VALUE AT RISK: DAX RISK MANAGEMENT

APRIL 12, 2015 STEFANO DI ROSA

LECTURER: BOB KOSTAKOPOULOS EUROPEAN SCHOOL OF ECONOMICS

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Table of Contents

Executive Summary……...……………….………………………….…….………………….2

1. Introduction……………………………………………………….….……….……….......3

2. What is the Value at Risk……..…………………………………….……….….…………..4

3. DAX index…………………………………………………………………..….…………..5

4. Methodology…………………………………………………………………..……………5

5. DAX analysis – Historical Method……………………………………………..…………..6

5.1 Variance-Covariance method…………………………………….………………11

5.2 Monte-Carlo simulations…………………………………………………………12

6. Advantages and limitations of the three methods…………………………………………15

7. Summary of findings………………………………………………………………………16

8. Conclusions……………………………………………………………………………......16

References......………………………………………………………………………………..17

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Executive Summary

Nowadays the volatility is one the biggest concerns for each kind of investor. The recent

subprime mortgages crisis in U.S and the Euro debt sovereign issue have demonstrated that

returns’ volatility and therefore investments’ risky is not limited to equity investors, but even

fixed income investors may undergo a lot degree of uncertainty.

This paper is going to apply the concepts of volatility and Value at Risk to DAX Index, the

most popular benchmark index for the German equity market.

The goal of the project is to calculate the Value at Risk of DAX using the three methods:

1. Historical

2. Variance-Covariance

3. Monte-Carlo simulations

The analysis proves the all three methods deliver similar results and investors are 99% sure

their daily loss should not exceed -2.91% or in EUR terms 291.015 by investing €10.000.000.

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1. Introduction

Value at Risk (VaR) is one of the major methods by risk managers to measure the financial

risk of an asset, portfolio, or exposure over some specified period of time. Its appeal

originates from its ability to deliver an outcome which tells investor what is the most he can

lose on its investment. Every investor, ahead of investing, would like to quantify the potential

loss which he could cope with. Obviously, it depends on what he invests in, how long he wish

to keep the investment and how this asset that he is investing in has behaved over its recorded

history.

VaR is a method to answer the question: how much can I lose with x% probability over a

given time horizon (JPMorgan, 1996).

The paper begins with an overview on VaR examining the history of its development and

applications. Then, will be provided a detailed description of the DAX index, that is the

object of the analysis. Next, will be calculated the VaR of DAX using three main methods.

The final section is going to provide a comment on findings evaluating how VaR can help

risk managers to handle with volatility.

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2. What is the Value at Risk?

VaR is an outline statistic which computes the exposure of an asset or a portfolio to market

risk, or the risk that a position loses value in case of adverse market price variations (Culp,

Miller, Neves, 2004). VaR can be also viewed as the lowest quantile of the potential losses

which may arise into a portfolio during a specific period of time. In other words, VaR enables

risk managers to assess what is the worst-case scenario or how much he could lose in an

adverse period. To a risk manager, VaR is the level of losses at which you stop trying to

guess what will happen next, and start preparing for anything.

VaR’s estimate depends on three key factors:

1. A time period (a day, a week, a month, a quarter, or a year, depending on the

investment horizon or regulatory requirements)

2. A confidence level (usually either 95% or 99%)

3. An estimate of investment loss (expressed either in dollars or percentage terms

The time period and the confidence level must be chosen according to the overall purpose of

risk measurement (Benninga and Wiener, 1998).

Moreover, it is essential to highlight some key elements to calculate VaR:

i. To determine the probability distributions of individual risks, along with a confidence

level, in order to estimate the probability of the loss. But also to compute the

correlation among those risks and the impact of such risks on value. This a reason

because the simulations are commonly utilized to calculate the VaR for asset

portfolio.

ii. VaR analysis is focused on downside risks and potential losses. In particular

commercial banks adopt VaR approach because they are afraid of liquidity crisis

which in worst-case scenario could create dangerous phenomena as bank-run and

panic selling.

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iii. VaR can be measured on a single asset, a portfolio of assets or on an whole company.

iv. The usage of the VaR is not limited to banks or investment institutions. For example,

VaR approach can be also adopted to measure the gold price risk for a gold mining

company (NYU Stern).

3. DAX Index

DAX index is one the most known index worldwide. Most analysts and researchers rely on

DAX to get a proxy of the German equity market. It is a capital-weighted index and it is

possible to get both its price and total return version, whereby total return includes the

reinvestments of dividends. DAX tracks the share performance of the largest 30 largest

German firms in terms not only of market capitalization but even exchange turnover.

Therefore, it is often considered as a leading indicator of the German economy’s status.

Furthermore the Index is utilized as an underlying instrument for derivatives and structured

products. It covers about the 80% of the market capitalization listed in Germany (www.dax-

indices.com, 2015).

