwin".

Starting with Malthus the cooperation metaphor - a key theme in eighteenth century economics, including in Adam Smith - is replaced by the competition metaphor.  The economic system shifts from one designed to co-ordinate (equitable) supply and demand to one designed to winnow out the weak and inefficient.  I have argued that this reflects society replacing a fear of the uncertain with a concern for scarcity (recall that there were a series of famines in Europe in the second quarter of the nineteenth century), enabled by Laplacian probability reducing distributions to expectations.  I do not suggest Darwin was responsible for this shift, but he does personify a general trend that included Mill, Spencer, Galton, Huxley et al., all now seen as pre-eminent figures of English nineteenth century thought.  I think the demotion of humans to be "just" another class of animal was a key feature of this period (and this is the opinion of a firm atheist) and was a central result of the nineteenth century biologists.

Something I had not thought of occurred to me on the final day of the workshop. At some point someone had asked if there was the concept of redundancy in economics.  I think I answered, but it might only have been in my head, that  there is not in economics but there is in finance, where, since the Scottish Ministers' Widows' Fund, at the latest, there has been the issue of reserves.  Listening to some of the ecologists I noted that there  were repeated references to optimising energy use as a key feature of ecological thought.  Similarly a key tool in the physical sciences is minimising energy or maximising entropy.  This thought was followed by the thought that physicists often try and see a conceptual relationship between money and energy which seems to completely mis-represent the nature of money.

Sitting in Heathrow's Terminal 5 I thought that the terminal was a masterpiece of efficiency but probably the most inhuman place I had ever been, worse than a nuclear power plant or production line.  Waiting for my flight I wondered if the relationship between ecology and economics was more profound that I had ever appreciated, and that the striving for efficiency in economics is in some sense the application of ecological principles.  Alfred Marshall is known for both improving the mathematical rigour of economics and advocating the adoption of biological techniques: “the Mecca of the economist lies in economic biology rather than in economic dynamics [i.e. mechanics].”  The problems of finance are that a financier's tendency to hold reserves in a stochastic market are challenged by the economist's desire to use capital efficiently, as an animal uses energy efficiently.  This is feasible because a Lapacian can reduce a distribution to an expectation and possible because a scientist separates facts and values and sees man as beastly, not virtuous.

I have come to conceive of markets as centres of communicative action reliant on the norms of reciprocity, sincerity and charity.  In this formulation mathematics is not used to determine (predict) the truth but as a language to facilitate discourse.  I believe, based on sociological studies of the markets,  that when market makers offer a price they can be seen as making a claim, which can be challenged by speculators and arbitrageurs, who either accept the claim/price and let it pass, or challenge it by taking the price or offering their own.  In this sense mathematical models are used to justify claims, furthermore I believe this practice of using mathematics in justifying claims in finance was adapted in both politics, to deliver democracy, and science: I do not believe in stochastic systems models can be used to predict, only to justify in a discursive process.

This might seem academic and irrelevant, the practical usefulness is that it enables me to identify why practices described by Farmer might be unethical, rather than relying on an instinct that speculation in complex financial instruments just can't be right (where as geo-engineering, genetic modification or nuclear weapons are OK).   I aim to show in my research, that such markets can offer growth without creating the inequality that Michel Loreau objected to, but I feel is a consequence of the aim at efficiency that a 'Darwinian' economics demands.

A popular tweet from the meeting was

Farmer ends on 'ecological approaches could play important role, but requires abandoning some fundamental assumptions in economics'
— Eco Squared 2014 (@e2c2o2) September 9, 2014

and I ask myself: is this a good thing?  Has society been served well by efficiency replacing redundancy in finance and competition metaphors replacing cooperation metaphors in economics?  Before ecologists invite economists to abandon some of their fundamental assumptions I would like them to explain how their approaches differ from Darwin's argument articulated in The Descent of Man

 My object in this chapter is to shew that there is no fundamental difference between man and the higher mammals in their mental faculties.

and that

 The great break in the organic chain between man and his nearest allies, which cannot be bridged over by any extinct or living species, has often been advanced as a grave objection to the belief that man is descended from some lower form; ...  At some future period, not very distant as measured by centuries, the civilised races of man will almost certainly exterminate, and replace, the savage races throughout the world. At the same time the anthropomorphous apes, as Professor Schaaffhausen has remarked, will no doubt be exterminated. The break between man and his nearest allies will then be wider, for it will intervene between man in a more civilised state, as we may hope, even than the Caucasian, and some ape as low as a baboon, instead of as now between the negro or Australian and the gorilla.

In contrast to Darwin, in An Inquiry into the Nature and Causes of the Wealth of Nations Adam Smith argues that humans are distinctive from other animals in the degree to which they are co-operative

Two greyhounds, in running down the same hare, have sometimes the appearance of acting in some sort of concert. Each turns her towards his companion, or endeavours to intercept her when his companion turns her towards himself. This, however, is not the effect of any contract, but of the accidental concurrence of their passions in the same object at that particular time

Humans are different to animals in that they exhibit

 the propensity to truck, barter, and exchange one thing for another.

Markets are not simply a technical tool to facilitate life, but they capture a key distinction between humans and  other animals.  I am not convinced that it is progressive to lose the 'civilised' aspect of humanity.

Pragmatism requires me to consider the practical implications of my aim.   Someone asked a BBC radio news program what would be the relationship between an independent Scotland and the BBC.  As with everything no one knows what will  happen post independence (because the nationalists have told us what they want not what they might be able to deliver).  The news was not positive for the lover of BBC's Radio 4.  It then struck me that much of the independence debate is focused on the unionists pointing out to people who "have"what they might lose in an independent Scotland: jobs, low mortgage rates, large research grants, a stable currency and the BBC.  Of course, if you have no job, no mortgage, no money and watch Sky but pay for the BBC, you have nothing to lose but the chance of winning in the nationalist vision of a wealthy and fair society.  For me the independence referendum, and the potential for a FN victory in France and the rise of UKIP are the consequences of basing markets on the theory of competition, that accepts the inevitability of extinction of those who can't keep up, rather than social cohesion.