# Why Convexity Matters: Efficient Downside Protection, More Upside Participation

By David Dredge

(4 min read)

By David Dredge

(4 min read)

‘If something cannot go on forever, it will stop’, states Stein’s Law. We keep this top of our minds. Clearly, central banks cannot forever maintain policy settings at one extreme as economic circumstances, whether growth or especially inflation, push to the opposite extreme. Something must give. It is the classic fat-tailed distribution: high expectations around the mean and very wide variance.

The investment solution to this dilemma remains the same. We should structure our portfolio for what we do not know (the wide variance), not what we think we know (the probabilistic mean). Compounding is driven in the wings (the extremes). It is just math. We want to participate in the unexpected upside – for example, central banks carry on and continue to drive asset values to historic levels without blowing things up. But we want to cut off the unexpected downside – for example, things blowing up. We want our portfolios to be convex!

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‘If something cannot go on forever, it will stop’, states Stein’s Law. We keep this top of our minds. Clearly, central banks cannot forever maintain policy settings at one extreme as economic circumstances, whether growth or especially inflation, push to the opposite extreme. Something must give. It is the classic fat-tailed distribution: high expectations around the mean and very wide variance.

The investment solution to this dilemma remains the same. We should structure our portfolio for what *we do not know* (the wide variance), not what we *think we know* (the probabilistic mean). Compounding is driven in the wings (the extremes). It is just math. We want to participate in the unexpected upside – for example, central banks carry on and continue to drive asset values to historic levels without blowing things up. But we want to cut off the unexpected downside – for example, things blowing up. We want our portfolios to be convex!

Even with the recent longest running bull-cycle in history, we can demonstrate the value of protection by looking at the returns of owning the equity index *and* buying put options versus the common risk premia strategy of selling put options since March 2009. In Chart 1, the blue line is the CBOE Put Protect Index (PPUT Index), while the red line is the CBOE Put Write Index (PUT Index). These indices do the following:

PPUT owns S&P and systematically buys one-month put options for protection.

PUT systematically sells one-month puts to earn risk premia.

They are, in a sense, mirror images. One pays risk premium (PPUT = buying puts), and one earns risk premium (PUT = selling puts). One is convex, one concave. Yet, over time, the convex one outperforms. That outperformance, unsurprisingly, happens in the wings or extremes.

This outperformance has nothing to do with alpha, timing or algos. It is literally just math. The PPUT Index has a compound annual growth-rate (CAGR) over this period of around 13% on a downside volatility of 6%. The PUT Index has a CAGR over the period of around 10% on a downside volatility of 8%. If we equalize the two on downside volatility of around 6%, the PUT Index CAGR drops to 7.6% over the period (Chart 2).

These returns challenge proponents of ‘risk premia harvesting’, or those academic papers that claim volatility is overpriced. Often, they compare apples and oranges. Likewise, anybody who discusses the ‘cost’ of owning hedges/volatility/convexity on a standalone basis is simply looking at the problem incorrectly. We must view these in the context of their impact on a portfolio through time and the ability to adjust upside risks due to superior downside protection. Not everyone misses penalties in a shootout; you need a goalkeeper.

It turns out there might be ‘alpha’ in the long volatility space, or maybe volatility is structurally underpriced. We can construct a simple “Always Good Weather” (AGW) theoretical portfolio. We use the CBOE Long Vol Index of Long Volatility Hedge Fund Managers and split the weightings in the compounding asset between SPXT and XNDX (Nasdaq Total Return). We can come up with something with the same downside volatility of the PPUT index of around 6%. This gives us something with a stunning bull market CAGR of around 19%, assuming yearly rebalancing (Chart 3).

On top of that, many absolute return strategies or hedge funds mimic some form of selling options to capitalise on apparent overpricing of options. Often, they underperform even a simplistic rolling of short S&P puts. Indeed, if we reduce the allocation to the Put Write Index to 40% to match the 3.1% downside volatility of the HFRX Index, a benchmark hedge fund return index, we find similar risk and return profiles (Chart 4). The PUT Index has a CAGR of 4.2% versus the HFRX of a mere 2.7%. There is no alpha.

How do we construct portfolios that perform in the wings, that, for example, will protect us from the excesses of central bank policies? We could package things that are convex in the wings and that are truly diversifying. For a compounding asset that is convex in extreme ‘up’ periods (the up-tail), we can use the Thompson Reuters Private Equity Buyout Index (TRPEI Index). For a strategy that is convex in the extreme ‘down’ periods (the down-tail), we can use the CBOE Eurekahedge Long Volatility Index (CBOE Long Vol), an index of the performance of active long volatility managers. Adjusting again to a 6% downside volatility and using a longer data series that starts in 2005, we get an allocation of 45% to the TRPEI on a standalone basis with the balance, 55%, in cash. We find that the TRPEI/cash portfolio and Long Vol hedge as two separate strategies allow the obvious convexity in the wings and true diversification to stand out (Charts 5 and 6).

Both strategies are clearly convex, and they are negatively correlated in the wings. They are perfect portfolio complements. We can package them together, a Barbell Portfolio strategy, and solve the weightings to again get us to something around a 6% downside volatility. That works out to be a 55% allocation to the TRPEI and 45% to CBOE Long Vol. As the comparison, we can throw in a Balanced Portfolio, similarly weighted to achieve an equivalent downside volatility of 6%, which leaves us with a 50/50 portfolio of S&P and US Treasuries. The CAGRs are 11.4% for the Barbell versus only 7.4% for the Balanced (Chart 7).

This is the magic of convexity. This is how to deal with the unstable equilibrium and massive tail entropy created by extreme central bank policy. This is what fiduciaries managing long-term capital (endowments, pensions, sovereign wealth funds, family offices) need to be doing. So much of that done in the name of diversification is just delusion. As we often ask, when it comes to building convexity into a portfolio, where is the cost?

An extended version of this article is available here

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