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Weekend reading: The exceptional mathematics of average returns

Weekend reading

Good reads from around the Web.

The clever bods at The Value Perspective have been doing a sterling job recently in tackling the weird world of the average and what it means for returns.

From writing here on Monevator, I know a lot of people get confused when it comes to how average returns work in practice – or more specifically about the different kinds of average – and what it implies for our investment decisions, whether it’s asset allocation or investing a lump sum or confronting sequence of returns risk.

And when I say “a lot of people get confused”, I’ll admit that can include me!

So most people could do with a refresher on the maths.

Start with this Value Perspective article on the law of averages, which begins:

Let’s play a game we will call ‘Russian dice’, the rules of which were invented by a physicist/economist called Ole Peters. They are pretty simple – roll a dice and, if it comes up ‘one’, I will shoot you.

Do you fancy playing? It does not sound very appealing but, if you were a nihilistic mathematician, you might be tempted because – in the very strictest terms – on average you will be absolutely fine.

The word ‘average’, however, can be somewhat misleading if not defined very precisely. If 100 people roll the dice instantaneously, the average result is ‘three and a half’ – that is, (1 + 2 + 3 + 4 + 5 + 6) ÷ 6 – so, if you are judging the game on the average result, everyone survives.

However, if I asked you to roll the dice 100 times in a row, it is extremely unlikely you could do so without at some point seeing a ‘one’ come up. Bang. You do not get a chance to see the result of the 100-roll average over time.

Intuitively, we all know there is a difference between these scenarios without any complicated maths or concepts. Mathematically speaking, it is known as the difference between an ‘ensemble average’ (the average of an event happening many times concurrently) and a ‘time average’ (what happens when you do something a lot of times consecutively).

However, this concept and its implications are not well understood in investment.

After that first article provoked a fair bit of debate and confusion, their follow-up tried to explain it with a diagram:

[This time] you simply start off with a stake of, say, £100 in the pot and we toss a coin. Every time the coin comes up heads, you increase what is in the pot by 50%; every time it comes up tails, you lose 40% of whatever is in the pot.

The coin is a fair one – so it is always a straight 50/50 chance – and there is not a firearm in sight.

Would you like to play?

At first sight, there appears to be no reason not to play – after all, if on each coin toss the only two possibilities are going up 50% or falling 40%, then surely, on average, it is a winning game. And indeed it is. No matter how long you play it for, on average, the expected return is positive – but, as we argued in the previous article, the word ‘average’, if not defined very precisely, can be misleading.

In this particular example, the average obscures a pattern where the majority of people who play the game actually end up losing.

Here’s the pattern:

Would you play the Value Perspective's counter-intuitive game?

Would you play the Value Perspective’s counter-intuitive game?

It reveals:

In the above example, 11 of the 16 possible sequences of coin tosses are losing permutations and, the longer the game goes on, the more money is lost.

In a game where the only possible outcome each time is a 50% upside or a 40% downside, that would appear counter-intuitive so what is complicating matters?

It is that difference between time and ensemble averages.

Read the whole article for a deeper explanation.

Go on – you’ve got Sunday to recover from the headache!

p.s. I was struck by the latest article by Carl Richards for the New York Times on the difficulties of investing in real estate. It’s aimed at US readers, and argues that most property investors will make a low return unless they have special skills and insights. How different from the UK, where almost anyone who bought a buy-to-let 5-10 years ago is an investing genius! Now it’s true that the US is a very different market when it comes to property. Still, I think the big divergence isn’t so much to do with the asset class as because their bubble burst and ours hasn’t. Would Richards have published this piece in the still-roaring US property market of 2006? I doubt it. (I wonder if he will be able to re-purpose his piece for a UK publication come 2016?)

From the blogs

Making good use of the things that we find…

Passive investing

Active investing

Other articles

Product of the week: Investing in alternative energy has an obvious appeal if you believe we’re not moving fast enough on climate change. So I’d expect the latest share offer from long-established Tridos Renewables to do well, especially since you can now put in as little as £50. With my hard-headed capitalist hat on though, I’d question whether the 5.8% return that The Guardian estimates an investor who is cashing out would have achieved over the past nine years is enough for the risk. The 9-10% touted by the company is more like it, but is it realistic? I suppose there’s also a good vibes dividend to take into account.

