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

What to put in the jar of financial independence

 Something struck me today. 

Six, seven years ago; financial security was so important for us. Getting to a work-situation that was sustainable, and then amass enough for financial independence was the most important thing to do.

Let's imagine that all things we wanted to do in a year would fit in a jar; something like collecting pebbles. 

Priorities shift for good reasons - so beware of the autopilot that keep going on old goals

Financial security was the first pebble to put in the jar of things to do in that year.

And that was a big pebble, pushing so much else out.

Only after we had put in the pebble of amassing as much as possible for financial security and independence; then it was possible for us to even consider other things. 

But now; we have passed financial independence. We have passed a possible monthly spending significantly higher than what we spend now - and what keeps us fairly happy.

We are probably on the path of having wealth beyond what we will be able to spend, or want to spend, and will end up in our 60ies and 70ies with more than we can understand.

So do we do what's most important today as well?

Or are we going on autopilot, and just put in the pebble of amassing more money in the jar of things we can do in a year? Do we really consider that what was a very important priority 7 years ago, has changed, when we have passed all those stations along the tracks to and beyond financial freedom?

Shouldn't we perhaps consider that other things are important, and need space over a year: health, for instance, why not go from decent health to top-health? Or friends? Or appreciate the shifting seasons and the small beautiful things in life? Visiting family?   

Everything is fleeting, and the moments will be lost if we don't attend to them.

How important is that, compared to but another bunch of euros in the barn? 

Have we set our priorities straight? 

Is it time to give up the feeling of financial insecurity, and understand that the pebble of amassing, now, hides other things; things that are more important to us.

Perhaps it's time to realize that we have arrived. 

Farewell,

antinous&lucilius 

The Moral Thing To Do

Not contributing productively could - and will - be seen by society, and the slave mentality, as something amoral. 

Not that it would be wise to argue.

But let's stop for a second and consider the argument.

Is not being productive ... amoral?

The Face of Arete in Ephesus, the Goddess of Excellence as in full realization of potential or inner function 

Carlos Delgado, CC BY-SA 3.0

What would be the purpose to agonize over how much one can produce; if it's one's insatiable thirst after recognition and plenty, that one is trying to quell: a more beautiful house, another car, another expensive pastime for one's children. 

Wherein lies the moral prominence in squandering one's life in the pursuit of any such insatiable appetite? Are these delights a worthy cause to worry over, and loose the little time that was given to us?

Or is it more moral to recognize what is enough, and instead of everlasting production pursuit health, wisdom and love?

Free people are far between. Shouldn't we, then, pursue what would only happen, if we are free to choose to do it.

That is the moral question.

Farewell,

//antinous&lucilius

Cash - The Strangest Asset Class

As we write this, inflation sores, markets tank, even hedges against inflation such as gold stay back.

We see our savings slashed as measured in euros and dollars, 

Day-to-day prices fall and fall.

Yet. There is one asset that behaves strangely. 

If another asset is down 50%, that means that cash is up 200% compared to that asset. 

 Le Radeau de La Méduse, sailing through strange waters
(Théodore Géricault, 1819)

And suddenly, cash - a perfect hedge when markets plummet - step out of the shadows with a quite impressive value explosion. 

Even when inflation rages through the economy.

These are months for cash.

How do you measure your wealth?

Farewell

//antinous&lucilius

Shifting priorities

When we started the journey, we were so focused on getting to financial freedom. We put our eyes on the money, and we didn't expect superior results if we behaved normal.

We still don't behave normal, and we still have our eyes on the money.

Yet, there is a shift in our priorities. It is as if we start to settle in, realizing that we have enough and will soon have abundance.

And now, other things come back to our cone of light. 

Friends. But not the friends from before. Instead our friends have grown closer. It's not the ones we want to hang with, like one would hang with a bottle of beer. Instead our friends are those we care about, who don't compare themselves with us, that don't give much for the normal, that inspire us and attract us, that we can spend a night and a day chatting with. 

Family. But not the struggles of the past, but the appreciation of in-laws, the happiness of our nieces and nephews, a good hour spent with an old grandpa or aunt.

Art. Art is developing as a concept for us a little as well. But here we are still searching. We have some enterprises that reach out, and we feel that art wants to reach out. We still don't know everything that hides here.

Travel? Not so much. Even though we realize that with tons of bonus points piling up it's basically free for us to go where ever we want. But we still can't think that much of where that would be. We start to realize that there are other factors that are important to us. We don't want to travel for the same reasons as before, with the emptiness of escaping, for we have nothing to escape from anymore. 

That doesn't mean that we're never going to change the scenery around us. But then travel is more a background, and not actually of much importance.

We start to realize that most important for us might be to help the sun wander over the sky; helping ourselves, helping those we care about, and helping humanity to be a little better. 

Hermes, messenger from the Gods and wanderer of light over the skies.

And in the end, when the sun settles in the west, we know that the most important thing for us might be too be able to say; it was a good day.

Farewell

//antinous&lucilius

Other Asset Classes, and a Fun Sex-life

Where might conventions limit us?

What is there, that we might not see or understand, because we stay limited in our thinking? 

For those with a more open mind there are deeper ideas to explore. 

'Scandalous' but beautiful, reminding us that there is more to this world than just convention
(The Warren Cup, 5-15 AD)

Like the conventional stock market and the stock-picking, there are more, broad and well-acknowledged asset classes, but they are out of the main stream and probably nothing you will hear of if you only attend a conservative congregation.

So of course this is taking things a little over the top. But being gay we sometimes marvel how purple around us seem to be locked in conventional thinking in not only who you could live and have fun with in bed, but also do much more; as if there's an built in value to studying conventional.

When you're not so straight at all, one really needs to consider that what's normal might not be what's good for you.

