As any regular readers will know, we are passionate about what we do and, along the way, hope we can also change how people think about fiduciary management of other people’s money. It sounds relatively straightforward, but there are multiple things that add up to one overriding problem, what we would call “bad convexity”. Most often we distil that to what we see as a simple truth; fix your convexity – change your life.

We have previously shown the below hypothetical return profile, and the impact that such concavity has on long term geometrically compounded returns. The impairment to terminal capital can come as quite a shock to people, and regularly leads to our favourite saying “it’s just math”.

One of the oft posed questions is; what is the end outcome if one were to invert the concavity into the equivalent positive convexity? Many make the mistake of assuming it would merely be the equivalent outperformance above the compounded returns of the linear benchmark as the underperformance of the concave realized returns. Of course, “the math” doesn’t work that way. The linear benchmark terminal value outperforms the realized concave returns by 533%, ie $100 becomes $800 on the benchmark, versus $100 becoming just $277 in the realized concave returns.

If we flip the concavity over to hypothetical positive convexity, with the same relative (but now positive) performance to the benchmark, the results look like Figure 3. The terminal capital becomes an extraordinary $2,171, a whopping 1,371% over the benchmark.

Again, it is just math. One might even go so far as to make the bold claim that the best way to improve your compounded return is to improve your convexity.

Of course, we can play this game with any historical return series. To bring some reality to proceedings we recall the famous Warren Buffett bet of 2007 where he claimed that simply 100% long the S&P Index would outperform any fund of hedge fund strategy. Only one party took him up on the bet, (starting January 2008 for 10 years), and Uncle Warren’s S&P Index fund handily disposed of the challenger, who conceded defeat after 9 years. Much like ourselves, Mr Buffett was and still is critical of the fee structure of the hedge fund industry and rightly claimed that his proposed strategy, effectively the linear line above but with a 1.0 beta, would outperform anything coming out of the 2/20 fee world. Mr Buffett, in his own way, knew of the convexity problem that both chasing short term performance and charging inappropriate fees, in particular performance fees on positively correlated returns, would create over the long haul. In amongst his various comments about investment acumen and belief in the US economy, he made this truly key statement “Performance comes, performance goes. Fees never falter.” Many people declare it to be his investment acumen, but we would claim “it is just math”. In making his bet he simply reduced things that cause negative convexity, ie performance fees linked to positively correlated returns.

Here is a link to an Investopedia story about the results: https://www.investopedia.com/articles/investing/030916/buffetts-bet-hedge-funds-year-eight-brka-brkb.asp.

The Fund of Fund competitor returned only 22% total over the 9 years, some 2.2% per year. The 100% in the Index Fund had a total return of 85.5% over the 9 years, a compounded annual growth rate (CAGR) of 7.71%. Over the full 10 years, by our calculations, the Index Fund would have had total returns of 126% and a CAGR of 8.50%.

We agree with Mr. Buffett, why pay fees to managers for correlated and short-term returns? The attendant incentives will drive such managers to do all those things that lead to a concave realized return dynamic. However, we would venture that one could have beaten Mr. Buffett in his bet had you focused singularly on convexity. All you had to do was own the same thing he owned, ie. S&P Index, then improve the convexity on either, or even better both, sides of the outcomes. In our simple Barbell examples a combination of SPXT Index and the CBOE Eurekahedge Long Volatility Index can be used to see what an even better hypothetical solution could have been.

Just a simple comparison for the period (2008-2016) that the actual bet ran of various weightings of SPXT Index and the CBOE Eurekahedge Long Vol Index gives some telling results. Amazingly, a 60% SPXT/40% Long Vol split, rebalanced annually, results in a total return of 104% and a CAGR of 8.25%, handily defeating Mr. Buffett’s equivalent total return of 85% and CAGR of 7.71%. We can represent the performance using our scattergram methodology above. In figure 5 below the blue dots represent 100% SPXT returns from each of the 9 years, while the gold dots show 60% SPXT/40% CBOE Long Vol.

Likewise we can show the compounded returns of the same two data series (Figure 6). It is obvious what makes the difference, the large negative compound event in Year 1 (2008), which is significantly offset by holding the Long Vol allocation.

In fact, we have worked out that any weighting in Long Vol from 1% to 90% and the balance in SPXT would match or beat Mr. Buffett’s bet. Quite obviously, this is driven by the large negative compound in Year 1, such that adding in any convexity/protection to the downside and slicing off some of the negative compound makes all the difference. All we have done is reduce the compounding impact of a sharp sell-off by adding some convexity to the downside.

Per our earlier charts, imagine if we could add convexity on both sides? Given the efficiency of the Long Vol allocation, ie. it has a major impact on the negative compound and surprisingly little drag on the subsequent performance, it might make sense to apply some leverage to it (we are assuming that, since we are on the “Hedge Fund” side of the bet, leverage is allowed). So instead of 60/40, what about 80/40 (i.e. we apply 2x leverage on the Long Vol side)?

The 80/40 Barbell increases the total return to 132% and the CAGR to 9.82%, on a cash basis. Cutting off the negative compound allows us to own even more of the SPXT. You can see ever so slightly the changes in convexity in Figure 7 vs Figure 5.

But why stop there? Let’s go back to the original 60/40 and apply 2x leverage to both sides. This leaves us with a 120/80 portfolio, and total return of 288% with CAGR of 16.27%. The convexity can be improved on both sides of the distribution!

We can play this game on and on. Improve the convexity, improve the geometric compounded returns. The big challenge in the investment world today is finding somebody that is even targeting the geometric compounded returns! If somebody is quoting you annual average returns, pretty safe to assume that they are not trying to solve the convexity problem in their investment portfolio.

We will forever keep coming back to this subject. Solving investors’ convexity problems is our obsession. We are absolutely convinced that convexity is the key to returning value to the end capital owners. We hear so often about ‘the cost’ of long volatility strategies making an allocation all but impossible, and yet “the math” of a barbell approach shows otherwise over almost any sensible period we have looked at. We would be delighted to discuss how we think you can improve the convexity of your investment strategies.

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