Mainstream media money

Some links are Google search results – in PC/desktop view these enable you to click through to read the piece without being a paid subscriber of that site.1

Passive investing

  • Farewell to the fund manager [No plug for us 🙁 ] [Search result]FT
  • New ideas about asset allocation in retirement – WSJ [w/ Mike]
  • Targeting the liquidity return premium – Swedroe / ETF.com

Active investing

Other stuff worth reading

  • Buy experiences, not things – The Atlantic
  • How to be rich with less money – Alan Roth/AARP
  • The optimal age to buy National Trust lifetime membership – Telegraph
  • The downsides of Scotland’s 12% stamp duty property tax – Telegraph
  • Beware this sophisticated Airbnb fraud – The Guardian
  • Author economics: The brutal truth [Search result]FT
  • Coming soon: (Happily) working until you die – CNBC

Book of the week: The Education of a Value Investor by hedge fund manager Guy Spiers is my new favourite book on active investing. It’s all about mindset rather than metrics and every active investor should read it. Prepare for regular plugs here.

Like these links? Subscribe to get them every week!

  1. Reader Ken notes that: “FT articles can only be accessed through the search results if you’re using PC/desktop view (from mobile/tablet view they bring up the firewall/subscription page). To circumvent, switch your mobile browser to use the desktop view. On Chrome for Android: press the menu button followed by “Request Desktop Site”.” []
{ 18 comments… add one }
  • 1 Passive Investor October 11, 2014, 12:42 pm

    The point that an mean average is a blunt and sometimes misleading summary is well made. Investment returns are very skewed as you say which makes the mean average less helpful than the median average (in the context of the example the median wound be represented by the 8th and 9th investor of the 16). The returns from premium bonds are subject to the same problem.

    I don’t understand why the returns of up 50% and down 40% are chosen though. After a loss of 40% you need a gain of 66.6% to recover your money. Using these figures 11 of 16 investors are up in the example. I haven’t done the calculation but using up and down percentages which more closely reflect say a 4% real return on equities ? Down 40% up 70% then relatively few investors would lose in the long-term.

  • 2 Aidan October 11, 2014, 12:44 pm

    Just one more iteration would give a 50/50 win lose ratio and then it rises there after. I guess the take home from this is that time in the market really matters to “average out” a few unexpectedly bad years. But you could still end up being really unlucky if you get landed with too many tails eating away your nest egg.

  • 3 Aidan October 11, 2014, 1:13 pm

    Oops my last comment is wrong, there’s 7 outcomes after 6 tosses I think. My brain hurts!

  • 4 Neverland October 11, 2014, 2:26 pm

    There was an interesting article on this very subject in Bloomberg personal finance section last week

    Two investors same on $1m retirement fund and exactly the same annual returns…. I think they used a 4% annual withdrawal rate

    Retiree 1 starts off with his investments falling in value and then experiences many years of strong returns

    Retiree 2 starts off with years of strong gains then afterwards hits years of losses

    Retiree 1s fund runs out after 19 years

    Retiree 2 still has $1.2m fund after 20 years

    It’s all luck in the end … Good luck…. and Bad luck

  • 5 P October 11, 2014, 2:40 pm
  • 6 John Kingham October 11, 2014, 4:25 pm

    “almost anyone who bought a buy-to-let 5-10 years ago is an investing genius!”

    I think you’d have to go back further than 10 years to be a genius (even a tongue in cheek genius). Houses haven’t really gone anywhere in real terms in the last decade, perhaps apart from London.

    Look back before 2000 to see the real super investing geniuses (more tongue in cheek). My first flat in London with 4 bedrooms cost £40k in 1995 (!), although I’m sure a few Monevator readers can easily beat my £10k/bedroom price; or even £10k for a whole house if they’re old enough!

  • 7 Graham October 11, 2014, 6:12 pm

    It’s an interesting illustration, but I think it tends to over-complicate.
    After 2 tosses, a simple one up, one down result (either way round) would result in a 10% loss. Without being expert on the more complex maths, this would be enough for me to feel I shouldn’t be playing.

    Great blog. Thank you.

  • 8 Matt October 11, 2014, 7:01 pm

    In an article in today’s Times, the active management industry fought back against the passive onslaught by challenging the use of “average”.
    “Some commentators are questioning the basis on which active funds are judged to underperform trackers simply because the average active fund performs poorly. Simon Evan Cook, of Premier Asset Management, says that a better way of evaluating the performance of active funds would be to give greater weighting to the performance of larger funds, because that is where the bulk of investors’ money is… Premier recalculated the averages of seven popular sectors for the five years to mid-2012, giving a heavier weighting to bigger funds, and found that in four out of seven cases the average active fund beat the relevant tracker”.
    I’d be interested to know when the proponents of passive investing state that the average active fund underperforms, are they weighting every active fund equally or do they consider the size of each fund?
    As a recent convert to passive investing, I’d be interested to hear the views of more experienced commentators.

  • 9 Greg October 11, 2014, 7:58 pm

    It’s quite neat, but I think the article is a bit confusing as the results are applicable in situations that it doesn’t really explain very well.

