Wikipedia’s definition of risk is rather straightforward: it is the possibility of something bad happening. As such, it seems like an inherently negative thing and yet, in a dialectical fashion that will be recurring throughout this post, it’s also the foundation upon which many people’s dreams of early retirement are built. More generally, in finance, risk is the main driver of returns, the threat of failure that makes success possible, no pain no gain.

The dual nature of risk prevents any one-sided solution from being functional (once again dialectics show up, it’s the complication of the matter that makes it more interesting), and calls for a comprehensive analysis of the apparent contradiction so that it can be overcome. We are thus presented with a common schema of German Idealism: thesis (initial statement: risk is undesirable, to be avoided), antithesis (contraposition: risk is profitable, to be sought after), synthesis (resolution: to be discussed).

In essence, most of finance can be summarized as answering the question: what to do with risk? The answer will be different each time, but below we propose a framework for orienting the process of finding an adequate response.

1. Introducing the toolkit: Kant’s categories

Thoughts without content are empty, intuitions without concepts are blind.

Kant

Our main objective is to facilitate the task of risk understanding so as to better guide our decisions. For that we will leverage Kantian categories, which can be thought of as a general tool for approaching any concept.

Categories motivation: some background

Individuum est ineffabile, literally, the individual is ineffable, is a saying commonly attributed to medieval scholasticism which points at the fact that no word can completely characterize the absolutely specific. This is because words, in order to be meaningful, must have some degree of generality. Equivalently, no map completely describes the territory, and rightly so: we could argue a scale 1:1 map is not a map, but a territory in itself.

In the same way that maps’ disregard for certain details permits having a portable object that can actually be used on the ground; words’ generality permits our mind to handle a reality that in its specificities would be too complex to deal with. Of course this doesn’t come for free, information is lost in this compression so one must be ready for surprises and periodically revise their conceptual framework.

In any case, orientation becomes easier with good maps and maps are better made with good cartography tools (words). With this in mind, already before the Middle Ages, Aristotle had set out to find the most general predicates, those that can be attributed to anything and thus constitute a framework for approaching any new concept (territory) one encounters.

This led to the idea of the categories, which he summed up in a list that would be amended subsequently, Kant being arguably the most prominent revisor. He appreciated Aristotle’s idea, but criticized the list for lacking systematicity, and reduced it to 4 types of categories, with 3 terms in each, following the aforementioned schema of thesis, antithesis and synthesis.

These 4 types of categories, namely quantity, quality, relation and modality, will be our main tools in drawing the conceptual map of risk which we will address below.

2. Risk through the lens of the categories

Since the generality of the framework makes it rather constraining, we will resort to a good amount of flexibility and depart significantly from the original intended meaning of Kant’s categories. This will be especially true for the categories of quantity and quality, that we have grouped together for convenience.

2.1 Quantity vs Quality – Time-varying risk premia

These two categories may seem opposite, but in this recurring dialectic spirit they turn out to be deeply intertwined. Math, the most quantitative of disciplines, often praises “elegant” demonstrations; Jim Simons, the founder of the most successful quantitative firm, in an MIT conference, included among the principles that had guided his professional career “beauty”; and here, in a site called Quantdare, we are writing a qualitative analysis of risk.

We left off the risk discussion with a contradiction between its inherently negative quality of potential failure, and its positive counterpart of profitability. This contradiction is synthesized by subjectivity, i.e., it’s subjective valuation of these two opposing qualities that allows for resolution into a single one of (un)desirability.

Now, this is a little bit trivial and hasn’t got us very far; let’s say we have given negative value to a certain risk, what to do then? In finance, it’s common practice to transfer it to someone else, who may be willing to accept it, but not for free. Intersubjectivity calls for objectivity, or more plainly, each participant has to convert their valuations (subjective) into prices (objective) to make negotiations possible. Hence, quality is converted into quantity, in the form of an exchange price, which in the case of risk is often referred to as premium.

Note there’s another contradiction here: price sets up a relation of equality (thesis) between two qualitatively unequal objects (antithesis), these objects being in our case risk and its premium (but more generally can be any two assets/liabilities). Here, the resolution comes in the form of temporality, price instability reflects the incapacity of a single quantitative measure to encapsulate certain qualitative differences without constraining the context.

Now this is important because much ink has been spilled regarding the time variability of risk premia (long discussions regarding the death of value come to mind) which are so focused on running regressions (no doubt a fundamental tool for scientific analysis) that forget to pay attention to the fundamental qualities of those risks, and how a different environment may justify a different quantification of them.

