**The Shape of Returns to Come **

*How volatility compounding changes the “shape” of long-term returns – and why shape may matter more than anything*

*How volatility compounding changes the “shape” of long-term returns – and why shape may matter more than anything*

*NOTE: My last piece, *__The Volatility Thief__, received thousands of views. I want to thank all of you for taking the time to read it. A few people asked if there was some more general theory or approach that need not assume binary return outcomes. Indeed, there is. This paper uses more realistic models of returns, and shows how volatility still saps returns no matter how you look at it. It’s a little longer read, but not a hard one (though there are puzzles and inconsistencies to scratch your head over.) If you enjoy this piece, pass it on. Also, sign up for the mailing list at www.LeeKranefuss.com so I can notify you of new posts, and progress on the book and software package I am developing for launch early this year. Don’t worry: I won’t clutter your inbox. I’ll only send you email when I have something new to say, which isn’t that often.

*NOTE: My last piece,*

__The Volatility Thief__, received thousands of views. I want to thank all of you for taking the time to read it. A few people asked if there was some more general theory or approach that need not assume binary return outcomes. Indeed, there is. This paper uses more realistic models of returns, and shows how volatility still saps returns no matter how you look at it. It’s a little longer read, but not a hard one (though there are puzzles and inconsistencies to scratch your head over.) If you enjoy this piece, pass it on. Also, sign up for the mailing list at www.LeeKranefuss.com so I can notify you of new posts, and progress on the book and software package I am developing for launch early this year. Don’t worry: I won’t clutter your inbox. I’ll only send you email when I have something new to say, which isn’t that often

There’s an old statistics joke. Nine people with no savings just lost their jobs at the same time and are in a bar drowning their sorrows. Then a billionaire stops in for a drink. Luckily for them, their problems are gone – since everyone in the bar now has an average wealth of $100 million. (I said it was a statistics joke, not a funny one.)

What is obviously wrong here is that the *average* is very different from what is *most likely*. For many people, most of their experience with uncertainty doesn’t recognize this distinction: the *average* and the *most likely* are the same (or close enough not to worry about).[2] But any logically consistent approach to markets and investing *guarantees* these will be different numbers. So we need very much to think about this difference, which can be seen in what I will call the “*shape*” of returns.

Why does shape matter? Quite simply, just like in the bar, it means that the *average return* (or *expected return*) will often be different from the *most likely return*. In the end, we must plan to live with the most likely outcome rather than the average (especially if what is most likely is smaller). And here is a spoiler alert to motivate you to keep reading until the end: *even an investment that has positive average or expected returns can end up with a near-certain loss unless its returns are high enough to offset the drag from volatility compounding*.

**Why Shape Matters**

Very often investment prognostication is framed only around *expected returns*. Most pundits make what are called point estimates – statements like “I expect the Dow to go back above 26,800 (where it stood in October) by March 1st”, or “I believe the Fed will raise short rates by another 0.50% in 2019”.

Some might even provide a sense of their uncertainty – i.e., should we expect drops and gains in the Dow of hundreds of points daily, or something less chaotic? Thus, in addition to returns we might get a “soft” point estimate of *volatility* so we know “how bumpy a ride” to expect.

But what we really do when investing is look at the range of possible outcomes over years and decades. And what gets left out is that even if we agreed entirely on the expected returns (unlikely, though possible), the *shape* of those outcomes matters more – especially if it makes the most likely outcome very different than the expected or average one.

And here is a bit of a surprise: it turns out that the critical ingredient driving this spread between* expected returns* and* most likely returns* is volatility itself – the variability in returns. This additional impact of volatility – above and beyond giving us an idea of how bumpy returns might be – doesn’t get talked about nearly enough. Or, for that matter, much at all.

**A Simple Example**

Suppose we have an index of the market that is currently at 1,000, and that everyone agrees that over the next year the expected (or average) return will be 10%, bringing the index, on average, to 1,100.

But that belief in a 10% average return could itself take different forms. One might be quite focused, delivering an average 10% like so (which has an annual volatility of about 5%):

However – at least as I write this – we live in volatile times: not many days ago the Dow dropped 653 points, and the next day gained over 1,000 points (its largest point gain in history). 5% is awfully low (around 17% has been the volatility of the S&P 500 over the past 50 years or so) so here is another forecast – which also has an average return of 10%, but an annual volatility of around 10%:

But now consider a third option that has a very strange “shape”. It has only two possible outcomes: an 80% chance of losing all your investment, with a 20% chance of it becoming 5.5 times larger in a single year:

I submit to you that if you have saved and invested well for decades, and have $1 million to invest, you are likely to do quite different things based on these three forecasts or investments.

If the shape is as in figure 1, you are more likely to go “all-in” since there is a very high chance of a positive return somewhere near 10%, with only a very small chance of suffering any loss at all of your capital over the next year (about a one-in-forty chance, to be exact). But it also has limited opportunity to have a great upside surprise of, say, 20% gains (also about one-in-forty odds).

