Expected shortfall and sigma


The concept of risk and its management in an effective manner is the core of all financial activity. Investors, individual or institutional are not just interested in returns from their investments but are also keen to get those returns in accordance to their risk preferences. Many measures of calculating the risk of an investment portfolio have been devised. The aim of this study is to learn about two important measures of risk of a portfolio, namely, Value at Risk and Expected shortfall. Once the background theories and techniques for their measurement have been studied in detail, an attempt is made to design an excel based tool to calculate their values for an equity portfolio of some UK stocks. This document serves as a design document for the software development process.


The word risk is derived from Italian word 'risicare' which means 'run into danger'. In the field of investments, it could therefore be said that by investing, the investor is running into the danger of uncertain returns. Why would then an investor ever want to put money in any venture that is risky? If we analyze the data from across the markets in the world, we would see that market has more often than less, rewarded the investors who were willing to take more risk; by giving more returns on their investments when compared to their more conservative peers [1]. This, in no way implies, that taking higher risk is a guarantee for higher returns. Investors who burnt their hands in the dot-com bubble in the beginning of the century or those who suffered massive losses in the Asian crisis in the late nineties had exposed their investments to very risky investments and had to pay dearly for the same. Risk therefore signifies the chances that the worst fears of an investor would come true. A safe option could be to invest in securities that are free of risk (theoretically at least). Consider for example, government bonds, believed to be almost default free, would be a good option. But, some events like Iceland going bankrupt in late 2008 and more recently Dubai crisis have shown that all investments are subject to risk at all times. An investor therefore should be able to quantify the risk and assess the maximum loss if an unfavourable event does occur. Given the risk of an investment alternative, the investor can thus demand the expected return on the investment [6].

Purpose of this study

The aim of this project is to understand risk management principles for practical scenarios by developing an Excel based tool. In this project, a study of two very popular risk measures, namely, Value at Risk and Expected Shortfall would be done. Both the measures would be compared to assess the benefits of one over another. The theoretical concepts will then be applied on a portfolio of equity stocks and calculation of VaR and expected shortfall would be then done for the portfolio. Monte Carlo Simulation would be used to simulate hypothetical portfolio values. Monte Carlo simulation has an advantage that many scenarios can be captured and it does not require any assumptions of parameters be made like in the variance-covariance method. Monte Carlo can be implemented using various tools like Matlab or Excel. Excel has been selected for this study because of its excellent data analysis capabilities, built in functions and ease in writing the code using VBA.

To develop the tool, waterfall approach to software development would be used wherein all the stages would be performed sequentially. At the end of each stage, the deliverables would be tested against the requirements and it will be ensured that desired functionality has been implemented.

The result of this study would be an in-depth study of risk management and the excel based tool

Theoretical Background

Risk is the degree of uncertainty associated with an action, such as project implementation within time and budget, profitability of a project, market returns on an investment etc.,[7]. It's an exposure to uncertain future outcome, i.e. risk is the chance that the actual outcome will be different from the expected.

Risk is directly proportional to level of uncertainty. Higher the uncertainty, higher is the risk. It is often the single largest factor determining the rate of return that an activity will provide. Annualized standard deviation of return is the generic measurement of risk in most markets, but asset and liability managers also look at the entire probability distribution of returns and the maximum cost of adverse developments to assess the risk.

Banks need to manage different kinds of risks, with seemingly opposing needs. For example, providing liquidity to the depositors on demand as well as credit to the borrowers with the ultimate objective to maximize returns.

It is therefore of paramount importance to mange risk in an efficient and effective manner [8].

The Concept of Risk Management

Risk management, as it is understood today, largely emerged during the early 1990s, in the wake of the extraordinary upsurge in the number of bank failures. UK small banks' crisis of the early 1990s raised basic questions about the effectiveness of the banks' risk management practices, regulatory and deposit insurance systems [4].

Official emergency liquidity support to UK banks occurred in the early 1990s, when the Bank of England had to lend to a few small and medium sized banks in order to prevent wider loss of confidence in the banking system. A rather larger group of small banks got into difficulty and was subject to intensified regulatory monitoring because of the misallocation of resources when banks and thrifts poured funds into high-risk commercial real estate lending and failed to recover their investments; About 25 banks failed or closed due to problems during this period. Risk management suddenly became the buzzword across the financial markets [9].

Many tools and techniques are available to measure and manage risk. Most of the techniques have been practical applications of theory of statistics.

Value at Risk

Every investor want to know what is the most that he can loose in an investment. Value at risk is the measure that tries to answer this question. It can be defined as the potential loss in value of a risky asset or a portfolio over a defined period for a given confidence interval. Thus, if the 10 day VaR on a portfolio is £100 million at 95% confidence level, then there is only a 5% chance that the value of the portfolio will drop more than £100 million over any given 10 day period. It must however be stressed that VaR measures "normal market risk" as opposed to all risk (normal and abnormal). VaR is very frequently used by commercial and investment banks to capture the expected loss in their traded portfolios over a specified period [2].

