Economics: A pool of uncertainty

"I suppose there exists an extremely powerful and, if I may so speak, malignant being, whose whole endeavours are directed towards deceiving me"
René Descartes, Meditations, II (1901)

Actuaries know a thing or two about uncertainty. As a profession, we are pretty good at harnessing the power of statistical inference while maintaining a healthy suspicion of our models and estimated parameters. Generations of actuaries have been taught to complement robust statistical techniques with a particular type of scepticism, termed ‘actuarial prudence’. Independently, economic decision theorists have been working hard at trying to develop better models of actual behaviour, some of which look suspiciously like models of actuarially prudent decision-makers.

So what exactly is actuarial prudence? Is it reasonable? How does it affect financial markets? And, why might people be actuarially prudent in their own affairs? My SA0 dissertation was motivated by insurance market failures in Africa. In this article I give a brief introduction to some of the theory I thought about along the way.

Uncertainty in microeconomic theory
In an idealised world with individuals who do not like risk, there is a simple notion from economics that uncertainty is pooled so that individuals’ fortunes rise and fall together. In the real world, there are institutions such as insurance companies, managed funds, large employers and public safety nets that help people to pool the shocks that life throws at them in the direction predicted by these simple economic models. Even in the presence of highly developed financial systems, however, the pooling of uncertainty is far from perfect.

Mainstream economic theory blames this failure on three features of reality that drive a wedge between buyers and sellers of uncertain cashflow streams. These market imperfections are well understood by actuaries:
1 Costs — firstly, there are costs of designing, writing and enforcing contracts that may be large enough to make some insurance unattractive. Fixed-claim costs mean that it rarely makes sense to insure against economically small losses. For example, iPod insurance may not make sense for an employed actuary, but neither might health insurance for a Ghanaian farmer.
2 Moral hazard — this happens where a party may change their behaviour to the detriment of other parties on being insulated from risk. Cheap insurance cannot be offered because insurers cannot force the insured to look after their property as if they were uninsured.
3 Adverse selection — Adverse selection occurs where one party knows more about a risk than other parties. When insurance companies are unable to distinguish between good and bad risks, it can be difficult to offer profitable insurance that anyone will buy.

My SA0 dissertation investigated a fourth constraint on risk pooling: neither people nor financial institutions like to trade uncertainty they do not understand, even if there is no moral hazard or adverse selection. Insurance actuaries would call this actuarial prudence over parameter or model uncertainty; economists would talk about ambiguity-averse decision-makers.

Risk, ambiguity and maxmin expected utility
Economic decision theorists often classify perceived uncertainty in two different types: risk describes situations in which events have obvious probabilities, and ambiguity (or Knightian uncertainty) describes situations in which they do not. A decision-maker averse to ambiguity does not like being unaware of the odds.

Perhaps the simplest model of an ambiguity-averse decision-maker, and one of the most important models in behavioural economics, is the maxmin expected utility (MEU) model. In this model, a decision-maker has a set of probability distributions they consider reasonable and, when comparing alternative courses of action, is as pessimistic as possible when deciding upon the probability distribution to use in a particular situation.

Actuaries appear to behave like MEU decision-makers. When pricing assurances, for example, actuaries prudently assume higher than best-estimate mortality and, when pricing annuities, actuaries prudently assume lower than best-estimate mortality. It is as if there is a set of mortality rates considered reasonable, and contracts are always priced assuming the most pessimistic rate is the true rate. Some of this spread in mortality assumptions can be explained by adverse selection and moral hazard — people buying an annuity may be different and act differently to those buying assurance. I argue, however, that some of this prudence is due to a fundamental dislike of not knowing what the odds are, as in the opening quote from Descartes.

Ellsberg’s urns
To encourage clear thinking about risk and ambiguity, economists are taught Ellsberg’s famous paradox. Consider the following version of his paradox, adapted to an insurance context.

You are using your own money to informally offer insurance to a friend where the payouts are dependent on a random ball drawn from each of two urns, A and B (see figure 1 below). Each urn contains 100 balls and every ball is either red or black. You start by choosing a colour to insure against — red or black.

An insurance contract pays £1,000 if the first draw from the insured urn is the colour you picked, and nothing if it is not. You have examined urn A and know that there are 50 red balls and 50 black balls, however, you are not sure of the colour of balls in urn B. You believe there to be a 50% chance that there are 60 red balls in urn B and a 50% chance that there are only 40. There is no adverse selection or moral hazard, so the buyer of insurance knows no more than you do and cannot affect the outcome in any way.