Another relevant note can be made on the value-price duality. Price, being objective, is unique; while value, being subjective, is multiple. Consequently, trades can be win-win situations in which no wealth is exchanged but both parties are benefited, as they prefer what they get to what they give.

This gives finance much higher chances of success: one need not be the smartest guy in the room (i.e., beat the market) in order to add significant value to their clients. This is not to say it’s an easy task either (that would make it boring); the contradiction between objective price and subjective value is inherently problematic and has no lasting solution, but may be properly handled with an adequate combination of quantitative analysis guided by qualitative understanding.

2.2 Relation

Here we will pay Kant a little bit more respect by following more closely (but still at a reasonable distance) his intended meaning for the categories of relation, which are substance/accident, cause/effect and reciprocity.

Substance and Accident – Risk profitability

We can consider substantial risk as essential risk, that is, risk that is not easily dealt with and, as such, it usually commands a premium. On the other hand, accidental risk is more superfluous and constitutes a weaker base for profits. An example might be clarifying.

A random SP500 stock will provide the same expected return as an SP500 ETF (having weighted sampling probabilities according to capitalization) and yet will have much higher risk (e.g., volatility). Hence, the individual excess risk is not translated into excess return. Because this risk can be easily removed by diversifying with more SP500 stocks, we can call it superfluous (accidental) and assume it will command no premium in the market. Note that the SP500 may have some accidental risk too, consider the possibility of worldwide diversification, although some may consider this more controversial and argue in favor of American exceptionalism.

On the other hand, the fact that both investments have positive expected returns implies that some of their risk is substantial, and the only way people can get rid of it is by trading it away in exchange for a premium.

There’s no consensus as to where to set the dividing line between substantial and accidental risks, which is of paramount relevance for efficient risk management. For instance, we may argue in a future post that a significant part of the risk of a passive FX exposure is superfluous, making the case for some sort of active hedging, but clearly this is not a unanimous view. In any case, the importance of adequately dissecting risks into these two components can not be understated.

Cause and Effect – Addressing time variability of premia

The idea of substance and accident, understood as core and periphery (centrality and superficiality), has some sort of spatial notion underlying. Contrarily, this second relational category is related to time.

As we have seen, one of the key defining features of risk premia is the time instability produced by the contradiction between quantity and quality. Effective risk management can adapt to time-varying risk premia by understanding the underlying drivers of that change. Here, it’s helpful to see premia as effects and switch our focus to their causes.

For instance, what’s the cause of the value risk premium? It’s often said that excess return comes from buying distressed (that is, riskier) companies which are thus cheaper. But what kind of risk are we talking about? Is it market risk, credit risk, liquidity risk, another kind of risk? All of them? In what proportion? One would expect these risks to have different premia and their share in cheapening stocks to be context dependant (maybe higher share of credit and liquidity risk in financial crises; higher share of market risk in economic booms). One could then imagine how static risk premia with time varying contributions to a broader investment strategy lead to seemingly unreliable returns; when in fact, if we switch our focus from the effect to the cause we would see a more navigable scenario.

Arguing for the staticity of fundamental premia is probably going too far. But, if instead of going after the stick, as a dog would do, we went after the thrower, as a lion would do (according to Milarepa), we may stand higher chances of ending the chase.

Reciprocity – Trading the right kind of risk

The two previous categories (which we loosely described as spatial and temporal), are one-sided. Here, reciprocity can be deemed synthetic via negation, as its symmetric character denies hierarchy. When applied to risk, the key aspect related to reciprocity is the convertibility of financial risk.

This serves to emphasize that one can be in both sides of a risk transaction, and probably most people already are (as long as they have some sort of insurance and investment vehicle). The key to good management is to always be on the right side of the trade: earning premiums where the market places more value on risk than you do, and hedging otherwise. This highlights the importance of being precise in risk dissection, as mixing desirable and undesirable risks in the same transaction increases hedging costs or reduces profitability. Along these lines, an argument can be made against FX passive hedges, for instance, but we will leave that for a future post.

2.3 Modality

We get to the last kind of category, composed of potentiality (something that may exist), actuality (something existing) and necessity (something that needs to exist and thus is both potential and actual).

Possibility and Existence – Slow and fast failure

The categories of potentiality and actuality fit very well with the idea of slow and fast failure described by Corey Hoffstein in this interesting post. There, a brilliant analogy based on Ulisses’ crossing of the Straits of Messina is drawn to illustrate a common trade-off between catastrophic but unlikely and digestible but certain risks.

In order to cross the straits, Ulisses has to choose between facing one of two mythological monsters, as attempting to avoid any of them will bring him dangerously close to the other. On the Sicilian side, Charybdis makes a giant whirlpool three times a day that would destroy Ulisses’ vessel and everyone aboard; on the Italian side, the six-headed Scylla will pick six people and eat them.