Figure #2 has more volatility. Your chances of getting over 20% returns are now about a one-in-six chance: the larger volatility gives you more upside. But that upside comes with a price, naturally, because you also run about a one-in-six chance of a negative return next year. With greater possible rewards comes greater possible risks. Some people would be willing to take that additional risk, and some would not.

Example #3 is the oddball in the group. Here you run an 80% chance of losing all of your investment next year – turning you $1 million into nothing. On the other hand, you have a 20% chance of seeing it rise to $5.5 million.

If we used expected returns alone to pick investments this option is the same as the prior two, since this investment has an average 10% return as well. But for most people this raffle ticket-like payout is simply too risky. Yes, the upside is phenomenal. But *the most likely thing to happen, by far, is losing all of your money*. It happens 80% of the time. Although it has the same average return (10%), it is nothing like the #1 or #2.

Now, you may say that #3 is unrealistic. Indeed it is contrived. But it was designed to show how the *average* (or *expected*) return of three investments could be identical, but the *most likely *thing to happen – which we can see from the shapes – could be radically different from the average. Soon we will see how “normal” returns can compound to something quite similar when no one is looking.

**Volatility Isn’t Just the ‘Bumpiness” Measure: It is what Drives Shape when Compounding**

So, what drives the shape of expected returns in the real world?

We are all taught volatility is a measure of the “bumpiness” of the ride in the markets. It is. In fact, some of you out there no doubt already surmised that investment #1 is the superior one based on finance theory, since its expected return to volatility ratio is the highest.

But what seldom gets discussed or covered – even in fairly advanced discussions of investing – is that volatility is more than just a measure of the bumpiness of the ride. In fact, volatility has another effect: volatility is what “shapes” compounded returns. Surprisingly, volatility can turn ordinary-looking return forecasts (like #1 and #2 above) into something more like the raffle ticket of investment #3 above when compounded over time. But very few people talk about volatility compounding and its effects.

Why these things are true takes some work to explain, but it can be shown by a simplified example quite easily. After the illustration that comes next, I hope you never look at volatility the same way again. Or return, either.

**Looking Backwards through the Telescope**

To make things very simple, let’s assume that the expected return of our investment choices every year are all exactly* zero*. (This simplification is only for convenience; the results will be similar even with positive average returns – but assuming zero makes the moving parts easier to spot).

Also, let’s assume all of our daily investment returns start off with the same sort of bell-curve shape we are all used to, but some are more volatile than others. Were that the case we might see three similar return profiles that look like so:

All of these daily return profiles have the same average return (zero), but they have, respectively, 5%, 10%, and 20% annual volatility (shown in blue, orange, and green, respectively).

Now let’s look at 20 years of compounded results based on investing with these initial shapes.

Since we expect zero average return – neither a gain or loss on average every year – you might guess that if we start with $1 million it is reasonable to assume we will see $1 million, on average, as the long-term value – as if the money were stuffed into a mattress (but with some ups and downs). Indeed, this is correct.

But something strange happens when you compound volatility over the years. If you look at the next graph of all three distributions after 20 years, you can see that the *most likely* result – the peak of the graph – is *always* less than $1 million. And the higher the volatility, the further below $1 million is that peak. Here is the shape of the three investments after 20 years, with 5% volatility shown in blue, 10% volatility shown in orange, and 20% volatility shown in green:

In addition, there is a red vertical bar at the expected value for all three – which is (as noted before) the $1 million initial value: with zero average returns, we don’t expect to gain or lose money over the long-run.

But clearly, the more the volatility,* the greater the chances that the most likely outcome, starting with $1 million, will be a loss of capital. * The impact is nearly negligible for 5% volatility (in blue). But, for the 20% volatility option, the odds are well over 50% you will lose at least half your original $1 million in 20 years.

Quite simply, volatility drives the “shape” of outcomes, and more volatility makes the most likely outcome much less than the average.

How big can this effect get? We can look at this another way to see it. Let’s now take option C – the one with 20% annual volatility – and see what the pattern of returns looks like if we invest in it for 10, 20, and 30 years (with the colors being blue, orange, and green, respectively):

With zero return and 20% volatility, after 10 years the most likely outcome is about a loss of 50%. At 20 years the most likely outcome is around a loss of 75%. And, over 30 years we are *almost guaranteed* that the most likely outcome is a near total-loss.

As we can see, higher volatility and longer investment horizon leads to an outcome that is an ever-growing loss based on what is most likely. Volatility makes the most likely outcome over time less than the average (or expected) outcome.

Yet, most investors consider only *expected* *returns*, not ever thinking about this shape-morphing effect from volatility. This is a serious problem, since the most likely outcome can end up being quite a bit less than the average. This outcome can, in fact, turn any investment into one that starts looking like a raffle ticket.

**What’s Going on Here**

As *Alice* said in *Wonderland*, “things keep getting curiouser and curiouser.” Which brings me to my main point.

All of these graphs have an average expected value in all years in the future of $1 million, because they have an average annual return of 0%. But with zero average returns, volatility, coupled with enough time, means we are guaranteed to have the *most likely* outcome become losing * all* of our initial investment – and the higher the volatility or the longer we go the more rapidly we get close to this result.

