*Does time increase or decrease the risk?*

An interesting and often misunderstood issue came up in Dr Stefan Lundberg’s interviews with Professor Zvi Bodie and me in his podcast Pioneering Pensions. It relates to the question of whether equity risk increases or decreases with an increase in the future time horizon you are considering. Zvi and I are both convinced that it increases. But I understand that most people think otherwise, that equity risk decreases as you lengthen the time horizon. In fact there’s a common name for this belief: it’s called “time diversification,” implying that time diversifies away (and therefore reduces) some of the risk of equities.

Let me explain my view. I’ll define risk, show that it increases with time, and then explain the fallacy underlying the common view.

Finally, I’ll get to the obvious follow-up question: what should you do?

Warning: Some of this is technical stuff, not plain common sense. Go to the final section for the practical implications.

***

I’ll start by defining risk. It’s the uncertainty of outcomes over the time horizon you have in mind. The bigger this range of possibilities, the bigger the risk. More accurately, you might use another measure of the range, such as the standard deviation, which takes into account not only the range but also the probabilities of different outcomes. Or even more complex formulae that consider both the outcomes and their likelihood.

No matter which measure you use, the risk increases as the time horizon increases.

A numerical (and unrealistically simplified) example will illustrate this fact.

Suppose you have $100 in equities.

If equities always return 10% a year, then after 1 year you have $110. After 20 years you have $673. (Hey, that’s the power of compounding! It has added 573% to the original $100.)

If equities always return -10% a year (that is, they lose 10% of their value each year), then after 1 year you have $90. After 20 years you have $12 (so you’ve lost 88% of your original value).

The one-year range of outcomes is from $110 to $90, that is, a range of $20.

The 20-year range of outcomes is from $673 to $12, that is, a range of $661.

It seems clear that the range has increased enormously as we’ve moved from 1 year to 20 years.

Case closed.

I could make it much more complicated, with all kinds of rates of return in the middle years and all kinds of variations, but you’ll find that the range increases with time regardless. And so too for other measures of risk like the standard deviation, which takes into account the fact that, with many possible outcomes each year rather than just +10% or -10%, extreme outcomes are less likely than middle-of-the-road outcomes. But whatever measure of risk you use, it’s bigger after 20 years than after 1 year.

***

So, why do people generally believe that time diversifies away part of the risk?

Well, notice one thing that hasn’t increased, in my artificial example: the range in the *average annual* returns.

In the one-year case as well as the 20-year case, the *average annual* return is either +10% or -10% per annum, so that range hasn’t changed.

What does change is the *total* return.

In the 20-year case, the *total* return ranges from 573% to -88%, as we’ve seen.

Suppose, instead, in the 20-year case there had been other returns in the various years, and that the *total* return had ranged between smaller amounts, say between 400% and -50%. So the ending dollar amounts would have been $500 and $50. (Just add the $100 starting base to the total returns.)

Well, that doesn’t change the fact that the risk is much greater for this second 20-year period than in the 1-year period.

But wait – suppose we now calculate the *average annual* returns that got us to $500 and $50 over 20 years. Answers: to get to $500, the average annual return is 8.4%; and to get to $50, the average annual return is -3.4%. Oh look, the difference between the *average annual* returns is now 11.8%, which is much less than the original range of 20%.

So, if you choose to measure risk by that strange measure, the average annual return (which removes the power of compounding), it appears that the risk decreased as we took the time horizon from 1 year to 20 years.

***

In a nutshell, that’s what misleads so many people.

They get so used to seeing average annual returns reported to them, rather than the total return over long time periods, that it seems to them that there’s a convergence of returns as the time horizon lengthens. There *isn’t* a convergence if what you look at is *total* returns, but there *is* a convergence of *average annual* returns. And that makes people feel (incorrectly) that their risk is decreasing over time.

Anything that reduces the range of outcomes is rightly considered to be a diversifying factor. But that’s the full range of outcomes, not the range of a derived measure that averages over time and suppresses the effect of compounding.

***

There may be another reason why intuition misleads, and Zvi pointed this out to me.

People say: What happens to the probability that there’s a shortfall, relative to some low level for the market that you fear? As the time horizon lengthens, that probability falls. Yes, it’s true. And if you measure risk purely by that probability, then it appears that risk falls as time increases. (This, Zvi reminds me, is the way that so-called “Monte Carlo simulations” often measure risk – just using the probability.)

But wait! What also happens is that the *size* of the potential shortfall increases. In fact, the size increases faster than the probability decreases. So, while there’s less chance that something extremely bad will happen, it’s not enough to make up for the size of the potential bad news.