Calculation of Value at Risk

There are three main approaches to measure VaR, namely, the historical method, the variance-covariance method and Monte Carlo simulations [3,5].

The historical method is a very simple way to calculate VaR for portfolios and can be implemented by creating a hypothetical time series of returns on the portfolio, obtained by running the portfolio through actual historical data and computing the changes that would have occurred in each period.

The variance-covariance method tries to derive the probability distribution of potential portfolio values in a given time period which is then used to compute VaR. In this method we estimate the variance and covariance values of all the assets in the portfolio by looking at the historical values of the portfolio. Value at Risk for the portfolio is then computed using the weights of each asset in the portfolio and the variance and covariance in these assets. This method is very easy to implement and is based on the assumption of normal distribution of returns. The downside of this simplistic assumption is that in almost all practical scenarios there is a considerable deviation from the normal distribution (fat tails) and therefore the results of this method may not always be reliable. One popular model for such scenarios is Generalized Autoregressive Conditional Heteroscedasticity (GARCH) which was proposed by Bollerslev (1986) [12]. The GARCH(p,q) model says that for any time series, the variance σt can be expressed as where a0>0, ai>=0 for i = 1,2,....q and bj >=0 for j = 1, 2,....p. σ is the standard deviation of returns, "t" is time, and ? is the random variable also known as the unpredictable component of the time series. The GARCH model is very good for calculations when thick tailed distributions are to be analyzed. However, it presents few practical implementation problems like a) a need for very large number of observations and b) being unstable for wild market fluctuations. Hence the Monte Carlo methods have gained popularity for measuring VaR [11].

The Monte Carlo Simulation method uses market risk factors that affect the assets in a portfolio and then computes the variance and covariance in the portfolio using simulation technique [11]. Thus, the portfolio manager can specify probability distributions for each market risk factor and also specify how the market factors move together. In each simulation run, the market risk variables take on different outcomes and the value of the portfolio reflects the outcomes. After a repeated series of runs, usually in the thousands, distribution of portfolio values is obtained that can be used to assess Value at Risk. To simulate stock prices, the most common model is geometric Brownian motion (GBM) [13]. GBM assumes that a constant drift is accompanied by random shocks. While the period returns under GBM are normally distributed, the consequent multi-period price levels are lognormally distributed. The formula for change in stock price using GBM is give as:

ΔS = S(μΔt + σε√Δt)

where S is the stock price, μ is the expected return, σ is the standard deviation of returns, "t" is time, and ε is the random variable.

Evolution of the concept of Expected Shortfall

The concept of Value at risk has been at the core of all risk management initiatives undertaken by financial institutions [4]. It is defined as the loss level that will not be exceeded given a certain confidence level during a given time period. For example, if the 10 day VaR at 99% confidence level for a portfolio is £1 million, then there is 1% chance that portfolio will loose more than £1 million in the next 10 days. While this is a very good measure of overall risk of the portfolio, it still allows for portfolios where the expected loss may be much greater than £1 million (say £5 million) even though the chance of that happening is 1%. If certain restrictions are imposed on creation of portfolio only on the basis of VaR, portfolios can be created with much greater risk even within the confines of VaR restrictions.

A better measure available to our disposal is expected shortfall; sometimes also referred to as conditional VAR, or tail loss. Where VAR asks the question 'how bad can things get?'; expected shortfall asks 'if things do get bad, what is our expected loss?'

Expected shortfall, like VAR, is a function of two parameters: N (the time horizon in days) and X% (the confidence level). It is the expected loss during an N-day period, conditional that the loss is greater than the Xth percentile of the loss distribution. For example, with X = 99 and N = 10, the expected shortfall is the average amount that is lost over a 10-day period, assuming that the loss is greater than the 99th percentile of the loss distribution.

Calculation of Expected Shortfall

Let us assume that we have the value of 99% VaR, which is about 2.67 Standard deviations from the Average (50%), Assuming Gaussian distribution, we would find the 50% point, and this will be our current unrealised Profit/Loss. In the next step, we would calculate the standard deviation from

(VaR - Unrealised P/L)/2.67

The next step would be to find out how many standard deviations the 90% point is (let's say x). Then the Expected Shortfall would be given by

E = x*standard deviation.

Concept of Coherence

Four important properties for a good risk measure were proposed by Artzner (1999) [4]. These are:

  • Monotonicity: If a portfolio always has higher returns than another portfolio, its risk should always be greater.
  • Translation invariance: If cash K is added to a portfolio, its risk measure should go down by K.
  • Homogeneity: changing the size of a portfolio by any factor (say x) while keeping the relative amounts of different items in the portfolio the same should result in the risk measure being multiplied by the same factor.
  • Sub-additivity: the risk measure for two portfolios after they have been merged should not be greater than the sum of their risk measures before they were merged. This is also known as principle of diversification.

