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Smooth is what we aim for

Let’s say I offer you a simple gamble.

You take £100 of your own money, then toss a coin. If it’s heads, you win 40%, if it’s tails you lose 30%.

Would you accept the gamble?

Simple maths says you should – the expected return is +5% (50% * 1.4 + 50% * 0.7).

But what if I said that if you accepted the bet, you had to repeat the gamble 100 times. Would you still accept?

The intuitive answer is to say “Yes”, as each time you play the expected return is +5%. The more times you play, the higher your overall expected return should be.

But that’s not the case. Once you start to repeat the bet, the maths changes.

Let’s say you made the bet once and lost – your £100 turns into £70 (£100 * 0.7). You then play again and win. Your £70 now increases to £98 (£70 * 1.4). £98 is less than your original £100 – you’ve ended up with less than you started with, despite having one win and one loss. The same thing happens with one win followed by one loss: £100 turns into £140, which turns into £98.

This is despite having a positive expected return – a 40% gain vs a 30% loss.

The key, and surprisingly little-known, idea here is that when you play the game multiple times, losses have a disproportionate impact on returns. Losses matter more because they require a larger percentage gain to break-even. The larger the loss, the larger the gain to break-even. Taken to the extreme, a 50% loss then requires a 100% get back to even. Taken to more of an extreme, a 100% gain is great, but a 100% loss means you can’t play the game anymore.

This is known as the volatility tax.

Why none of us are average

 

As a reminder, our game was: starting with £100, you gain 40% if you flip heads, but lose 30% if you flip tails.

For a simple example of how this game starts to play out with multiple flips, we can plot 2 rounds of coinflips in a simple decision tree, which looks like this:

Coinflipping decision tree

By weighting the 3 results by their probabilities, we can see that the average ending result is £110.25 – a return of +10.25%. After 2 flips, the expected return on average is 10.25%. This is consistent with our +5% expected return per flip (after 2 stages, the expected return would be 1.05*1.05 = 1.1025).

But the most likely scenario for a single person journeying through the decision tree is to lose money. If you were to start at the far left of the decision tree and flip the coin twice yourself, you’d have a:

  • 50% chance of ending up with £98
  • 25% chance of ending up with £196
  • 25% chance of ending up with £49.

The most likely result for one person travelling through the tree is £98, but the average result was £110.25.

Why the difference?

It’s all to do with averages. The average outcome is pulled up by the top branch – there might only be a 25% probability of it happening, but because it’s a much higher value, it pulls the average up.

The top branch quickly gets larger the more stages the decision tree has. If we expand our decision tree past our 2 stages, the top branch grows faster at each stage, as each win increases the balance by a higher monetary amount, which further grows after each win. As long as there’s a positive expected return (ours is +5%), this will always be the case.

Just look back at the decision tree above – in the best possible scenario we’ve gained £96 – in the worst, we’ve only lost £51. And that’s only after 2 flips.

A good way of looking at how this effect becomes more extreme over time is looking at the difference between the mean, median, and mode results. In our example above, the mean is £110.25, the median and mode are both £98.

The mean is skewed by the top branches which do incredibly well, but the median the mode are not.

If we carry on our bet through for 40 flips, keeping everything else the same, and playing the game 10,000 times, we come up with the following:

Simulations of coinflips

The mean result gets more and more divorced from the median and mode over time, as a result of the few branches which end up with extraordinarily large balances.

We can predict this with the same maths we used above. We’d expect the ending balance for the mean to be somewhere around £700, as that represents a 5% expected return compounded over 40 flips (£100*1.05^40). We’d expect the median to be somewhere around £67, as that represents a 2% loss every second round (thinking back to our 2-stage decision tree: £100*0.98^20). And that’s exactly what happens – our mean is £740 and our median and mode are both £67.

Because very few people will experience the top branches, most individuals journeying through the decision tree will experience something towards the middle of the tree. As an individual with only one opportunity to play the coin flipping game, we’re more concerned with the median and mode returns (those in the middle of the tree and those that are seen most often), than the mean.

