Newcastle grabbed the headlines during the January transfer window, outspending every other club in Europe – at least if we discount transfers between clubs owned by the Pozzo family. As an analyst and longtime member of the Toon Army, I can’t endorse all of the club’s signings. But I do know this: Mike Ashley, the club’s owner, was completely correct to splash the cash.

There are two big “cusps” in the payouts attached to final positions in the English Premier League. One is between fourth and fifth, with entry to the Champions League on the line. The other is between 17th and 18th, on the edge of the relegation zone. There are some smaller cusps, too, of course. But if we look at the historical value of an extra goal scored or conceded, it is nowhere higher than when teams have a goal difference that puts them on one of these big cusps. Relegation happens near a goal difference of -20, and the Champions League at around +25.

Going into the January window, Ashley found himself on the big cusp of relegation. After 19 matches, Newcastle sat in 18th place with 17 points and a goal difference of -15. My most basic simulation of the rest of the season suggested that the chance of relegation was 77%, providing nothing changed on Tyneside (or anywhere else). But 17th place was only two points away, and the club would only need to perform a little bit better in the second half of the season to have a good chance of being safe. So Ashley began to spend.

His first signing was Henri Saivet, a midfielder from Bordeaux who was rather better as a defender than as an attacker, for £5 million. He was the sort of player who could make an incremental improvement in the team’s performance, if not a game-changer. Ashley’s next move was a little riskier, but it was a masterstroke: bringing in Jonjo Shelvey from Swansea City for £12 million.

I couldn’t believe that a club battling relegation would sell a talented player, even a flawed one, to another club in the same situation. Sure, the Swans needed a striker, and they duly spent two-thirds of Shelvey’s fee on Alberto Paloschi. But they could have sold Shelvey to any other team in the world except for the handful at the bottom of the Premier League. Why Newcastle? If Shelvey happened to fit better there than at Swansea, the Welsh club would only have compounded its problems.

By January 24, the Magpies had played 23 matches and were still two points away from safety. But according to my simple simulation – which didn’t take account of the effects the new signings might have over the rest of the season – the chance of relegation had fallen to 74%. How much money would it take to push that probability to 50%, and how much would that be worth?

With the Premier League’s enormous new broadcast deal kicking in next season, the cost of relegation has never been higher. Parachute payments cushion the blow, of course, and there’s always the chance of winning immediate promotion. But the size of that chance is hard to estimate, so let’s say – conservatively, I think – that relegation this season would cost £100m worth of today’s money in expected value.

For a risk-neutral investor, lowering the chance of relegation from 74% to 50% would therefore be worth £24 million. Typically, investors act in risk-neutral ways when they have a long time horizon and make repeated decisions about how to allocate their funds. Ashley does seem like an owner who looks to the long term, but he doesn’t get too many opportunities to shift his allocation. If this makes him somewhat risk-averse, then lowering the chance of relegation could be worth even more*.

The question was whether that amount – £26 million or maybe more – would be enough to buy players who could reduce the likelihood of relegation to 50%. For a club deep in the cellar of the Premier League, it might not have been. They would have had to do a lot to change their fate, as the charts in the post linked above suggest; they might have had to revamp the whole squad. But for a club right on the cusp, like Newcastle, there was a strong chance that the money would indeed be enough.

And so Ashley continued to spend – and quite rightly. On January 26, the club agreed to pay another £12 million to Spurs for Andros Townsend. Then, just as the window was about to shut, he nabbed Seydou Doumbia on loan from Roma.

Not only was the spending justified; it was also intelligent. For one thing, Ashley diversified his spending across several positions. He also bought different types of players. Saivet was solid enough, and Shelvey could do a job if he kept his head and didn’t shoot too often from distance, yet players like these wouldn’t necessarily be enough to lift Newcastle clear of relegation. For that, Ashley had to take a couple of bigger risks. Townsend was a streaky winger, and Doumbia had struggled with injuries. But if either of them caught fire, safety would be assured.

In the end, Newcastle spent £29 million in fees, much of which would be amortized over the years to come, and likely added millions to its wage bill as well. To me, that sounded just about right. I might have chosen different players, but the strategy was on target. Now let’s see if the players are, too.

* UPDATE: Colin Trainor asked for clarification of this point. Spending in the January transfer window may lower the probability of relegation, but it also uses up money that may not necessarily be recouped by selling players later. As a result, both the bad and good outcomes (relegation and safety, respectively) may become somewhat less attractive. The question is whether shifting the likelihood toward the good outcome still makes a risk-averse investor better off. For risk-neutral investors, there is no difference; they care as much about the first pound they lose as the last. But for other investors, this depends on the shape of the their utility functions at the level of wealth where the decision is being made.

Let’s say Mike Ashley starts with £100 million. Relegation knocks him down to £50 million, and safety raises him to £150 million. If the chance of relegation is 74%, then his expected payoff is £76 million. If the chance of relegation falls to 50%, then his expected payoff is £100 million. A risk-neutral investor would pay up to £24 million for this; with the maximum payment, the investor would have a 50% chance of ending up with £26 million and a 50% chance of having £126 million – the same £76 million in expectation as before.

But a risk-averse investor might or might not want this deal. The payoffs are lower in both outcomes (which a risk-averse investor won’t like), but the chance of the worse outcome has fallen (which a risk-averse investor will like). As I wrote above, the choice whether to spend the same amount, more, or less than a risk-neutral investor will depend on the shape of the risk-averse investor’s utility function.

A final point here is that I’ve been assuming none of the money spent by Newcastle on transfers will be recouped. That’s obviously an oversimplification. Wages and loan fees for the remaining portion of the season are gone forever, but transferred players retain and can even gain value. So the net expected spending in today’s money is actually less than the sum of the transfer fees.