# Difference between risk and uncertainty

I'm reading the paper "Risk Measures in Quantitative Finance" by Mitra and Ji (2010). On page 2 it reads:

We note that some Academics distinguish between risk and uncertainty as first defined by Knight [Kni21]. Knight defines risk as randomness with known probabilities (e.g. probability of throwing a 6 on a die) whereas uncertainty is randomness with unknown probabilities (e.g. probability of rainy weather). However, in Financial risk literature this distinction is rarely made.

Let's say I have a portfolio optimization problem: $$L=\text{argmax}(C, X),$$ s.t. $$AX \le B, X \ge 0.$$ In a general case elements of vectors $C$, $B$ and the matrix $A$ can be random numbers. But we can have two cases: a) distibution characteristics of random variables are known (this is the case of risk), b) distibution characteristics of random variables aren't known (this is the case of uncertainty).

Many of the tags at Quant Stack Exchange contain the word "risk" and only one tag contains the concept "uncertain".

I'd like to know an opinion of the QF community on this problem:

Do you distinguish these concepts: risk and uncertainty especially in statistical sense?

References:

• F.H. Knight. Risk, uncertainty and profit. Houghton Mifflin Company, 1921.
• Mitra, Sovan, and Tong Ji. "Risk measures in quantitative finance." International Journal of Business Continuity and Risk Management 1.2 (2010): 125-135.
• Nick, I would advise you to not get too deep into the weeds about this. Note that the trader’s definition of risk is related to margin calls, and approximately zero progress has been made via academic hair-splitting since Frank Knight through Russell, Savage, Wiener, the Vienna Circle, …. Trust me, this is not a productive question. – isomorphismes Jan 27 '17 at 17:46

I am one of the people (a minority in Quant Finance) who think there is a difference between Risk and Knightian Uncertainty, and Alan Greenspan is another. Uncertainty explains what is meant by Financial Crisis: a situation (such as 2008) where the information is so confused that people find it impossible to assign probabilities to different scenarios. In this case investors generally stop what they are doing (they refuse to transact) and the standard methods of Risk Management become less applicable (if at all). What were previously thought of as No Arbitrage conditions may be violated. The markets essentially stop functioning temporarily and we rely on outside intervention by central banks, governments etc. to get by. It is a dangerous and scary situation, although t typically does not last long.

However, as Sovan Mitra said, most people in Financial Risk do not agree with this view.

• How would you rigorously define risk vs uncertainty? – SMeznaric Nov 30 '16 at 10:06

In classical finance you only work with risk (according to the abovementioned definition), which you define in a certain way (e.g. volatility), try to estimate and work with the resulting probabilities.

Having said that you will discover that in practice there is often a certain spread priced into financial instruments (like derivatives) which accounts for the unknown unknowns (like e.g. black swans), e.g. when you look at deep OTM puts you will find that they are priced higher than standard pricing theory would suggest - this is uncertainty priced into these instruments.

Another thing are concepts that are called risks although in reality they are uncertainties according to the abovementioned definition, like operational risks. These are notoriously hard to quantify, see e.g Power, M.: The invention of operational risk (2003).

It is easier to talk about these concepts by moving past Knight a bit. There has been a lot of discussion since then. The easiest way to start talking about this is in discussing the three main schools of thought in probability and statistics. The schools are, in order of discovery, the Bayesian, the Likelihoodist and the Frequentist. We are going to ignore Fisher's Likelihoodist school because it is epistemological in nature and implies no logical rational behavior.

In the Bayesian school there is no such thing as chance. It doesn't exist. There is only uncertainty. To give an example, imagine you watch someone toss a coin twenty times and it comes up head every time. Now, imagine the same person tosses it H-T in perfect alternation in ten pairings of twenty tosses. The person tossing the coin is the Amazing Randi. He lets you inspect the coin and you have a physicist friend test it and it is a perfectly fair coin.

