Getting average returns right - The Merchant of St Petersburg

Many, even those that think they know mathematics, will do this simple but misleading mistake.

We've even seen experienced investment people getting this wrong.

The mistake of mixing up what kind of average to use. 

If you're interested in estimating growth, historical growth, comparing performance of different strategies (and we all are interested in those things) - then avoiding this mistake become supremely important.

Let's go through three examples: The Merchant of St Petersburg, Chekov's gun (an easy example), and an Investment Portfolio. 

We will see that the arithmetic average is misleading for evaluating an investment, and yet, this is a prevalent measure that is commonly presented with investment advice, and as we will see, the arithmetic average brushes risk under the carpet of miscalculation.

1.  Slightly trickier example: The Merchant of St Petersburg

A paradox borrowed from Mark Spitznagel that in turned borrowed the example from Daniel Bernoulli.

Here's an adapted version of Bernoulli's paradox.

A 18th century merchant wants to ship merchandise from Amsterdam to St Petersburg, over a pirate-infested Baltic sea. 

Wrong century, but still seems looks like a very savvy merchant. 
Jan Gossaert, Portrait of a Merchant, c. 1530

Our merchant has 11 000 florins, buys merchandise and pays shipment costs for 8000 florins, and sell the merchandise in St Petersburg for 13000 florins, making a 18% gain on the trade.

Unfortunately, every 20 ship is lost to pirates, which means that all the invested 8000 florins are lost, and the merchant is left with 3000 florins. Our merchant makes a -73% loss. So, there is a 95% possibility of making a 18% gain, and a 5% risk of making a 73% loss.

The arithmetic average of this trade would be 95% * 18% + 5% * (-73%) = 13.45%

But pirates are frustrating. Is there a way to get around these infrequent setbacks? The Merchant ponders his options, and finds an insurance company that can insure his cargo for 800 florins.

This would change his expected gain to only 12200 florins each time the ship arrives in St Petersburg, changing the profit to only 10.9%. When the ship is captured by pirates, he only loses his insurance money, that is, 800 florins, which, on that trade, limit the loss to 7,3% instead of 73%.

Let's do the arithmetic average once more: 95% * 10,9% + 5% * (-7.3%) = 9,9%

So the expected average outcome is lower than going without the insurance. 

What should our rational merchant do? 

The insurance seems expensive, but is it really?

The crux here is that one can't add percentages. A loss of -10% is not the same as a win of +10%. It's not meaningful to add them together. What our Merchant is really after is how his wealth is expected to grow.

So if he does the trade between Amsterdam and St Petersburg a hundred times, reinvesting what he earned, he could expect, without insurance, a growth of 1.18 of his wealth if the ship arrives, and otherwise a setback to only 0.27 (=100%-73%) of what his wealth was when he sent that unfortunate shipment.

To evaluate the uninsured case then, we need to look the trade occuring again and again, multiplying the trade to the power of 95, and the negative pirate-outcome to the power of 5: 1.18^95 * 0.27^5, which corresponds to a growth of 9674 times his initial investment after 100 shiptments. 

As this corresponds to 100 times the trip, we need to take the 100th root out of 9674 to see what it would correspond to as growth per shipment if there were no pirates. The 100th root of of 9674 is 1.096, which is called an geometric (not arithmetic) average, which is then 9.6% per shipment.

This is clearly lower than the initial 13.45% we calculated!  

There's something going on here!

If we evaluate the insured case, the return is 1.109 in the 95 shipments when things go well, and 0.927 in the 5 shipments when things go bad. So his wealth will increase with 12498 after 100 shipments, corresponding to an expected 9.99% growth per trade (close, but not exact to the arithmetic average). 

That's a 30% higher return in the end for our Merchant. 

So our Merchant will be a lot more wealth over time with the insured alternative, in stark contrast to the first simplistic calculation using averages as we learned to calculate them in, well, kindergarden? 

It seems that, especially for large losses, the arithmetic average gets things dangerously wrong and underestimate the risks. 

Interesting. 

2. Easy example: Chekov's gun

Another example.

Three russian frenemies, Alexei, Boris and Chekov, play russian roulette (with a three barrel gun, to make it simpler). They all put 100 roubles on the table, and after the game, Chekov ends up dead. 

Their average outcome on the investment of this would be (+50% + 50% - 100% ) / 3 = 0 %

A zero sum game when it comes to wealth distribution, so no surprises so far. 

So there would be, on average, no expected change in wealth playing this game, but would Chekov agree? 

Let's look close at what happens. 

Once again the arithmetic average shows up. As in our Merchant example, it assumes - almost always erroneously for the kind of questions that we want to study - that units can be compared one-per-one, that is that 1% up is the same as 1% down, so they can be added together and divided to create an average.

But of course, that is not the case, especially not for our frenemy Chekov. He will suffer an even more dire fate than our Merchant from St Petersburg.

