Risk Update: September 2020 – Ergodicity.


Above is a link to an entertaining little note which echoes conversations that we have every single day, including past ones with the very antagonist of the story. Our experience in said discussion would lead us to agree with the comment in the article that “Meng, like many finance professionals, also never understood tail hedging”. We have a certain degree of empathy to the concept of it being us versus the world.

We talk about it over and over, it is geometric compounding that matters to the end capital owner. The two relevant ingredients, in a recipe to maximize terminal capital value, are time and downside risk mitigation. Once you are competent at stopping, going fast is pretty easy.

In empirical terms, it looks like Figure 1. In this example we’ve taken performance going back over 40 years, so 480 months, and shown the geometric compounded value with different versions of excluding the worst and/or best 9 months of those returns.

Figure 1: SPX Index Returns Oct. 1980 – Sept. 2020

Source: Convex Strategies, Bloomberg

This is what generates our bar chart view of the contribution to Long Term Compounded Annual Growth Rate (LT CAGR) in Figure 2. The blue box represents the contribution of the best 9 months, the orange box the contribution of the worst 9 months, and the grey box the 462 other months. Each of the 9 month boxes represents a mere 2 percentile of monthly returns, yet between them they contribute 70% of the LT CAGR. Thus, the middle 96th percentile of monthly returns, 462 months, contributes a mere 30% of the LT CAGR.

Figure 2: SPX Index Contribution to LT CAGR

Source: Convex Strategies, Bloomberg

It seems, at some point, the then managers of CalPERS had the good fortune of meeting Messrs. Taleb and Spitznagel, and the above point was sufficiently elucidated. Unfortunately, new (now also former) management was perhaps unable to understand the concept, or at least not sufficiently so to sustain it against how they were defining their objectives. Sadly, maximizing terminal capital is a much rarer objective than most would think.

As usual, it simply comes back to the math. It is the flawed practice, throughout the industry, of applying ensemble averaging as though wealth were an ergodic progression, and therefore using simplistic short term, probabilistic, expected returns as a decision-making tool. Properly evaluated, investment returns must be evaluated as non-ergodic and evaluated with time averages. Each of us can only get the results that we get, we can’t get the average of results of a large group and cannot go back in time to change the results once received! Thus, the old coin toss game where you pitch up with $100 and are offered a 50% return if the toss comes up heads or a 40% loss if it comes up tails, but you have to play for some minimum numbers of tosses. While the expected return of any one toss might look attractive at +5%, we can all see that the time-average returns, at the median, result in your capital approaching zero. Sadly, the +5% expected return of any one toss might look attractive to the fiduciary agent, who doesn’t share in the negative compound of capital, but starts fresh with their average arithmetic return target each year.

We can run a simple coin toss simulation to look at these outcomes. If we do the simulation with 200 players doing one toss of the coin, sure enough the average will come out with capital rising to $105, the 5% expected return. If we flip it and play the game with one player and 200 tosses of the coin the player goes to zero every time, of course. We can use this simple simulation game to see how various investment implications impact the median outcomes.

Setting the game at 200 players and 40 tosses we can see how the calculated median of outcomes tracks to the theoretical median, starting capital of $100 becomes terminal capital of $12 for 40 tosses, for this payout ratio.

Figure 3: Coin Toss Simulation +50% vs -40%. 200 Players. 40 Tosses

Source: Convex Strategies

This round of the simulation results in a median terminal capital of $12, precisely in line with the theoretical median, and an average terminal capital of $404. In other words, most people are losing everything, but a few people are getting lucky, creating a Pareto type wealth distribution (very much like the real world!). The solution to this dilemma is to reduce your exposure to the bet, ie. to deleverage, and is in effect what leads to the concept of the infamous Balanced Portfolio. If we adjust the bet such that the players only risk 60% of their capital in the game and hold the other 40% in (supposedly) risk-free bonds that yield 2.5% per toss (think of each toss as a year), the calculated median of terminal capital now rises to $115, and in this case an average terminal capital of $580. Now not everybody is trending to bankruptcy and a few more people are getting lucky. We note though that of the $115 median return a full $107 of it comes from the 40% invested in bonds, which says something about where long-term performance has come from for a lot of balanced portfolio benchmark followers.

