This isn’t an individual stock discussion – I haven’t added any new positions recently. And contrary to what the title suggests I’m not going to add to the moaning about the current lack of returns from value strategies. Instead I’ve put together some thoughts about calculating projected returns based on something I’d never heard of until a week ago (maybe I’m slow on the uptake) – time averaging expected returns instead of using expected value.

For example, say I find a stock selling for $1.00 per share where I think the intrinsic value is $1.70 per share, no dividends are paid. I think the probability of it re-rating to full value in one year is 50% and the probability of the price declining (albeit temporarily) to $0.50 in one year is also 50%. If I work out the expected value of buying one share of this stock then I get 50% x $1.70 + 50% x $0.50 = $1.10. Sounds like a reasonable bet – right? Read on.

The above example is what the two famous elderly value investors that everyone loves to quote mean when they speak of ‘the mathematics of Pascal and Fermat’. What they’re referring to is known as expected value (or more correctly the ensemble average) and it applies to situations where many bets are made in parallel such as you would in a very diversified portfolio (hundreds or thousands of positions).

When you make a smaller number of bets in series then the more appropriate formula for the long-term average of the change in the value of the principle with each bet is the time-average:

V = P x r_{1}^{p1} x r_{2}^{p2} x r_{3}^{p3} x…

where V is the amount returned at the end of the bet, P is the amount paid for the bet ($1 in the above example), r_{1} is the amount returned to you if the 1st outcome happens ($1.70 above) and p1 is the probability of the 1st outcome happening (50% above). So on for outcome number 2, 3 etc. If we plug the numbers from the above example into the formula we get:

V = $1 x $1.70^0.5 x $0.50^0.5

= $0.92

Which means that if you allocate a chunk of your portfolio to stocks with similar characteristics to the example and hold it (perhaps rebalancing periodically) for the rest of your career then that portion of your portfolio will vapourise at a rate which approaches 8% per annum, guaranteed.

You could argue that the number of possible decisions that would be rejected by using the time series formula instead of expected value would be negligible. But if we consider a bet of $1 with a binary outcome where we either lose $0.50 or win some amount then consider the outcome for each of the combinations of probability and return from winning we can produce Fig. 1 (using the expected value method) and Fig 2 (time average method). I’ve assumed we only want to invest in scenarios where we expect the return to be more than 15% and highlighted these in green.

Comparing Fig. 1 and Fig. 2 you can see that at 90% probability of winning the minimum return needed to make the bet worthwhile is $1.30 for both cases. Using either calculation method in this area doesn’t change things much. But as you move towards lower probabilities the minimum return needed under the time average calculation skyrockets compared to the expected value method. You need a massive possible return to make low probability bets worthwhile. And the expected value calculation will encourage bets in this area that shouldn’t be made unless you have a highly diversified portfolio.

I take away a few practical insights from this:

- In times where the market presents few high probability or extreme upside opportunities there is a temptation to look at opportunities with average upside and middling probability of winning. This is a worse idea than holding cash.
- I currently sell when stocks reach an expected return of 11%. The probability of getting that 11% needs to be better than 90% to make the position worth holding as it approaches 11%.
- The potential upside and downside returns are likely the most reliable inputs to estimate. The probabilities are harder to estimate but have the greatest impact on the result. To me this creates two extreme categories of stocks which can be held in a concentrated portfolio – those where the probabilities can be estimated with high confidence (which necessitates a well-researched, concentrated portfolio) and those where the stock is trading ridiculously below intrinsic value (not 50% off, more like 70% off).

Most of the above fits with the value investing dogma – only bet when the odds are heavily in your favour, only buy companies where you can predict the outcome etc. And I expect there is a book out there somewhere that explains this better than I can – please let me know if there is. But it is nice to have a another method for ruling out dud opportunities. I can never get enough of those.

Disclosure: No individual stocks are discussed but I recommend doing your own homework and reading anything you can find on this subject by Kelly or Ole Peters. A few details to note:

- To have a high probability of achieving the returns calculated by either formula you need to make a lot of bets, probably more than an investor picking individual stocks will make in a lifetime. So neither formula is going to predict your results – this is only useful as a filter to remove bad bets and to size individual bets.
- Worked examples of the Kelly Criterion that I’ve seen use expected value in the calculation, for example: https://www.oldschoolvalue.com/blog/investing-strategy/kelly-criterion-investing-portfolio-sizing/. Kelly used the time average in his original paper so it’s odd that others (including Buffett & Munger) have adopted the ensemble average.