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Wednesday 12 December 2012

Invert, Always Invert

I Wouldn't Start From Here
“All I want to know is where I’m going to die, so I’ll never go there” - Charlie Munger
Carl Gustav Jacob Jacobi was a nineteenth century mathematician famous for his work on elliptic functions, amongst other accomplishments.  Oddly he ends up being frequently quoted by Charlie Munger and Warren Buffett, despite having no known connection with the investment world.

Jacobi's great contribution to investor thinking was his maxim “man muss immer umkehren”: invert, always invert.  Of course, Jacobi was actually making a statement about mathematics, not investment thinking, but we shouldn’t much care where we get our models from, as long as they have the distinct advantage of being useful.

Pythagorian Inversion

Charlie Munger uses the Pythagorian proof that the square root of 2 is irrational as an example of inversion – because they started by trying to prove it wasn’t irrational and wound up with a contradiction that could only be resolved by assuming it was.  This idea – that we should try to prove the opposite of what we really want  – is the key concept behind inversion.

It hopefully goes without saying by now (c.f. The Big List of Behavioral Biases) that we’re all biased in various ways, many of which are beyond our conscious control, which leaves us with the tricky conundrum of knowing that we’re making mistakes but being unable to control them.  Dealing with this is obviously difficult, but one of the recommended techniques is to try and take multiple viewpoints on the same problem – which is easy enough to state but fiendishly hard to put into practice, since it tends to be quite hard to build Chinese walls in our minds, short of having some rather brutal and irreversible brain surgery.

Backwards Thinking

Technically there is a way of finding the optimal path to achieving the best solution, and we’ve discussed it in Dividends Keep You Anchored and (more subtly) in Games People Play: it’s called backwards induction.  At its simplest this involves identifying where you want to end up and then working out the best route to getting there by reversing the sequence of events.  Think of it as playing chess in reverse; at each move you work out the best next move and then go back in time to consider the previous step.

Of course, life and investing isn’t really like chess, which is one reason why it’s quite hard to apply game theory in the real world – although you can create artificial situations where it does apply, as in the wildly successful auction for cellphone bandwidth carried by the UK government, which was expected to raise £5 billion ($8 billion) but ended up bringing in £22.5 billion ($36 billion) – see The Biggest Auction Ever.  Nonetheless taking an inverted view of the investing world can often provide illuminating results. 

1999 and All That

Consider Warren Buffett’s analysis of the growth prospects for the markets back in 1999.  Instead of rosily projecting forwards the astonishing growth rates seen in the previous two decades, as most investors were happily doing, he looked at how investment returns were fundamentally dependent on GDP growth and interest rate fluctuations and argued that:
“I think it's very hard to come up with a persuasive case that equities will over the next 17 years perform anything like--anything like--they've performed in the past 17. If I had to pick the most probable return, from appreciation and dividends combined, that investors in aggregate--repeat, aggregate--would earn in a world of constant interest rates, 2% inflation, and those ever hurtful frictional costs, it would be 6%. If you strip out the inflation component from this nominal return (which you would need to do however inflation fluctuates), that's 4% in real terms. And if 4% is wrong, I believe that the percentage is just as likely to be less as more.”
Which, of course, has turned out to be more or less correct.  Buffett was inverting the then current focus on the wonderful dotcom growth story to look at the end game: stock prices ultimately depend on earnings to grow and earnings depend on GDP growth.  

Inverting the Future

Inversion is a favourite technique of Buffett and Munger and was set out in detail in the Berkshire Hathaway Shareholder letter of 2009 (where the starting quote from Munger is taken from):
“Just because Charlie and I can clearly see dramatic growth ahead for an industry does not mean we can judge what its profit margins and returns on capital will be as a host of competitors battle for supremacy. At Berkshire we will stick with businesses whose profit picture for decades to come seems reasonably predictable. Even then, we will make plenty of mistakes.”
So if we apply inversion, when we come to a conclusion we should then work backwards from that conclusion and try and figure out what it should mean in the real world.  So, let’s say you’ve decided to purchase some stock or other – Facebook, say.  You believe strongly that Facebook is destined for many years of great growth because the power of social networking is frighteningly strong.  Of course, we don’t know exactly how social networks will monetize but let’s put that to one side for the time being.

Inverting Facebook

So at IPO Facebook launched with a market capitalization of about $100 billion and a price earnings ratio of over 100.  These are staggering numbers but let’s not worry about that – we’ve decided that this a a go-go-growth stock and the temporary post-IPO setback is nothing to worry about.  But let’s invert a little and compare Facebook to the world’s biggest company, Apple which has a market cap of about $500 billion and a price earnings ratio of around 12 – it earned around $40 billion last year.

Now let’s assume that Facebook will become the world's largest company - on that growth rating it probably needs to, to justify it.  To do this let's assume it grows its earnings to half the level of Apple, but has a price earnings ratio of double – so it has to grow its earnings by around 20 times to $20 billion.  If it achieves that and becomes the world’s biggest company it will have grown its share price, from IPO, by a factor of 5.  

Assuming Apple doesn't grow then Facebook would need to compound earnings at 17% to catch up by 2032.  To catch up by 2022 then earnings need to grow by 35% a year.  Now given the caveats outlined above by Buffett in terms of new business sectors does that seem like a reasonable risk-reward balance?  Do we think the barriers to entry in social networking and the likely sources of monetization will make this sort of growth achievable?  Even if we think it's possible do we think it's probable?

Beauty Contests

Of course, many people will have bought Facebook with the calculation that others would buy it out of excitement, and we saw a lot of evidence of what Keynes called “beauty contest” investing – where everyone tried to outthink everyone else (see: A Keynesian Theory of Mind).  And in this particular case failed dramatically as Facebook’s post-IPO price tanked – possibly because of the well-publicised technical problems, but more likely because the price was simply too damn high.

In an investing sense inversion can be a great way of testing your analysis – it can force you to think through the problem in a different way.  All too often investors start their analysis with a purchase in mind – the issue becomes justifying the decision.  If you invert and start by thinking about why you shouldn’t buy the stock you often get interesting, and different, answers.  If a company is growing fast look at the competition and see if they are also.  If they’re not then why not?  Remember that the invisible hand will drag back the flyaway winner unless they have something that can’t be copied.

Jacobi's Rule

Investing inversion is a powerful tool, if used honestly: it’s all too easy to ignore confounding factors, but at least it pushes investors in the right direction.  Always look for reasons to do the opposite of what you’re considering, because then you’re pushing against your biases, even if you don’t realize it.  

Remember: always invert before you invest.

Game Theory: A Very Short Introduction (Very Short Introductions)   
Seeking Wisdom: From Darwin to Munger, 3rd Edition  

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  1. Good article but proofs by contradiction are of the weakest kind. Still we need proof that investing path is an invertible function and that has not been presented. Many functions in math are not inverticle.

  2. Learned this important lesson too late in life :-)
    Decide to do something about it, as I felt that the next generation should not make the same mistake. So made up a story to help them learn this concept of inversion, you can read it here,

  3. I heard you mention your book on The Motley Fool's Money Talk and the Amazon preview looks good, but I'm not a Kindle kind of guy-- I like a physical book. Or at the very minimum a .pdf I can print out. Will it be available for purchase that way at some point?