“Invert, always invert”

Boeing B-17GA good investment process is really difficult to achieve and even harder to sustain. The variables are many and the problems challenging. Charlie Munger borrowed a highly useful idea from the great 19th Century German mathematician Carl Jacobi that provides a helpful way to deal with the myriad problems investors face in trying to establish a good investment process.

Invert, always invert (“man muss immer umkehren”).

Jacobi believed that the solution for many difficult problems could be found if the problems were expressed in the inverse – by working backward. As in most investment matters, we would do well to emulate Charlie here.

During World War II, the Allied forces sent regular bombing missions into Germany. The lumbering aircraft sent on these raids – most often B-17s – were strategically crucial to the war effort and were often lost to enemy anti-aircraft fire. That was a huge problem, obviously.

One possible solution was to provide more reinforcement for the Flying Fortresses, but armor is heavy and restricts aircraft performance even more. So extra plating could only go where the planes were most vulnerable. The problem of where to add armor was a difficult one because the data set was incomplete. There was no access to the planes that had been shot down. In 1943, the English Air Ministry examined the locations of the bullet holes on the returned aircraft and proposed adding armor to those areas that showed the most damage, all at the planes’ extremities.

The great mathematician Abraham Wald, who had fled Austria for the United States in 1938 to escape the Nazis, was put to work on the problem of estimating the survival probabilities of planes sustaining hits in various locations so that the added armor would be located most expeditiously. Wald came to a conclusion that was surprising and very different from that proposed by the Air Ministry. Since much of Wald’s analysis at the time was new – he didn’t have sufficient computing power to model results and didn’t have access to more recent statistical approaches – his work was ad hoc and his success was due to “the sheer power of his intuition.”

Wald began by drawing an outline of a plane and marking it where returning planes had been hit. There were lots of shots everywhere except in a few crucial areas, with more shots to the planes’ extremities than anywhere else. By inverting the problem – considering where the planes that didn’t return had been hit and what it would take to disable an aircraft rather than examining the data he had from the returning bombers – Wald came to his unique insight, later confirmed by remarkable (for the time, and long classified) mathematical analysis (more here). Much like Sherlock Holmes and the dog that didn’t bark, Wald’s remarkable intuitive leap came about due to what he didn’t see.

Wald realized that the holes from flak and bullets most often seen on the bombers that returned represented the areas where planes were best able to absorb damage and survive. Since the data showed that there were similar areas on each returning B-17 showing little or no damage from enemy fire, Wald concluded that those areas (around the main cockpit and the fuel tanks) were the truly vulnerable spots and that these were the areas that should be reinforced.

The more useful data was in the planes that were shot down and unavailable, not the ones that survived, and had to be “gathered” by induction in that instance.  This insight lies behind what we now call survivorship bias – our tendency to include only successes in statistical analysis, skewing or even invalidating the results. Inverting the problem allowed Wald to come to the correct conclusion, saving many planes (and lives).

The key investment application (beyond the benefits of inverting the problems we face generally) is that in most cases we’d be better served by looking closely at the examples of people and portfolios that failed and why instead of the success stories, even though such examples are unlikely to give rise to book contracts with six-figure advances.  Similarly, we’d be better served examining our personal investment failures than our successes.  Instead of focusing on “why we made it,” we’d be better served by careful failure analysis and fault diagnosis. That’s where the best data is and where the best insight may be inferred.

As I have noted before, investing is a loser’s game (using the famous expression of Charley Ellis) much of the time – with outcomes dominated by luck rather than skill and high transaction costs.  If we avoid mistakes we will generally win.  By examining failure more closely, we’ll have a better chance of doing precisely that. In other words, invert, always invert.


21 thoughts on ““Invert, always invert”

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  8. Wald’s inversion did not go far enough. Why were so many U.S. bombers lost over Europe early in the war? Not that they were being shot in vulnerable areas by German fighter pilots but they were being shot at all. How to reduce lost bombers? Shoot down Nazi fighters attacking bombers. The bombing sorties were too long for fighter escorts, which lacked enough fuel to return. Installing extra fuel pods below the fuselage (and a better engine) gave the Mustang the range it needed to meet German fighters coming up to attack bombers. The skies were soon clear of Nazi aircraft. This meant the bombers could target German aircraft manufacturing to prevent lost fighters from being replaced. What could not be replaced were Nazi fighter pilots. So the skies remained clear for the duration of the war.

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  13. Interesting article. I was reminded of Abraham Mazlow and his Hierarchy of Needs. I thought perhaps he followed the wrong tack by looking at success. One of the people he studied most was Einstein. Would Mazlow have been better off by inverting? Then I realized he did invert. Freud studied people with psychological pathologies but Mazlow, later on, inverted that process and studied who he referred to as the “self-actualizers,” thus developing his now famous hierarchy. Thanks for helping me think today.

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