Because what he, Alexei and Boris have in common, is that they are interested in the growth of their investment (and perhaps the sudden exit of one of their competitors). 

So, what is interesting for the individual is the growth one can expect from playing the game, and especially playing the game several times. 

Let's say that Chekov, while still alive, decides to play the game three times, always with three players that always match what Chekov puts on the table, and, unfortunately, our friend Chekov dies the third time he pulls the trigger.

His growth will be: 150% the first game, then 150% the second game, and 0% the fatal final round. He's wiped out.

So the growth of his investment here would be 1.5 * 1.5 * 0 = 0, with a complete loss of wealth and no possibility to rejoin the game. 

The total expected growth to do Chekov's little game would thus be 0, even if he earns 50% on the first two games.  

The worse the outcome of a single game, the more misleading the arithmetic average becomes.

3. Stock market example

A final example. 

Let's take the US stock index for the years 1970 - 2022. The arithmetic, inflation adjusted average, is around 7.5% - a number that we have seen many times, and that is being touted as the expected inflation adjusted return from the US stock market. 

This inflation adjusted expected average of 7.5%, we're sorry to say, is wrong.

As we see above, it's really growth we are concerned with, and the geometric inflation adjusted average is only 6% of the US stock market. So if one invested those 52 years, one would earn 6% per year, on average, known as the cumulative average growth rate (shorted CAGR) - inflation adjusted.

It's just harder to get up when one has been fallen down, and this is more accurately reflected in the geometric average.

The very common number that is floated around is unfortunately wrong, if one wants to stay invested over time and consider one's growth.

Growth is hard. 

Can we hedge?

Yes. We can hedge. If one mixes 20% gold into the mix and rebalance annually, we actually - and very counterintuitively from most investment advice out there, get 6.3% geometric average return.

So actually slightly better, and this is more astonishing and counterintuitive than one might think. 

And we probably sleep better, because the swings are smaller with this 80/20 portfolio.

So adding gold doesn't cost us as the common thinking might have it - it has actually given us a higher return - just like the insurance for the St Petersburg Merchant above.

We earn more money over time. There's no other way to put it. 

Another way to hedge is using options, though this requires serious expertise. Options can give insurance that is quite similar to our Merchant of St Petersburg above. If we buy an option that costs us 2% of our capital each year, but has a payoff that offset the loss each time the loss is more than 15%, we get a geometric average return of 7.3% for those 52 years. 

There are actually ways to hedge an investment, that makes it less volatile, and gives it higher return over time. 

What's going on?

There are three common ways to calculate means. 

Arithmetic, geometric and harmonic.

Arithmetic concerns what can be added and subtracted linearly (such as adding or removing apples in the shopping cart), geometric concerns growth, and harmonic concerns velocity. 

For any given set of changes, the arithmetic average will be the highest, the harmonic the lowest, and the geometric will be in between. 

Almost everywhere we look we see how this is presented wrong when we see assumptions on return. 

We are concerned with growth, not apples and oranges.

This has misleading consequences when making assumptions about the performance of different portfolios, the true cost and impact of downswings and risk, and what numbers people unfortunately put in when trying to calculate different "rich-by-Excel"-numbers, and misleadingly indicates especially high volatility or large drawdowns as less dangerous than they are. 

Geometric averages more correctly assess the impact of growth.

And the risk of ruin and the impact of large drawdowns are very dangerous risks with big impacts, that the arithmetic average gets very wrong as we saw with our friend Chekov and Merchant above.

Still, one should not only rely on geometric averages. 

One still need to be prepared for the black swans, ergodicity - ensuring that one can live through and recover also from the very bad scenarios - and understand that a close shave with an absorbing barrier is more common, likely to hit at some point during a lifetime, and hard to get out of. 

Never put your life savings at unacceptable risk, thinking that the barrell will always be empty when  playing russian roulette.

Remove hope out of the equation. Look at scenarios. Simulate. Be careful when seeing an average being presented. What will happen in the worst and the bleak scenario. Do you get stuck in an absorbing barrier?

And will you be able to get up again? 

Farewell,

//antinous&lucilius

Hope is not a good strategy

When we started to think about investing, one of our reservations - and a reason that we avoided investing - was that we felt that there was, well, far too much hope involved.

And hope felt like speculation.

Who would like to put one's hard-earned money up for something as fickle as that?

Hope is not a strategy

Yet, if one can make a decent return again and again, consistently, even if one doesn't end up on top every year, as time accumulates, one will have a very good return over one's investment lifetime.

John William Waterhouse, Pandora, 1896.
According to Hesiod, Elpis - the Goddess of Hope - was hiding as the last item in the box.

When thinking through a strategy, as we felt intuitively when we were young, the best path is to try to figure out a way to remove hope. 

If we can look at the strategy without hope for any particular scenario over another, then we're on to something.

Removing hope

For a small-guys investor, there are some ways of removing hope.