Would you charge more to insure your friend against urn B than urn A? What if you were advising your company how much to charge? What if you were acting as a consultant, advising another company on how much to charge?

For both urns, the overall probability of picking a red ball is 50%; however, urn B is risky and ambiguous whereas urn A is only risky. If you would charge more to insure urn B, then you are behaving as though you were averse to ambiguity, or actuarially prudent under parameter uncertainty. Ellsberg’s paradox lies in the fact that if you prefer to insure urn B then you cannot be represented as an expected utility maximiser, contradicting a fundamental building block of classical economic theory.

My suspicion is that people care about weight of evidence, as opposed to just best estimates, particularly in professional settings. The first justification for this is that financial professionals and institutions are unlikely to be blamed for outcomes from well-understood uncertainty (i.e. risk), but do get blamed for bad outcomes from poorly understood uncertainty (i.e. ambiguity). That professionals and institutions do not like ambiguity may, therefore, be an unintended consequence of professional and regulatory incentives.

There is another explanation for ambiguity aversion, however, ignored by economists but developed in my dissertation. Imagine you have to set a price at which you will sell many small policies, each based on independent draws from the urns, where balls are replaced between draws. For example, imagine N balls were drawn independently from each urn and insurance pays £(1000/N) each time the pre-agreed colour is picked. As N gets larger, the law of large numbers guarantees that your payment from urn A tends towards £500. Your payment from urn B, however, tends to £600 with probability 50% and £400 with probability 50%.

Abusing terminology, the ambiguity is correlated between draws and does not diversify in the way the independent risk does. In this case, risk aversion over your aggregate portfolio is enough to justify ambiguity aversion over pricing of an individual policy.

If a financial institution specialises in trading a particular type of uncertainty with a common source of ambiguity (such as with life insurance business), then ambiguity will not diversify in the way that independent risk will. A limited-liability financial institution is right to act as though it was averse to aggregate portfolio risk and, following the above line of reasoning, it is perfectly right and proper to add prudence margins when pricing to allow for correlated ambiguity.

Similarly, if an individual has to make multiple choices where beliefs are subject to the same source of ambiguity — if they have to make the same sort of decision many times, for example — it is perfectly reasonable for them to act as though they were averse to ambiguity when faced with an individual decision.

Note that this correlation in ambiguity is not the same as correlation in risk. Separate draws from urn B are independent and uncorrelated, no matter what my beliefs are. It is the ambiguity in my beliefs — my inherent lack of knowledge about the true probability distribution — that is correlated between draws.

Financial institutions do not maximise expected profits
It is fairly easy to extend the above reasoning to show that a financial institution that charges more to insure urn B than urn A is not maximising expected profits. Briefly, an expected profit maximiser would compare the expected profits arising from alternative strategies using a single probabilistic belief, whereas an ambiguity-averse financial institution would prudently use different beliefs depending on whether it was buying or selling any given ambiguous cashflow stream.

That financial institutions prefer to trade uncertain cashflow streams of which they have a better understanding is pretty obvious to practitioners. It is only just starting to be thought about rigorously by economists, who typically think about financial institutions being expected profit or utility maximisers.

If instead we start from the assumption that financial institutions are averse to ambiguity, we can explain many features of financial markets and make some unusual, but defensible, prescriptions.

Further reading
A follow-on article by Daniel Clarke on the implications of ambiguity aversion theory in areas such as rainfall insurance, credit ratings and cat bond markets will appear in a later issue. 
>> For Daniel Clarke’s full SA0 dissertation visit http://www.actuaries.org.uk/__data/assets/pdf_file/0018/102456/
ClarkeDaniel_AmbiguityAversionandInsurance_20070914.pdf  
For further reading on behavioural economics, Clarke recommends Advances in Behavioral Economics by Camerer, Loewenstein and Rabin 
>> Thinking Strategically by Dixit and Nalebuff introduces the insights economists have gleaned from using game theory to think about moral hazard and adverse selection.

Daniel Clarke recently became the first actuary to qualify through SA0, the research alternative to a fellowship exam. He is currently a doctoral candidate in economics and part-time lecturer in actuarial science at Oxford University.