You are currently viewing Insights – Convex Returns: The Key to a Diversified Portfolio

Insights – Convex Returns: The Key to a Diversified Portfolio

Table of contents

  • Return distribution of S&P 500 and identifying the fat left tail
  • Definition of convex returns
  • Analysis of diversification benefits from assets with convex return relationships
  • Summary and conclusion for including assets with convex diversification profiles in an All Seasons Portfolio

A short version of this post was originally shared on my eToro feed in July 2023. Make sure to follow me there as well and to allocate a part of your portfolio to copytrade my All Seasons Portfolio that I share on eToro? Create an account today, copy my portfolio by searching for user “Allseasonsport” to automatically and effortlessly copy my All Seasons Portfolio strategy.

Diversification forms a fundamental part of portfolio construction. Especially portfolios that follow a framework of allocating risk to each of the four macroeconomic environments (made up of rising/declining economic growth on the one hand and rising/declining inflation on the other hand), diligently carrying out the diversification is key to ensure the portfolio is well balanced.

At its core, diversified portfolios gift the investor smoothed returns by cutting off the largest peaks and troughs for a more stable growth of capital. While you in the short term may give away some seemingly hyperbolic (and lotter ticket-like) returns from isolated financial assets, you will gain in the long term by ensuring your wealth grows in a more stable manner.

Namely, diversification significantly reduces the risks of deep drawdowns. If your portfolio’s value declines by 50%, you will need 100% return to break even. That is a challenging spot to be put in, as it takes about 10.25 years to achieve a doubling of your wealth if you manage to compound at an average rate of 7% per year (which is the stock market’s average annual return over the long term). You are thus wise to avoid such deep drawdowns in the first place by diversifying.


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Another way to put it is that most people are (whether they admit it or not) risk averse. In the choice between receiving a certain sum of $50,000 dollars (no strings attached) on the one hand or a 50% chance of $25,000 and a 50% chance of $75,000 on the other hand, most will chose the sure $50,000 over the coin toss. A loss feels worse than a profit of an equal amount feels good, which has been studied extensively in the field of behavioral finance. Therefore, one of the most important characteristics of portfolio management is drawdown control.

Diversification is however, not just about adding any asset to the portfolio and then calling it a day. Rather, it is about adding assets which have a low or negative correlation to the assets already included in the portfolio. There’s a huge difference between a portfolio with four different stocks and one with four different asset classes.

It is important to ensure that there is always one sleeve of the portfolio that does well when other sleeves do poorly. True diversification also means that there will always be (at least) one line item that underperforms at any given time, but looking at the portfolio holistically and not focusing on each individual asset, we will benefit from much more stable returns over time.

While we have been covering the benefits of diversification many times on this blog before, we will today be looking at it from another perspective, namely by illustrating the convexity of cross-asset return relationships and how this benefits a diversified portfolio. And don’t worry if that description is a mouthful, it will become clear as we go along.

In short, what we want to achieve with diversification is thus to identify and add assets to the portfolio that outperform when other assets underperform.

Return distribution of S&P 500 and tail risk

To keep things as simple and easy to grasp as possible, we will today not cover the whole spectrum of asset classes that are traditionally contained in an All Seasons Portfolio. Instead, we will use just two major asset classes to illustrate the diversification effect of uncorrelated assets.

Let us begin by introducing our first major asset class: stocks – or more specifically the S&P 500 index which tracks the performance of the largest companies in the United States. Stock market exposure is also one of the most common investments among retail investors, and it therefore poses a good starting point for our discussion about diversification.

As you will see from the below (regular and logarithmic) charts, there have been several occasions when the S&P 500 has declined sharply in the last 60 years, and with several lengthy periods where the drawdowns from the previous all time highs have lasted for several years. (Note, though, that these charts are not including dividends, as the S&P 500 is not a “total return index”, or trading fees, why the actual investor returns could differ).