• Long-run. One strategy is to go for the long run (read: Welcome to the 1825 day-year)
  
  - If you don't need the money for 20+ years: invest in the stock market (but we have some reservations)
  - If you don't need the money for 5-10 years, invest in some kind of asset allocation

• Creating a well-devised money machine (read about our pathfinder portfolio, or the permanent portfolio), put the money there, and trust the mechanics that a certain withdrawal rate (3% or 4%) should work in the future as well. 

The stomach and the trade-off

The trade-off here is: will one stomach the lower returns when other assets are booming? When the less prudent investors chase hope and gets the fickle rewards as fortune sails in their waters?

Betting on both

The mistake in our youth was to think that there was no other strategy than hope.

But there is, call it the Kelly-criterion, the safer bet, or winning the war - not the battle.

Bleed a little, and win a little, all the time.

Yes, a strategy contains an element that bets on a good outcome.

But it also bets on protection, so even a bad outcome becomes good.

Don't chase the risky bet.

Or as Howard Marks has it: Take care of the downside, and the upside will take care of itself. 

Farewell.

//antinous&lucilius

More reads:

- Amor fati. The art (and Stoic habit) of loving whatever fate has in store for us. 

- Don't predict. The ego-defending little-sister of Hope is the Fortune Teller. 

- How long is the long run? Read: The speed and the destination

- What is asset allocation? Some thoughts here: How we dared to start investing

When the knifes are falling

It has been a rough spring for our open societies, not to mention the people in Ukraine where we have friends and acquaintances that have caused many a white night for us. 

A stress-free portfolio

Despite all that, we have not been particularly anxious about our portfolio. Sure, it has fallen somewhat, but not with more than we can brush it off. We also remember all the simulations and back-tests for the "bouncing back factor" of our portfolio, which is one of the reasons we've chosen it. 

Over a three-year period the portfolio has been back where it began in all cases, during the last 52 years. 

So it's more bombs falling than the portfolio falling that keep us up at night. 

In short: we felt prepared when the financial world started to seemingly fall apart during this spring.

,
In Ciceros original telling of the story, there were boys (twinks?) at Democeles' party. Just saying. But a debauched same-sex orgy (with additional food and wine to satiate all appetites) was a little too much in 1812 when Richard Westhall imagined the impeding fall of the sword and gory end of the party, so the twinks became ladies instead. 

The markets price everything in

In the last few weeks, we've read doomsday forecasts for all asset classes we own. 

Allegedly, Putin would be sitting like an old dragon on a ton of gold, and what happens with that pile, one way or the other, might completely perturbate the price of the shiny metal. We'll soon see kitchenware in pure gold instead of steal at Ikea, according to the most negative predictions.

Inflation eats bond yields, and it's going rampant and then central banks and governments will not able to control inflation, or so it goes, so treasury bonds and toilet paper are soon to be equivalent investments. Actually, toilet paper might be an investment with a better outcome if the wars in Europe get severe enough. 

And the world economy will never be the same, with supply chain disruptions, a scared populace that refuses to consume and shrinks demand, and shortages of all kinds. 

Perhaps. Perhaps not.

Don't forecast. And with enough time, everything happens. 

What these fortune tellers seem to forget is, in our opinion, a very fundamental thing. 

All assets above correspond to financial contracts, traded on open markets in anonymous transactions, by intelligent agents - mostly institutions - with access to much information, and much more than the alluring stories presented above. 

Which means that all ideas about what will happen in the future is already priced into the current asset prices. There's no "natural laws" or "safe bets" that haven't already been baked into a (very refined) average assessment of the situation - an assessment that we normally call the current price.

For instance, bond prices already anticipate what the future payment stream (coupons) will be worth today, in today's money, inflation and all, with expected real returns, in the net present value in relation to existing bonds, buy backs, expectations of future quantitative measures, money printing and issues of new treasuries. It's all there, in the price, already.

So what one is saying when trying to see anything as "doomed", is that one is more intelligent than the market, or perhaps that one has figured out a bias that no-one else is exploiting. But beware. Markets are learning machines, and they are smart. 

As good stoics, we prefer to lean back instead of trying to outsmart people that, truth be told, probably are much more intelligent than us. 

We rely on the method, and we have pre-meditated that the sword may fall.

Come year's end, we will follow our strategy, and as usual pour our hard-earned money into the worst performing asset of the year (whichever that might be). That is probably then the lowest priced bet possible between long term treasuries, gold and stocks, and hence, also the bet with most upside if the market expectations are surprised.

So yes, we bet, but according to a pre-meditated and simple plan.

And the markets are always surprised, but not in ways that the stories above indicate - but genuinely surprised and one, at least not we, will not be able to predict why, when or how. 

For instance, in our own risk assessments for our future FIRE-life, we hadn't even really written out war explicitly (it was implicitly there, but more like "Sweden becomes impossible to live in"). 