Sadly for us, because we’re more likely to end up with the median/mode than the mean, our return actually decreases the longer the decision tree goes on – even though the payoff is an even chance of 40% gain or a 30% loss, which is a positive expected return. And that’s because, as we’ve seen, losses are disproportionately harmful for compounded results. Because there’s a 50% chance of a loss at each stage of our decision tree, the most likely result (the median/mode result) for any individual travelling through the decision tree decreases the further through the tree they go.

This shows that as the mean increases over time, the median and modes decrease over time.

Which brings us to an interesting point. The result of one person’s journey through the decision tree is very different to the average result if a million people all journeyed through the same decision tree.

This is a concept known as ergodicity.

What is ergodicity?

 

Consider the following thought experiment offered by Black Swan author Nassim Taleb.

In the first example, 100 people go to a casino. Some may lose, some may win. Now assume that gambler number 28 goes bust – will gambler number 29 be affected? No.

You can safely calculate, from your sample, that about 1% of the gamblers will go bust. And if you keep playing and playing, you will be expected have about the same ratio, 1% of gamblers over that time window.

Now compare to the second case in the thought experiment. One person (Taleb calls him Theodorus Ibn Warqa) goes to the casino 100 days in a row, starting with a set amount. On day 28, he goes bust. Will there be day 29? No. He has hit an uncle point; there is no game no more.

Ensemble versus time probability

No matter how good he is, you can safely calculate that he has a 100% probability of eventually going bust.

The probabilities of success from the collection of people does not apply to the individual. Taleb calls the first set ensemble probability, and the second one time probability (since one is concerned with a collection of people and the other with a single person through time).

Let’s take another example.

Assume a collection of people play Russian Roulette a single time for £1,000,000. About five out of six will make money. If someone used a standard cost-benefit analysis, he would have claimed that one has 83.33% chance of gains, for an “expected” average return per shot of £833,333. But if you play Russian roulette more than once, you are deemed to end up in the cemetery. Your expected return is… not computable.

Repeated exposures to a risky activity mean that eventually the risky event will happen. If you gamble enough, eventually you’ll go bust. If you play Russian Roulette enough, eventually you’ll shoot yourself.   

Taylor Pearson, whose excellent writings on ergodicity I’ve just discovered, provides the following set of rules for determining whether a system is ergodic or not:

In an ergodic scenario, the average outcome of the group is the same as the average outcome of the individual over time. An example of an ergodic system would be the outcomes of a coin toss (heads/tails). If 100 people flip a coin once or 1 person flips a coin 100 times, you get the same outcome. (Though the consequences of those outcomes (e.g. win/lose money) are typically not ergodic)!

In a non-ergodic system, the individual, over time, does not get the average outcome of the group. A way to identify an ergodic situation is to ask do I get the same result if I:

  • Look at one individual’s trajectory across time
  • Look at a bunch of individual’s trajectories at a single point in time

If yes: ergodic.

If not: non-ergodic.”

How does all this relate to investing?

 

So far we’ve seen that losses have a disproportionate impact on a repeated activity, and that probabilities of success from the collection of people do not apply to the individual. Now let’s see why they’re relevant for investors.

Over the long term, the stock market has risen at roughly 7% per year, on average.

But, as we’ve seen, the probabilities of success from the collection of people do not apply to one person. No person can get the returns of the market unless he has infinite pockets and infinite time.

To provide a few examples, these are a few reasons why you might not get the same return as the average:

(Given what we know about the power of losses, we’ll focus on how our returns could end up worse than the average – although we could equally focus on the upside.)

1) Life events

If you’re forced to reduce your stock market exposure because of divorce, or retirement, or because you had to pay healthcare costs, your returns will be different from those of the market.

(On the flip side, you might receive an inheritance, sell a business, receive a redundancy package, or land on Go and collect £400. These are non-ergodic, but in a good way.)

