When is a Risk Premium Earned?

This is a difficult question.  One that challenges our accounting systems.

Think of two insurance contracts.  One lasts for one minute, the other for an hour.  They are both sold for residents of Antartica.  They cover the risk of being hit with a snowball.  On the average, residents of Antarctica are hit with a snowball 3 times per day.  The contracts pay out $10 every time the insured is hit with a snowball.  Premiums are $5 for the one hour policy and $0.10 for the one minute policy. The policies are renewable.

On Antarctica it is the custom of insurers to publish quarterly financials, that is every quarter hour.

Right now, there are 4 people on Antarctica and two of them have policies.  One bought the one minute policy and the other bought the one hour policy.

After the first quarter hour, there have been no claims.  What are the profits from the two policies?

The answers are completely different depending upon whether you decide to look at earning risk premium over the maximum holding period for the policies or the minimum holding period.

Our accounting systems tend to take the minimum holding period approach.  However, the maximum holding period approach might give a more useful answer.

The maximum holding period approach would be to think of the risk over the effectively infinite maximum holding period.  To determine profits, look forward and consider the amount needed for future losses.  You expect to have lumpy losses that average to a certain level.  The maximum holding period approach would then tend to reflect the excess over the expected losses as the profits in a period as profits and allow a build up of reserves to take care of the lumpy losses.

The minimum holding period approach suggests that at the end of the minimum holding period, you are done and reflect any revenue not needed to pay losses as profits.

So if no snowballs were thrown that quarter hour, all premiums (less expenses, which are very high for insurers operating in Antarctica) as profits.  If during any quarter hour there are snowballs, then there will be losses.  Very lumpy results.

This is not just an insurance consideration, it applied to any risks, such as credit or any far out of the money derivatives where losses are not expected in every period.

The minimum holding period approach will tend to encourage risk taking.  The maximum holding period approach would tend to make risk takers realize that they do expect losses sometimes.

Then if someone wants to recognize all of their profits from exposure to an infrequent risk during no loss periods, they would need to totally exit that position.




The following chart is referenced in the comment from Robert Arvanitis…

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One Comment on “When is a Risk Premium Earned?”

  1. Robert Arvanitis Says:

    Very nice post, showing the challenges in expressing business results.

    Financial accounting is driven by the need of absentee owners to monitor their professional managers. Shareholders must know enough to judge the performance of those who run the company, but not so much that reporting costs are prohibitive, nor so much that shareholders lose the benefit of delegating and BECOME the managers.

    Those compromises in reporting make it very difficult to deal with concepts like “release from risk.” If a contract has uniform risk and short reporting lag (think fire insurance), then “release from risk” is uniform. Spreading the premium uniformly over the contract term makes sense. The same is true for simple options.

    But for insurance with a long reporting lag (think excess and umbrella liability), or for compound options (knock-in or other multi-factor), then the release from risk is prolonged and far from linear.

    The chart below is a generic premium cascade:

    (see inset above)

    On day one we get $110 of written premium, an asset against which we post an immediate liability of $110. The green represents the profit. (Any guesses on what the pink wedge represents?)

    Assuming claims are $100, we have $10 of profit, which under current accounting rules is earned pro-rata over one-period term of the contract.

    At the end of the term of the contract, we post a liability for the “incurred but not reported” claims. Over time, claims are reported, set up as case reserves, and then settled as paid. At the end of the process, we may recover salvage and subrogation, reducing the final liability.

    The key point here is to show that we “earn” all the profit in period 1, but there is still substantial uncertainty (read “risk”) for as long as there are potential unpaid losses. Those losses are subject to inflation, judicial innovation, and potential bad-faith settlement risks, among many others.

    We need a better release-from-risk framework. For example, recall that in general insurance, we may price risk by separately estimating frequency and severity of claims. If we have in mind lag patterns for reporting, and another set of development patterns for settlement, then we can amortize part of the profit over the term of reporting-uncertainly, and another part of profit over the term of settlement uncertainly.

    The green wedge over period 1 would be elongated until there was no more reserve (albeit ever-thinner as we gain knowledge of ultimate losses).

    There are two open issues to such any such reform of profit reporting:

    1. Technical trade-offs among the considerations of (i) costs of implementation, (ii) credibility and value of the information to absentee owners (shareholders), and

    2. Weaknesses of human nature. No management will ever willingly defer reported profits to shareholders. (Reporting to the IRS is a different matter altogether…)

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