Once more: with enough time, everything that can happen will happen, and now war is raging in Europe, despite (at least) us not foreseeing it. 

The best is to be aware that the knives might be falling at any time, and take precautions in advance and not hope that one will be able to do a last millisecond rescue when the unforseen actually happens. 

Be prepared in advance, prepare for all eventualities, consider a strategy that works for the human you are and not the hero you wish to be, so you are able to stick to the plan when the party ends.  

Farewell,

//antinous&lucilius. 

More reading:

Ergodicity - everything that can happen, will eventually happen.

Amor fati - love what destiny has in store for you.

Our portfolio - the pathfinder, bringing us to our goal. 

Don't predict

We want to avoid speculation. To avoid speculation, we need to avoid predicting. But how can we earn a good return and not predict?

It's really hard not to make a prediction. News, twitter and much commentary in general are predictions. Prediction creeps into our thinking whatever we seem to do, even if we actively try to avoid it. 

• "Let's hope this will be a good year"
• "Small cap, investment trusts or tech stock are overvalued. The P/E ratio seems high, don't you think?"
• "Inflation will destroy all returns in the next decade!"
• "Interest rates can't go down more"
• "I'm sure this stock is undervalued"

Why shouldn't we predict? Because there is no reliable crystal ball for the future. Predicting has a notorious bad track-record, obscured by winner bias.

The price on the market is the average prediction of all the market's participants. For us, the best model of the market is that it follows an entirely random process - it's perhaps not always true but it's the best approximation for how the market behaves for, we dare say, any small investor, like us.

The Oracle of Delphi, perhaps reading the financial news. 
Delphic Sibyl, fresco painted by Michelangelo, Sistine Chapel Ceiling (1508-1512) 

But what should one do if we don't want to predict?

First, we should refrain from predictive narratives that sound logical. Because that's what narratives do (sound logical) but it doesn't make those narratives into better predictions.

Examples:

• Reached the bottom? Perhaps we, or the 'experts', are sure that we're at the bottom of a stock market cycle - but in reality it's very hard to say when and how fast the recovery will come. And when prices are low, developments can be very quick. One can loose 50% or 90% of one's money in a day going in or out of the market when it moves around the bottom of a boom- and bust cycle. So it's better not to predict and stay the course with a strategy that was thought-through before the stock market cycle went downhill. 
  
  
• Reached the top? Perhaps we are sure that the stock market will stop going up, because any number of good-sounding reasons. Yet - when will that happen? Over time the market goes up, and that means that it beats its all time high again and again. And whatever theory one picks to predict the top, there are nuances that the theory will not encompass. The markets are a learning-machine, that already contains all known theories. For instance, a simple point: there's a denominator in the P/E-equation.
  
  
• Cash and gold. Perhaps we think that "only productive assets" will survive, and discover that market price appreciation for other assets also delivers real money, especially when those productive assets are at bargain prices. Who realizes that cash increased 500% in value between 2001-2003 instead of seeing the other (but same) picture - that the stock market went down 80%?
  
  
• Interests are low or high. Perhaps we think that "interests can't go down more" and discover that long term interest is powerfully controlled by the long-term expectations on inflation and interest rates by the markets. And as the central banks and politicians discover, markets are much more adaptable learning machines that are much more powerful then what a 'sovereign' state wants to admit or can fully control. 

We might hear what we want to hear when we think we should rely on a prediction. The easiest person to fool is ourselves, with our desire to hear what we want, and avoid to hear what we don't want to know. We're prisoners to our own delusions. 

Temet nosce as it was said in Delphi. Know thyself before you try to understand a prediction about the future; and the ancients knew with many a cautionary tale how a prediction tend to fool the listener.

We've tried as well. We've been rock sure that we're at the bottom of a stock cycle. And just as sure that gold can't go up. What we've learned is that we are usually 180 degrees wrong when we try to guess what is going to happen. 

Perhaps you are better than us; which shouldn't be too hard, but to really outperform the market one needs to be better than average, which doesn't mean better than any average Joe, but better than professional investors.

It's a tough game to play.

So what could one do?

A solution to not predicting

Well, build a strategy that doesn't predict. 

One popular way is just to buy the stock market on average, by buying an index fund.

We don't think this gives a good price on the risk one is taking on, but it's a strategy with less prediction. 

Another closely related strategy is to focus on dividends, not stock market prices.

Betting on many futures at once

Our way is to combine assets that performs well for most scenarios of an economic and credit expansion and contraction. 

The idea behind asset allocation is to buy into different likely scenarios and have assets that perform well in different futures and hence not bet on any future in particular.

What we started our journey with, was the hardcore solution of strategies like this and when it comes to erring on the safe side: the permanent portfolio, which has much better returns and performance than most newbie investors think.

Read more here: A-well kept secret: Our portfolio for accumulation.

Not even we are that hardcore as the permanent portfolio anymore. But it's worth knowing that it exists and how it behaves, and also ponder using it. We did so for many years.