2) Investment decisions

If you make poor investment decisions, you won’t receive market returns.

For example, if you:

      • Concentrate your portfolio in very few stocks/funds,
      • Pay too much for your investments, in terms of investment fees, trading costs, or taxes,
      • Engage in performance chasing, by piling into the best performing stocks/funds,
      • Invest over too short a time period, having to sell out at a loss, or
      • Take on too much risk, through the allure of higher returns, or the use of leverage

In all these scenarios, you’ll likely receive less than the average market return.

3) Sequence of returns

If you retire and see the market drop 50% on the day of your retirement, your portfolio won’t last for as long as you originally thought.

(Conversely, if you’re just starting out investing and see the market drop 50%, then you’ll be buying the market at cheaper valuations and are likely to have excellent future returns.)

The sequence in which we receive our returns matters hugely in how quickly we’re able to build and preserve wealth, and can lead to very different returns to the average (for more on sequence of returns risk, see here).

All that’s to say that we shouldn’t confuse the long-term market return with our expected returns. The market returns apply to the collective, whereas we each experience our own individual returns.

Long term asset class returns

Source: Credit Suisse

Because we only have one lifetime to invest in, we can only tread the decision tree once. This means that risk matters. Ergodicity in an investment context therefore forces us to focus on the downside.

It’s great if our experience in the market one year is non-ergodic and we receive a return of 100%. We might be able to afford a nicer house, a better holiday, an earlier retirement. But the flip side of that is much worse. If we receive non-ergodic returns and lose 100% of our investments, it’s devastating.

Warren Buffett famously said, “In order to succeed, you must first survive.”

We’ve seen from our gambling and Russian Roulette examples that repeated exposures to a risky activity mean that eventually the risky event will happen. Given that investing is a risky business, if we invest for long enough, we’ll one day experience massive losses. That’s for certain.

Understanding that returns aren’t ergodic is important, as it governs how much risk we’re willing to take. Given what we know about the volatility tax, we know that losses are disproportionately bad. Given what we know about ergodicity, we know that averages can be misleading and that we need to be focussed on making sure we can keep playing the game – avoiding the bone-crushing drawdowns we know the market can inflict.

Ergodicity and the volatility tax go hand in hand. Ergodicity shows that because we don’t get average returns and only have one route through the decision tree, we could very well end up with periods of large losses. Moreover, because we’re playing an inherently risky game (the stock market), over time the likelihood of experiencing large losses rises to certainty the longer we invest for. The volatility tax shows that when we do experience these periods of losses, those losses will be disproportionately painful (from a mathematical point of view – but also from an emotional point of view). As a result, we should seek the smoothest possible return – i.e. make our returns ergodic as possible.  

How can we smooth our investment returns?

 

How we construct our portfolios after understanding ergodicity and the volatility tax depends on how we view risk. Do we assume that a Black Swan event has the potential to wipe out equities (or come close enough to wiping out that recovery is made incredibly difficult), and so take an ultra-conservative approach designed for survival? Is the stock market like the casino, where repeated exposure eventually leads to ruin?

Or do we assume that equities – even after huge losses – will recover given enough time, and that by taking an ultra-conservative approach designed only for survival, you’re failing slow rather than failing fast? (See this post by Justin Sibears of Newfound Research for the concept of failing fast vs failing slow). By not taking enough investment risk in your portfolio, you’re creating the new risk of not having enough to meet whatever you’re investing for. In the words of Corey Hoffstein from Newfound Research, “Risk can neither be created or destroyed – only transformed”.

For me, the answer lies in another way of thinking: not being stupid beats being smart. This idea was first introduced to me by Bob Seawright. The concept is massively useful for decision-making in general, but is also useful here. Its focus is to stop trying to make the most ‘optimal’ decisions, the absolute best decisions, but instead focusses on avoiding making stupid decisions.

Nick Maggiulli of Ritholtz Wealth Management, in his typical style, has also written a far more eloquent and insightful post on this topic than mine, highlighting how powerful it is to ‘Avoid the Zeroes‘.

