This work (that actually is the main part of my recent internship experience) was motivated by references to risk management of futures portofolios in various multifactor commodity models related papers.

Goal: Find a model that estimates correctly the Value at Risk of commodity future’s portofolios

But, what is Value at Risk? and what is risk in general?

Well according to wikipedia we have this definition:

**Risk** is the potential that a chosen action or activity (including the choice of inaction) will lead to a loss (an undesirable outcome).

Okay, but we are operating in markets.

We have a portfolio of various financial instruments.

What is this potential and how do we quantify it in order to make it useful?

Well, that is the reason why we need risk measures and though the term might seem fastideous, in fact we do derive heuristically such measures when required (as many things we do without generalizing it or putting a complex scientific name on it). For example you need to take the plane for a job/deal or holiday (I’ll retain the deal), airline companies will try to prove me wrong but there is always a risk that the plane might crash. In order to take a decision whether to take the plane or not, you need to compare the gain with the risk, so we might quantify the risk as the lowest amount of money you will get so that you will choose to jump into the plain and close that deal (Actually the notion of Value at Risk is not so far from this one). We can also see from this, that the perception of risk is different between people, the lowest amount of money I need to take to get in the plane is not the same as Bill Gate’s for example, so everyone’s way to measure this potential is different.

The notion of risk measure might seem not-that-useful in this trivial case but we can make it more useful by comparing it with the notion of gain (that’s actually our first risk measure) and then solve problems of conditional gain maximization/risk minimization. Actually a famous investment criteria broadly used throughout the world of finance is the Sharpe ratio, that is nothing but the ratio: gain/risk. There, the risk is expressed as variance but for many reasons variance is not a good measure of risk. For example, your gains might be only positive and have a larger variance than when you might have potential negative values, but a bigger problem is the fact that variance does not take into account extreme events losses (for example a financial crisis). In the plane example if risk is the magnitude of perturbation in the plane trajectory, variance would not be a good indicator of the probability of it crashing, so the risk would be 0, great! Let’s take the plane.

*Though the Sharpe ratio as an investment policy might seem problematic, from this standpoint it still remains the reference in the field for its simplicity and the fact that it requires basic the knowledge of basic concepts that people that work in finance are familiar to.*

We all see the necessity to create a better measure of risk. Okay now if I gave you the magnitude of perturbation so that the probability that it is surpassed is equal 1%, wouldn’t that be more useful and give you more information in order to decide whether to get in the plane or not?

Okay enough with the torture let’s throw the mathematical definition:

**Definition (Value at Risk)**

*If X is a random variable expressing the losses of a portfolio the Value at Risk of level a% is the a%-quantile of its distribution (So its nothing but a quantile).*

The historical background behind its notion can be found in Charpentier’s notes on risk measures (in French unfortunately), and surprisingly it comes from vaccination problematics at the ending of the 17th century. Then it was used formally after the 80s petroleum crisis and published by JP Morgan in a risk measurement procedure under the name of Risk Metrics. Today, it is considered to be the most used risk measure in banking and its usage is further backed by banking regulation *(Basel *II)

Okay, history is nice but from a mathematical viewpoint we learned not much, I will remediate to that in later posts.

PS: this risk measure still is imperfect, we will detail why in another post.

ref: Arthur Charpentier, polycopié Mesures de risque

see: Samuelson’s coin experiment