When you’re teaching somebody a new subject, simplifications can creep in. Rules of thumb at best. Outright untruths in the extreme.
For example:
- The simile “as blind as a bat” isn’t true – many bats see better than we do. (Maybe they’re also better than us at similes?)
- Christopher Columbus didn’t think the world was flat. The notion combines scientific and terrestrial exploration into a neat historical parable, but even the Ancient Greeks and Romans knew the Earth was probably a sphere. (Columbus owned books that told him so.)
- Teaching children the classical laws of motion wouldn’t be made any easier by telling them they’ll eventually learn the whole shebang is a gross simplification – that Newtonian physics is a shadow on the wall approximation of the statistical weirdness of quantum mechanics. (Yes, I know I’m oversimplifying here, too!)
- The “i before e except after c” rule often works – but not enough that foreign students can seize the weird exceptionalism of the feisty English language. (Spot the rule’s deficient idiocies there?)
So it is with investing. We say higher returns come with higher risks. That assets that go up and down a lot in price such as shares should to be held with a long-term view, and that braver investors can eventually capture higher returns this way.
But reality isn’t quite so simple. Those higher returns are only expected – not guaranteed – and not all risk is rewarded.
For starters, some risks don’t even come with the expectation of higher rewards:
…the relationship of more risk, more reward does not always hold.
In some circumstances you take greater risk, yet you can’t expect the market gods to reward your chutzpah with greater returns.
Academics call these lousy bets uncompensated risk.
Read our previous article on uncompensated risk to learn if you’re gambling for nothing.
Not every stock market has read the textbooks
It’s also important to realize that even the ‘right kind of’ risk can go unrewarded.
You might expect higher returns, but higher returns are not guaranteed.
For example, we say investing in risky equities can be expected to deliver higher returns than super-safe government bonds. But there’s no guarantees, and no timescales.
Indeed there have been long periods where the return from bonds beat shares!
Over the ten years to the end of 2008, for example, the annualized returns from US and UK shares were negative. In contrast, bonds soared.
So much for risk and reward over that decade.
And in case you’re thinking you can handle a ten-year duff stretch – and you will have to over a lifetime of investing – some have had it much worse.
How would you feel if your well-founded risk-taking wasn’t rewarded for half a century?
In 2011 Deutsche Bank reported that:
…for three members of the G7 group of leading industrialised nations, Italy, Germany and Japan, returns from equities have been worse than those of government bonds since 1962.
Indeed, the Italian stock market has even managed to deliver a negative real return over the past half-century, -0.38 per cent on an annualised basis versus +2.64 per cent for bonds, “a remarkable statistic in a world where we are all used to seeing equity outperformance increase the longer you expand the time horizon”, wrote Jim Reid, strategist at Deutsche.
In Japan, government bonds have delivered a real return of 4.17 per cent a year, beating the 2.72 per cent of equities, while in Germany bonds have won by 4.28 per cent versus 3.46 per cent for equities.
Academics – and professional investors – struggle with findings like these. They go against the theory of efficient markets I discussed earlier.
For the market to get it wrong for 50-odd years might suggest:
- Those markets were unusual for some reason.
- We don’t have enough data. (A tossed coin coming up 10 times in a row doesn’t disprove probability theory. Try tossing it a million times.)
- The efficient market theory has limitations.
Personally I’d plump for a mix of all three in the case of Germany, Italy, and Japan.
But I’d also point out that all the leading efficient market academics hailed from the US, a country that has had the strongest, most consistent, and least ‘anomalous’ markets – with the best data, tracking a period including two World War victories, or three if you count the Cold War, and a transition from emerging market to sole global superpower status.
Could this very positive North American experience have biased the research or the conclusions? It seems feasible, but we’ll leave going down that rabbit hole for another day.
The important point here is expected returns are not guaranteed returns. Real-life investors in some countries never saw a sniff of them over a lifetime.
There are several important practical takeaways. For example, somebody on the point of retiring should not have all their money in equities, despite the higher expected returns.
Stock markets usually crash once or twice a decade, and that can chew up your higher returns in the short-term. That’s a big risk, especially for an imminent retiree or a newly-retired one. Statistically you might think it’s unlikely, but if it’s you, and you had no back-up plan, you’re somewhat stuffed.
