On starlings and casinos

Prospect theory, complex systems research and the work of two physicists can help investors make better decisions
Ruffer Review
Andrew van Biljon
Andrew van Biljon
Research Director

How do people make decisions when faced with uncertainty? Academics have advanced various ways to explain how we make decisions in the face of an unknowable future, both as individuals and collectively, as is the case in a market. They can shed light on how humans approach risk.

One theory has us weighing up the likelihood of different outcomes, and calculating the possible effects on our happiness. This is the expected utility paradigm of economics. While this may sound sensible, probabilities can be difficult to estimate, particularly across a number of outcomes. And the concept of happiness or utility may not be consistent enough to be of much use.

In the late 1970s, two psychologists, Daniel Kahneman and Amos Tversky, set out to explore how we actually make decisions under uncertainty.1 In their experiments, people revealed foibles, contradictions and irrationality when choosing between a series of carefully-framed bets. From this work, Kahneman and Tversky developed prospect theory, a central pillar in the now-established field of behavioural finance.

In a nutshell, prospect theory states that people make decisions based on potential gains and losses, rather than any resulting level of wealth. These gains and losses are evaluated using rules-of-thumb and mental shortcuts. This leads to something novel: people tend to be risk averse when faced with gains, and risk-seeking when faced with losses.

What follows is a crash course in prospect theory, straying into the role of time in decision making, and on to accounting for complexity in financial systems.

A taxonomy of attitudes

Once the reasoning is spelled out, prospect theory feels intuitive. Losses hurt more than gains feel good. People tend to overweight unlikely events, and underweight likely ones. This gives us an extended taxonomy of attitudes towards risk.

Avoiding risk when faced with likely gains – through fear of disappointment. Avoiding risk when faced with unlikely losses – through fear of losing large. Seeking risk when faced with likely losses – because we’re desperate to avoid loss. And seeking risk when faced with unlikely gains – say, buying a lottery ticket.

Prospect theory suggests that as individuals we seek out stocks that have a small chance of a large return, in an attempt to find the next big winner.

For investors, becoming risk-seeking in the face of losses, and risk averse in the face of gains, may sound familiar. Selling winners too early. Holding losers too long. Then there’s the downtrodden portfolio manager. Taking one last big bet in an attempt to save his fund, only for the bet to prove to be just that – his last.

After the financial crisis in 2008, this type of analysis caught the collective imagination. Economic orthodoxy was judged to have failed – to have failed to predict or explain events, a failure linked to neglect of human psychology. Down with the staid approach of mathematics and rational utility maximisers! Up and onwards… into the brave new world of human behaviour.

Addressing financial puzzles

Any theory of individual choice faces a challenge – can it be applied more generally, in aggregate, while still describing the world well? In economics, expected utility can be aggregated into the Capital Asset Pricing Model (CAPM), a model that aims to account for the relationship between the risk and return of financial assets. The model has a number of weaknesses, including its handling of extreme (tail) events and the inconsistency of results through time. Does accommodating behavioural considerations make for a better pricing model?

Prospect theory suggests that as individuals we seek out stocks that have a small chance of a large return, in an attempt to find the next big winner. In a number of studies, the evidence has supported the theory.2 Investors do seem to care about the tendency of an asset either to have a tail of more positive returns or of more negative returns (skewness). Therefore, including these preferences in a CAPM-style model can improve the model’s performance under certain assumptions.

The equity premium puzzle3 is another part of the finance landscape that has been subjected to the behavioural lens. The puzzle is that investors generally demand a higher return for investing in risky equities over safer bonds; this excess is higher than conventional economic theory would suggest. Prospect theory helps – because investors’ loss aversion combines with myopia.4

Take Bob, an imaginary investor. Bob checks the performance of his portfolio several times a week. This regular checking means he sees more volatility, up and down. Like most people, Bob feels losses more keenly than gains of the same size. Over time, a sort of emotional deficit builds up: for an equal number of gains and losses, the losses hurt more. For this reason, Bob demands a larger-than-predicted equity premium.

While a behavioural angle provides some appealing solutions to market conundrums, there are shortcomings.5 There is also the danger we create our own Just So stories of how the world works – extrapolating observed or postulated behaviours and assuming investors behave in a certain way all the time. Are we using our stories to explain away consequential features of finance and markets?

Consider the battered world of the short-volatility exchange-traded fund. In effect, these products were a bet the market would keep going up smoothly. They stood to do well for as long as this was the case. They also stood to suffer badly if markets fell. In the jargon, their return profile had a highly negative skew – a tendency to produce gains, punctuated by infrequent but sizeable losses. In an era of steady equity returns and rock-bottom interest rates, this return profile proved irresistible. Retail investors piled into the products, chasing the momentum of positive gains. This all ended in tears in February 2018: volatility spiked, the negative skew showed the sting in its tail, and short-volatility products lost most or all of their value.

