Risk Update: April 2021 – Shannon’s Entropy.

A question we often get asked, as we discuss building convexity into investment portfolios with large Pension Funds/SWFs/Insurance Companies/Endowments/Banks, is something along the lines of “How much can we realistically do? Can we do enough to make a difference?”. The honest answer is plenty enough to make a major difference!

In last month’s update https://convex-strategies.com/2021/04/16/risk-update-march-2021/ figures 10-17 showed the nearly identical concavity of a range of familiar investment strategies. We observe, regularly, how the near universal incentive around short-term arithmetic returns, conveniently paired with flawed risk methodologies, leads unsurprisingly to fiduciaries in all forms targeting the mean of short-term probabilistic outcomes and ending up with near identical results. If you hire defensive midfielders (diversifying strategies) then pay them to score goals (performance fees for arithmetic annual, correlated, returns), what do you think you are going to get? No fiduciary is likely more relevant to gaming the immaculate combination of incentive and risk methodology than the banking industry with its unique (though increasingly expanding to others) blessing of moral hazard and accountability avoiding regulatory construct.

[As a quick aside, per the above, we refer you to the just released May 2021 Financial Stability Report from the Federal Reserve https://www.federalreserve.gov/publications/files/financial-stability-report-20210506.pdf. We would not dub it a “great read” but do find it a fine example of the ongoing effort to justify the endless extension of Moral Hazard. As we have noted frequently, the broadening of the umbrella to include nonbank financial institutions (NBFIs) continues with a couple of explicit notes;
“Although markets for short-term funding are now functioning normally, structural vulnerabilities at some nonbank financial institutions (NBFIs) could amplify shocks to the financial system in times of stress” “Measures of Hedge Fund leverage are somewhat above their historical averages, but the data available may not capture important risks from hedge funds or other leveraged funds.”]

The financial system, dominated by people managing risks with other people’s money, is wrought with near endless provision of uncapitalized tails. Archegos is an obvious recent example of this sort of behaviour by banks protected by their regulatory accountability shields. Let’s take Credit Suisse (CS) as an example, and please bear with us as we simplify a fair bit. CS is reported to have lost $5.5bn on their exposures to Archegos, versus previous estimated profits from that client of $17.5m. If we assume CS is targeting 1% return on risk-weighted-assets (RWA), we can peg that they measured their exposure to Archegos on an RWA equivalent basis at $1.75bn, which at a 10% Tier I BIS Regulatory Capital level implies capital against the exposure of $175m, so a 10% return on capital. In the end, on something that was presumably all within their regulatory requirements, CS was targeting a 1% return on assets and a 10% return on capital but ended up with a loss of -3,142% on the allocated capital. That is quite a regulatory friendly provision of uncapitalized tail!

A like-minded friend, Memphis based no less, kindly reminded us of the relevance of Shannon’s Entropy to the opportunity that the financial world provides us. Claude Shannon was an American mathematician and is known as the father of Information Theory, in essence for adapting the Physics concept of entropy to information. His thesis “A Mathematical Theory of Communication” is pure genius.


Entropy is far from an easy concept to grasp, but for our purposes we think of it as a measure of uncertainty or chaos. Entropy is high when something unexpected, low probability occurs, and the information value of knowing it is significant. Using a lottery as an example, there is a very high probability that any given number will not win the lottery, thus knowledge or possession of a non-winning number has little information value and no real impact on one’s financial circumstance (the scaling issue in fat tails), ie low entropy. There is an infinitesimally small probability that a given number will win the lottery, thus knowledge or possession of the winning number has significant information value and would have a massive impact on one’s financial circumstances, ie high entropy. The mean and variance of the wealth effect of gazillions of lottery draws is meaningless until you draw the one number that is the winner. As such, Shannon shows, and again we are simplifying, that entropy is the log of the inverse of probability (log1/p). As Shannon puts it, “..is thus approximately the logarithm of the reciprocal probability of a typical long sequence divided by the number of symbols in the sequence”.

Entropy aligns very logically to the concept of fat tails and to the value of tail risk hedging. Remember, fat tails are about the scale of their impact on the parameters of the distribution (mean and variance), not that they happen more often than Gaussian probability distributions would indicate. In fact, just the opposite. Likewise with entropy, the lower the probability, the greater the entropy. As a simple visual, we picture it this way.

Figure 1: Entropy vs Binomial Probability Distribution

Source: Convex Strategies

We think the commonalities in logic behind the likes of fat tails, Jensen’s Inequality, and Entropy are reasonably clear.

This brings us back to where we started: virtually the entire financial system is functioning in a probabilistic risk methodology way and endlessly providing attractively priced tail risk protection. Fiduciaries in the banking and investment world are mandated to account for the returns of underwriting tail risk, but not required to account for the capital to absorb the potential losses. All the while, participating through incentive pay on the accounting returns but passing the losses onto the end capital holders. Given the current balance in the world of those managing for short term arithmetic returns (managing for the benefit of the fiduciary), versus those managing for terminal compounded capital (managing for the benefit of the capital owner), there is nearly unlimited availability of attractively priced tail protection.

