By Tostne Kutalia
If you are reading this blog, this means that you are aware of Value at Risk, or simply VaR which is the largest loss that can be incurred by a portfolio with a given confidence level (i.e. a pre-specified probability level).
However, there is a significant drawback which the VaR as a risk measure carries. In particular, it tells us the threshold of potential loss but not how bad the loss can be beyond that threshold.
For example, if a portfolio’s 1-day VaR is $1 million at a 95% confidence level, you don’t know how much the loss could be in the remaining 5% of the cases. In addition, it does not account for “tail risk" or extreme events that may lead to larger losses. This brings us to yet another statistical risk measure called Expected Shortfall (ES), which fixes this problem to some extent.
As seen in the preceding paragraph, to be fully equipped to understand the topic of Expected Shortfall, first one needs to have the knowledge of VaR. What follows is the list of prerequisites needed alongside their descriptions.
Prerequisite knowledge needed to make most of this blog post:
- To understand the basics of Value at Risk (VaR), refer to the introduction in this Value at Risk blog. It explains how to compute and interpret portfolio VaR, along with its limitations and advantages, in a simple and concise manner.
What is Expected Shortfall (ES)
Simply put, Expected Shortfall is the average loss beyond VaR. It measures the expected loss in the tail of a distribution beyond a certain quantile level (e.g., 95%). It provides insight into the potential losses exceeding the Value at Risk (VaR). Obviously, we specify the confidence level by which ES is computed. Below is the geometric illustration of ES alongside with VaR. Here are the steps for the computation of ES:
1. Define Parameters
- Confidence Level (c): Choose the confidence level, typically 95% or 99%.
- Loss Distribution: Have the historical or simulated distribution of portfolio returns or losses.
- Calculate portfolio value at risk (\( \text{VaR}_p \)): This is also computed by the same confidence level.
where z stands for standard normal quantile corresponding to c probability, p is the standard deviation of portfolio returns and W represents the portfolio value in dollars.
2. Calculate Expected Loss Beyond (VaR): This is in effect the Expected Shortfall and is computed by the formula
where \( L \) denotes the loss as a random variable (i.e. the possible value of actual loss). The above formula is read as the expected (average) loss given that this loss is greater than \( \text{VaR}_{p} \).
The formula of Expected Shortfall is a theoretical one. We take empirical (historical data based) and analytic approaches to compute this quantity.
Empirical Approach:
The simplest way to compute the ES from historical data is to find VaR using the non-parametric method. As long as VaR is computed, one needs to simply average out all historically observed values beyond this quantity.
Example:
We extend the example given in the section “Non-Parametric VaR” of blog Value at Risk. In particular, as long as the annual VaR (in this case we use VaR (zero), however VaR (mean) would also work) has been computed, we compute the average value of losses beyond this level. Note that these quantities are computed in the blog on the given link above. See the attached Excel file for full computations as well.
The given portfolio in this example is initially worth $1,000,000 and it consists of three assets. First, the portfolio values are computed.
Then we compute the loss incurred compared to the initial value of the portfolio.
The VaR turns out to be $326,554.42 (computed in the blog above) while the Expected Shortfall is $337,559.84. This quantity is computed by (2) as illustrated below
We interpret this quantity as the expected loss in extreme circumstances. To make it clearer and simpler, $337,559.84 is the expected amount if the actual loss surpasses $326,554.42 (the VaR). This is a self-explanatory number in the sense that it provides an impression about what to expect in extreme cases, i.e. if the 5% probability of loss exceeding VaR is realised.
Analytic Approach:
Instead of computing ES by non-parametric method, as long as the parameters of loss distribution are estimated, we can compute it via the analytic method. In particular, the ES given by (2) is equivalent to
where \( l \) denotes the loss. So this formula just gives the average of the losses beyond \( \text{VaR}_{p} \) adjusted to the confidence level \( c \). If the distribution is known, the computation becomes simpler. In the simplest case, assuming the losses are normally distributed (which is quite a basic assumption, not necessarily true though), we compute the ES by the following formula:
where L and L denote the mean and standard deviation of losses respectively. Note that these quantities have the indexes to differentiate them from portfolio return mean and standard deviation. z usually represents the standard normal quantile corresponding to c confidence level and is the standard normal density function. (Would be great to have a link for probability distributions). The computations are given in the example below
Example cont’d:
Now we compute the Expected shortfall using the analytic, a.k.a. parametric method. Estimated mean and standard deviations are respectively.
we can compute Expected Shortfall by (4) to be $328,130.07.