Who knows, other ways might be something to consider.

And with finance, the real beauty is that we can bet on all at the same time. 

Let's consider some alternatives.

Long term government bonds.

Long term bonds are bonds that promise a fixed payment far off into the future, as far as 25 years from now. 

The price today is the market's valuation of what that future promise is worth today. 

That means that the value of a long term bond today contains the markets consensus of long-term expectations on inflation, monetary policy and interest.

Small changes in those expectations has a large impact on the current price of the bond.

Why do we prefer government bonds? Well, because governments - and we're talking governments from stable countries here - are rather reliable when it comes to paying back their debts. At least, they own the money printing press. 

And hence, the price of bonds are dictated by something very different than stocks.

It's a hedge out of the conventional.

Cash.

Cash? Can that be an asset class? 

Yes. But many are blind to cash as an asset class, because of convention; almost an illusion - that keeps it the asset characteristics of cash out of sight. 

Let's make a thought experiment to see the blind spot.

We usually measure wealth in a currency, that is, as if it was cash. 

And so, when the portfolio drops in value, we have a hard time to see the situation from the other way around.

The drop of value in prices is exactly the same thing as the cash increasing in value.

Stocks falling 50%? That's one perspective. 

Another perspective is that cash just increased 200% in value against stocks. 80% drop in the stock market? Or was it the other way around, and cash just increased 500%?

What about an option that is guaranteed to raise 500% in value if the stock market drops 80%? An asset that is guaranteed by the money printing press. And an asset that will be lovely to go on a shopping spree with when prices are low. 

Don't underestimate the value of cash as an asset. Available cash when there is blood on the streets will build a vast fortune. 

Gold

Why does governments keep gold in their reserves? 

Gold is a value storage through space and time. 

It keeps a piece of the long-term increase of value and production of all humanity, and it's much independent on political whims, short term booms and busts of stocks and unreliable monetary policy. It's even anonymous, if it is melted down.

And we know that we will always keep a reliably exchangeable piece of value with gold, and, when political turmoil and trust in the economy is low, it's more than unlikely that gold will have lost its value - quite the opposite, in most cases - and we will be in for a good deal. 

The states know this. 

When trust is low, gold might still trade.

Bet on everything: The Art of Re-Balancing.

As with all good things, there is balance to be found.

By doing a little bit of all, one also always have what the market wants. 

Are you into Gold? We've got gold aplenty. Cash? A credit crunch and you need a bail out? We got you covered, just hand us some of those stocks in exchange.

Are you crazy about stocks? We've earned a lot while the markets went up and you realized you want some of the fever. We will sell of ours to you.

By knowing more asset classes than the most common kind of love, we're on a sure, steady spiral upwards.

For a better experience: mix it up

And just like life might be better by broadening the perspectives; the financial life might benefit, even find a new stability and security, if one mix in more dimensions.

Farewell,

//lucilius&antinous

More to read:

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

- Our thoughts on portfolios.

We have already arrived, and why we carry on

We knew it at the beginning of the year.

But in all the tumult in the world during spring, we forgot it again.

We have already arrived.

We could spend all our time sipping on soft-drinks from now on.

We have low, probably even very low, monthly expenses. And atop of the 'must have to live', we still add around 30% of "pleasure spending", which makes us happy with life (in a stoic way). 

What dawned upon us now as summer goes over in fall, is that the safe withdrawal rate gave us, already in the beginning of the year, that we could withdraw all we needed + our small "pleasure spending" and another 50% in safety margin, and still be safe in the worst historical case.

That's a 50%-100% safety margin if one does the math. 

That's what we had already at New Years Eve earlier this year. 

And since then, half a year has elapsed, and we've been adding to our investments since then.

So on top of the safety margin, which is already a worst-case-scenario, we've been adding what correspond to several years of expenses. 

So, yes. We have arrived. 

Why carry on?

Fundamentally, we don't believe that more money from this point on will a big difference when it comes to if we feel content with what we have. 

If we're not happy, the biggest problem will probably not be if we can or can't spend a little more per month.

So why go on?

So why don't we stop then, and savor what we have? 

Well, one thing is that we don't really know our future preferences. Even if we think we're stoic today, why not give our future self the gift of actually being somewhat wealthy as well. A money-machine on top of the money-machine, so to speak.

And it's pretty easy for us to earn money right now. We're in the middle of our careers. We're relatively well-payed. Our jobs don't imply freedom for sure, but we're also in positions that are interesting, we perhaps even allow to delude ourselves that society is slightly better off if we show up for work. So it's not freedom, but it's not directly painful neither to build that money-machine on top of the money-machine right now. 

A third thing is the black-swan-factor. We only know the expenses we know, or can forsee. It's the unknown unknowns that will bite us. And we don't want to keep looking over our shoulder and wonder if what quacks (or whatever swans do) is that proverbial black swan.

So hopefully, we'll soon be free with a margin, and soon thereafter, wealthy as well.

Are we right? Are we wrong? Are we just greedy? Are we not greedy enough with the time that we have left? We don't know.

We'll see how it goes.

As Seneca has it in his second letter to Lucilius:

"[quotoes Epicurus]: "Contented poverty is an honourable estate." Indeed, if it be contented, it is not poverty at all. It is not the man who has too little, but the man who craves more, that is poor. What does it matter how much a man has laid up in his safe, or in his warehouse, how large are his flocks and how fat his dividends, if he covets his neighbour's property, and reckons, not his past gains, but his hopes of gains to come? Do you ask what is the proper limit to wealth? It is, first, to have what is necessary, and, second, to have what is enough. "

We're poor, in Seneca's eyes. And perhaps he's right, because we hope of gains to come. And this part of the journey will continue for some time more.

Farewell

//antinous&lucilius