When it comes to avoiding ruinous drawdowns, it’s all about avoiding the stupid decisions. 

In the previous section we saw some examples of what causes our returns to deviate from the average. Now let’s dive into them a little deeper, and look at how not being stupid can help avoid them:

1) Life events

Parkinson’s law of triviality states that the amount of attention a problem receives is the inverse of its importance. The more important a decision, the less is taken to make it. In my anecdotal experience, this is true. I’ve seen plenty of people “fall in love” with a house after viewing, and complete the purchase in less time than it takes to decide on what to watch on Netflix.

But one of the most important things you can do in building and preserving wealth is getting the big decisions right.

Your choice of partner, how much you spend on your house, and how much you spend on your car are going to be huge determinants of your future returns. No matter how high your returns are, a divorce cuts your wealth in half, a house which you need to liquidate your investments for in order to afford cuts all your compounding to zero, and a car bought on finance means your cash is funnelled into a depreciating asset rather than building wealth.

These are all life events inside your control. In addition, there will always be unexpected life events that derail your plans. Maybe a relative falls ill. Maybe you lose your job. Maybe you’re forced to take early retirement. Maybe your house needs repairing. Morgan Housel has an excellent article on this concept, titled “Risk is what you don’t see”, which I thoroughly recommend reading in its entirety (along with everything else he’s written).

The bottom line is that even if you get all the big decisions right, there’s always going to be unexpected things that punch your portfolio in the face.

One common way to deal with life’s unexpected potholes is to maintain an emergency fund in cash. How much you should hold depends on your individual circumstances, but it should be enough that you don’t have to dip into your portfolio to cover those expected-yet-unexpected expenses.

Another way to protect your portfolio is to maintain a margin of safety. If you’ve figured out you need 5% returns a year to achieve your goal, assume you’ll achieve 3%. If you think you’ll have enough to retire in 5 years, assume you’ll have enough in 10.

In summary, not being stupid beats being smart. When it comes to getting the big decisions right:

    • Don’t buy a house you can’t afford
    • Don’t buy a car you can’t afford
    • Don’t marry someone who’s goals/values are different to yours

When it comes to preparing for the unexpected life events:

    • Don’t neglect an emergency fund
    • Don’t neglect a margin of safety

2) Investment decisions

Making consistently good investment decisions is probably the most difficult way of keeping returns ergodic, because it relies on us mastering our own behaviour. We all know the basic rules of investing, but rules tend to go out the window when we’re faced with the opportunity of getting rich quick.

We need only remember the Bitcoin craze to see how many investors made terrible decisions in search of easy money. When all our friends are “getting hilariously rich and we’re not”, not being stupid can sometimes be the hardest thing to do.

There are plenty of academic studies showing how investors’ decisions reduce their returns. While some of them have been called into question, the message is broadly accepted – investors underperform the funds they’re investing in. They chase performance, overpay, speculate, and invest outside their risk tolerance.

Investing is simple, but not easy. Mastering our own behaviour is difficult even for the geniuses among us.

Off the top of my head, here are some examples of very smart people who lost a lot of money (or lost a lot of money for their clients) because of their behaviour:

Isaac Newton

Long Term Capital Management

Neil Woodford

Ben Graham

Warren Buffett

Bill Ackman

For those that are interested, Michael Batnick of Ritholtz Wealth Management has an excellent book called ‘Big Mistakes: The Best Investors and Their Worst Investments’ which provides a series of short stories chronicling how even the best can fall victim to their own irrationality.

If those genius-level investors make bad investment decisions, then we surely will too. For those of us that have spent any time in the stock market, I’m sure we can all remember a time where we’ve lost money as a result of becoming too emotional. When it comes to how our behaviour affects our investment returns, “We have met the enemy, and he is us”.

