# Sigma and value at risk

### Abstract

All financial activity is subject to risk of getting lesser returns than expected or even loss of principal. For institutions like banks, it is imperative to measure and manage the risk of their entire portfolios such that they may serve liabilities whenever required and at the same time maximize their returns on investments. The most common measure of risk is Sigma or the standard deviation of expected returns from the mean return. It gives a fair idea of how much the returns may vary from the mean historical return for the portfolio. Another popular measure is the Value at Risk for the portfolio. This project is aimed towards gaining an understanding of these concepts in the field of risk management and then creating a tool to measure and compare their values for an equity portfolio.

### Prologue

ISO 31000, the family of standards related to risk management defines risk as the effect of uncertainty on objectives (whether positive or negative) [1]. 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.,. 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.

The risk of an investment is equivalent to the distribution of potential outcomes, where the distribution consists of all possible outcomes weighted by their relative probability of occurrence. The more extreme the distribution of outcomes, the riskier the project. Two projects could have the same expected return (the weighted average of all possible outcomes) but differ in their risk, if one project had a broader range of outcomes or a higher probability of extreme outcomes than the other. Risk 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.

### The Evolution of Risk Management

The importance of risk management at the bank level or at the industry level can't be more emphasized in the wake of the recent financial crisis that took the world on the brink of recession. Risk management has become a very important functional are in any banking organization and encompasses:

- Risk reduction through safety, quality control and hazard education,
- Alternative risk financing, including self-insurance and captive insurance, and
- Purchase of traditional insurance products, as suitable.

Derivative dealers have been promoting "risk management" by the use of derivatives to hedge market-risk exposures. This is the reason why derivative instruments are sometimes referred as "risk management products."[2]

### Role of Risk Management

Risk management is the practice by which a firm optimizes the manner in which it takes financial risk.

The objective of risk management is to

- Measure and manage risk
- Allocating appropriate capital
- Avoid failures or huge losses

It includes monitoring of risk taking activities, upholding relevant policies and procedures, and distributing risk-related reports.

Risk management is controlling the probability, and/or the severity, of a potential adverse event so that the consequences of that event are within acceptable limits. All risks have, the potential to generate losses, and preserving the invested capital is the underlying basis of risk management, it is sometimes meant to be equivalent to managing solvency risk. [3]

Risk management requires a sound understanding of the bank's operational environment, the risks to which it is exposed to and the techniques available to measure and manage these. Risk management is also affected by the infrastructure put in place to implement the chosen risk management framework. There are many different elements to this infrastructure, including organization, management, procedures and controls, all of which have an impact on the effectiveness of the implemented framework. One of the key factors is the effectiveness of the risk management system, whether an integrated treasury and risk management systems or a specialist risk engine.

Some of the key decisions and policies that needs to be formalized for the risk management purposes includes

- To what extent the losses needs to be avoided and to what extend is risk of loss a necessary part of the banking activity.
- Formalization of standard procedures for measuring risks and risk exposures
- The extent of decentralization for underwriting the loans and the incentives structure to align individual goals with the banks overall goals
- Capital allocation across various types exposures and whether there is excessive exposure towards particular risk
- Calculation of consolidated risk for asset types and allocation of appropriate capital
- Various risk mitigation techniques to reduce the risk.

Risk can be reduced by bring down the level of uncertainty to whatever extent possible. Institutions can reduce some risks by thoroughly researching on them. A bank can reduce its credit risk by conducting due diligence of its borrowers. A brokerage firm can reduce market risk by understanding the markets before it begins to operate in it.[4]

### Concept of Standard Deviation (Sigma)

Standard deviation or sigma (σ) is an important metric derived from probability theory and statistics. For a dataset, it is defined as the "square root of its variance". In simple terms, it is a measure of on an average how far away the values in the data set are from the mean. A low sigma would indicate that all the values in the population are very close to the mean and a high sigma would imply that the values are distributed far away from the mean.

The concept of sigma has a great significance in the field of risk management. It can be used to evaluate the risk of a portfolio by measuring the variation of returns of the portfolio with the average return of the portfolio over a period of time. The larger the variation, the larger is the possibility of extreme outcomes. While investors wouldn't mind positive extremes, negative extremes would certainly deter them from investing in portfolios that do not match the risk appetite of the investors.

### Calculation of Standard Deviation

Suppose the portfolio value for days 1 to n (1 being the most recent day) is x1, x2.....xi,....xn. Where xi is the portfolio value on ith day. Now, the daily return would be given by

Ri = (xi-1 - xi)/xi

So we will have a series of daily returns denoted by R1, R2,......Ri,.....Rn

The mean return of this portfolio will be calculated by the following formula:

Rm = (R1 + R2 + ......Ri + ......Rn)/n

The standard deviation of returns would be given by

The larger the σ, larger is the variation in returns from the mean and therefore greater is the uncertainty of getting the desired returns from the portfolio.

### Understanding Value at Risk

VaR is a powerful tool for assessing market risk. Unlike market risk metrics such as duration and convexity, or beta, which are applicable to only certain asset categories or certain sources of market risk, VaR is applicable to all types of assets in general and is based on the probability distribution of market value for a portfolio. [5]

The uncertain market values of all liquid assets can be characterized with their probability distributions. All sources of market risk serve as a contributor to those probability distributions. VaR is an all-encompassing measure of market risk because it is applicable to all liquid assets. [6]

To measure market risk in a portfolio using VaR, it is imperative to find some means for determining the probability distribution of that portfolio's market value. This becomes a challenging task for more complex portfolios that have more asset categories and sources of market risk.

