Mathematic formulas, Photo by Antoine Dautry on Unsplash

3 Maths Mistakes That Drive Me Nuts

When you’re messing around with statistics, it’s all too easy to come up with something that’s horrendously misleading. Investing stats are no different. Here are three that really set my teeth on edge.

In truth, with all these three, the maths is actually correct. Technically anyway. And the accompanying messages are often made with the best intentions. It’s the interpretation and the conclusions drawn that I take issue with.

#1 Missing out on the best days

Over the ten years to 31 August 2018 (from 15 September 2008), the FTSE All-Share produced a total return of 122%. Had the ten best days been missed, the return would have fallen to 21%. Missing the best 20 days would have resulted in a negative return of -15%.

The ‘best days’ argument is often trotted out as a good reason for staying invested all the time. I’ve no issue with staying invested, no sirree. But the likelihood of any person missing out on just these best days seems vanishingly small.

Looking at data back to 1994, there are 19 days where the market has gained by 3% or more.

The 3 best days were 19 September 2008 (8.5%), 10 May 2010 (5.2%), and 24 January 2008 (4.6%). The other 16 fell between 3% and 4%.

Overall, there are two days in 1998, one in 2007, six in 2008, two in 2010, four in 2011, and two each in 2015 and 2016. When there is more than one in any particular year, they are often just a few days or weeks apart.

Now the likelihood of anyone just selling out for just 10 days over the course of a decade and then buying back in the day after is absolutely miniscule.

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Admittedly, at this point, my ‘A’ level Maths gets a bit shaky. I reckon the chances of anyone choosing just those 10 worst days are somewhere in the region of 4,000,000,000,000,000,000,000,000,000 to 1. Even if my maths is off considerably, those seem like rather long odds.

What’s more, a lot of these best days actually come in bear markets. If you missed out on a few more days either side, you probably missed out on some big down days as well.

Staying invested is definitely a good thing in my opinion. But the fear of missing out on the elusive best days is not the reason why.

#2 Dividends account for almost all equity returns

The Wharton professor Jeremy Siegel calculated in 2005 that, over the previous 130 years, 97% of the total return from stocks came from re-invested dividends. $1,000 invested in 1871 would have been worth $243,386 by 2003. Had dividends been reinvested, the figure rises to $7,947,930!

The numbers get a bit easier for this one.  Yes, $243,386 represents just 3% of $7,947,930, but hang on a minute:

  • Price rises get you from $1,000 to $243,386. That’s a gain of 243 times.
  • Dividends get you from $243,386 to $7,947,930. That’s a gain of about 33 times.

So, you could say price rises actually account for a much greater proportion of the gains.

I’d go even further though. Going from $1,000 to nearly $8m sounds impressive, but it actually represents an average annual gain of 7.15% over those 130 years. Break that up into price gains and reinvested dividends and you get 4.32% and 2.83% respectively.

Therefore, I think you should say that price gains account for around 60% of total returns (being 4.32/7.15) and reinvested dividends the other 40%. In the UK, I think it’s a similar ratio of 55% and 45%.

There’s a variation of this which looks at real returns, i.e. those after the effects of inflation. Here, you’ll often find that inflation (around 4% a year typically) is taken off stock price gains while the returns from dividends are left untouched. Miraculously dividends then account for pretty much all returns once again!

Don’t get me wrong. Dividends and their reinvestment are very important if you want to harness the gains that the stock market has to offer. They are not the entire shebang, though.

#3 Down 50%, up 100% to break-even

Losses really hurt you. Don’t you know that if a stock falls 50%, then it has to rise 100% just to get you back to where you started from!

Again, this is true. The implication, as I read it, is that a gain of 100% is twice as unlikely as a loss of 50%. Not so true.

I reckon we’re better off considering this as halving and doubling. So, if something halves it then has to double to get back to where it started from. Look at this way, and it’s easier to see that these two things are just opposites.

The fact that one percentage is twice as high as the other merely reflects the different starting point for that calculation. It doesn’t reflect what is an appropriate valuation for the company in question.

Now, a 50% loss is certainly serious and not something anyone wants to experience. But decent investments often recover, and kneejerk selling reactions every time you experience any significant loss rarely work I have found.

Maybe I am wrong as well

I reckon my maths is decent, but it’s been a long time since I was taught this subject at school.

It’s entirely possible that I’m thinking about these things incorrectly myself. If you do think I’m barking up the wrong tree with any of them, please let me know in the comments section so I can look less of an idiot learn more.

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2 Replies to “3 Maths Mistakes That Drive Me Nuts”

  1. “Had the ten best days been missed …”: and had the ten worst days been missed?

    Is it easier to predict the best days or the worst days?

    Best days (I have read) usually correspond to announcements of macro-economic policy. How about worst?

  2. Interesting question dearieme. I don’t have the data to hand right now, but I seem to recall the big down days were largely in the depths of bear markets as well. They were of the same sort of order, so missing the 10 worst might have added 100% to your returns in the same way as missing the 10 best knocked 100% or so off.

    It’s nothing something I have looked at it in any detail but I suspect many hedge funds have algorithms looking for this sort of stuff. The problem, I suspect, is that you would get a lot of false signals and the costs of trading in and out on these would eat up any gains you made by missing out the worst days.

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