To help mitigate basic investment makes, it’s about sticking to the basics. Not being stupid beats being smart:

3) Sequence of returns

A soon-to-be retiree with an equity portfolio who sees the market drop 50% on the day of their retirement is going to have a bad time. Conversely, a new graduate who’s just started his first job and sees the market drop 50% is going to have a fantastic time saving into his pension scheme. 

The sequence in which we receive our returns matters hugely in how quickly we’re able to build and preserve wealth.

There’s some excellent research been done on how to mitigate sequence of returns risk at retirement, including using a dynamic equity glidepath, trend following, dynamic withdrawals, and using cash reserves.

As ever, not being stupid still applies:

    • Don’t have a high equity weight when you start making withdrawals from your portfolio.
    • Don’t get scared out of the market when it starts falling. Have a read of this post, which is one of the most popular on my blog, and talks through steps to take when the market is crashing.

I’ve talked a lot above about not being stupid, and how it can help make our returns more ergodic. Not being stupid is simple in theory, but figuring out whether a decision is stupid can be incredibly difficult in practice.

How can you tell whether a decision is stupid?

 

This is a difficult one to answer, but I’ll share my own method.

Personally, I like to take a regret-minimisation approach. Before making an investment, I’ll ask myself “Could I look back on this decision and severely regret it?”

This generally helps me weed out most of my stupid decisions.

For example, if I’m considering allocating 50% of my portfolio to Tesla – the answer to “Could I look back on this decision and severely regret it?” is a definite “Yes”. If I’m allocating 1% – the answer is probably “No” (NB: I don’t own Tesla).

If I’m trying to figure out how much I should hold in cash for my emergency fund, the answer to “Could I look back on this decision and severely regret it?” is a “Yes” for both 1 month’s expenses (too little – I could regret not having more) and 3 year’s expenses (too much – I could regret not investing it). So I settle somewhere in the middle.

Notice that with all decisions, there is likely to be at least some amount of regret. I could allocate 1% to Tesla and it then triples in price. I’d regret not allocating more. I could save a sensible amount in an emergency fund and the market could crash. I’d regret not having saved more in cash.

The key is to minimise, not eliminate, the potential regret. Trying to eliminate regret leads to big all-in or all-out decisions. This then runs the risk of being completely wrong and can end up causing the regret you were trying to avoid in the first place.

Because minimising regret usually leads to settling “somewhere in the middle”, you accept a certain level of regret. But because you’ve accepted that potential regret in advance, that small amount of regret is expected and therefore easier to live with. You’ve essentially swapped a large potential for regret in the future for a smaller amount of known regret now.

Thinking this way helps smooth your returns.

Through avoiding the 100% regret or 100% no-regret scenarios, you avoid big swings in outcomes. By accepting a small level of known regret, you reduce the chances that you’re making a completely wrong decision.

None of your decisions will be the absolute best decision in hindsight. But that’s exactly the point.

None will be 100% correct, but more importantly none will be 100% incorrect either. As we’ve seen, avoiding a completely incorrect decision is much more important than making a completely correct decision.  

As John Maynard Keynes said, “It’s better to be roughly right than precisely wrong.”

By minimising potential regret, you’re minimising the chance of making stupid decisions. Minimising stupid decisions help smooth your returns over time, making them more ergodic, and minimises the effects of the volatility tax.

Conclusion

 
  • The volatility tax is the idea that losses have a disproportionate impact on returns.
  • Ergodicity is the idea that probabilities of success from the collection of people do not apply to the individual.
  • Ergodicity shows that because we don’t get average returns, and only have one route through the decision tree, we could very well end up with periods of large losses.
  • Moreover, because we’re playing an inherently risky game (the stock market), over time the likelihood of experiencing large losses rises to certainty the longer we invest for.
  • The volatility tax shows that when we do experience these periods of losses, those losses will be disproportionately painful.
  • As a result, we should seek the smoothest possible return – i.e. make our returns ergodic as possible.
  • Smoothing portfolio returns can be achieved by focussing on not making stupid decisions.
  • Stupid decisions can be avoided by minimising potential regret.
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