Excerpt

## TABLE OF CONTENT

I. INTRODUCTION: WHAT I AM GOING TO TALK ABOUT?

II. DEFINITION OF VALUE AT RISK

III. HOWTO CALCULATEVALUEATRISK-THECOMMON APPROACHES

III.1. Variance-Covariance model

III.2. Historical Simulation

III.3. Monte Carlo Simulation

IV. SUMMARY-PROBLEMS AND CRITICS

IV.l. Problems of Value at Risk

IV.2. Value at Risk: And beyond?

IV.3 Overview: Advantages and problems of the VaR calculations

V: FINALLY: WHICH WAY IS THE BEST?

LIST OF LITERATURE

LISTOFTABLES

LISTOF FIGURES

## I. Introduction: What I am going to talk about?

This seminar paper is divided in the following chapters:

1. Definition of Value at Risk: What is VaR, several definitions of this figure

2. The three common approaches for calculating Value at Risk: Historical simulation, Monte Carlo simulation, Variance-Covariance model

3. The critical view: Problems and limitations of Value at Risk. Which approach can when meaningfully used and when not? Why is Value at Risk not the “only truth” in financial institutions? What are the strengths and weaknesses of the several approaches in calculating Value at Risk?

For better understanding, all chapters and approaches will be visualized with examples.

## II. Definition of Value at Risk

Nowadays the Value at Risk has become the central figure in financial institutions in calculating several risks, like credit risk. In practice, the objective should be to provide a reasonably accurate estimate of risk at a reasonable cost.^{[1]} It was pioneered by J.P. Morgan. Value at Risk is a function of two parameters: the time horizon (N days) and the confidence level (X %)^{[2]}. It is defined as the value than can be expected to be lost during severe, adverse market fluctuations.^{[3]} It asks “How much could we lose today given our current position and the possible changes in the market?”^{[4]}

John C. Hull defines Value at Risk in the statement “We are X percent certain that we will not lose more than V dollars in the next N days.”^{[5]}

Philippe Jorion defines Value at Risk as followed: "[...] VAR describes the quantile of the projected distribution of gains and losses over the target horizon. If c is the selected confidence level, VAR corresponds the 1 - c lower-tail level. For instance, with a 95 percent confidence level, VAR should be such that it exceeds 5 percent of the total number of observations in the distribution. "^{[6]}

Anyway, it does not matter how many definitions you take: All say the same. This figure tells me my expected maximum loss of my portfolio with a certain level of certainness, called confidence level, during the holding period of my portfolio.

illustration not visible in this excerpt

Figure 1: Illustration of Value at Risk under normal distribution

## III. How to calculate Value at Risk - the common approaches

### III.1. Variance-Covariance model

In this approach, we assume a model for the joint distribution of changes in market variables and use historical data to estimate the model parameters. This model was developed by Harry Markowitz.^{[7]}

The main assumptions of this model are:^{[8]}

- Normal distribution of the probability distribution

- Changes in instrument values are linear with respect to changes in risk factors

- Expected change in the price of the market variable over the time period is zero.^{[9]}

In this assumption volatility of an asset is measured in the time horizon of one day. The relationship between volatility of one year and a day is: [illustration not visible in this excerpt].^{[10]}

In a single asset-case, where my portfolio consist only a single stock/position, it is very easy to calculate the Value at Risk. It is given with the formula: [illustration not visible in this excerpt], and [illustration not visible in this excerpt] where a is the inverse value of standard normal deviation, σ the one-day volatility of the asset and w the portfolio value.^{[11]}

illustration not visible in this excerpt

Table 1: 95 % VaR in a single asset case^{[12]}

As we see, it is very easy to calculate the Value at Risk. We just need to know the volatility and the value of the assets. But now what is the Value at Risk in a portfolio which includes both shares, JoWood and OMV? In this so called “two-asset case”^{[13]}, we need additional information about our assets: the correlation between themselves. At first it is important to calculate the whole portfolio risk with the given formula [illustration not visible in this excerpt] Alpha is the amount of the share (here JoWood 0.33 and OMV 0.66). We will assume a correlation coefficient between OMV and JoWood of 0.25. Our portfolio risk is then 3.81 or $ 1,244. For the one-day 95 % Value at Risk we only need to multiply our last result with the inverse value of standard normal deviation: $ 1,244 * 1.645 = $ 2,048. The 10-day 95 % Value at Risk would be $ 6,476.

In comparison to the single-asset case we could reduce our risk by combining two stocks into one portfolio. As shown in the following table is the Value at Risk in our portfolio with two stocks even lower than the single Value at Risk of OMV in the single-asset case.

illustration not visible in this excerpt

Table 2: Comparison between VaR in the single-asset- and two-asset case

Like shown in this example, the variance-covariance model for calculating is very easy and fast to implement. But we know, no model has only advantages. When a portfolio includes options, the linear model is an approximation.^{[14]} Another problem is the existence of fat tails in the distribution of returns on most financial assets and Value at Risk attempts to capture the behavior of the portfolio return in the left tail.^{[15]} Further, this model uses covariance matrix, and this implicitly assumes that the correlations between risk factors is stable and constant over time.^{[16]}

### III.2. Historical Simulation

Historical simulation is the most simple Value at Risk technique, but it takes significantly more time to run than the first model.^{[17]} It involves past data to predict future events. At the first step we identify the market variables affecting the portfolio. After then we collect data on the movements in these market variables over the most recent 500 days. This provides us to generate scenarios for what can happen between today and tomorrow.^{[18]}

This simulation takes the market data for the last 500 days and calculated the percent change for each risk factor on each day. Each percentage change is then multiplied by today’s market value to present 500 scenarios for tomorrow’s values. For each of these scenarios, the portfolio is valued using full, nonlinear pricing models. The 25th-worst day is then the selected as being the 95 % Value at Risk.^{[19]}

A very detailed example can be found in Hull, 2007, page 218 ff.

**[...]**

^{[1]} See Jorion, 2002, p. 205

^{[2]} See Hull, 2007, p. 196 f.

^{[3]} Marrison, 2002, p. 98

^{[4]} Marrison, 2002, p. 102

^{[5]} Hull, 2007, p. 196

^{[6]} See Jorion, 2002, p. 22

^{[7]} See Hull, 2007, p. 233

^{[8]} See Marrison, 2002, p. 104

^{[9]} See Hull, 2007, p.234

^{[10]} See Hull, 2007, p. 234

^{[11]} See Hull, 2007, p. 235 ff

^{[12]} Figures taken from my personal portfolio on www.wienerboerse.at

^{[13]} SeeHull, 2007, p. 235 ff

^{[14]} See Hull, 2007, p. 244

^{[15]} See Jorion, 2002, p. 221

^{[16]} See Marrison, 2002, p. 105

^{[17]} See Marrison, 2002, p. 116

^{[18]} See Hull, 2007, p. 218

^{[19]} See Marrison, 2002, p. 116

- Quote paper
- Alexander Melichar (Author), 2009, Problems of Value At Risk - A Critical View, Munich, GRIN Verlag, https://www.grin.com/document/162499

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