You are uncertain as to whether the coin is the same coin that was tossed. You are uncertain as to whether the Amazing Randi has sufficient control to cause the pattern of tosses. You are even uncertain as to whether the Amazing Randi had complete control the entire time or whether some of the tosses were really accidents that worked out despite a loss of control.

In Bayesian models there is only an absence of information. Probability exists in the mind of the person making decisions. It is not a physical property of the universe. This is even true in quantum mechanics which is usually interpreted in terms of chance. The goal is to use data to construct a distribution of "plausabilities," to use Cox's terminology.

In the Bayesian framework, data is fixed and not random. There can be no such thing as a "random sample." It is non-nonsensical in meaning. Because of this, the Bayesian definition of an expectation is $$E(\theta|X)=\int_{\theta\in\Theta}\theta{f(\theta|X)}\mathrm{d}\theta.$$

If you would gamble on an uncertainty and an expectation exists, it does not exist for heavy-tailed distributions, then you integrate over the set of all possible models of the world.

In the Frequentist world $\theta=k,k\in\Re$ and there is no uncertainty at all. Rather than use the word "risk" at this point, we will discuss chance as the word risk has an additional meaning since Knight. Events can happen only due to chance and when used in the calculus of variations, then all parameters are treated as perfectly known. Chance is a physical property of the universe.

Mechanically, this is no different than treating the null hypothesis as strictly true. In the Frequentist school there is no such thing as a p-value. That is only part of the Likelihoodist school. In the Frequentist school there is a value, $\alpha$, that creates an acceptance region and a rejection region. If a sample implies that the null cannot be rejected, then it is treated AS IF true. The Frequentist school is behavioral. You do not know if it is true, but the only rational behavior is to act as if it is. If it is in the rejection region, then you are are to behave AS IF false and AS IF the minimum variance unbiased estimator is the true value.

In the example, for the first test for a null of $\theta=.5$, the null would be rejected. In the second sample, despite the weird pattern and the past observations, it would be accepted as true. The samples cannot be combined due to the axioms. This creates a weird to irrational response where one moment you are behaving as if false and the next as if true.

Finance has almost entirely dropped the Bayesian meaning and almost entirely adopted the non-Bayesian meaning. You rarely see a serious discussion of uncertainty in finance because all mean-variance finance models are Frequentist models. They strictly depend upon Frequentist axioms.

The Frequentist definition of an expectation, where one exists, is $$E(X)=\int_{x\in\chi}xf(x|\theta)\mathrm{d}x.$$ Frequentist models live in the sample space, which is random.

Risk, in the Bayesian sense is the amount of resources exposed to uncertainty. Risk in the Frequentist sense is the amount of resources exposed to chance. Both definitions could have the phrase "of loss," appended to the end.

This focus on Frequentist methods is problematic because it blocks good modeling. Any function of the data is a statistic. Therefore $\sum\sin(x_i)$ is a statistic. Intuitively you may know it is a pointless statistic, but how do you show that mathematically? It turns out that the problem was solved with what is known as the complete class theorem.

Any solution inside the acceptable set is valid and any solution outside the set is invalid. The sum of the sines is an invalid estimator of the sample mean of the normal distribution. You can know this because it is outside the complete class of solutions.

All Bayesian solutions are inside the complete class of valid solutions. Frequentist solutions are valid either to the extent they map to a Bayesian solution for a specific sample, or at the limit. This is problematic because there is a recent proof that there does not exist a Frequentist solution for stocks. It is the reason the CAPM has never been validated and the heavy tails exist in the tests. Frequentist solutions are always outside the complete class of solutions. They are valid for Vegas style gambles and for corporate bonds with fixed coupons, but not for stocks.

I would recommend reading "Probability, the Language of Science," by E.T. Jaynes and "Decision Theory:Principles and Approaches," by Parmigiani. Models like the CAPM cannot work in a Bayesian framework, though it is far from obvious, which means they cannot work. It is unfortunate that Knightian uncertainty fell out of discussion for elegant models.

You can find papers on stocks and uncertainty at https://ssrn.com/abstract=2828744 and https://ssrn.com/abstract=2656681