Mathematics of investing and life

A few ideas, involving a wee bit of math, which keep me out of trouble

I never imagined that I’d write an essay with this title, since there’s very little math in investing. At best middle-school level, that too state-board. My ageing brain can’t handle more anyway. However, news-flow over the last few months has been plagued by an overdose of dubious math and meaninglessly scary numbers. These raised some interesting questions that tie in with a few mathematical concepts that have been useful in my day job. At the least, they’ve helped me avoid dodgy inferences. Here’s an arbitrary list of some such questions, the concepts they tie in with and how I’ve applied them in an investing context.  

Why is there extreme dispersion in outcomes, from similar starting points?

My guess is that the answer has to do with multiplicative probabilities. Such dispersion is surprisingly common across the messy world. Buggy humans like tying these down to single causes (“My genius at spotting next-big-thing led to multi-bagger”). However, reality is fuzzier, with a bunch of factors that just happened to line up in a certain direction. Multiplicative probabilities can have an especially severe effect at the opposite end. As an example, Indian companies have had a disastrous experience with overseas acquisitions. Going in, these adventures didn’t seem particularly egregious. However, a whole host of seemingly small differences between acquirer and acquiree – context, people, culture, distance, distribution setup, weaker competitive position, business profile – made a big difference to the final outcome. While each difference, in itself, might have reduced odds of success by a small amount, their multiplicative effect was disastrous. An appreciation of this concept results in paranoia about avoiding ‘zeroes’ anywhere in the chain. Big-bang M&A, high debt or dodgy promoters can nullify a lot of other positives in a business.

How do we know we’re on the right track?

A peculiar type of decision tree can help. Academia teaches decision trees in a forward looking sense, to determine expected values along different branches. This has limited utility, as real-world probabilities are indeterminate. Decision trees are more useful in a retrospective sense. Messy world is entirely about process, as individual outcomes are unknowable and uncontrollable. How do we know whether a process is doing more good than bad, especially relative to an alternative process? As an illustration, one investing process is to own good businesses for good. There are instances when this backfires, as valuations correct. So, is this process flawed? The answer requires a retrospective decision tree, with two branches, actual and counterfactual: (a) Own for good, (b) Sell on a valuation rule. Then, this tree needs to be comprehensively applied, over every action and deliberate inaction. Both costs and benefits, including opportunity costs, need to be incorporated along both branches. While I may not complete the tree numerically, merely going through the exercise indicates that bigger damage may not have been on the way down, when an occasional 100 becomes 50. It was on the way up, in prematurely exiting every potential 100 at 20. One of my thumb-rules for ignoring a columnist or researcher is use of a partial decision tree. Any analysis that simply looks at costs of a chosen path, without considering cost-benefit vis-à-vis an alternate is misleading. My checklist for a useful decision tree: backward-looking, actual & counterfactual, comprehensively applied, costs & benefits on both branches, include opportunity cost.

What can you infer about normal times from a highly abnormal time?

Absolutely nothing! Due to a lack of representativeness. It’s as if there’s no lockdown in my part of the world. Dozens of calls are organized every day to provide running commentary on the red, orange and green of 100s of companies. I don’t know what to make of these. What matters is where things end up well after we’ve settled into (new) normalcy. It’s unclear how a highly abnormal, transient period would be even remotely representative of steady state. More generally, representativeness is a key filter in separating signal from noise. It’s the reason I prefer secondary data over primary research. In many industries, with some effort, secondary data covers the universe and isn’t limited to a biased sample. Further, it’s the outcome of millions of actual customer decisions made with real money. It’s not contrived opinions expressed by a few buggy humans who rarely know their real motivations. In times when reporting is based on moving anecdotes cherry-picked to grab eyeballs, representativeness is the first filter for any incoming data point.

Why are shockingly large numbers mostly meaningless?

Without a denominator, there’s no frame of reference. I’ve never figured why buggy humans hate the denominator so much. It practically deserves its own victimhood narrative. It’s not just media saying “Investors lost 1 lakh crore” instead of “Market fell a bit”. Professionals misleadingly claim “This business makes money”, referring to EBITDA rather than return on capital. The most important thing in investing – risk – is cleverly hidden as an innocuous ‘cost of capital’ in, wait for it, the denominator. Those who spend endless hours clicking and dragging the numerator fill out this crucial item as an afterthought, albeit with a few decimal points thrown in. More generally, numbers make more sense on a denominator-adjusted basis. Market share matters more than revenue growth. Ratios convey more than raw financials. Profit figures are to be normalized over a business cycle. Valuations are best viewed in context of long run percentile data. ‘Denominator’ is really a frame of reference rather than a place under a line. Without it, numerators fall somewhere between meaningless and dangerous.

How do we decide on a course of action?

Guided by asymmetric odds. When I was young and stupid, I calculated ‘expected value’, with scenarios, probabilities, outcomes and all that jazz. Now that I am no longer young, I have moved away from such dangerous nonsense. Assume there’s enough evidence to suggest that a downward spiral of doom is unlikely (i.e. no ‘zeroes’). Act if odds appear favourably asymmetric. Yes, I did say it’s impossible to figure out precise odds. That doesn’t imply that good and bad are equally likely. Occasionally, a 90th percentile business is available at its 30th percentile valuation. No brave assumptions are needed for an acceptable outcome and particularly nasty ones are required for a terrible outcome. Fuzzy as it is, this favourable asymmetry is as good as it gets, when it comes to decision-making. Why not wait for 10th or 20th percentile? We return to our decision tree, comparing an approach of reasonable cheapness to one of unreasonable cheapness. Latter entails significant opportunity cost due to missed opportunities. Former only entails temporarily looking stupid on the rare occasion when things get spectacularly cheap.

These ideas – multiplicative probabilities, decision trees, representativeness, hidden denominator, asymmetric odds – have been quite invaluable in investing. I have a feeling they’re more generally applicable. Then again, I wouldn’t know the odds on that.

[Originally published this May at]