This is called sequence of returns risk. It’s not a reason to have no shares or own only gold, or any of the other dramatic things people write in the comments on blogs. It’s a reason to own fewer shares, and to diversify.
Risk in real life
I was set thinking about this recently by a family friend.
Having come into some money, she bought me that quintessential millennial brunch of avocado on toast and picked my brains about what to do with it. (The money, not the toast.)
My first step when this happens is usually to send over a bunch of Monevator links, and wait to see if they get read.
If the person doesn’t do their homework (and they usually don’t) then that helps inform where I steer them next.
But in this heartening case my friend read all the suggested articles, and she was keen to let me know so.
For example she explained to me that she now understood that she should never sell, that stock markets always come back, that you have to take risks to win… right?
Er, right, I said. Sort of.
It’s complicated!
Investing is like that. You have to learn a lot to realize there’s a lot you don’t or even can’t know for sure.
One excellent if rather gnomic definition1 of risk is:
“Risk means that more things can happen than will happen.”
Whereas my friend had taken risk to basically mean “what goes down will come up.”
We can have expectations, given time, but there can be no certainty. If there was certainty, there’d be no extra risk. And if there was no extra risk then there’d be no expected higher returns – because they’d have been bid away by the market, at least in theory.
As blogger Michael Batnick says, you are owed nothing:
This is how stocks work. The stock market doesn’t owe you anything.
It doesn’t care that you’re about to retire. It doesn’t care that you’re funding your child’s education.
It doesn’t care about your wants and needs or your hopes and dreams.
Batnick stressed in that article that he still believes shares are “the best game in town” for long-term investors.
But you must have realistic expectations about your expectations.
Shit happens
To conclude, I’ve long wanted to include this graph in a post. It’s from Howard Marks’ wonderful investing book The Most Important Thing:
The right way to think about risk.
Source: The Most Important Thing.
What this graph shows us is that expected returns do indeed increase with risk – but there’s a range of potential outcomes along the way. Some are dire. Plenty are bad.
It is a good graph to sear into your brain.
This graph is why most people are advised to use widely diversified stock funds, not try to find the next Amazon or Facebook.
It is the reason to hold some money in cash or bonds even when savings accounts pay you nothing and bond yields are negative.
It’s why we should stay humble and diversify our portfolios across asset classes, even ten years into a bull market. (Or make that especially ten years in…)
As with many things, expect the best outcome when investing – but be prepared for the worst.
- I first heard it from Elroy Dimson of the London Business School. [↩]
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Great post, all “rules” in life have subtleties, exceptions, and grey areas. You have to really learn and deep-dive into a subject to get to the truth.
That is a truly stupendous graph, and provides a neat visual interpretation of the maxim that increased risk increase both the realm of possibilities and the potential magnitude of their impact, both positive and negative.
When I look at the two main investment forums on MSE I am horrified at the high proportion of people who have no idea of how variable equity returns have been historically. They also seem to feel that taking equity risk entitles them to high returns.
Other beloved and inter-related factoids include (i) cash always loses against inflation, (ii) over any reasonable period equities will defend you from inflation, and (iii) over any period of several years equities will outperform cash.
It’s a bit old but I find this useful:
https://www.bondvigilantes.com/blog/2011/02/01/inflation-hedging-for-long-term-investors-the-most-important-academic-paper-you-will-read-this-year/
Ding dang dong…..what a treat that was! Thank you! Every day is a school day! Even on the summer hols!! 🙂
I totally agree with you. A lot of people really don’t understand the subtleties of risk. In fact I’m not even sure that investment analysts fully understand the risks. Often times risk is defined in terms of standard deviation of average returns from a particular asset. However, this isn’t quite the risk that you and I think of in terms of investing. Someone can gain a return from equities without every realising what an incredible risk they were taking in the process!
One way to think about investment risk is that if higher risks guaranteed higher returns, they would not be higher risks.
Great post TI – in fact, I’ll probably print this out and read it to my mum (she has learnt about investing through Monevator).
That Howard Marks chart is one of my all-time favourites as well. 🙂
Nice article, a follow up on capacity for risk may put this in context. I always find it difficult explaining to people that they should not do what I do because that risk (to them) would be disproportionate to their circumstances. Investing is a very individual game that is hard to talk about… Keep up the great work.
A good Investing 101 course would consist of students looking at that Howard Marks chart for two hours straight without blinking.
“It’s why we stay should humble” Words up mixed?
