To track patterns in the number of shares on loan for major German stocks, ProPublica requested the data for companies in the DAX 30, which is similar to the 30-stock Dow Jones Industrial Average. We swapped out two recent additions to the DAX, Vonovia SE and ProSiebenSat.1, for their predecessors, Lanxess AG and K+S AG, which were in the index for most of the period covered by our analysis , from January 2010 through December 2015.

German companies typically pay one large dividend in the spring of each year. Shareholders on the official “dividend record date” (abbreviated above as "dividend date") are entitled to the payments, which are usually made the next day.

To avoid taxes on the dividends, banks and non-German investors structure short-term loans around these record dates – what’s called “dividend arbitrage.” Stocks are typically loaned over two to 14 days to German investment funds and banks that pay no dividend tax or can claim refunds. This is why demand to borrow shares spikes around the record date.

To estimate the annual tax loss to the German government, ProPublica measured the spike in borrowed shares between the dividend record date and 20 days prior. The increased number of shares on loan was multiplied by the dividend payment and a 15 percent withholding tax assumed to be reclaimed by those investors and shared with other parties. The resulting estimated tax loss was about $1 billion annually, the bulk of it for DAX 30 stocks and a smaller portion from arbitrage on non-DAX issues.

Demand for borrowed shares is measured by short interest, which is the number of shares on loan for each security.