This buying behaviour arguably went against one of the key findings of prospect theory. Investors were buying an asset experiencing healthy gains, participating in a momentum trade. Instead of selling winners – as per the original findings – investors were chasing winners.

What is clear is that investor behaviour is far from consistent over time and under different conditions. There is always potential for a tweak to be made to a model’s preference curve or reference point, but the hallmark of an enduring model is that its foundations don’t need to be altered to cope with differing environments. The path of time shouldn’t change our fundamental description of the markets.

A brief exposition on time

Fittingly, it is to time that we turn in search of a more consistent perspective on investor decision-making.

In Study A, 100 gamblers spend a day at a casino. Each starts with £100. At the end of the day, the gamblers will have either made money or lost money. Some might lose their entire £100 and go bust.

Say only one gambler lost everything. We might reasonably infer there is a 1% chance of going bust, particularly if we re-run Study A on several occasions.

Now let’s tweak our study. In Study B, we give one gambler £100 and force the poor soul to gamble for 100 days in a row. What are the chances of going bust? Because this is one gambler betting repeatedly for 100 days – rather than 100 gamblers betting for only one day – the odds are certainly higher than 1% and possibly even a certainty. Furthermore, once the money is all gone the gambler cannot play anymore. Whether on day five or 55, our experiment is over.

This example draws out a key distinction. In Study A, we are considering risk across outcomes at a point in time – a parallel universe view. In Study B, we are considering risk through time. Study B captures the notion of risk of ruin: any decision is vastly different if someone faces the prospect of losing everything along the way. Risk of ruin scenarios are difficult to analyse, because a hard ending is lurking. The sequencing of events is vital when assessing risk. Any sequence that results in disaster at zero renders the rest of the sequence irrelevant. You can’t make money if you’re out of the game.

Unfortunately, modern finance is not very good at accounting for any of these phenomena. The current state of decision theory suggests that if a gamble is expected to be favourable on average, then playing it repeatedly will be advantageous. In Study A, even if there were more winners than losers at the end of the day, repeatedly gambling at that casino is likely to be dangerous, given that one player went bust.

Insights from physicists

How then do we reconcile the parallel-universe approach and the sequencing-through-time approach? Two physicists, Ole Peters and Murray Gell-Mann, argue much of decision theory has been misinterpreted by failing to distinguish between the two approaches.6 If we understand decisions as being taken with reference to outcomes through time – rather than across hypothetical parallel universes – the results fit much better with real-world observation. And they can help address discrepancies in existing theory.

In simple finance terms, Peters and Gell-Mann argue that people seek to maximise the growth rate of their wealth through time. The outcome of such an approach has the expected positive implications for overall wealth, but it has also been shown to be the best way to allocate across uncertain investments.7 No ethereal notion of utility is needed. And, notably, the results start to look like those proposed by prospect theory.

It’s worth unpacking why this is the case. Recall that prospect theory describes people as loss averse – more affected by losses than gains of an equal size. In seeking to maximise the growth in her wealth, an investor is impacted asymmetrically by down periods: a 10% fall must be followed by an 11% gain in order to get the investor back to where she started. What’s more, the threat of bankruptcy, or at least a substantial dent in starting capital, always lurks in the shadows.

The flock appears to move through the air as one organism, forming myriad shapes and flowing in different directions, almost like a liquid.

Negative shocks therefore affect the overall growth of wealth more than an equivalent positive – either because they must be followed by even better outturns to get back to zero, or because they drag us closer to the bankruptcy line, at which point we’re out of the game. With this in mind, it becomes natural, even necessary, to protect small gains and to avoid losses, especially large ones. But the reasoning doesn’t require any allusion to behaviour, preferences or emotion. The mere focus on growth in wealth is enough to prescribe this approach. We are now shown a way to address market conundrums without resorting to a behavioural crutch. Maybe we are all more rational than we thought.

Is this focus on growth rates through time the answer to all of our decision-making woes? Alas not, because the information needed to make accurate assessments is normally very difficult to come by. There are already problems with how risk is characterised in finance, with CAPM-style thinking still dominant. Even the entire history of modern markets as we know them only stretches for a couple of hundred years at best: this is unlikely to be enough to have witnessed the full range of possible outcomes.

Accounting for complexity

For investors, one of the deeper insights from prospect theory links to people’s inability to evaluate small probabilities accurately. If we are trying to maximise the growth of our wealth through time, but are not good at accounting for unlikely events, we may come a cropper. What can be done?