Note that this quantity is slightly lower than the ES value computed by non-parametric method, however both certainly are greater than VaR.
Pros of using Expected Shortfall as a risk measure:
1. Accounts for Tail Risk
- Unlike Value at Risk (VaR), which only considers the threshold loss at a given confidence level, ES averages all losses beyond the VaR, providing a more comprehensive view of tail risk.
2. Better risk sensitivity
- Unlike Value at Risk (VaR), which only considers the threshold loss at a given confidence level, ES averages all losses beyond the VaR, providing a more comprehensive view of tail risk.
3. Regulatory Acceptance
- ES is increasingly used in financial regulations (e.g., Basel III) because it overcomes some of VaR’s limitations, such as ignoring losses beyond the VaR threshold.
4. Flexibility
- ES can be applied to various distributions and risk scenarios, including heavy-tailed distributions and non-normal data.
Cons and Limitations of using Expected Shortfall as a risk measure:
1. Estimation challenges
- ES requires accurate modeling of the tail of the loss distribution, which is difficult in practice due to the scarcity of extreme loss data.
2. Data and dependence
- Historical and simulation-based methods can produce inaccurate ES estimates if the data does not adequately capture future risk scenarios.
3. Computational intensity
- Calculating ES can be computationally demanding, especially for complex portfolios or when using Monte Carlo simulation.
4. Interpretation complexity
- The choice of the confidence level (c) significantly affects ES, introducing a degree of subjectivity.
5. Sensitivity to model assumptions
- The accuracy of ES depends heavily on the assumed distribution of losses. Mis-specification of the distribution can lead to unreliable estimates.
Conclusion
This blog covered an interesting quantity – Expected Shortfall (ES). ES is the expected loss surpassing the VaR level. It gives a valuable information in the sense that during the realization of extreme losses, one knows what to expect on average. VaR as a risk measure, fails to account for this loss. There are several ways to compute Expected Shortfall. Examples were given illustrating computations of ES by non-parametric method, which gives an estimate using the historical data and an analytic method, which is more robust assuming the distribution of losses is known and the parameters of the distribution (like mean and standard deviation) estimated. Finally, the topic is summarized by providing the list of pros and cons of using ES as a risk measure.
Files in the download:
- Excel file
The excel file illustrates the computation of Expected Shortfall for a portfolio consisting of three stocks. Portfolio VaR is computed first. The Expected Shortfall is computed by parametric and non-parametric approaches.
- Python notebook
The python code snippets illustrate the computation of Expected Shortfall step by step. The portfolio VaR is computed first on top of which the portfolio Expected Shortfall is calculated.
Bibliography:
- Jorion, P. (2001). Value At Risk: The new benchmark for managing Financial risk. New York: McGraw Hill.
Continue learning:
- For practical, step-by-step tutorials on computing VaR, see
Calculating Value at Risk in Excel & Python. This guide offers hands-on examples and code snippets to deepen your understanding of VaR calculations. - Learn quantitative portfolio management techniques such as Factor Investing, Risk Parity and Kelly Portfolio, and Modern Portfolio Theory using Python.
- Learn, create and backtest various position sizing techniques such as Kelly, Optimal f, and volatility targeting on a trading strategy, using Python.
- For a structured, hands-on learning experience, consider Portfolio Management and Position Sizing using Quantitative Methods on Quantra. This course covers position sizing techniques like Kelly Criterion, CPPI, and Volatility Targeting, along with Mean-Variance Optimization and the Fama-French Three Factor Model. You'll also explore factor timing, beta, covariance, and performance ratios, all while implementing strategies using Python libraries like NumPy, Pandas, Matplotlib, Seaborn, Sklearn, OLS, cvxpy, and TA-Lib.
All investments and trading in the stock market involve risk. Any decision to place trades in the financial markets, including trading in stock or options or other financial instruments is a personal decision that should only be made after thorough research, including a personal risk and financial assessment and the engagement of professional assistance to the extent you believe necessary. The trading strategies or related information mentioned in this article is for informational purposes only.