The bottom line is: we still pay extreme attention to avoid trying to foresee the future.

Farewell.

//antinous&lucilius

Where to go next? Perhaps some thoughts of why it's not overly wise to rely on an expert. 

Ergodicity: Anything that can hit us will, eventually, hit us.

Ergodicity is an interesting property. 

The idea of ergodicity is that in a stochastic process, a point will eventually visit all parts of the system it moves in.

Another way of phrasing it is that given enough time, everything that can happen will happen, with a probability approaching one.

We're by no means mathematicians, and even less experts in probability theory. 

Yet, what we've understood (or misunderstood) about ergodicity might be interesting for how one looks at the world, and how one looks on investments and an investment strategy in particular.

Let's start with examples.

Two Sides of A Pet Example

And where better to start than with a gun. 

Mikhail Yuryevich Lermontov aged 33, four years before he was shot through the heart during a duel. He was, allegedly, the inventor of the morbid game of Russian Roulette.

Russian roulette is a favorite game of all amateur game theoreticians. 

1 gun, 6 chambers, 1 bullet in one of the chambers. We spin the barrel, and then the game begins.

So, in which of the following game settings of Russian roulette would we like to participate?

Game A: Ensemble probability

6 persons walk into a bar (in Novosibirsk). The first one puts the Russian roulette-gun to his head, and pulls the trigger. If he survives, he hands the gun to the next person, and so on.  

What is the a priori expected return from participating in this game?

Game B: Time probability 

Now a much more, for the individual, deadly version of the game. One person walks into a bar to play Russian roulette. In this version of the game, she puts the gun to her head, probably has an good glas of vodka, and pulls the trigger 6 times.

What is her expected return from participating in Game B?

Let's conclude that whatever the return is for game B, it's not good. 

What does this mean for us?

In life, one might easily believe that one is playing Game A. When a yearly expected return is calculated, it gives the illusion of Game A. 

It's as if we participated in one year only, and, like our six Russian Roulette-players above, we cross our fingers when we pull the trigger and hope that it's a good year.

We hear ourselves say things like 'Let's hope that the portfolio goes up this year'.

In Game A, hope is part of the equation. We can hope that we are on a good run. We can hope that we will pull out of the game before that fatal bullet. 

When we look at expected return, like in Game A - it's ensemble probability we see; an ensemble of years as if they happened at the same time, not as if they where happening one after another. 

Of course, years doesn't work the way Game A does. 

It's not ensemble probability that is a good model for designing a strategy. 

Like our unfortunate player in Game B, we must survive time probability. 

Which means acknowledging that bad events will hit us as well, due to ergodicity. What is unlikely to happen in a year might very well be very likely during a lifetime or over the timespan of our strategy, or for the unfortunate lady playing solitary Russian roulette.  

In investing, we are playing the long game, year after year. So the mechanisms of our strategy and the behavior of the game table we're at, are very important indeed. 

During a life time, a really bad year WILL hit us. Really bad events WILL happen. Then our strategy better be wiser than the one fool hoping about the average outcome of Game A above. 

Our strategy to increase and protect our wealth must be built in such a way that we don't end with a gory mess.

There's no use in having a strategy on the assumption "as long as nothing bad happens", or even worse, a strategy that leads to ruin if a bad event or year hits us. 

Then we might permanently be out of the game, and our strategy doesn't matter much anymore.

Real life examples

What does ergodicity mean for us, practically?

To sum up: if we are to stay in the game, ergodicity means that in the long run, our strategy need to be able to survive anything that can possibly happen.

Application 1. Return.

The diagram above shows the same run for the Pathfinder portfolio.Why does it look so spread out if it's the same run? We've just varied the start date with three year intervals and repeated the same series of returns on the same starting point. 

Look at the diagram again. The green, the red and the blue line all come from the same run of years. The green line sure looks lucky. But even a "lucky strike" as the green line, also has a "bad run" as can be seen around 2031 in this simulation, and what looks like hopeless laggards will overtake the initial good run. Remember that this is the same series, the variation comes from the starting year only.

The same strategy gets hit with every event; with everything that happened, and luck and misfortune even out. 

So this would be as example of a strategy that can do well both if we're lucky or unlucky with a run of years.

A side-note: Ergodicity also puts some lights on the thinking around the FIRE-number itself; the amount of money invested needed, to reliably cover one's expenses. If one hits the fire number early, one should probably be cautious. On the other hand, if one never seems to hit the fire number, one might be on a lower trajectory, with more upside potential. More about that in another article, perhaps.

Application 2. Risks.

When contemplating exiting the work life, we've set up a list of risks, consisting of things that might derail our future freedom. Socialism (this is Europe, after all), large unexpected costs, family members faring bad, our relationship taking a bad turn, and of course death. With risks, it's tempting to assign impact and likelihood and care about the high probability, high impact ones.

