# Value at Risk (VaR) introduction

In this article we’re going to discuss what value at risk is. Basically what we’re trying is to estimate the level of possible losses over a given period of time with a certain probability. We have loss given certain probability.

## The basic principle of Value at Risk

Value at risk by default assumes that in a specific time horizon, for example one month, we can calculate the 95% value at risk confidence interval loss using historical data or some assumptions. So value at risk is a 95% (if we’re using the 95% confidence interval) the maximum loss in your portfolio.

But you can see also the other way around. Using the 5%, the 5% is the minimal loss that you’re going to have on your portfolio during that month.

So we have 5% or we have 95% confidence interval. Usually we work more with the 5%, because this is associated with 1 / 20, that is, once a month. There is a probability that you once a month will lose the value at risk or more.

The problem with the value at risk, and you will see that weakness later, is if you have that number, you don’t know really what is going to be your maximal loss. If you have this value at risk that is, for instance, minus 3%, if that’s your value at risk, you have a 5% probability of losing minus 3% or more.

But you can lose 10%, or you can lose 20%. But the value at risk is not mentioning anything about that. You’re going to see some of those extensions in the rest of the article.

## Building a graphical understanding

To understand a little bit this concept, let’s have this graph. We have the Dow Jones index. This is the index from January 2000 and August 2015. This is daily data. And we have, as you can see, fluctuations on this Dow Jones index, along with some trends.

There are trends in this image. For instance, we see clear upward trends from around 2009 to 2014. On the other hand, there are downward trends from around 2007 to 2009.

So what happens with the drops we observe? How big are these drops? Are you concerned if you have a portfolio with this asset if you’re facing this kind of probabilities that you will lose a big part of your portfolio? This is what you can solve with the definition of value at risk.

To understand better this, we use to build or analyze assets not using the prices, but using the returns.

So in the image above here, we have the same index, but we have the returns.

What is the definition of return? We have basically two definitions.

1. One traditional definition is that return is equal to: $$r_{t}=\frac{P_{t}-P_{t-1}}{P_{t}}$$ That’s the arithmetic definition. Then you multiply by 100.
2. Or you have the logarithm definition, where the return is equal to: $$r_{t}=\log \left(\frac{P_{t}}{P_{t}-1}\right)$$

I’m using the second definition. And why is that? Because basically what we are dealing with here is we’re trying to understand the evolution of the level of the price of the stock or the price of the bond.

We need some behavior from $P_{t}$ . And usually this has a log-normal distribution. If we have a log-normal distribution, then we can prove (we won’t do that here) we can prove that the returns are a normal distribution.

So if the returns are normally distributed, then we can do a lot of things that we will see in the following paragraphs. Let’s work with these returns.

Let’s go back to the image of the returns. So as you can see, we have positive and negative returns. So for instance, we have some rare positive returns around 10%, 11%, but also returns around 6%.

But here we also have negative returns. So if you have a portfolio, you’re concerned about these drops in the stock prices or in the price of bonds, or in the increase of the interest rates if you have a bond inside your asset. So how do we deal with this volatility?

That’s why we have this concept of value at risk. So the basic definition will be to just leave 5% of the sample of the historical events– for instance, just to explain basically what is value at risk– below some line.

This line will be the value at risk at 5% if we have these samples, this number of events that belongs to the 5% of the sample. That’s the 5% of the sample. So if we have 100 data points in returns, we have five data points below that value at risk. One way to analyze this is to work with a histogram. Most of the series here, most of the values are around 0.

So you can build a histogram, flip it 90% on the side, and you can build a histogram here, or a histogram the other way. So you have here the area that belongs just to 5% of the sample. This number will be the value at risk. So this number here will be the value at risk for 5% of the sample. So in the following minutes, we’re going to cover this histogram with actual data. And you’re going to work in Excel so you can be familiar with the concept of historical simulation and another kind of measure of value at risk.