Another way of illustrating the performance of the S&P 500 index, which adds a practical perspective over the performance of the investment, is by looking at how the index performs over shorter periods of time. What we mean here is that while the long-term average annual return for the stock market is about 7%, that has only been true for the last ~110 years. However, in practice, retail investors have much shorter investment horizons than that (20-40 years at best), while our psychological horizon before we feel a stomach ache is even shorter. For the latter, it is enough that we see our returns be down over the last 3 months or so before we are feeling poorer.

Therefore, here below, you find a distribution of the S&P 500’s returns over all rolling 3-month periods from 1981 to 2023. Each bin is 0.25% wide, meaning that for example for the 10% bin, we count all instances when the S&P 500 has returned -10% ±0.125% over a 3-month period (which was 22 times out of all ~11,000 observations).

There is one particular thing I would like to draw your attention to. From the chart, we see that it is not a symmetric distribution. Actually, the mean 3-month return was 2.37% and the median 2.97%. This means that all in all, we can expect positive returns from stocks, but because the median is larger than the mean, the distribution is negatively skewed, i.e. that the S&P 500 has a fat left tail. This is the area marked in red, where we find that there is a large amount of rolling 3-month periods with negative returns of up to -30%. Even though most returns are clustered around the mean, we as investors really want to avoid experiencing the returns in the red box.

Rolling 3-month returns of S&P 500 from 1981 to 2023, with bin size of 0.25% (e.g. 10% +/- 0.125%). (Click to enlarge)

Diversification benefit from assets with convex return relationships

So how do we make sure that our portfolio does not decline by up to 30% in such a short period as only 3 months if we invest in stocks? Considering that this is an article about diversification, it might not have been too difficult to answer that rhetorical question…

But to spell it out: you counter the left tail risk in stocks by adding assets to the portfolio which have low correlation to stocks.

The next piece of the puzzle is to identify assets which are actual diversifiers that make a difference. Typically, these are assets that have low or negative correlation with the other portfolio assets (the S&P 500 in our example).


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While this article is not about gold per se, we will anyway use gold as an example for an asset that can act as a counterweight to stocks.

Let us return to the chart set from further up, but now adding the line for gold (note that both have been indexed to 100 per 22 March 1968, which is the start date for gold in the data set I found when researching this topic). Note, though, that until 1971, the gold price in USD was pegged, so that it is only in the years after the end of the Bretton Woods system that gold as an asset has been able to live out its life as an uncorrelated diversifier.

At first glance, it is quite obvious how stocks and gold have moved independently from each other over the whole, even though both have seen almost identical returns over the last 55 years (again, excluding dividends and trading costs, etc.).

Diving into the diversification aspect of including gold in a stock portfolio, we will continue to use rolling 3-month returns for our analysis.

As er can see from the blow scatter plot of R-3M returns, we see that the relationship between stocks (x axis) and gold (y axis) is very low. In fact, the correlation of the R-3M returns between 1981 and 2023 was 0.00223, i.e. as low as it gets.

Rolling 3-month returns of gold (y axis) and stocks (x axis) over 1981-2023. (Click to enlarge)

But how can we draw any valuable conclusions from this? Remember, with this post we set out to investigate assets with convex return relationships. What this means is that we want gold to perform well when stocks perform badly. In other words, we want the left tail protection and something that excels when stocks experience huge drawdowns.

If gold is such an asset class, we want the dots on the left hand side of the chart above to cluster on the upper half of the chart. Ideally, we would also see as many dots on the upper half across the chart as well, and we can kind of see a hint of this.

But let’s make the relationship a bit easier to read, so that we are no longer staring at 11,000 data points.

We return again to the binning of stock returns from -30% to +30% to bins with sizes of 0.25%, as we did with the distribution chart further up. Then, we review how gold performed on average for each of those stock return bins. For example, when stocks returns -10% (±0.125%) over a 3-month period, the average return of gold over all such periods was +2.5%.