But ergodicity introduces something that normal risk-thinking doesn't quite comprehend. The longer a game is played, the more likely all events become. 

In the long run, we need a strategy for everything. Nothing can really be avoided.  

So we must be prepared that we will have to perform all the mitigations for all risks. We will at some point have to pay that unexpected cost. 

There will be a run of socialism with high taxes and a wealth tax during the roughly 50 years we will live from our portfolio. There will be family problems and relationship problems, illness and tragedy. And finally, one of us will die and leave the other one behind. 

Conclusion

If we at any time think it's meaningful for us to "hope" for a certain outcome, then we have probably fooled ourselves into believing that we are playing Game A.

Our strategy needs to be adopted to reality, and the long run. Amor fati; love what fate has in store for us. Or as Mark Spitznagel of Universa fame has it. He makes a parallell to Nietschze for a good investment strategy - being able to exclaim "Thus I willed it" for whatever fate throws at us. 

In real life, hope is not a good strategy.

Farewell,

//antinous&lucilius

Portfolio Stability And A Good Night's Sleep

If one goes to a place like portfoliocharts (strongly recommended for the interested asset allocator), there's a concept that we have struggled with.

It's the idea of Start Date Sensitivity.

For us, this is not at all intuitive, so let's try to go all the way to see where we are now in our understanding. We have discovered that it's key to quite a lot of insights around investment portfolios.

It's a funny and unusual measure. It took us quite some time to start to understand this way of looking at a portfolio. 

One way to define it is as a measure of how good a guide the 10 years last history has been for the 10 years that lay ahead. By pointing out when this difference is as large as possible (both in the postive and negative sense), it focuses on the year when the previous time period was as maximally misleading for the upcoming time period, and how big that effect was. 

Is history a good guide for the future?
(Reconstruction of the West Pediments of the Parthenon, CC BY-SA 2.0 Tilemahos Efthimiadis)

Going with one asset class

Let's look at some examples:

• Total US Stock Market: Luckiest 10 years = 14.9 percent points better, per year, than the preceding 10 years, Unluckiest 10 years = -18.1 percent points worse per year then what the preceding 10 years annual average return hinted at.

So this mean that a happy US investor could sit and look at the stock market and think that: well, this is sure looking good. And at some point, the investor would say that, what the heck, the last 10 years have been going really, really well. Let's jump in and put my savings in the stock market.

The most unlucky the investor could have been when doing that decision, was the period when the stock market performed on a yearly average -18.1 percent points worse per year, for 10 years, than it had done the last 10 year period.

So that's a measure of how bad a guide for the future the last decade was.

For the US stock market, that would have happened in 1999, back when we were in our teens. The market had had a return on around 14 percent per year for ten years, but that would not continue. The average yearly return for the next 10 years would be -4 annually. This gives us the most unluckiest start date sensitivity for 1999, with those 18 percent points .

As both of us experienced and at least vaguely remember 1999, this negative experience colored our view of the stock market and probably at least partly explains why we treat it with caution. 

Of course, time would have partly fixed it for our unlucky investor, but it would take a very long time indeed. Still today, that 1999 investor would have got only a modest 5% return per year from that initial decision, and a lot of volatility on the way. Not that much of an issue if one is a teen and just started to accumulate money. More nerve-wracking if one is deeper into one's career and accumulation, and put substantial money on the table.

On the other hand: the luckiest investor would have jumped onboard the stock market in 2009 and then been in for ten tremendously good years, a lucky strike we're still part of.

Just for fun, let's have a look on a much more volatile asset class. No, not bitcoin. Gold, of course. 

• Gold (USD): Luckiest 10 years = 21.0p.p. , Unluckiest 10 years = -30,7p.p.

One would probably be close to insane to put a significant part of one's money in an asset class that is as volatile as gold, but let's play with the idea. 

In that case, an investor, after witnessing a terrible performance for gold during 10 years and then by some miraculous inspiration buying into it anyway, could be 21 percent points better off, per year, for the upcoming ten years. 

So the start date sensitivity goes both ways, and the bigger it is, the less guidance we seem to get from the last ten year period.

What happens if we mix two assets?

Let's instead mix a healthy portion of the stock market with a substantial amount of gold. Let's say by using our hypothetical 80/20 split.

• TSM (80%), Gold (20%), looking on performance in USD:  Luckiest 10y= 10.6 percent points, Unluckiest 10y = -11.3 percent points difference per year

What does that mean? Well, our investor would still be in for a surprise if she hit the unluckiest year, but after yearly rebalancing in and out of that gold, she could comfort herself that she would be much better of than with going totally into the stock market, and she would have come out quite ok 10 years later and would be back in black numbers much quicker.

Good for her.

Several asset classes

What happens if we go to our more conservative portfolios that we've been using ourselves, where we mix gold, long term bonds in different currencies, cash and small and large cap stocks, both domestically and abroad?