This contraposition between potential catastrophic failure and actual (certain) limited loss is ubiquitous in finance. For instance, one may choose to invest their savings in the equity market or leave them as deposits. Which one is riskier? It depends. If you don’t have enough saving power, the constant drain of inflation may be a guarantee of unmet financial objectives by the time you retire, while equities provide higher chances of success. In summary, failure to accept higher fast risks sometimes leads to assuring a worse outcome.

Interestingly, on the way forth, where the ship still has a sizeable crew, Odysseus avoided Charybdis at all costs. Let’s see the reasoning behind this choice. For the sake of simplicity, it’s assumed that each individual only cares about their lives, that there are 60 people on board, and that the probability of Charybdis forming a whirlpool is 1/3. Then, each crew member has a probability of dying of roughly 1/10 approaching Scylla and 1/3 approaching Charybdis; the former seems the rational choice.

However, later on, after Odysseus has lost his crew to Polyphemus and is driven to the straits again, this time, being alone, chooses Charybdis, as now a 1/3 chance of death seems a good deal compared to the 100% offered by Scylla. Same preferences and logic but different context lead to different conclusions.

In summary, any attempts to reduce potential failure require the acceptance of actual costs, which may themselves be unbearable. The optimal choice is context-dependent, which helps make the case for active management and close monitoring of changes in the environment.

Necessity and Contingency – Are risk premia reliable?

A necessary event is one that not only could have happened and has happened, but had to have happened (we may say it would occur in every repetition of the same generating process). We could have concluded the earlier section saying that the prospect of loss is necessary and thus cannot be eliminated, only transformed into a different kind (from slow to fast failure and vice versa).

Moving back to the bright side of risk, premia, we could ask ourselves whether these are necessary or, as it is often thought, contingent on people’s risk preferences. The case for the latter is certainly strong, but there are some subtleties to it.

In a recent review of risk premia, Asness, Moskowitz, et al. argue that all but the market factor are contingent. More specifically, they state risk premia are subject to crowding out effects, meaning that if many investors exploit them they will become less profitable. This is quite straightforward, if premium is the price or risk, the higher the offer with respect to demand (people willing to accept others risk), the lower the price.

However, this argument is general, it makes no reference to the nature of that risk, and so it should also apply to the market factor too. Imagine everyone holding bonds sold their assets and bought equities, stock prices would go up and they would be so ridiculously overpriced that earnings would represent too small a fraction and, speculative bubbles aside, they would be hardly profitable.

But could this actually happen? Again, if premia are contingent on risk aversion and investors change their preferences to risk-seeking it could, but it’s not that easy. Market risk aversion is not only driven by participants’ subjectivity, it also rests on more solid foundations. Regardless of a manager’s taste for risk, known cash flows are more easily manageable than uncertain ones: one allows to maximize every asset’s productivity, the other requires to hold some reserves as provisions. So ceteris paribus and unless a compensation is offered, certainty will always be chosen over unpredictability.

To sum up, there’s an economic cost to uncertainty that renders the market’s customary risk aversion more objective than it may have seemed at the beginning. This pushes premia closer to necessity and away from contingency, and makes them a more solid foundation upon which reliable financial products can be built.

3. Concluding Remarks

We have used a broad set of abstract categories from philosophy as guiding tools for a discussion of the central role that risk plays in finance along with the main aspects that need to be considered for its proper management. We can group these into three categories

1. Risk analysis.
   • Value-price divergence should be leveraged, tailoring risk exposure to client’s subjectivity.
   • Accurate risk dissection is key to separate substantial risk from superfluous risk and optimize hedging costs and/or risk profitability.
   • Investigating the cause of premia facilitates the management of its time variability.
2. Risk choice: navigating the straits of Messina successfully requires a comprehensive understanding of the trade-off between fast and slow failure and its implications.
3. Risk monitoring: as the environment changes, same tools yield different results, stressing the need for adaptability.

The generality of this discussion may make some of it look too abstract. Indeed, these are closer to guiding principles than to specific advice. In order to bring down this concepts to risk management practice, the other side of the coin, quantitative analysis, is indispensable. Science begins with concepts, ideas (one could even argue there’s some creative component to it), but it always ends with some imperfect quantification of qualities, that is the necessary ingredient for objectivity, which in turn is the basis for productive discussion, systematicity and discovery.

Rather than having to bet on heads or tails, we can more safely predict that the coin won’t land on its edge. Once again, thesis, antithesis, synthesis.