There are three distinguishing concepts of VaR:

VaR is an algorithm with which we calculate a portfolio's exposure to risk.

A VaR model is the financial theory, mathematics, and logic that is used to measure VaR.

A VaR metric is the interpretation of the output of the VaR measure.

### VaR Calculation concepts

Value-at-Risk (VaR) is a measure used to estimate how the value of an asset or of a portfolio of assets will change over a certain time period (usually over 1 day or 10 days) under usual conditions. VaR has two parameters:

- The time period we are going to analyze (i. e. the holding period) and
- The confidence level at which the estimate is being made.

The typical holding period is 1 to 10 days and popular confidence levels usually are 99% and 95%. [7]

Value-at-risk (VaR) is an estimate of the maximum potential change in the value of a portfolio over a period given the historic pattern of movements in financial markets.For example, "95% of the time losses will not exceed $10 million over a two week period."

### Calculation of Value at Risk

All practical VaR measures accept portfolio data and historical market data as inputs. They process these with a mapping, inference and transformation procedure. Output is the value of a VaR metric.

Broadly, there are three different ways of calculating VAR: -

- Historical Simulation
- Variance - Covariance
- Monte-Carlo Simulation.

### Historical Simulation

Historical simulation requires relatively few assumptions about the statistical distributions of the underlying market factors. In essence, the approach involves using historical changes in market rates and prices to construct a distribution of potential future portfolio profits and losses and then reading off the VaR as the loss that is expected at a specified (for example, 5%) percentage of the time. The distribution of profits and losses is created by taking the current portfolio and subjecting it to the actual changes in the market factors experienced during each of the last N days. These hypothetical values of the market factors are then used to compute N hypothetical mark-to-market portfolio values.

The use of actual historical changes in rates and prices to compute the hypothetical profits and losses is the distinguishing factor of the historical simulation technique. Once the hypothetical mark-to-market profit or loss for each of the last N periods is calculated, the distribution of profits and losses and VAR can be determined.

The variance-covariance method This approach is intuitively very simple. All you need to do is calculate the volatility of an asset (at a specified confidence level) and use this volatility figure to calculate the value at risk.

This approach has two major limitations. One, it assumes that asset returns are normally distributed - this is a serious limiting constraint in most real life cases. And two, it assumes that the portfolio consists of assets whose deltas are linear - this makes this approach ineffective for handling portfolios with derivative instruments.

The Monte Carlo Simulation method The Monte-Carlo simulation methodology has a number of similarities to historical simulation.

Here, one chooses a statistical distribution that is believed to capture the possible changes in the market factors. Then a pseudo random generator is used to generate thousands of hypothetical changes in the market factors. These hypothetical changes are used to construct thousands of hypothetical profit and losses on the current portfolio. Finally, the VAR is determined from this distribution.

The various steps involved are:-

- Step 1: Identify the appropriate market factors
- Step 2: Assume an appropriate distribution that captures changes in the market factors. For example, you might choose the normal distribution for capturing stock price returns. You should understand that you are not restricted to any particular distribution type - you are free to choose any distribution that you think reasonably describes possible future changes in the market factors.
- Step 3: Use a pseudo random number generator to generate N hypothetical market factors. These hypothetical market factors are used to compute the N hypothetical mark-to-market portfolio values. Finally, subtract the actual portfolio value on a given date to obtain N hypothetical daily profit and losses.
- Step 4: Rank the profit and losses. Choose the one that corresponds to the specified confidence interval.

### Evaluation of VaR as risk measure

Over the years, VaR has become a very important tool for risk management because of many advantages associated with it.

- It represents financial exposure to loss in a portfolio. It gives a loss figure in terms of dollars.
- VaR methodologies take into account potential risk mitigating factors like diversification and correlation.
- Regulators have endorsed VaR as a representation of market risk and better representation of exposure.
- There are several different approaches to VaR calculations, depending upon certain assumptions about loss probability distributions that are built into the models.

Value-at-Risk type measures are increasing in popularity because they give an indication of the potential gains or losses in value that may occur but there are some disadvantages of the risk measure;

- VaR figures can be difficult to interpret for many people and tend to emphasize the exchange rate component of the risk associated with a portfolio when the bank has limited scope for action.
- VaR does not give a direct indication of the actions that need to be taken in order to reduce risks.
- VaR figures can also be subject to bias, particularly as a result of the way in which the position is broken down into its component parts.
- Due to the high computational demands that VaR calculations place on systems, these figures may not be available in real-time or may only support a limited level of analysis.
- VaR calculations are also sensitive to the quality of the input data, particularly correlation and volatility data.

### Software Creation

To augment the study of theoretical concepts of VaR and sigma, 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 sigma, 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 Specification

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 sigma for the portfolio.
- The results should be summarized on a separate sheet in a clear and concise manner.

### Design of the Software

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.

### References

- http://en.wikipedia.org/wiki/Risk_management
- http://www.riskglossary.com/articles/riskmanagement.htm
- http://www.afs-seminars.com/glossary-r.html.
- Managing Risk in Financial Sector: an article by Adeel Mirza:http://ezinearticles.com/?Managing-Risk-in-Financial-Sector&id=251370
- Directory of open access journals. VAR Methodology Used for Exchange Risk Measurement and Prevention by Ion Stancu and Florentina Balu. http://www.doaj.org/doaj?func=abstract&id=214213.
- http://www.books.iupindia.org/overview.asp?bookid=IB1100584
- http://www.economicexpert.com/a/Value:at:Risk.htm.
- 'The Financial Stability Conjuncture and Outlook', 2004, Financial Stability Review, June, pp. 46-60.