This naturally aligns with our oft stated premise that, in the self-organized world of financial markets, the biggest risk resides where and when the return for underwriting the risk is the lowest, or inversely the price to protect against it is the cheapest. Uncapitalized tails. You could say that Shannon’s Entropy is the mathematical equation for the famed quote (often attributed to Mark Twain) “It ain’t what you don’t know that gets you into trouble. It’s what you know for sure that just ain’t so”.

It is not making high risk weighted loans to small business borrowers that leads to broad swaths of banks needing to be bailed out. On the contrary, it is the supposedly “safe” assets, that attract the smallest risk weighting from the all-knowing regulators, that lead to the subsequent bail-out necessities. It is the application of leverage to the thing they “know for sure that just ain’t so”. It is the flawed application of assumed Gaussian distribution over far too small sample sizes.

The current regime of historically and artificially low interest rates provides an interesting visual of this dynamic. We have taken 10yr swap rates for various countries and extrapolated their historical series into Normal Distributions, one short term perspective and one longer term perspective.

Figure 2: Normal Distribution of USD 10yr Swap. Shorter Perspective

Normal Distribution of USD 10yr Swap. Longer Perspective

Source: Convex Strategies, Bloomberg

For the ‘Shorter Perspective’ (27 months of daily samples in this case, still longer than the standard series used by most entities for their VaR series) we get a mean of 1.50% and a standard deviation of 0.69%. On the ‘Longer Perspective’ (389 months), the mean is 4.88% with a standard deviation of 2.36%. The recent period has much lower mean and a much tighter distribution, which leads to some interesting dynamics. For example, a move from the current level of 1.65% back to the long term mean of 4.88% would, by the long-term distribution, be well within the probabilistic expectation. Even a move to one standard deviation above the longer-term mean, to 7.24%, would seem unspectacular. On the shorter-term distribution, however, those levels are 4.68 standard deviations and 8.10 standard deviations from the current level of rates. Those are outcomes that, by the incorrect application of a Normal Distribution, are considered virtually impossible.

Just for fun, we have created the same views for EUR, AUD and JPY. Again, based on the properties of these Normal Distributions, you can see that applying the properties of the shorter-term versions leads to probabilistic impossibilities of ever seeing levels of rates that are well within the norms of the longer-term perspectives. You can do the math for yourselves but leave it to say that applying the shorter-term standard deviations to the likelihood of returning to longer-term historical norms, comes quickly to the point of “it can never happen”. One could say the entropy is high!

Figure 3: Normal Distribution of EUR 10yr Swap. Shorter Perspective

Normal Distribution of EUR 10yr Swap. Longer Perspective

Source: Convex Strategies, Bloomberg

Figure 4: Normal Distribution of AUD 10yr Swap. Shorter Perspective

Normal Distribution of AUD 10yr Swap. Longer Perspective.

Source: Convex Strategies, Bloomberg

Figure 5: Normal Distribution of JPY 10yr Swap. Shorter Perspective

Normal Distribution of JPY 10yr Swap. Longer Perspective

Source: Convex Strategies, Bloomberg

We might conjecture that, if in any way the market is using their short time series and probabilistic based risk models, the pricing to protect against a return to longer-term historical norms is probably pretty attractive. Or, looked at from the other side, the low yield and supposed risk-mitigating benefits of fixed income holdings could just be a ticking time-bomb. Even worse so if you factor in the self-organized criticality, cum reflexivity, of leverage applied to “low risk” assets in the fiduciary world. If you think the leverage provided by highly regulated banks to the likes of Archegos (on things like Viacom shares with historical realized volatility of circa 30%) was crazy, just imagine the leverage the same guys are giving on fixed income type structures where the underlying has realized volatility in the low single digits. The Self-Organized Criticality, cum reflexivity, of this exposure is truly the greatest fire risk in the system today. As we have said so many times before, the system is one big LTCM now.

Figure 6: 30day Realized Volatility Viacom (orange) vs HYG ETF (white)

Source: Bloomberg

As usual, it all brings us back to dead capital tied up in investment portfolios allegedly with the purpose to diversify and risk mitigate yet measured by a return metric. Broadly speaking, the bulk of this is in fixed income and carry strategies. Again, if you think there is a lot of leverage backing beta, you can hardly imagine how much there is behind carry. In a recent speech, one of the RBA’s Deputy Governors threw out the following gem as regards their upped intervention to keep bond yields from moving higher: “We were not concerned about the fact that bond yields were rising nor the volatility per se, but rather market functioning.” Aside from wondering what “functions” they are worried about that do not encompass direction and volatility, one can only imagine what happens if ever the market truly tests yields out in the tails. Maybe that is why such officials make speeches that sound more like wistful guidance than any sort of valued and thoughtful analysis.