***

I’ve saved the best argument for last. And this time it is simply common sense. Zvi has the killer argument.

If the risk goes away in the long run, then the risk premium has to go away. Otherwise there is an arbitrage situation. In other words, why would you get extra money for taking no risk?

***

Stefan adds another thought. The reason why so many people believe in the time-diversification of returns is that it’s a convenient view of the world that allows them to follow a recipe without having to think or take decisions. It makes it easy to sleep at night, when you have that sort of strong belief.

Nobel Prize winner Paul Samuelson wrote a piece1 on this subject in 1997 in which he said exactly that: “It makes for good sleep at night and comfortable retirement living … [But if the whole world believes this dogma] only the Tooth Fairy can then fulfill your dreams.” And actually in that piece Samuelson specifically cites Zvi in connection with the pricing of portfolio insurance to demonstrate that, as the time horizon lengthens, it’s necessary for the insurance premium to rise, for protection against a return lower than the risk-free rate.

Which is the very point that Zvi makes in his killer argument.

***

And now the obvious follow-up question. Given that risk increases as time increases, what should you do?

Simple (but not easy) answer: Do whatever suits your attitude to taking investment risk. (The jargon expression is “your risk tolerance.”) In other words, there’s no right or wrong answer. There’s only an informed attitude as relevant input.

In fact Stefan wrote an article on this after he interviewed Zvi and me. He very generously called it “Two Great Minds,” and his theme is that, though Zvi and I are in agreement that equity risk increases with the time horizon, we take different paths with our personal investment approaches and in our recommendations. As Stefan expresses it: “To deal with uncertainty, Zvi prefers buying protection. Don advocates adapting as the main approach for dealing with uncertainty. It is a traditional trade-off between earning risk premium and tail-risk protection.”

He asks the reader: Are you a Don or a Zvi? And he answers the question this way: “The conclusion is that each of us must think about how to manage the trade-off between earning risk premium and dealing with tail risks. Personally, I am more of a Don than a Zvi when it comes to dealing with uncertainty, so for me the trade-off is relatively straightforward. But this is based on my personal preferences. So while it is right for me, you might come to another conclusion based on your own situation.”

I would add that my approach of adapting requires action from time to time. If risk starts to materialize in the short term (meaning low or negative equity returns), I reduce the withdrawal from our pension pot, so that I’m cutting off some of the further risk that comes with the time horizon. If instead reward, in the form of a high return, comes early (as in 2021), I take advantage of the situation to take some future risk exposure off the table. (These actions are explained in my previous blog post.) In both cases, I am systematically cutting down the expansion of the naturally increasing risk exposure.

Two hackneyed phrases sum this up. One is that knowing is not the same as doing. The other is that mine is not a “set it and forget it” approach. Both phrases are aimed at showing that monitoring and action are essential ingredients.

***

## Takeaway

*Artificial or incomplete measures of risk are what cause many people to wrongly believe that time diversifies their equity risk exposure. Once you understand that the risk increases with time, what do you depends on your risk tolerance.*

1 Samuelson, Paul A.: Dogma of the Day: Invest for the Long Term, the Theory Goes, and the Risk Lessens, *Bloomberg Personal Finance*, January/February 1997, pp 33-34.

### 4 Comments

I have written about retirement planning before and some of that material also relates to topics or issues that are being discussed here. Where relevant I draw on material from three sources: The Retirement Plan Solution (co-authored with Bob Collie and Matt Smith, published by John Wiley & Sons, Inc., 2009), my foreword to Someday Rich (by Timothy Noonan and Matt Smith, also published by Wiley, 2012), and my occasional column The Art of Investment in the FT Money supplement of The Financial Times, published in the UK. I am grateful to the other authors and to The Financial Times for permission to use the material here.

Interestingly enough, dismissing the “killer argument” is exactly the error that Kevin Hassett and James Glassman made in “Dow 36,000” around 25 years ago – they argued that stocks should command no risk premium, as they had not underperformed bonds over long periods. Markets have since put the theory to rest, but the error crops up repeatedly, and in different guises.

You have a wonderful memory! To remind myself of the main themes of that book I just checked it on Amazon, which carries these bullet points: “Why the financial establishment is wrong” and “Why stocks are actually less risky than bonds.” What stuns me is a WSJ quote from Burton Malkiel: “… a provocative and well-written treatise that cannot be dismissed …” Thanks for this reminder!

Looks like you are heading towards ergodicity, Don?

We have this issue at the intersection of climate science and economics. My narrative there is only half-formed, for now!

Best, Mike

Gosh, I had to look up the word! Yes, I think so, but not consciously. I’ll look forward to the time when your narrative is complete!