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Let's compare like with like. As at the 2009/10 annual report, the seven board directors whose roles remain active today were earning base salaries totalling £2,925,000. In the 2012/13 annual report, published today, the forward-looking total of the corresponding base salaries comes to £2,460,000. So on that basis, the BBC is saving £465,000 a year.

The BBC is also saving the salaries of the two board members who left and weren't replaced – Sharon Baylay (£306,000) and Caroline Thomson (£335,000). But the corporation is intending to replace another leaver, editorial director Roger Mosey, with someone who will likely be paid at least as much as Baylay.

And how much have they cost? According to a recent National Audit Office report, Byford was paid £1,022,000 in severance and accrued holiday pay. Baylay took £395,000. Caroline Thomson got £681,000.

There's also a strong argument that if poor George Entwistle had had wise heads like Byford's and Thomson's around him last autumn, he would have made a better fist of the Savile crisis – and, potentially, would not have had to leave. His severance was £470,000.

The severance costs for those four individuals alone come to £2,568,000. Divide that by a forward-looking predicted cost saving of about £800,000 a year, and Midgley's fag-packet calculation predicts that licence fee payers will – if everything goes to plan from now on – break even sometime in the middle of 2016.

Of course, this is a simplified version of a complicated story, especially with the intervention of the Savile crisis last year. But this largesse with severance payments, in order to make relatively modest ongoing salary savings, is hardly the story that Mr Thompson was selling when he started all of this in 2010.

And it just goes to show something that seasoned BBC watchers know instinctively: when the BBC tries to save money, it always seems to end up costing an awful lot.