@W Neil — Eek, thanks! Hopefully you’ve caught that just before the email goes out. 🙂
@all — Glad the article hit the spot. That chart is the greatest, agreed. I felt my brain move with a sort of ‘clunk’ sensation when I first saw it, and understood risk a little better, forever.
Why would Columbus think the world was flat? We were always taught he set out to find Asia – i.e. to prove the world was round.
@Pete — Hmm… I wrote that off-the-cuff and perhaps I misremembered, or perhaps you went to a better school than me! 🙂 That said a quick Google now seems to show quite a range of preconceptions out there (and a smoking gun pointed at the author Washington Irving?)
Thanks for this carefully thought through article. I still don’t think I have properly got the proper perspective on risk and my tendency to underestimate it persists even after lots of reflection. For example, the Marks graph is great in highlighting the key issue of how risk inflates into the future, I agree, but I suspect it is far worse than even this graph suggests. The graph still maintains the impression of a gradual rising tide that can somehow be relied upon. I think that is probably false – or rather a lot weaker than implied by the graph. In reality the upward trend is a meandering line that can end up anywhere.
Here’s a tentative insight (and I might be wrong on this and could stand to be corrected) that shows how insanely risky the outcomes illustrated are. Looking at that last, most widespread bump to the right. Most people would probably say that although widespread, the most likely outcome will be at the middle of the bump. However, if you take a 1% slice of outcomes around the middle of the bump, and compare this to the worst 1% – what then? I think it is the case that it is equally likely that you wind up in either region (one of which includes totally busted). Now that’s a real eye opener isn’t it?
I think perceived risk (rather than actual risk) does determine the price of an income, so the only way to (probably) get more income is to buy a cheaper (ie less certain) one – and perceptions can of course be wrong, so it can all be mispriced. I think definitely diversify between different sectors and countries and asset classes, but the asset classes are so wildly different animals.
(It’s not the country specific example but) For global gov bonds to outperform global equities for a long period of time (ie a negative equity premium) something would have to be very amiss, ie ww3, or global deflation. Whilst it’s possible to do well with gov bonds in unusual circumstances, returns aren’t the primary motivation for that part of the portfolio
For the avoidance of doubt with anyone reading, this isn’t what the graph shows. 🙂
The left hand side, the Y axis, shows increasing return. The bottom axis, the X axis, shows increasing levels of risk (i.e. *not* time or the future at all).
The smooth line running from bottom left to top right is the traditional risk vs return graph. This line shows that as risk is increased, expected return increases (and vice versa).
This is the traditional risk/reward graph we’ve all seen, and those of us who have written about investing have used to demonstrate the relationship. (We certainly have, and will do so again. 🙂 )
The genius of the graph though is how it unpacks visually what goes into that smoothed line.
At any point on the risk/reward graph, you can see there’s a range of outcomes. If you’re lucky, you get a high return for the amount of risk you took. If you’re unlucky, you get less that you expected, perhaps a lot less. On average you get a positive return for any level of risk, which is indicated by the horizontal line in the centre of each ‘risk return bell curve / bubble’ that touches the smoothed line.
You’ll notice this horizontal line is always above the smoothed line. That shows there is a positive *expected* return at each level of risk. But you might not get this average return.
Also, importantly, and contrary to the slightly confused comment that I’ve excerpted above, if you are making a risky enough investment it might *not* “be relied upon”. You can see this because at the rightmost point on the risk graph here, the range of outcomes goes negative. i.e. You lose money / see a negative return.
So it’s really a brilliant visual in its complexity yet simplicity.
It also explains quite simply why, for example, casual comments like “bonds are riskier than shares right now” can set someone like me off so much at times. 🙂
Right now, if you buy a UK bond and hold to maturity in all cases you’ll see a positive return, presuming you can reinvest the income back into that bond. In other words, the ‘potential return bubble’ doesn’t break the 0-bound of the Y axis.
In contrast, shares are quite capable of delivering a negative return, not least because you can’t hold them to maturity. (They don’t have a maturity.) Over a sufficiently long period for various reasons the chances of losing money with shares held and dividends reinvested becomes slim, but as cited in the article it has happened in some countries.
The *probability* of losing money with shares over say 10 years may be small. But the possibility exists. Not so with a ten-year bond today, in nominal terms.