So, what to do? Same old, same old. Replace lousy defensive strategies with explicit hedges and take more risk. We show it over and over again. Replace Fixed Income allocations with a simple 50/50 bucket of Long Vol and Beta. Replace Absolute Return Hedge Fund strategies (notorious hiding places for levered carry) with a simple 50/50 bucket of Long Vol and Beta.

Figure 7: SPXT and Long Vol 50/50 vs US Tsy Index (Jan 20 – Apr 21)

Source: Convex Strategies, Bloomberg

Figure 8: SPXT and Long Vol 50/50 vs HFR Hedge Fund Index (Jan 20 – Apr 21)

Source: Convex Strategies, Bloomberg

The basic 60/40 Balanced Portfolio, therefore, becomes an 80/20 Barbell Portfolio (ie. the 40% in fixed income gets split to 20% to Long Vol and 20% to more equities). In the long run view that could look like this:

Figure 9: 80/20 Barbell vs 60/40 Balanced

Source: Convex Strategies, Bloomberg

As we always say, 2x lever the Hedge. Thus Barbell 80/20 becomes Barbell 80/40.

Figure 10: 80/40 Barbell vs 60/40 Balanced

Source: Convex Strategies, Bloomberg

The point of an efficient hedge is not to reduce your exposure to the upside, it is to allow you to enhance it! Take advantage of the increased risk mitigation in your portfolio to dial up your exposure to the up-tail and move half your beta allocation to something more juiced, eg. Nasdaq. So, the Barbell becomes 40/40/40 across Nasdaq, SPX, and Long Vol – our so called “Always Good Weather” portfolio.

Figure 11: 40/40/40 Always Good Weather vs 60/40 Balanced

Source: Convex Strategies, Bloomberg

Remember, these backward-looking histories are all over a period where interest rates went steadily lower, and in particular during periods of sharp equity declines. In other words, the period where Fixed Income was as good a portfolio compliment as it could possibly have been. The Balanced Portfolio has been great during the era of the central bank reaction function of forever cutting rates when equity prices fall, but still not nearly as good as a portfolio with explicit asymmetric hedging and more topside participation to growth assets we would argue. We know from our Kelly Criterion analysis that deleveraging your risk (ala the Balanced Portfolio) is the right way to avoid the wipe out of negative compounds, but not nearly as good as buying insurance and participating more on the potential upside. If your “deleveraged” capital is sitting in Fixed Income and Levered Carry at a time when the tail risks of rising interest rates have never been higher, it may be time to rethink potential risks to that view.

On the one hand they (central banks et al) cannot let rates go higher, but equally, can they stand by and do nothing as inflation, true and measured, tears society apart? The April CPI numbers in the US give some idea of the potential building problem. The Fed and their mouthpieces continue to swear it is all transitory and due to base-effects. However, looking at CPI and Core prices, both year on year and month on month, it is hard not to reach the conclusion that we have not seen anything like this since mid-2008. Funny how most people, when thinking about 2008, will not recall that just prior to the big cleanse, the world, US included, was experiencing a measured inflation spike. Just saying.

Figure 12: US CPI YoY and MoM % Change

Source: Bloomberg

Figure 13: US CPI Core YoY and MoM % Change

Source: Bloomberg

The current bout of measured inflation, along with the far more severe actual inflation of things like lumber and copper and gasoline and corn and houses and cars and cryptos and equities, may or may not be transitory. Regardless, it would seem, at the very least, to raise questions as to whether we should be on the most extreme accommodative monetary policy setting, along with most aggressive fiscal policy support, in known history. Might some sort of hypothetical “neutral” setting make some sense? What is neutral nowadays? How do you transition to neutral when you have created such exorbitantly fat tailed distributions that the slightest move towards historical norms enters a realm of unfathomable entropy? All good questions! (even if we say so ourselves!) The answer, as ever we believe, is convexity. Insure against risks that cannot be managed. Protect against the pain of a thousand little cuts of ongoing inflation (this is the pain that has been and continues to be greatly suffered by most people we speak to) by owning more of the things that participate in it (put more strikers on the pitch). Protect yourself from some inevitable future cleansing of the risk with explicit hedging strategies (hire as many reputable goal keepers as you can find). There is more than enough out there to make a difference.

One last comment on Fat Tails and Shannon’s Entropy. Another common phrase that seems to fit well in that space is “too good to be true”. One of the things we regularly find when markets enter a down cycle, economies have a downturn, is a pervasiveness of frauds. We have thrown in a few links below that might relate to this issue. LTCM was the biggest such “too good to be true” that the industry had seen, up to that time. Madoff put that to shame. Looks like if/when the current cycle of endless liquidity and moral hazard comes to its inevitable end, we are likely to see a whole bunch of things that set all new standards. If anybody knows the folks at the Pennsylvania Public School Employees’ Retirement System, we would love to have a chat with them.


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