• Permanent portfolio, our take on it, in SEK: Luckiest = 5.1 p.p. , Unluckiest = -4.9 percent points difference per year.

The results are roughly the same for the US market, but with even smaller sensitivity. You can read more about our take on the portfolios here, here and here.

What does this mean? Well, there was a  year, where the last 10 years had an annual average return of 10%. If our unlucky investor jumps in that year, she would be in for 10 years where the average annual return would be 5.1%, or 4.9 percent points worse than indicated by the preceding 10 year period. All this after inflation.

And that was the most unlucky difference that ever happened during the last 50 years.

So in the worst case it would be roughly the same as for the stock market, but with much lesser volatility. And the permanent portfolio is back in black much quicker.

That's a glimpse of why we, who both were into our careers and had a bunch of money, decided to start our investment journey with the permanent portfolio. 

The permanent portfolio with its more narrow start date sensitivity, in contrast to the stock heavy alternatives, almost entirely avoids the question of 'is this the right time'? 

That's an important, stress-inducing question that can be avoided.

Finding our path

But of course, one pays a price with the permanent portfolio when it comes to where return. So now the fun part. 

• Our Pathfinder Portfolio, our take on it, in SEK: Luckiest= 6.5 percent points, Unluckiest= -7.3 percent points difference annually
• Our Pathfinder Portfolio, our take on it, in USD: Luckiest = 4,1 p.p. , Unluckiest = -4,2 p.p.

Our pathfinder portfolio of course has a higher start date sensitivity, measured as the difference between the luckiest and unluckiest points, than the permanent portfolio.

But still, it's much, much lower than the stock market. For our domestic set-up, the average return for the pathfinder portfolio was 9.2%, and a US adaptation was 7.3%. 

The total US stock market for the same period had an average return of 8.3%. 

So by mixing a combination of individually high volatility assets, one gets a lot more safety, quite small sensitivity in regards to when to buy into the portfolio, and an average return on par or even higher than the stock market. 

Just saying.

A Good Night's Sleep

That could be the end of this article. But it's not. 

What we then slowly realized was that the question if one should put all one's money in this or that portfolio is not only a question about start date.

It's not a one-time question.

It's a question that comes up all the time, nagging with these little thoughts that can keep anyone who is not a Stoic God awake at night.

• Have the last years been booming? How much could we realistically loose by sitting still?
• For how long should we accept that the portfolio is lagging behind? 
• Should we sell tonight? Or should we sit still and hope that the boom continues for a little longer?
• Do we dare to buy into an asset that has been lagging for a long time?

All investors take the decision if we should stay in an asset allocation or leave it every night, irregardless of if we try to pretend that we don't. There's no real way of avoiding this, as taking no decision is still a decision.

At the bottom of it, start date sensitivity is about the stability of an asset mix. Having a good stability in the mix of assets means that one has to think much less about time periods, paradigms, bull and bear markets and timing in general, and all those little nagging questions that timing entails.

Because the effect of being lucky or unlucky with timing has much less impact on the performance of the portfolio.

The decision to stick with the portfolio can then truly become permanent, with less second-guessing, and better sleep at night. 

Farewell,

//antinous&lucilius

Where to go now?

• Go over to portfoliocharts.com and play around for yourself.
•  Read about the Pathfinder Portfolio
• ... or the Permanent Portfolio: A Portfolio for Accumulation Part 1 and Part 2.

Note on the numbers: As usual, this is back-testing from 1970, cumulative annual growth rate, with inflation removed. 

Why the Expert We Follow Will Go Bust Tomorrow

There are no experts that will make us rich. Here are some thoughts why the strategy one copies is surprisingly likely to go bust tomorrow.

The Problem with the Expert

How can we fool ourselves by following an expert?

It feels good to let someone else do the thinking. And we could sure enjoy some high returns while we follow an investment-wiz, instead of the painstaking 5-8-10 years' slow road of frugal living until we achieve financial independence, right?

Let's say that we have been following some portfolio-wizard for a long time; a stock-picker or investor in some more or less exotic assets.

And by the Gods: is she good! Our guru outperforms the market with 20-30% year after year. Not so much that it's obvious that its a fluke; no - just so good that she Amounts to Being A Very Gifted Investor. 

And, in this hypothetical world, we can't help but to dream away. 

If we keep up with a 20-30% growth year after year we would already be deep into financial independence, sipping sublime pink champagne from a golden bathtub in a glade with the Gods since many years.

Poolparty for the Gods
Diana and Actaeon, Titian 1556-1559 

So we start to get more interested by the guiding light of this sage, and we read ourselves into the details of her thinking.

And indeed, she has a theory. Her ideas are based on bright and piercing observation. It just makes sense. How could such clarity, perhaps tinted with refreshing cynesism, be false? She might even be like us. How wonderful.

After several years of observation - we are overly cautious and conservative, after all - we decide to copy her portfolio.