This is the right way to think about risk and reward — ranges of probabilities and potential returns and the trade-offs between them. 🙂
Thanks for the explanation – I had got the wrong end of the stick thinking it was over time. Wot a nana. A graph of returns over time would look similar, I guess. I think the second part of my original comment is accurate in both contexts though (or maybe not!).
There is another point: the fact that “in the long run” equities have grown (in most countries) is fine. But as Keynes said, in the long run we are all dead.
@Ianh — You’re welcome! 🙂
–“Why would Columbus think the world was flat? We were always taught he set out to find Asia – i.e. to prove the world was round.”–
There was a guy in Ancient Greece – Erastothenes? I believe that he calculated the circumference of the earth to within a fraction of the known value. Did it by looking at the times when the sun shone right down wells in Egypt and Greece.
Likewise Pythagoras nicked his theorem from the Egyptians who knew that pyramids could have sides with lengths 3,4,5.
To continue on the tangent, I too have come across the notion that Columbus thought the world was flat. As Pete points out, that makes no sense given that he was trying to find a western passage to India. Despite Eratosthenes’ calculations, Columbus also underestimated the distance to India. Investment writer William Bernstein comments in his fascinating book “A Splendid Exchange: How Trade Shaped the World”: “[…] no vessel of the period could stock supplies adequate for so long a journey. […] Had Columbus not collided with America in his quest for the Orient, he and his men would certainly have vanished like the Vivaldis [two Genoese brothers who tried to find the western passage in 1291, 200 years earlier].”
Make that graph continuous along x and it reminds me of the risk profile of the accumulator bets from my matched betting days—where I first encountered and learnt to stomach this kind of risk/volitility.
I can just picture a third axis of time into the graph to make a three dimensional graph. I wonder what it would look like. I am hoping the returns will generally move upwards with time, perhaps the distribution of returns will be similar or perhaps even wider with increasing risk profile, but I am guessing the returns will less likely to be in negative territory. One modelling challenge for the statisticians.
-FIREplanter
One note, the title says that risk doesn’t “guarantee” reward – guarantees are very expensive because so many people require them – in most other areas of life if somethings not guaranteed it’s probably not happening (ie not guaranteed a promotion, not guaranteed a parking space, not guaranteed to be paid back for money you lent to a friend, etc) – people even think things that aren’t guaranteed will make them a victim of either fraud or exploitation (ie gambling)
We have to settle for probability, and what the graph does indeed show is the increasing probability of ever greater loss with risk, as well as reward. It changes the time frame required to get a median result
I imagine as time progresses those distributions gradually narrow around their mean values and as t -> infinity converge into delta functions (i.e. you eventually get the simple gradient). It would be a cool visualization.
@Mathew ” It changes the time frame required to get a median result” and @DriftGlass “as time progresses those distributions gradually narrow” At the risk of digging myself even deeper into my hole, be careful you are not making the same assumption I made in my first comment – the Marks graph is not related to the time invested, but shows the risk and reward profiles of increasingly risky investment options.
It must be me but that graph tells me absolutely nothing.
What am I missing?
We’re just musing on what a third time axis might look like.
As you drag that whole diagram along it maps out a 3D shape whose far end is the conventional straight line.
Point being you need an awful lot of time (infinity) to be able to guarantee extra return for more risk (i.e. get rid of the tails on those distributions). And that’s only if the risk premium (slope) is there to begin with!
@TI Shares do NOT have infinite duration (as I am sure you already know), although if you only consider dividend payments in your estimate of duration, the duration of shares that do not pay dividends would be infinite.
Stock market returns can be approximated by a lognormal distribution in which the standard deviation (volatility) grows with the square root of time and mean return grows exponentially. The result, is that the variation in rate of return outcomes from stock markets narrows with time. To put some numbers on it, if we assume a real mean annual growth rate of 5% and annual volatility 15%, then after 1 year there is a roughly 2 in 3 chance that the growth will be between -9% and +21%. Quite a big range. After 4 years the mean growth would be about 22% and volatility 15% times the square root of 4 = 30%, leading to a 2 in 3 chance of the observed growth rate being between -1.1% and +12%. After 9 years, the volatility goes up to 45%, with a 2 in 3 chance that the observed growth rate will be between +1.9% and +8.7%. After 25 years there is a 2 in 3 chance that the observed growth rate will be between 4.3% and 6.2%, even though volatility has risen to 75% (15% times square root of 25).
