Why mean outcomes are often meaningless


You have spent £2 on a lottery ticket. On Saturday evening you may be a millionaire. Or, more likely, not.

But meantime, the auditors arrive. They must confirm that your accounts show a true and fair view. An old-fashioned auditor might allow you to record the lottery ticket at its historic cost of £2. A modern one would want to assess its fair value.

But there is no market in second-hand lottery tickets. The auditor might allow you to treat it as a “level two” asset which can be valued by reference to the price of other traded items and use a discount to the primary market price. Or the accountant might encourage you to “mark to model”: multiply the payouts by their probabilities and compute an expected value, £1.20 say, though good models attach different values to different tickets because some numbers are more popular than others.

But no one buys a lottery ticket to trade it. The purchasers seek the thrill of winning, but expect the disappointment of losing. You cannot sensibly value an asset at £2, or £1.20, when you can be certain that on Sunday morning its value will be completely different.

There are lessons for accounting practice from this silly example. It is misleading to value assets as if they were held for trading when there is no likelihood or intention to trade. But the more subtle and more fundamental problem is that there is no good method of accounting for transactions whose outcome is binary. You are either a winner or a loser. To assess a value from the mean outcome is as meaningless as the observation that the average person has 1.99 legs.

The return on a bank loan has a similar binary property. The majority of such loans are repaid as expected. A few result in substantial losses. It is rare for creditors to be repaid 99p in the pound. And that is why there has always been difficulty in accounting for losses on bank loans. Pricing them with reference to expected value describes a mean outcome which will almost certainly not become reality.

If banks had large portfolios of uncorrelated loans, it might make sense to value that portfolio at 99p in the pound: but, as financial institutions discovered yet again in 2008, the outcomes of a portfolio of loans are generally closely correlated. It is not just the performance of the individual loans, but the overall portfolio, which can have a binary outcome.

Quantum physicists have long struggled with the problem of Schrödinger’s cat, at once dead and alive, and accountants have not been more successful in resolving the paradox. The epitome of nonsense created by the problem of accounting for binary outcomes is the well-known absurdity that makes a decline in the credit standing of a bank’s source of profit because it reduces the fair value of the bank’s debt. The bank is either solvent, in which case the debt is repayable at par, or it is not, and there is no middle case corresponding to an expected value.

Banks traditionally dealt with this problem by obfuscation. They were allowed to smooth their profits almost at will through the use of hidden reserves. But that approach has given way to demands, which have not in practice been fulfilled, for greater transparency in the accounts of financial institutions.

An amended International Financial Reporting Standard 9, a collection of rules that dictate new principles for accounting for loan losses, was issued last week by the International Accounting Standards Board; the latest attempt to elide rather than acknowledge the problem. It has taken a long time to agree even this proposal, and American and European standards setters are still at loggerheads.

There is no “right” answer to the problem of accounting for these kinds of uncertainty; only a need to acknowledge that there is never such a thing as a single true and fair view, only a range of possible outcomes. When a business has many long-term contracts, or teeters on the verge of bankruptcy, that range may be very wide.

I can see the difficulty a bank chief financial officer will encounter if they tell depositors, shareholders and regulators that annual earnings are something between a loss of $5bn and a profit of $10bn; but such a statement may be the only view of the company’s affairs that is genuinely true and fair.

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