For risks associated with average values,

We have been grappling with the challenges of applied statistics for quite some time now. 

As consultants and auditors, we routinely analyze estimates made by management that are presented as indisputable, backed up by mathematics and data. And while math and data are usually essential for telling opinions apart from analysis, they can also be tricky. This is true even for something as simple as mathematical averages. Since this topic keeps popping up, let's look at two examples.

“Stage migration” (the “Will Rogers” phenomenon) is about analyzing averages over time. Let's say you're a portfolio manager and you have two portfolios, A and B. Portfolio A has two classes of securities with returns of 2% and 3% (so the average return is 2.5%), while portfolio B has only one class of securities with a return of 1% (which, not surprisingly, is also the average return). If your remuneration is linked to the performance of your portfolios' average returns, and you have no influence over the actual returns, there is a simple way to bring you closer to your goals: You take the security class with a 2% return from Portfolio A and place it in Portfolio B. As a result, Portfolio A's average rises (from 2.5% to 3%) and Portfolio B's average rises as well (from 1% to 1.5%). In other words, both portfolios have become more profitable without having earned a single additional cent. This effect is also found in various analyses of time series, as data is often aggregated using averages or other statistical measures and interpreted over time. In the case of financial instruments, this may involve, for instance, deriving credit risks for trade receivables using historical data (e.g., in the expected credit loss model of IFRS 9). 

It is even more common for errors to occur when applying the effective interest method (as a subsequent measurement concept for financial instruments, such as financial liabilities) due to a failure to distinguish between arithmetic and geometric averages. This distinction is particularly important in cases where updated and modified cash flows must be discounted using an effective interest rate that has already been calculated (e.g., when changing the repayment expectation in accordance with IFRS 9 B5.4.6 or when performing the “substantial modification” test in accordance with IFRS 9 B3.3.6). This is tricky because if a threshold is crossed, it must be recorded in profit or loss possibly right away, and the impact can be significant (IFRS 9 B3.3.6). Also, the calculation has to be done on the date of the contract change, which isn't always the same as when interest is due. Since the effective interest rate is an average interest rate calculated from all cash flows under the contract (i.e., it also takes into account a discount and transaction costs, for example), various methods of averaging are conceivable in principle. If, however, an arithmetic mean was used in the initial calculation of the effective interest rate, inserting an additional calculation date would result in an additional compound interest effect that is not present in the geometric mean. This means that the necessary discounting in these cases does not correspond to the derivation of the effective interest rate. 

Just to be clear, if you have a lot of steps in between, the arithmetic mean will naturally become the geometric mean, so given enough time, the roundabout way will also lead to same result eventually.

To sum up, it's a good idea to make your assessments “objective” by analyzing data. It's also helpful to have a guide that points out the typical detours and danger spots. 

Source: KPMG Corporate Treasury News, Edition 154, May 2025
Authors:
Ralph Schilling, CFA, Partner, Head of Finance and Treasury Management, Treasury Accounting & Commodity Trading, KPMG AG
Felix Wacker-Kijewski, Senior Manager, Finance and Treasury Management, Treasury Accounting & Commodity Trading, KPMG AG