So, if the most likely outcome is losing more and more money over time until eventually it is all gone, how is it that the expected (or average) value is always $1 million?

To explain what seems like a logical impossibility you need to look at the shapes very carefully. What is not at first obvious are some not-so-easy-to-see “raffle ticket” payoffs – much like our initial example #3 – that are also being generated through compounding.

Look carefully at the first graph of returns above (the one showing 20 years for each investment plotted together). If you look very carefully at option C – which has the highest volatility – you will see that somewhere just above $2 million it becomes the top line, with the blue line already very near zero and the orange line headed there fast (here it is again, for reference, with option C in green):

it has the largest chance of delivering a good payoff, say, $4 million after 20 years – even with zero yearly average returns – simply from possible random luck. If we extended the graph, we would see that it could even pay off $10 million, or $20 million – or more – with much higher odds than the other investments. All of these spectacular payoffs are more likely for the 20% volatility investment (the green line) than for the lower volatility ones.

The problem is these big payoffs are *not very likely at all*. But, much like our raffle ticket earlier, these very low-probability but high value outcomes keep the average value constantly at $1 million – even though we will over time, with certainty, be guaranteed to lose virtually *all *of our initial $1 million investment with any of these investments. The average always stays at $1 million, but the most likely outcome keeps dropping.

Sounds confusing? Think of it this way, if it helps. The average lifespan for someone in the U.S. is around their mid-80s. Suppose someone comes up with an elixir with which we can inoculate babies, and one-in-a million will stop aging at age 60 and live to be 100 million years old. However, we can’t tell who will respond until they hit age 60 and stop aging.

Were this true, the expected lifespan would suddenly increase to about 185 years of life. But for 99.9999% of people this average would be irrelevant: nothing has changed for them, and their most likely lifespan is still around 80. Clearly most people need to plan around the *most likely* outcome, not the *average* one. (In fact, if you think about it, no one needs to plan around the average result in this life-extension story).

So what should an investor focus on? In the end, you need to weight what is *likely* to happen much more than an abstract, mathematical “expected” or “average” outcome. Because, what is most likely is, quite simply, what is most likely – and is not always the average or expected result.

Yet few investment tools give you this option, and few investors ever consider this difference between what is the most likely return over time versus what is the average or expected return.

**Summary**

We usually think of volatility as measuring only how bumpy or wild a ride we might get from an investment. Indeed, it does this. But we seldom, if ever, consider how volatility shapes the possible future outcomes. But it does. Very strongly. And, in the end, you must focus on the *shape* of forecasted results, and what is most likely.

This is a lot to ponder, but I will leave you with two observations.

** First**, this is why stocks must, over time, deliver returns greater than zero (and greater than bonds, by a healthy enough margin). Remember, for simplicity we assumed zero average annual returns in the example above. This let us see the “drag” from volatility compounding clearly.

In the real world the stock market has some expected positive return every year. It still suffers volatility drag, but one applied against growth. But for stocks it is a perpetual battle: stocks must generate enough average annual return over the long-run to overcome the perpetual decay that volatility compounding causes. Otherwise, no one would be able to profit from investing in stock due to how volatility drives the shape of returns over time. (And, as you might imagine by now, that hurdle is a pretty big one – but that is the topic for another post.)

** Second**, it goes to show you the importance not just of getting good returns, but also of reducing volatility – something that gets far too little attention.

One of the most tempting things for most investors is to chase returns – through buying “hot stocks”, or trying to find new and more aggressive managers, getting into hedge funds, etc. But looking at average or expected returns alone without accounting for the shape-shifting effects of volatility can lure someone into an investment strategy that looks promising – but becomes guaranteed to lose money over time as the most likely outcome.

In fact, quite often investors make poor choices here, not even aware that the increased volatility they are taking on with some investments might mean the most likely outcome is that they are doomed to a high chance of losing money if they stay invested long enough – even with what seem like high expected returns. The simplest way for this to happen is to place money with a manager who has had a run of high returns but with a high volatility strategy. Now, if his or her hand “goes cold” – and the returns drop closer to market levels, but the volatility doesn’t – there may not be enough returns to overcome the high volatility.

In the end, we don’t know what the future ever holds for stocks. So we have to develop assumptions and models. To do that right we need to look at the right estimates in making plans – which usually means focusing more on the *most likely* outcome than the *average* one. Unfortunately, this is seldom done right in today’s world with today’s tools.

*Opinions expressed are current opinions as of the date appearing in this material only. While the data contained herein has been prepared from information that The Kranefuss Group believes to be reliable, The Kranefuss Group does not warrant the accuracy or completeness of such information. This communication is for informational purposes only. This is not intended as nor is it an offer, or solicitation of any offer to buy or sell any security, investment or product.*

*Copyright © 2019, The Kranefuss Group and/or its affiliates. All rights reserved.*

*[1]** Lee Kranefuss is CEO and Founder of The Kranefuss Group LLC.*

[2] If you have ever taken an introductory statistics course they probably assumed this all the time, even when it wasn’t true, to avoid issues like ones in this paper.