# Value at Risk or Expected Shortfall

### Pablo Sánchez

#### 26/09/2018

2

Value at Risk and Expected Shortfall are related to the risk taken by a portfolio but… Which one is the best? Let’s learn together the differences between these two measures.

## Risk measures

Coherence is really important when defining a risk measurement. If the measure is not coherent, it will not give us adequate information about the risk of our portfolio. For that reason, there are four properties required by any risk measure:

1. Monotonicity: if a portfolio achieves higher returns than another in every state of the world, then it will have lower risk.
2. Translation invariance: if an amount of cash is added to our portfolio, the risk will be reduced by that amount.
3. Homogeneity: maintaining the weights, if the size of a portfolio is increased by a factor, the risk will be multiplied by the same factor.
4. Subadditivity: the risk measure of two merged portfolios should be lower than the sum of their risk measures individually.

### Value at Risk

#### Definition

The Value at Risk (VaR) is a statistic used to quantify the risk of a portfolio. It represents the maximum expected loss with a certain confidence level.

We use this measure to answer the following question:

What value of a given portfolio is at risk?

#### How is it calculated?

Given a confidence level (α), the VaR is the αth percentile of the portfolio’s return distribution. For example, the VaR 95 of a portfolio is the 5th percentile of its return distribution.

### Expected Shortfall

#### Definition

The Expected Shortfall (ES) or Conditional VaR (CVaR) is a statistic used to quantify the risk of a portfolio. Given a certain confidence level, this measure represents the expected loss when it is greater than the value of the VaR calculated with that confidence level.

This measure is used to answer the following question:

If things go bad, what could the expected loss be?

#### How is it calculated?

Given a confidence level (α), the ES is the average of the portfolio returns that are lower than the value of VaR calculated with the confidence level α.

## Value at Risk vs Expected Shortfall

The Value at Risk measure always satisfies the first three properties but it will only satisfy the fourth one if portfolio returns follow a normal distribution. On the other hand, the Expected Shortfall measure satisfies the four properties in any circumstance. I strongly recommend reading the following paper where the subadditivity property in the Expected Shortfall measure has been proven (some maths background may be necessary):

Seven Proofs for the Subadditivity of Expected Shortfall

For that reason, it is said that the Expected Shortfall measure is more adequate than the Value at Risk.

### Example

We are going to see an example of why the Expected Shortfall measure is useful. We have two assets (both from the S&P 500 close prices time series) and we have to decide which is the riskiest investment. As we can see in the previous plot, both daily returns’ distributions are quite similar. The main difference between them is the shape in the left tail (the risky area). Then, we can use the Value at Risk measure to make a decision.

When we calculate the VaR with 5% of confidence level (VaR 95), we see that both assets have the same result. We can compare VaR using another confidence levels (3%, VaR 97 or 1%, VaR 99) to help us but we are going to use the Expected Shortfall with the same confidence level (5%).

Using this measure, we conclude the turquoise distribution is riskier than the black one. I let you the code I used for this post.