Gareth’s Big Mistake: Distributions vs. Averages

Gareth is 50 years old. Retiring in 15 years time, he realises that he needs roughly £1.5M to live comfortably in his retirement. If he wants the Mediterranean cruises, frequent golf outings, and gifts for grandchildren, this is the total sum he must accumulate.

Gareth is in a bit of a pickle. He only has £500,000 in his investment account. He calculates he will need a fairly aggressive 7.5% rate of return to meet his goal. Maybe he should have saved more earlier in life. Maybe he should have listened to that financial adviser who told him to do so (but the man was only retuning 4% when all his friends were bragging about double-digit returns- he clearly knew nothing).

Disregarding the recommendations of his current, new, financial adviser, Gareth comes up with a plan. “If I can just make a 7.5% return”, he thinks, “then I can get to my goal! What am I paying this financial adviser for? This investing business is easy!”. So it would seem. He selects a few assets (he has heard vaguely about the benefits of diversification) each with average historical returns of over 7.5% and commits his capital.

What could go wrong?

Gareth’s big mistake

This is an incredibly naive approach, not uncommon among retail investors. On a personal level, I have seen friends think and invest in this way (I have given up trying to convince them otherwise). There are countless things wrong with the way Gareth is thinking and his subsequent actions. I won’t get into most of them here (see notes) but I will talk about what, to me, is his most egregious error.

Gareth neglects the distribution of possible outcomes and only considers single-point estimates.

 

Now, this approach can be valid with certain distributions and in certain situations. For example, looking at the normal distribution, we can see that the mean is representative of the bulk of the distribution. Single-point estimators can be useful. But the fact remains that when you reduce the distribution of possible outcomes to a single number, you lose information.

Sometimes lots of information.

For example, looking at averages specifically, if a distribution exhibits a significant degree of kurtosis then the bulk of the distribution may actually lie away from the mean.

                PDF of normal distribution. Red line is the mean.

          PDF of lognormal distribution. Red line is the mean.

  

Hence, the mean will be misleading as most of the observations you are likely to see will be far away from this mean.

If I draw 10 balls out of a bag and the first 9 have the value 1 but the last ball has the value of 10 then the average of the balls is 1.9. Investors in Ball Bag Ltd. think they’re getting a value of 1.9 when they invest but, in reality, they’re probably going to get 1.

Looking at probability distributions, glancing at the lognormal distribution we see that the mean is not representative of the bulk of the distribution. Single-point estimators can be misleading.

Consequences for poor Gareth

What is one to do? How should Gareth act with this new information in hand?

He should take all the “possible versions of history” into account. He must focus on the distributions of outcomes, rather than just, in this case, the average. This is the main point that Sam Savage makes in his book The Flaw of Averages. Simply put, one must consider the full distribution of potential returns, rather than focusing on just the average. Looking at just the average return of an asset or group of assets is an egregious error. The primary and most basic error retail investors make.

But this is easier said than done.  Firstly, it’s hard to conceptualise alternative versions of history. This is the point Nassim Taleb makes in Fooled by Randomness. It’s just not natural to think of what might happen or what might have happened. We only see causes and effects as if no other version of reality was possible. As if no randomness existed.

Secondly, the distribution may not be always easily identifiable. We can have a good guess but we will always be stuck with that age-old problem: the problem of induction. Whenever we use empirical analysis for discovery rather than disconfirmation, we find ourselves on shaky ground. In most cases, we can only give our best estimate and acknowledge that we could very well be wrong. Still, this is a whole lot better than taking a simple historical average rate of return. This, at least, I can say with confidence.

Notes

Gareth’s other major mistakes

  1. Over-reliance on inductive reasoning. The market is a continual process, not a series of experiments in a lab. This is talked about in more detail here.

  2. Not taking time into account. He will have to liquidate these investments in 15 years time. The price could be anywhere at this point.

  3. Not factoring-in correlation. Although the concept of some intrinsic correlation existing between assets is flawed, one must account for the possibility that assets can move together. Hence, whilst Gareth thinks he is sufficiently diversified, he may not be.

  4. The implied use of “required risk”. The concept that you should “adjust your risk to your goals” is completely backwards and deeply dangerous.

  5. Not listening to financial advisers. I have my issues with them but they generally know a little more than the average retail investor about lifetime wealth accumulation. Probably best to listen to them in most instances.


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