Figure 4: Coin Toss Simulation +50% vs -40%. 200 Players. 40 Tosses.

Balance Portfolio 60% in Game/40% in 2.5% yield Risk-Free Bond

Source: Convex Strategies

Figure 5: Standalone Contribution of 40% in 2.5% yield Risk-Free Bond

Source: Convex Strategies

Of course, there are very obvious implications to terminal values in the balanced portfolio model should the yield on the risk-free investment decline to lower levels, say for example 0.75% instead of 2.5%. Now the calculated median terminal capital flips back below the breakeven line of $100 to a disappointing $61. Obviously, the compounding benefit of the risk-free rate disappears as yields approach 0%.

Figure 6: Coin Toss Simulation +50% vs -40%. 200 Players. 40 Tosses.

Balance Portfolio 60% in Game/40% in 0.75% yield Risk-Free Bond

Source: Convex Strategies

Clearly, as risk-free rates approach zero (and lower!) the balanced portfolio ceases being a particularly effective solution. This, necessarily, has the same implication for Risk Parity type strategies, and is something we will come back to in this note. Another alternative might be to hire a super talented manager that can somehow garner higher returns (sometimes claimed to be alpha, but generally just levered beta and carry) for the same risk. At -40% risk of the toss coming up tails, to get to a theoretical breakeven terminal capital median of $100, the upside payoff would need to be 66.7%, but should said talented manager claim such skill, he would likely charge some fees for it. Conservatively, let’s set those fees at 1% Management Fee and an interest aligning Performance Fee of 10% on the winners. Those level of fees would take your calculated terminal capital median back down to a mere $11. At those fee levels, you would need to adjust the upside to +77.75% to get the calculated median back to breakeven of $100. At that sort of payout ratio, you have the lucky few that are doing very well, the calculated average in this simulation is $2,167, and in particular the talented manager that is getting 1% per toss plus 10% of all of the winnings, while losing nothing when tails drags down the bulk of the players.

Figure 7: Coin Toss Simulation +77.75% vs -40%. 200 Players. 40 Tosses

1% Management Fee. 10% Performance Fee

Source: Convex Strategies

As you have likely already guessed, the solution to this problem is insurance. If one can cut off the downside, the impairment of the negative compound, and stay in the game to participate in the upside, even the terrible math of the +50%/-40% payout ratio can be fixed. If we add to our game “insurance” that cuts off the -40% loss after having incurred it for three consecutive tails, despite the theoretical median of the +50%/-40% being $12, we get calculated terminal capital medians like $141. Far superior to the alpha manager with 1/10 fees at a payout ratio of +77.75%/-40%. This “insured” portfolio also generates superior terminal capital medians over the 2.5% yielding balanced portfolio, let alone in the world of 0.75% yields.

Figure 8: Coin Toss Simulation +50% vs -40%. 200 Players. 40 Tosses.

100% Loss Insurance on 4th Consecutive Tail

Source: Convex Strategies

Apologies for the long-winded step by step build-up, but that gets us to, as promised above, the meat of what we wanted to get at. As Mr Spitznagel says so succinctly in the linked article, “Risk mitigation should raise your returns.” Diversification with strategies that have low correlation to markets on the upside, and often bounded upside returns, but high or uncertain correlation to the downside, are destroyers of compounding. We agree with Mr. Spitznagel that efficient, convex, asymmetric risk mitigating strategies should be the next Risk Parity. We would go one step further; they should have been the old Risk Parity as well!

We showed in last month’s update https://convex-strategies.com/2020/09/21/risk-update-august-2020/ the superior compounded returns and risks of what we termed the “Dream Portfolio” over a Risk Parity index.

Figure 9: “Dream Portfolio” vs S&P Risk Parity Index

Source: Convex Strategies, Bloomberg

In the July Update https://convex-strategies.com/2020/08/27/risk-update-july-2020/ we had shown an updated version of our Hypothetical Concave Portfolio scattergram and noted four key reasons that lead fiduciary managers to impede compounding.

The compounding effect of the Dream Portfolio over Risk Parity can also be shown in the scattergram format and makes quite clear where the differentiation in compounded outcomes occurs. As a reminder, the scattergram plots S&P 500 monthly returns on the X-axis and the designated portfolio returns on the Y-axis.