    – The game has a positive expectation on every play (win 50% vs lose 40%).
    – History does not affect this.
    – The distribution of end points is a simple binomial one. (i.e. symmetric)
    – However, the distribution of values at the end is very much not symmetric.
    – After a large number of rounds, the large majority of people will be clustered around the middle, some with moderate losses, some with decent gains, averaging to a reasonable return. There will be people at the extremes, including some with fantastical gains but even excluding those, the ensemble is net positive. (Even if most people lose, they lose small.)

    What it does highlight, which is the point of the article, is that:
    a) When considering an individual, where they only care about their lifetime, a geometric mean of each year’s return is correct and an arithmetic mean is misleading.
    b) A lifetime is not long enough to make it a ‘large’ number of plays, so people are subject to huge risk of deviating from the ‘average’. It’s just that each yearly cohort plays as one! (Sort of…)

  • 10 Greg October 11, 2014, 8:34 pm

    Thinking about this a bit more, I suppose one should care about the distributions really quite a lot as far as society is concerned. I like this avenue of thought. I will ponder on it some time. Thanks for bringing it up!

    I think it also dictates we should heavily tax the guy who flipped a load of heads, no matter how clever he thinks he is…

  • 11 The Investor October 11, 2014, 11:21 pm

    @Matt — That statement doesn’t make any sense. As much discussed in the recent videos I’ve been featuring — so have a re-watch if you need a reminder — active investing is a zero sum game. For every winning trade, there must be a losing trade. Winners and losers equal out, and investors in winners/losers get their returns, minus costs.

    So mathematically, I can’t see how the largest funds could also be the best performers. It’s essentially impossible.

    It may be that some subset of the largest funds (i.e. a subset of retail funds) has done better than the market. But that’s a different thing. There will always be some way in which some identifiable set of funds has done better over the past 5 years. There always will be. If chasing around trying to predict them in advance — the opposite of what that study does, any fool can see them in hindsight — appeals to you then go for it. These guys are masters of marketing, never forget that.

    Incidentally, I am giving them the benefit of the doubt and assuming they are looking at the largest funds five years ago, not today.

    But that may be charitable.

    If they are looking at the largest funds *today*, then all that proves is the well-known truth that funds flock to recent performance (which is why investors do worse than the funds they invest in — because they buy in near the top).

  • 12 David October 12, 2014, 9:19 am

    @Investor: “Incidentally, I am giving them the benefit of the doubt and assuming they are looking at the largest funds five years ago, not today. ”

    Too charitable I’m afraid, my reading is that these are current sizes:

    So hypothesis should be “funds that have outperformed attract more money” not “funds that had more money have outperformed”

    Something the very same person is then quoted around here:

    I still think the key issue here is “can you identify an fund that will outperform in advance?”, and I’m still struggling to see any evidence that you can. In my mind the analysis they’ve presented doesn’t even get close to that question (although to be fair they admit some of the shortcomings in the article).

  • 13 long-term-passive-investor October 12, 2014, 1:51 pm

    Excellent article, but please — roll a die (NOT roll a dice)!

    You don’t say click with a mice, so why say roll a dice?

    Now if you have two of the cubical, dotted objects, you could say throw two dice, or throw the dice. But the words ‘a’ and ‘dice’ DO NOT belong together.

  • 14 oldthinker October 12, 2014, 6:10 pm


    ‘Die’ and ‘dice’ are equally correct as the singular form.

  • 15 Greg October 12, 2014, 6:22 pm

    Heh, if we’re complaining:
    1) Can you dump the telegraph Scottish property article? It’s misleading drivel. (12% is the marginal value and according to one commentator, will affect about 100 houses…)
    2) What’s with ‘gotten’ appearing everywhere?

    Oh and I see IG are now doing share dealing, including ISAs with low exchange rates. Perhaps they deserve a mention in your epic table as they will fill a niche?

  • 16 Rob October 12, 2014, 7:08 pm

    An important issue in the active v passive debate is that active funds rarely confine themselves to their asset class. They get additional returns by buying assets that are ot included in the index. One famous income fund used to invest in unlisted companies as well as overseas shares.

    At a stroke that makes comparions with the index flawed, let along with passive funds that no not go off-piste.

  • 17 ZD October 13, 2014, 5:32 pm


    I’m sure it’s been posted on Monevator previously, but the excellent http://www.firecalc.com/ lets you simulate those sort of scenarios using real world (US) historical data.

    Somebody retiring in, say, 1970 with a 25-year retirement ahead of them could prosper, whereas somebody retiring in 1972 with the exact same sum could run out of money prematurely if the market had a bad year in the meantime. FIREcalc allows you to work out the average chance of success (with the obvious caveat that it is extrapolating from historic – and US – returns). Very useful and interesting resource.

  • 18 rajkanwar October 15, 2014, 3:44 pm

    I wonder whether there is a any theoretical work underpinning the folly of trying to pick future wining funds.

    Beyond’s Heisenberg’s uncertainty principle off course.

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