# The Winner’s Curse

Although the term may be unfamiliar, the Winner’s Curse is a subject that many actuaries have come across in one form or another and implicitly understand without having investigated in depth. Understanding the subject of the Winner’s Curse is important in some actuarial fields, in particular, Group Risk.

## The Concept

The Winner’s Curse was developed as part of auction theory. It is where, given the right circumstances, there is a tendency for the winning bid to exceed the true value of the item obtained through the auction process. In 1971, the Winner’s Curse as a concept, was first described by Capen, Clapp and Campbell in their paper ‘Competitive Bidding in High-Risk Situations’. The paper examined the sale of oil drilling rights. The background to the original idea is simple: consider a number of parties who are bidding for the drilling rights of a particular parcel of land. If the drilling rights are worth the same to each party, then it is a common value auction where each party is trying to estimate the value of the land. Given the difficulty of valuing the drilling rights, some estimates are likely to be higher than the true value while some estimates will be lower. The winner of the auction will be a party whose estimate was too high and can be described as ‘cursed’.

The ideal circumstance for the Winner’s Curse to operate is when it is difficult to determine the intrinsic value of an item up for auction, the item has a similar value to each party and the feedback on the true value of the item is slow or inadequate. While this concept can be relevant to actuaries in other areas (e.g. reinsurance tenders, acquisition of insurance companies etc.), the remainder of this article will examine the Winner’s Curse in the context of the Group Risk market.

## Group Risk

For Group Risk schemes, the selection of the insurer is usually determined through a tender process. In any tender, a large weighting is given to the price offered by the tendering insurers. The winning insurer is often the one that offers the lowest premium.

In the tender, each insurer will determine a premium based upon their best estimate of the future claims cost by using historical claims information. A variety of methods can be used to determine the best estimate of future claims. Each insurer will use their own internal models, take into account a wide range of information and involve considerable application of judgement.

As pricing is carried out independently, no two estimates that are determined by insurers will be the same. Assuming that each insurer’s best estimate of the expected future claim costs is unbiased, then some estimates would fall below the expected claims cost, whilst others above. If the insurer that wins the tender offers the lowest price, then we would expect that their best estimate would be below the true expected claims cost, all other things being equal.

The implication is that, unless tendering insurers build in margins to take the Winner’s Curse into consideration, then, all other things being equal, the winning insurer is likely to underprice and their premiums will not return the expected profit margin.

## A Simple Model

A simple model with a few basic assumptions can show that a tender can result in the lowest price being materially below the true expected claim costs. A tender based only on price, will result in a premium that is likely to be too low.

Consider a situation where the unbiased best estimate of each insurer is normally distributed about the mean with a given standard error. In this situation we obtain the outcome shown in Chart 1 from the simulations of a hypothetical tender.

The expected loss ratio in Chart 1 is based upon the winning price. The loss ratio will be higher than 85% when the price is below 100. Chart 1 demonstrates the main premise of the Winner’s Curse. That is, in a market where participants price using best estimate loss ratios, the expected long-term outcome will be worse than the best estimate.

It can also be seen that, as the number of participants increase, the expected loss ratio increases and the average margin falls well below the 15% used by each company for pricing.

## Parameter Risk

One of the reasons that no two insurers will calculate the same best estimate claims cost is parameter risk. Parameter risk is the risk that the true mean of the expected future claims costs are not estimated accurately. It includes mis-estimation arising from the choice of model, calculation of parameters used in the model, assumptions relied upon and the required use of judgement.

In Group Risk pricing, some of the sources of parameter risk are:

- interpretation of data;
- choice of the experience period;
- segmentation of data;
- credibility formula to blend with an insurers standard book or risk rates;
- the book or risk rates themselves;
- IBNR methodology and parameters;
- allowance for IBNR tail;
- use of subjective information and judgement;
- allowance for product design changes; and
- costing of profit shares.

## Other Factors

As well as parameter risk there are other factors that we need to consider that may result in the price or best estimate departing from the true mean. Some of these are:

- cognitive biases;
- third party pressures;
- market dynamics; and
- operational risks.

The extent that the above factors increase variability in the best estimate, will increase the impact of the Winner’s Curse.

## Magnitude

The magnitude of the Winner’s Curse is partly determined by the likely range of the best estimates determined by insurers. The range of best estimates will depend on how accurately each insurer can estimate the expected claim costs.

To see how the level of uncertainty affects the magnitude of the Winner’s Curse, we can use a simple model to vary the standard error whilst holding all other factors constant.

Chart 2 shows that, as the standard error of the best estimates increase, the expected future loss ratio also increases whilst the expected margin reduces. This is because an increase in the standard error reduces the bottom end of the range of best estimates from insurers. Similarly, as the standard error reduces, the expected loss ratio reduces and is closer to 85%.

“The magnitude of the Winner’s Curse is partly determined by the likely range of the best estimates determined by insurers – which depends upon how accurately each insurer can estimate the expected claim cost.”

## In Conclusion

Possible standard error or range of prices is difficult to determine, however, from experience I have seen some tenders with a very wide range of prices. From this, and other experiences, I conclude that the

Winner’s Curse is a material issue in Group Risk that needs to be considered.

During periods of strong competition, actuaries need to bear in mind the likely consequence of the Winners’ Curse, which, on average, will result in outcomes being worse than expected. For now the Group Risk market is very different to what it was a few years ago when there was strong competition for business. However, the insurance cycle will turn and become more competitive again and we will need to balance the impact of strong competition with the difficulties of pricing.

## Further Reading

- Yellowlees (2013), The Winner’s Curse in Group Risk, Actuaries Summit 2013
- GIRO (2000), The Winner’s Curse: The Unmodelled Impact of Competition, Report of the Winner’s Curse GIRO Working Party, August 2009
- Thaler (1988), Anomalies: The Winner’s Curse, Journal of Economic Perspectives

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