So the characteristic returns from stock markets means the observed rate of growth is likely to become smoother the longer you hold shares. Some say this smoothing means the risk of holding shares reduces with time. Personally I think this is a complete misunderstanding of risk, but you can sort of see the point.
I should add that after 25 years, with a 2 in 3 chance of the growth rate being between 4.3% and 6.2%, that does represent a huge range of actual growth, between 186% and 350% and there is still a 1 in 3 chance of being outside that range.
The version of the graph which other posters have referred to with an extra axis for time could be manufactured using a 3D printer. It would make a great Monevator branded stocking-filler this Christmas if the infamous investing book still isn’t quite ready.
@Naeclue — Yes, good catch, that was hasty/clumsy language of me. Might go back and modify it so I don’t confuse anyone. The trouble is it’s hard to state in very simple / quick terms how the uncertainty of equities’ future pricing and dividends (coupon equivalents), as well as potential infinite cashflows, makes them so different from bonds, especially over short to medium time periods. 🙂
Undated bonds potentially have infinite cash flows, but the duration is very simple to calculate if the coupons are fixed. It is just (1+y)/y, where y is the running yield. With equities duration is complicated as dividends, earnings and growth rates have to be estimated, so duration really needs to be a probabilistic value.
@beta it’s visualizing the fact that as risk increases, the distribution of probable returns doesn’t just increase, it expands. In both directions. That may be completely obvious to you already 🙂
“Right now, if you buy a UK bond and hold to maturity in all cases you’ll see a positive return, ”
I think this is a school boy error. That positive return is only in nominal terms. In real terms you are likely to suffer a loss.
@a beta investor — Yeah, well aware. 🙂 I wrote real as it happens, then deleted it. I think when talking about the maths in this context it makes more sense to talk nominal. 🙂
Gilts (for diversification) would be a much harder sell for TI I think if we admitted the expected real terms loss (could profit if you didn’t hold till maturity), although it is still less of a real terms loss than a typical savings account
Anyway I wonder if ti worries about everyone having too much risk going forward
“It must be me but that graph tells me absolutely nothing.
What am I missing?”
(i) It shows that if you take the professional money manager’s view of risk (which is a measure of volatility of returns – let’s say the standard deviation) then higher risk implies a wider distribution. That’s tautological.
(ii) It also shows that the higher the risk the higher the expected return i.e. the higher the mid-point of the distributions. That’s a sort of empirical result, with added “it really ought to”, but there was much discussion here some months ago of a counterexample. It seems that residential property tends to give a slightly higher return than equities with a noticeably smaller “risk” i.e. standard deviation.
(iii) You are free to reject that definition of risk. You might prefer, for instance, something like “risk = the probability of an outcome that would result in a permanent reduction in my income”. That would incorporate the notion that “permanent” ends with my death.
Roughly speaking, for a professional investment manager the risk that matters is career risk i.e. the probability that he’ll do so much worse than his peers that he’ll be sacked. That would indeed reduce his income. For the investor the risk is to his investment income. It’s quite different from the pro’s risk.
For example, for the pro it’s a big risk to be out of equities. For the investor it can be a big risk to be in equities. That’s why I would never leave a pro in charge of my asset allocation. Economists dignify the clash of interests with the name “the principal/agent problem” but it ought to be the sort of thing you learnt at your father’s knee, irrespective of what he called it.
@dearie – I suppose you could say pros don’t get rewarded for being safer than their benchmark, so the bias is to take extra risk, especially since the risk of a crash is less than the risk of getting sacked
Note regarding property, if there is leverage then potentially (to quote daytrader adverts) “losses could exceed deposits” as they say. A narrow standard deviation and good bull run might hide a sinister risk
Hah… I’m not here to sell anything. 🙂 I think people should be diversified, and not only own equities, but I’m agnostic about how you get it. If somebody wants to use say cash savings instead of gilts (and take a different flavour of real terms loss) then that’s more than fair enough. That’s been a component of my strategy for several years, versus gilts.
That said, one thing that I have noticed is people talk about real terms as if they’re very important with gilts, but less so when you talk about equities.
From a diversification perspective, it doesn’t really matter when making comparisons if you’re comparing 1% nominal return from gilts versus say 5% nominal from shares, or -1% real terms from gilts versus 3% real terms from equities.