Figure 10: “Dream Portfolio” vs S&P Risk Parity Index: Jan 2005-Sept 2020

Source: Convex Strategies, Bloomberg

Figure 11: Dream Portfolio vs Risk Parity: Jan 2020-Sept 2020

Source: Convex Strategies, Bloomberg

We have drawn the respective parabolas making the convexity, or lack thereof, standout. What you can see, and as discussed per Figures 1 and 2 above, it is performance in the wings that matters. During the bulk of market realizations, ie. the 96th percentile, the two strategies track with little differentiation in their return profiles. It is in the wings, two of which have occurred already in 2020, that the convexity shows its value and the superior “Dream Portfolio” distances itself. It is, of course, worth remembering that the bulk of that steady performance during the 96th percentile months, for the Risk Parity portfolio, would have benefitted from years of higher, then falling, interest rates. That is less likely to be the case with today’s yield levels as the starting point looking forward.

The scattergram is a nice simple tool to look at the convexity of returns and confirm the impact of such on compounded capital through time. As another example, we can go back to our contribution of long-term returns bar chart and construct a portfolio based on those percent contributions. No doubt, overly simple, but when you are positively convex, you can get away with simple. For the 40% contribution to LT CAGR of the worst 2nd percentile of months, we can allocate 40% of capital to the Long Vol Index, but as we always say “lever the hedge” so let’s take advantage of that and make it a 2x allocation for 80%. For the 30% contribution to LT CAGR of the best 2nd percentile of months, we can allocate 30% capital to juiced-up Beta, say Nasdaq 100. For the 30% contribution of the middle 96th percentile of months, we can just go ahead and allocate 30% capital to vanilla Beta, SPX Index. Let’s call it the “Always Good Weather” Portfolio. Just for comparison, we can compare that against the HFRX Global Hedge Fund Index, which as you might expect is concave in its historical returns. In the scattergram version, that looks like this;

Figure 12: Always Good Weather Portfolio vs HFRXGL Index: Jan 2005-Sept 2020

Source: Convex Strategies, Bloomberg

Figure 13: Always Good Weather Portfolio vs HFRXGL Index: Jan 2020-Sept 2020

Source: Convex Strategies, Bloomberg

On a compounding capital basis, and assuming annual rebalancing, it looks like this;

Figure 14: Always Good Weather Portfolio vs HFRXGL Index: Jan 2005-Sept 2020

Source: Convex Strategies, Bloomberg

The parabolas sort of speak for themselves. Once again, we would refer you back to our July Update and the four items that lead to concave returns. Absolute Return Hedge Funds are hotbeds for those sorts of issues.

Before we wrap it up, we would like to show the scattergram view of the oft claimed return dynamics of Hedge Funds as being no different than just short a put on the S&P. We will substitute in the HFR Fund of Fund Index this time. To the credit of the Fund of Fund Managers in the index, it does outperform the HFRX Global Hedge Fund Index (HFRIFOF Index). For the comparison on the Short Put leg we use the CBOE S&P Put Write Index (Put Index).

Figure 15: Put Index vs HFRIFOF Index: Jan 2005 – Sept 2020 scattergram

Source: Convex Strategies, Bloomberg

Figure 16: Put Index vs HFRIFOF Index: Jan 2005 – Sept 2020 returns

Source: Convex Strategies, Bloomberg

Neither of them is particularly attractive, either in scattergram or compounding form, but to compare Hedge Fund returns to short put strategies is being unfair to the short put strategies. We’ll take the long convexity approach over either of them every time.

Our opinions remain steadfast. Participate on the upside, while cutting off the downside. There is no more important task for those who care about terminal compounded capital. Commonly preferred and supposed risk mitigation cum diversifying strategies like Fixed Income, Risk Parity, Absolute Return Hedge Funds, levered carry, etc. tend to do a good job at neither of those. This has long been the case, but becomes in-your-face critical as yields settle in at all time lows, increasingly in the general vicinity of zero. Convexity is now, and has always been, the answer.

Figure 17: SGD / JPY ‘Seasons’ Chart

Source: Convex Strategies, Bloomberg

Figure 18: Convex Vol ‘Comet’ S&P500 Realised Correlation & Implied Volatility

Source: Convex Strategies, JP Morgan

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