We've done our homework. And we have the facts to prove it, or so we think, with increasing significance with the evidence accumulated of each successful year. It's a strategy that have been going strong for so long. What could possible go wrong?

The month after we buy into her portfolio composition, the losses are up to 90%. Her webpage and blog disappears, and she is nowhere to be found. And, before we understand what is going on, and because losses are fractal and can happen over and over again, we lose another 90%. 

Our life's savings are now obliterated.

The Expert We Copy Will Loose Everything Tomorrow

When we first read about the expert fallacy in Harry Brown's book about financial safety, a chapter entitled "Don’t expect anyone to make you rich", it seemed contra-intuitive, almost mystical. It smelled like a believe in foresight, a believe in faith; something to be taken as serious as a fortune teller armed with a crystal ball or a quack selling a cure against upset bowels, ill temper and social media addiction. 

How could one possibly know what will fail tomorrow? 

Now the funny part: It's not just the Gods machinating against us for their pleasure. There is math and logic behind this. The example above with the wunderkind investor wasn't as simple as us being unlucky. Just like a magic trick, where reality and our own dreams are the magician; a magician that turns one's expectations inside-out, a trick made by our own brains by wanting to cling to a good narrative.

But there is another, truer perspective. The failure of the expert is much more probable than it might seem at first glance.

The answer lies in the realm of our good ol' friend probability theory, and how she can sneak up on us in unexpected, opaque and subtle ways.

Winner Bias, once again

Apart from the opacity with an investor itself (do we know all about her investments? What are here motivations ? How oblivious to Fortune changing course is she?) But in a larger picture, this fallacy is about winner bias in one of its many disguises and reincarnations.

Many of the 'experts' we see, are just those that happen to still not have blown up. 

We might have been watching a few gurus, and semi-unconsciously lost interest as this or that 'expert' blew up with his or her portfolio. By forgetting about evidence - not to mention all evidence that never reach us - we masquerade the likelihood for ourselves if our a single expert is succesful or not. 

We see only Her, the one that survived, and it's Her that we fall in love with.

In reality, it was never much special with the portfolio of the wunderkind. It just happened to have survived a little longer than the others that went out of the game. And we happened to fashion a narrative around it, a constructed explanation why we liked the portfolio, or the person, or the made-up 'theory', or all of it.

But there was nothing special with any of it. We just got lost with the direction time moves. What one has seen is not what will be in the future.

When we act in the now, we loose this advantage of hindsight, and like a Heisenberg equation around an electron, the probability wave collapses to the observation - or rather, to our action. The strategy that we had been able to cherry pick in a cloud of possibilities, that strategy now become concrete, real. And we no longer has a possibility to cherry-pick. 

And this mountain of self-delusion is build on a truly, non-linear, high price for the risk of the strategy's over-achievement. Hence the dramatic downfall.

There is always someone standing on the battleground of life, and she might seem clever, but all things considered - it might be a question about luck, and it will be very ill-advised to copy her behavior.

Shouldn't We Never Listen to Advice?

What can we do against this? Can we never trust anyone?

Well, we think that it's just hard - we're so sorry to say. And as said over and over again, the easiest person to fool is usually ourselves.

Here are some rules of thumb we try to use:

1. We don't pick individual stocks. We just don't. 
2. We ask ourselves if an idea we have is actually just about chasing higher returns instead of balancing and protecting the downside. 
3. We think that it's very hard to reach above 10% annual growth consistently. And when one does, the risks behind the return are not linear. Then we believe that the hidden risks are much more dire, and can very quickly get us close to ruin and a loss of all our savings. 
4. We try to catch ourselves when we are retrofitting an explanation to past performance. In science, that would be very bad. In investing, it might be even worse.
5. We try to imagine if there might be dead, silent evidence that we are missing.
6. We try to look at similar strategies; did they leave blown-up investors in its wake? 
7. We ask ourselves; would this be good advice if history unfolded differently? Paradigms shift, what would happen if there was a new paradigm tomorrow? 
8. We don't believe in going all in in a single strategy. 
9. We don't tie our savings to a single asset class, and barbell the risks.
10. We try to construct a simple rule or algorithm, linked to the bouquet of strategies we use, and try to figure out if there's true return under different paradigms behind our idea, independent on past performance or a certain future playing out. But even then we don't trust our idea.
11. We test our thinking over a long run of past data. We really do remember the downturn in 1871 here, no kidding.
12. We try our thinking in many different countries, as a proxy for different paradigms and scenarios.
13. And we test even more scenarios that even never really happened, by doing Monte Carlo simulations; by using tools on the net and just building them ourselves.

We will not be the smartest ones out there. In all likelihood, no individual retail investor ever will, even if they might seem to be able to pick stocks or a fancy strategy for a while, even a long while. All that will change as soon as we invest.

So rather than chasing the higher return and dreaming about the divine pool party, we think it's better to waterproof our strategy before trying to join the Gods.