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10th June 2019 | Michał Karol Ejdys | Portfolio Quantitative Analyst

# How to measure risk, what are key risk factors?

## From praying to gods to buying insurance, as a species we have come a long way in methods of addressing risk. In this article we try to define risk as we understand it in financial markets, together with some core sources of it.

We intuitively understand risk as a possibility of loss, or of an adverse or unwelcome circumstance arising. In that sense, it is a narrower term than uncertainty, which we understand as a possibility of more than one possible scenario occurring in the future. Even though, usually, the former is more of interest, we cannot understand or predict it without the tools, that were originally developed for the latter.

#### Humble beginnings

It is assumed that the first step to measure uncertainty, and hence, risk, was made in 1494 by an Italian monk, Luca Pacioli. Being a man of many talents, to say the least, he wrote Summa de Arithmetica, a mathematics book that contained the first probability riddle on record. It said: assume two gamblers playing dice. Five turns, in each the higher value wins. Now assume they were interrupted after three turns, with one leading two to one. The riddle is: what is the correct (fair) way of splitting the pot now?

It is this riddle that instigated the development of concept of outcome probabilities. Now it is probably somewhat intuitive to follow the path of reasoning that starts with: the die has six sides, there are 4 throws in total left, so the number of scenarios is six times six times six times six. Count the ones in which player one eventually wins, divide by the total – that’s his share in the pot.

It took us fifteen decades and two great mathematicians of the time (Blaise Pascal and Pierre de Fermat) to come up with the result (it is 75% for the leading player, by the way). The advance might seem small and painful in hindsight (150 years to multiply some numbers?) but on the other hand it enabled us to numerically assess any situation where the number of outcomes is finite (think: Brexit or no Brexit? Trump or Hilary? etc.). Such association of probability value with each from a number of scenarios is called a discrete probability distribution.

#### To infinity and beyond

But what about the case where the number of scenarios is… infinite? Will the GDP of the US be 1% higher than last year? How about 1.1%? 1.01%? For such questions involving real numbers we require a tool that enables us to calculate the probabilities for indefinite number of scenarios. We have to go from discrete probability distribution to a continuous one, meaning one that associates every possible number (rather than a fixed number of scenarios) with its probability value.

By definition, there is no fixed number of scenarios that we can simply count to estimate the distribution. Fortunately, Jacob Bernoulli discovered the so-called law of large numbers. Random sampling of items from a population has the same characteristics, on average, as the population. Meaning – if we analyse a large enough number of possible scenarios, we can chart their probability distribution. As people started to experiment with it, it turned out most of the experiments yield a similar, bell shaped distribution. De Moivre, Laplace and Gauss helped refine these empirical results to what we now know as the normal distribution. Normal distribution

#### How does this apply to financial markets?

As early as in 1900 a post-grad student of mathematics at the Sorbonne found no evidence supporting correlation of price movements over time. In other words, price increase today tells us nothing about the price behaviour tomorrow. It took the academic world nearly sixty years to catch-up, but now we all agree – price movements are random. And thus, we can use the Bernoulli’s aforementioned law of large numbers to infer their probability distributions.

From the probability distribution we can infer the standard deviation of returns, a proxy of volatility, which is the core risk metric introduced in another article.

Another interesting feature that is derived from probability is conditionality. Similar to the example with the game of dice, where we assumed a certain score following our set of random outcomes (finishing the game), we can assume, for example, that a stock’s value is related to the dollar. Dollar’s exchange rate would be a risk factor here. Now, if we calculate the stock’s exposure (think about it similarly to a correlation: how much thing 1 moves, when thing 2 moves by one?) to such risk factor, we will know the importance of analysing the dollar, when analysing the stock.

#### Risk factors

We can define many risk factors and calculate each asset’s exposures to all of them. This way we can understand the asset better and make more informed decisions. Common risk factors include:

• Currency risk – the effect the change in currency rates will have on the asset
• Country risk – the exposure of an asset to the economic situation of a given country
• Systematic risk – the exposure of the asset to the broader market
• Idiosyncratic risk – the risk inherently built-in in the asset, that cannot be diversified by holding other assets together in a portfolio

Let’s take the large US IT company known for their phones and computers, that's a part of the S&P 500 index and assume it has large exposure to the dollar, small exposure to the Gibraltarian economy, large exposure to the S&P 500 index (the U.S. blue-chip index) and medium idiosyncratic risk. That means we should expect large price swings in case the dollar moves, we shouldn’t care much about the economic outlook of Gibraltar, watch other large U.S. companies from the S&P 500, and finally after taking it into account, we are still left with some risk that is connected only to the internal situation at large US IT company known for their phones and computers, that's a part of the S&P 500 index.

#### Putting it all together

What we are basically doing with these risk factors, in a way (purists of probability please stop reading here), is multiplying probability distributions, a more advanced way of multiplying the number of possible scenarios in a game of dice. This, however, yields concrete information that influence investment decisions. It makes all the difference to invest in large US IT company known for their phones and computers, that's a part of the S&P 500 index, knowing that in fact you invest only a part ‘in it’, and the other part in a mix of the dollar, US stock market, and all the other risk factors that you might have contradicting opinions about.

Golden Sand Bank ("Bank") exercised due diligence to ensure that the information contained in this publication was not incorrect or untrue as at the date of publication.

All Investment products are at risk, as their value can go down as well as up. The tax treatment of your investment will depend on your individual circumstances and may change in the future. If you are unsure about whether investing is right for you, please seek financial advice. This publication is not an investment recommendation or investment advice in connection with any services provided by the Bank to the Client.

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