# Winning systematically with Fortune’s formula

When timing is everything

When Ed Thorp, the famous billionaire money manager, wanted to test his gambling theory of how to beat the dealer, he started out by card counting at the casinos.

In those days casinos only used a 52-card deck, and when the deck was exhausted the house shuffled and started again.

This allowed Thorp and his partner Claude Shannon to understand that if they knew which cards were already ‘dead’ then the odds on the current hand they were dealt would change.

If all four Aces were already out of the pack, for example, then there was no point calculating the odds of a win based on the appearance of another Ace.

Thus the probability of success was slightly altered.

Playing the probabilities

A simple example: let’s say you’re playing Blackjack, and you’re dealt a King, and so to get 21 or Blackjack, you need an Ace.

Now imagine it’s the first hand of a fresh deck and so there are 51 cards left (52 minus your King) and so there are 4 Aces in 51 cards.

The probability of drawing an Ace is low.

Now imagine the same hand but it comes nearer the end of the deck, and you know that there are still 4 Aces among the 10 remaining cards.

This increases the probability of drawing an Ace and thus winning the hand.

Thorp understood that in these cases where the probability was higher, it was profitable to bet a larger amount.

How much to bet, though?

The Kelly Criterion

Thorp went on to help create and modify what is now often known as the Kelly Criterion.

Put simply, when you know as an investor the odds are heavily in your favour (a bit like card counting) then you should bet more accordingly (but not so aggressively that you damage your account!).

It doesn’t mean you will win every time, but Kelly and Thorp developed a mathematical formula which demonstrated that if you bet according to the your odds (or probability of winning) then your wealth would end up far in front…and by a greater and greater amount over time.

Why is this important?

Because if you think of your capital as a series of bets, and bet only when the odds in your favour – with appropriate position sizing – your wealth in real terms will accelerate much faster than if you blindly bet the same amount regardless of the odds.

Aiming for each and every dollar invested to work harder is a key reason Buffett has generated such enormous wealth over the decades.

Buffett famously invested 40% of his partnership’s capital in American Express in 1963 after its stock had halved during the ‘Salad Oil’ scandal.

Buffett doesn’t remain fully invested at all times, and he doesn’t bet the same amount each time he invests, which is one of the ‘secrets’ to his enormous wealth.

In many gambling situations, such as over a long series of Blackjack hands and only betting when the odds were in your favour (generally better than 50/50), you would, on average, succeed in winning more than losing.

Once the casinos wised up to what Shannon and Thorp were up to they created larger decks to make card counting harder.

Why were the casinos forced to respond?

Because card counting worked by tilting the odds in the gambler’s favour.

Not just once, but systematically.

Card counting is banned because it gives the gambler an ‘unfair’ advantage.

It worked because you could use updated knowledge (the ‘new’ number of cards left in the deck to calculate a new probability of success after every hand) to connect the present to the past.

As with card counting, what has happened in the past affects your current and future prospects in stock markets.

Fortune’s formula

After making some money, Thorp sought other fields of gambling where he could test out his theories.

He developed a mathematical formula and a way to wager using knowledge and probabilities to determine a rate of success.

Thorp developed what is now often known as the Kelly Criterion or ‘Fortune’s formula’.

After a few years of winning and developing the formula further, he came to the stock market, and was tremendously successful as a hedge fund investor, obliterating the index return.

Thorp was able to compound his wealth dramatically faster than the index, by using the Kelly Criterion.

Back to card counting

Gamblers counted cards because they realised that the outcome of previous hands influences your probability of success in the current hand.

Imagine being dealt a hand with a high probability of success.

If you don’t know what cards are already out of the deck through previous hands you can’t really calculate the odds of drawing a card you want.

However if you’ve been card counting and realise the odds are in your favour then you should place a larger bet.

Why bet a larger amount on this particular hand?

Because you know when there are 10s, Jacks, Queens, and Kings left in the remaining cards and when the probability of a win is higher.

But that’s looking at the odds in isolation…

Thinking in 10-year timeframes

Investing doesn’t quite present the same dynamic.

You must account carefully for what has happened previously, a bit like listening to the echoes from the Big Bang.

Benoît Mandelbrot showed that stocks have a long memory, and in turn that markets are not always efficient (see Long Term Dependence, and the Hurst exponent).

If you’re an investor seeking favourable bets then you need to look at the expected returns given what has gone on in the past.

10 years of history is a somewhat subjective figure, but it does have an accord with the business cycle, and so Ben Graham and later Robert Shiller invented the CAPE ratio to account for the previous ‘echo’, and the influence it has on future returns.

So investing now is not quite the same as playing the current hand and ignoring the cards that are no dead and no longer available.

What’s happened in the past significantly influences the present – like echoes of the Big Bang it fades, of course, the longer it goes on, but still it pays a key influence.

3 key rules

If investors in US stocks have made a wonderful 400% of gains since 2009, how long will it be before they start living like they have made 400% and cash in their gains, driving prices lower?

You’d be surprised at how alike we are in our behaviour.

As Robert Shiller stated: ‘the folks who are responsible for the bull market don’t realise they are the same folks that cause the bear market’.

There are 3 important takeaways here:

1. Be patient – you can’t and won’t win every hand. But there will be opportunities when the odds are heavily in your favour. As Buffett is fond of saying – you don’t have to swing at every pitch;
2. Bet accordingly – Following rule #1 above means that when the odds do come in your favour then you should bet accordingly. This is why we like using the CAPE ratio, because it tells us when the odds have swung in our favour; and
3. It does mean that there will be hands where we miss out on a ‘win’ – but still we always think in probabilities.

Probability in investing

There are two fields in probability.

Some forms of gambling are games where the probability and the odds can be calculated – like cards, horse racing, and lotto, for example.

But investing is a little different.

It’s more to do with uncertainty – you don’t know with any degree of certainty what the payoff will be if you outlay a bet today.

Bayesian probability

When investing you must update your knowledge and calculate the ‘new’ odds given what has happened previously.

The CAPE ratio gives us a good reading on what to expect because it uses a smoothed 10-year timeframe (you can use others if you wish, but the 10-year has a good correlation with actual investment returns).

But what else you can do to improve your chances of a positive return, such as when card counting?

Look at the situation you are currently faced with and see what has happened in the past.

In investing we know that history won’t repeat, but it does rhyme.

We argue that investing in stock markets is far more favourable given that you can’t easily lose your entire stake on one hand as you can in a card game.

Cards are more independent in the sense that each hand has only a small influence on the odds for the next hand.

We know stock markets around the world rise and fall, however, and so we can look at history (our informational advantage) and determine that they don’t all go to zero (losing every hand).

Therefore you can simply adjust your allocation to ensure that you survive to play the strong Aces hand each time it comes up.

And we know from history that the bigger opportunities will come up, thanks to the patterns of stock market behaviour (market cycles and mean reversion).

At the time of writing we believe the odds are not very favourable in many developed markets.

Buffett, for example, is holding US\$128 billion in cash at the moment – a huge percentage share of his capital – and that tells you he’s waiting patiently for the fat pitch.

He wants to make his money work harder.

It’s Fortune’s formula.

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