There is one area of research that has shown great promise – if not in solving the problem of market uncertainty, then at least in understanding it. Complex systems research has been developing since the 1950s. It looks at systems that share a number of characteristics: simple components that interact; adaptive dynamics that respond to conditions; information being shared between the components; and no central control.8

This may not sound particularly special, but systems with these characteristics can give rise to wonderfully rich dynamics, often unexpectedly. Good examples include seismic activity, weather systems, traffic conditions and the different states of water. Or consider murmurations of starlings. The birds flock together when arriving at or leaving their roost site. They do this to confound predators, to socialise, to share information and to group together for warmth. The flock appears to move through the air as one organism, forming myriad shapes and flowing in different directions, almost like a liquid. At the individual level, each bird is following simple rules to keep close to, but not crash into, its neighbours. On a large scale, the results are surprising, richly complex and beautiful.

Back to investing. Studying complex systems is insightful, especially when compared with how the markets tend to be conceptualised. The dominant view tends to consider risk versus reward in rather static terms. Prevailing relationships are used as a guide to the future. Assessments of trade-offs don’t take full account of the potential actions of others.

A current example is the proliferation of investment strategies that aim to react dynamically to any market weakness. These include most strategies labelled “crisis risk offset” or “risk mitigation”, many funds with “systematic” or “dynamic” in their titles, and any approach that scales its degree of risk using the past volatility of prices. These strategies often look wonderful when tested over historical data, but can fail spectacularly when applied in real time. A signal the strategy uses may fail if history doesn’t quite repeat, or even rhyme. Or the strategy may become too widely used, resulting in a self-fulfilling failure: if too many people rush to the fire exit at once, the fire exit ceases to function.

Critical states

The complex system label seems to fit the structure of markets well: investors interacting; prices evolving through time and responding to the environment; information being shared, often rapidly; and no central control (in general).

One concept from complex systems that seems to be particularly useful is that of the critical state. Complex systems can undergo something called a phase transition, which describes a significant and often rapid change in the state of the system. Traffic is an example. A free-flowing motorway can suddenly back up to a standstill, seemingly without cause, before freeing up again a short while later. It’s impossible to predict exactly where and when such a jam might occur. But we know jams are a consequence of the conditions of the motorway, primarily the density of vehicles and the speed at which they’re travelling. If too many cars are trying to travel too fast, any small perturbation can turn into a cascade of brake lights, and a blockage that travels upstream.

If markets can be monitored for signs of a critical state, then we might be able to identify when the odds of a phase transition – here, a large move in markets – are higher than normal. What sort of indicators fulfil such a role? The science is evolving rapidly, and advances in computing power have made modelling complex systems more viable. So far, studying the manner in which prices move relative to each other, changes in the pricing of insurance against large market moves, and even the nature of the oscillations of a single price through time have all been informative in specific scenarios.9

Towards a path through risk

For investors making decisions, the verdict seems mixed. We aren’t all that bad at optimising basic investing choices through time. But we struggle with probabilities, especially in the extreme, and markets don’t always behave how we think – or are led to think – they do. Prospect theory says we overestimate the odds of more unlikely events, while some studies10 suggest we underestimate the odds if the parameters are poorly defined, as they often are in investing.

Turning from human limitations to the market as a whole, a complex systems perspective can help us. Markets are not wise discounters of economic prospects, divining truth through time and susceptible only to occasional hiccups. They are living complex systems, with uncountable feedback loops. They undergo severe and rapid changes in character, presenting patterns we haven’t seen before. In such an environment, we should be wary of our estimates of the likelihood of risks; these risks are likely to be poorly or insufficiently defined.

The better the complexity of the financial system is understood, the closer we can get to appreciating the true nature of what’s involved when investing.

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  1. Initially (1979), Kahneman and Tversky
  2. Boyer, Mitton, and Vorkink (2010); Bali, Cakici, and Whitelaw (2011); Conrad, Dittmar, and Ghysels (2013)
  3. Mehra and Prescott (1985)
  4. Benartzi and Thaler (1995)
  5. Attempts to reproduce experimental results have not always been successful (Gal and Rucker (2018), Yelchiam (2018)). And while the approach provides a descriptive framework for observed human behaviour, it doesn’t extend to any sort of psychological explanation, making it incomplete as a model of human motivation (Staddon, Taylor, and Francis (2017)).
  6. Peters and Gell-Mann (2016)
  7. Kelly (1956)
  8. Towers Watson (2013), Thinking Ahead Institute (2016)
  9. Sornette and Cauwels (2015)
  10. Hertwig et al (2004)

This article originally appeared in The Ruffer Review 2019The views expressed in this article are not intended as an offer or solicitation for the purchase or sale of any investment. The information is fact based and does not constitute investment research, investment advice or a personal recommendation, and should not be used as the basis for any investment decision. 

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