This will seem obvious to many readers, but still, the strong sense I get is some people were happy enough with gilts as diversification if/when they were getting say a 1-2% real return, but now they’re beyond the people because the expected real return is negative. But they are not in the mix of a traditional portfolio to deliver a strong positive real return. And as we discussed at length the other day and I’ve said above, if real return expectations from gilts are negative, that does not imply an environment where you’d expect years of storming returns from shares. It suggests a low return world, across the board, including correspondingly lower real returns from shares. At least in theory.
Again though, people need to do what they feel comfortable with. I rarely hold gilts myself, and tend to use cash combined with a changing menagerie of all sorts of exotic fare. But as I’ve mentioned before, because I’m an active investor, my decision to favour say cash over gilts or to play around with various slightly exotic fixed income alternatives isn’t really going to move the dial compared to my active bets (10% of my portfolio is currently in one single company share…!)
But for sensible passive investors, I can only ask you consider the standard question: Why do you think you know better than the trillions deep government bond market, but not the relatively smaller equity market? If one thinks one can time government bond markets, feel free, but that’s an active investing decision.
We’ve had more than five years now of readers bemoaning not only the prospects for gilts but the wisdom of even considering them. In the meantime, they’ve generally done at least as well as cash and most could have bought a modest allocation, forgot about them, and then saved themselves years of worry. 🙂
(Apologies if any of the above is a bit muddled. Friday night, I’ve been out and I’ve had a couple of ales. 😉 )
@TI
“Why do you think you know better than the trillions deep government bond market”.
Maybe the bond market is “fixed” though? As I understand it there is legislation called “Mininimum Funding Requirement” that forces pension funds to hold gilts. If pensions funds are forced to hold hundreds of billions of gilts its bound to depress gilt yields.
There are other investments that have the same characteristics as gilts (ie guaranteed income over a long time) that yield a lot more than gilts. For my money I’d rather buy property rented out on long leases or ground leases. Heck, I’d even prefer to buy a wind farm than buy gilts.
Is an office rented to a government department on a thirty year lease really that much different to a 30 year gilt?
That graph is still a mystery to my despite many erudite explanations.
As long as we are talking about long only investments the maximum amount you can lose is the amount invested, the size of the loss does not increase with the riskiness of the investment.
The only thing that changes is the potential for that loss, there is a higher probability of losing it on a higher risk investment, but a higher chance of bigger return. Anyone with $100 in Enron and $100 in Apple in 2000 would appreciate that.
Going back to the risk free argument that government bonds were the “safe” investment people, other than historians, forget that in days of yore merchants were much more nervous of lending to sovereign states than other merchants. The king or similar could simply refuse to repay his or her liabilities and, for a while at least, could get away with it. A merchant would have no such luck as the Rothschild’s knew very well.
Finally, over the last 100 years you had a greater chance of suffering a real loss holding gilts than equities over a ten year period.
Great article that has helped me to crystallize risk vs reward is not necessarily proportionate. Thanks.
@Gordon, there are other assets than gilts that are lower risk than equities, but they are not as risk free, hassle free, nor as liquid as gilts. Physical assets such as wind farms may burn down for example.
I still hold a 6 through 12 year gilt ladder in my SIPP, along with US Treasuries, via ETFs in my SIPP and ISA. Outside my SIPP I now use FSCS/NS&I protected cash deposits instead of gilts, although cash is a pain as you have to keep moving it as special offer rates and term deposits end. I am not expecting to make anything in real terms from cash or bonds, but they will be very useful when the next stock market crash comes along.
In engineering terms risk = severity × probability.
In financial terms it might be different and for personal finance it might be something completely different. People’s attitude to money is a lot more complex than you might think.
@matthew, the paper we discussed some months ago compared ungeared equities to ungeared housing across sixteen national markets. On average the housing gave slightly larger returns than equities with markedly less volatility. The authors themselves remarked that their discovery ran contrary to accepted doctrine.
Mind you, I have no idea how I might buy a portfolio of diversified housing across sixteen countries. Maybe that difficulty/impossibility explains the deviation from doctrine; maybe the doctrine works only for diversified, liquid assets.
Anyway the paper did make me realise that there could be more than one reason that the filthy rich enjoy boasting of multiple “homes” in multiple places.
https://www.frbsf.org/economic-research/files/wp2017-25.pdf