Now, the convex relationship starts to show, as the pattern of the below chart clearly has the shape of a smile with upward sloping corners of teh mouth.

The bottom of the smile is centered near (but just above) zero, which is a result of the extremely low correlation between the asset classes (the relative 3-month returns more or less cancel out on average).

More importantly, when we review the left hand side of the chart – the region when stocks decline by -15% – -30% over any 3-month period, on average, gold sees positive returns. This is the characteristics – the return convexity – we want to see from a diversifier that we add to our portfolio. Drawdowns will be shallower, and we can earn a rebalancing premium by selling gold at its dearest and buy stocks at their cheapest.

Average rolling 3-month return of gold for each 0.25% bin of rolling 3-month return for S&P 500 from 1981 to 2023. The chart is clipped at -30% and +30%, so that these data points include all data below and above these marks, respectively. (Click to enlarge)

Why is the relationship not a perfect U shape?

This is one of the questions I decided to study more closely, by looking at the data points which are found in the bottom left quadrant, i.e. the 3-month periods when both stocks and gold are down significantly.

I found that all of the worst of these data points stem from mid-October to mid-November 2008 at the height of the Global Financial Crisis, when stocks were down more than 20% over rolling 3M periods, while gold too failed to act as a risk off hedge during that volatile time.


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If we adjust for this period, the convexity of the relationship becomes even more clear. Here below, you see the same chart as above, but with the most affected data points from the fall of 2008 marked in red (to the left) and a chart which removes the approx. 30 days from the 11,000 day long test period (to the right).

In practice though, we shouldn’t be making these kinds of adjustments to our historical analysis – even if it is relevant to find out why the data looks like it does. Instead, we need to accept the flaws and add additional assets to the mix that did well also in this particular environment to cover all our bases (for example cash, managed futures, government bonds, etc.).

Whilst we have focused mainly on rolling 3-month return relationships, for the sake of completeness, I also looked at a shorter time frame, namely rolling 1-month returns. A similar pattern emerges also for this time frame, albeit the left and right tails are a bit more erratic. This was expected, as larger declines over a shorter time are rarer and often more volatile events. Still, the convex shape can be distinguished also here.

Summary – Why you should diversify with return convexity

Diversification is key, but this is hardly any news for someone who reads a blog about All Seasons Portfolio strategies. With this post, however, we have looked at diversification through another lens by focusing on more on the area around the tails of the return distribution of the stock market. As the S&P 500 is skewed with a longer and fatter left tail, we as investors better make sure to protect our portfolios for when those tail events occur.

One way of viewing diversification is to find assets that have low or negative correlation with the other assets in our portfolio, as well as which outperform when other assets underperform. For this review, you can analyze the rolling 3-month returns for both assets and plot how the diversifying asset performs against each bin of returns of e.g. the stock market. The diversification has been successful, if the relationship is convex (has a shape of a smile). Even a smirk with a higher slope on the left-hand side can be attractive.

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In our example, we looked at how gold fits in this line of thinking and we found that the plot (when averaging gold’s returns for each rolling 3-month return bin of the S&P 500) indeed has the shape of a smile. The return relationship is thus convex, in addition to gold having extremely low correlation with stocks. Thus, we can conclude that gold is a valuable diversifier that protects against the deeper drawdowns of the stock markets.

However, we also found that gold cannot take this role alone, as there were times when both gold and stocks fell sharply in tandem, more precisely during the autumn of 2008. Thus, one should also include other assets in the portfolio to enhance the protection against drawdowns, ensuring the portfolio returns also over shorter time frames remains stable. This is what the All Seasons Portfolio aims at achieving, by also including exposure to e.g. bonds and commodities (and other asset classes you deem appropriate to include, such as managed futures).

With this, I thank you for your attention, and I hope that you have found this alternative take on diversification enlightening and thought-provoking. Let me know in the comments or shoot an email to, and I’ll get back to you as soon as I can.

All the best,

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