The first three conditions are simple and straightforward. The fourth condition states that diversification helps reduce risks. When two risks are aggregated, the total of the risk measures corresponding to the risks should either decrease or stay the same.

Risk measures satisfying all four of the conditions are referred to as coherent.

VAR satisfies the first three conditions, but it does not always satisfy the fourth.

Risk managers sometimes find that, when two different portfolios are combined, the total VAR goes up rather than down. It can be mathematically proved that expected shortfall measure satisfies all the four conditions of coherence and is therefore a better measure of risk.

Software Development

To augment the study of theoretical concepts of VaR and expected shortfall, excel based software tool would be developed that would calculate these values for an equity portfolio with multiple stocks of UK companies. Since it is a relatively small project of academic nature, the requirements are not too complicated and are well defined. A waterfall approach to software development is proposed. There would be three sub parts to the complete project and all of them would be done in a VaR and expected shortfall, 2) Software requirements specification and design and 3) Software development and testing. In each part again there would be three to four sub parts and the activities for each part would also be taken up sequentially. At each stage of the development life cycle, the milestones would be evaluated to see if the progress is as per the plan. There would be a provision for testing the software at each stage in the development process. The results would be tested against the requirements defined initially.

Software Requirements Definition

The following requirements have been defined for the software:

  • Simple and intuitive user-interface so that even a user unfamiliar with these concepts can use the tool and gain an insight into the functionality of the tool.
  • Externalization of execution parameters such that changing these does not require change in code. This would separate the functional logic from code logic and future changes would be easier.
  • Minimum effort to be required from the user like just entering the raw pricing data and the calculation parameters. Once that is done everything should be calculated on click of a few buttons.
  • Division of code in small modules that communicate with each other so that maintenance of code is easier and parts of code could be reused wherever similar functionality is required.
  • The tool should allow for entering historical end-of-day pricing data for one or more UK stocks.
  • There should be a provision to enter quantity/weight of each stock in the portfolio in order to calculate VaR and expected shortfall for the portfolio.
  • The results should be summarized on a separate sheet in a clear and concise manner.

Software Design

The basic framework of code was designed keeping in view factors defined in the software requirements definition. The idea was to keep a simple workflow for end user; something similar to shown below:

Using the above premise, the tool was built. The parameterized design would help in giving more flexibility to the tool and user could evaluate the tool in many scenarios by setting the appropriate parameters.


  1. Anonymous 1991, 'Risk and Return', The Economist, pp. 1-2.
  2. Larsen, N., H. Mausser and S. Ursyasev, 2001, Algorithms for Optimization of Value-at-Risk, Research Report, University of Florida
  3. Coronado. M 2002, A Comparison of Different Methods For Estimating Value-at-Risk (VaR) For Actual Non-Linear Portfolios: Empirical Evidence, Universidad P. Comillas de Madrid, Madrid, Spain.
  4. Expected Shortfall: a natural coherent alternative to Value at Risk. Carlo Acerbi & Dirk Tasche May 9, 2001 http://www.bis.org/bcbs/ca/acertasc.pdf
  5. Jorion, P 2001, Value at Risk: the new benchmark for managing financial risk, 2nd Edition, New York, McGraw Hill.
  6. Mathis, R 2004, Corporate Finance Live: Risk and Return, Prentice Hall, Upper Saddle River, New Jersey.
  7. McCracken, M 2004, CAPM, viewed 10 March 2009 .
  8. 'The Financial Stability Conjuncture and Outlook', 2004, Financial Stability Review, June, pp. 46-60.
  9. Zakrajsek, E 2003, 'Recent Developments in Business Lending By Commercial Banks', Federal Review Bulletin, December, pp.1-30.
  10. Christopher Ian Marrison, 'The fundamentals of risk measurement', New York, McGraw Hill, pp.102-130.
  11. Guglielmo Maria Caporale, Christos Ntantamis, Theologos Pantelidis, and Nikitas Pittis - The BDS Test as a Test for the Adequacy of a GARCH(1,1) Specification: A Monte Carlo Study Journal of Financial Econometrics http://jfec.oxfordjournals.org/cgi/reprint/3/2/282?maxtoshow=&HITS=10&hits=10&RESULTFORMAT=&fulltext=monte+carlo&searchid=1&FIRSTINDEX=0&resourcetype=HWCIT.
  12. The Use of GARCH Models in VaR Estimation. Timotheos Angelidis, Alexandros Benos and Stavros Degiannakis,June 2003 http://statathens.aueb.gr/~sdegia/papers/HFAA2003ft.pdf
  13. http://www.investopedia.com/articles/07/montecarlo.asp?viewed=1

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