4. Methodology

In order to calculate VaR’s DAX index three main methods have been deployed:

1. Historical

2. Variance-Covariance

3. Monte-Carlo simulations

The time period object of the analysis goes from January 2, 2014 to March 31,2015. For each

of those methods, 90%, 95% and 99% level of confidence have been used. Moreover, it has

been assumed an investment of €10.000.000. The sample size consists of 314 observations.

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The analysis is based on daily index value, therefore the purpose of this paper is to answer to

the following questions:

With a 90%, 95% or 99% level of confidence, what is the daily expected loss in euro

investing in DAX index?

With a 90%, 95% or 99% level of confidence, - what is the daily expected loss in

percentage investing in DAX index?

These questions can be also viewed in a different way. In fact, VaR also allows to figure out

the return higher than VaR, given specified time period and confidence interval.

The final part of the analysis involves the estimate of the conditional VaR (also called

expected shortfall), which provides the expected loss conditional on being on the left tail.

5. DAX analysis – Historical Method

The historical method is a simple way to calculate the VaR. According to this method, the

VaR for a portfolio is estimated by re-organizing actual historical returns, putting in order

from worst to best. Therefore, it assumes that history will repeat itself, from a risk point of

view (www.investopedia.com).

The first step of the analysis is to calculate the daily rate of change from January 2, 2014 to

March 31, 2015. It has been calculated as: – ln . Then, the analysis of sample data

may proceed. The 10%, 5% and 1% size have to be computed in order to interpolate the VaR.

Next, the Excel function “SMALL” returns the k-th smallest value in a data set. Given a

confidence interval of 90% the VaR for DAX is -1.34%. This means that we are 90% sure the

daily returns will be higher than -1.34%. Moreover, investing €10.000.000, we are 90% sure

the daily loss will not exceed €134.283. Using a level of confidence of 95%, the expected

daily loss will not exceed -1.87% or in currency terms, the expected daily loss will not exceed

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€187.292 with an initial investment of €10.000.000. Finally, given a level of confidence of

99%, the analysis shows the daily return will be higher than -2.91% and investing

€10.000.000 the expected daily loss will not exceed €291.015.

The analysis goes on focusing on the characteristic of the distribution. This is a fundamental

step because highlights the key statistics of the distribution, as mean and standard deviation.

Moreover, two statistics deserve a brief comment, skewness and kurtosis. The skewness,

which describes departures from symmetry, represents the scaled third moment of the

distribution. From the DAX analysis, the skewness is negative (-0.1429). In general this is a

dangerous signal, as negative skewness indicates that the distribution has a long left tail,

which means high probability of observing large negative values (Jorion, 2009).

Another useful moment of the distribution is the kurtosis. It outlines the degree of “flatness”

of a distribution, or width of its tails. Examining the DAX distribution, kurtosis is 0.7099, so

the distribution is platykurtic. In fact, as kurtosis represents the scaled fourth moment of the

distribution, large observations in the tail will have a large weight and therefore create large

Analysis of sample data

Sample size 314 Investment 10,000,000

VaR 10% Size 31.4

VaR 5% Size 15.7

VaR 1% Size 3.1

VaR 10% 32 -1.30%

VaR 10% 31 -1.35%

VaR 10% Interpolated -1.34% We are 90% sure the daily return will be higher than -1.34%

VaR 10% Interpolated (134,282.64) We are 90% sure the daily loss will not exceed €134.283

VaR 5% 16 -1.79%

VaR 5% 15 -1.88%

VaR 5% Interpolated -1.87% We are 95% sure the daily return will be higher than -1.87%

VaR 5% Interpolated (187,291.63) We are 95% sure the daily loss will not exceed €187.292

VaR 1% 4 -2.75%

VaR 1% 3 -2.91%

VaR 1% Interpolated -2.91% We are 99% sure the daily return will be higher than -2.91%

VaR 1% Interpolated (291,014.75) We are 99% sure the daily loss will not exceed €291.015

Exhibit 1. VaR historical method - Analysis of sample data

DAX analysis - VaR Historical Method

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kurtosis. In that case, the distribution is called leptokurtic, that is characterised by fat tails. A

kurtosis of 3 is considered the average, so values above the average stand for higher

probability of extreme movements.

Another important tool to outline the distribution is the Histogram. It essentially shows

frequency data. Bin numbers are the intervals used to measure the input data in the data

analysis. The Histogram unveils as the bulk of the distribution lies within the range -1% and

+1%, as shown below.

Mean 0.08%

Standard Error 0.0006 Standard Error of the average, SE(mean)= STDEV*(1/n)^0.5 0.0006

Median 0.0011

Standard Deviation 1.08%

Sample Variance 0.0001

Kurtosis 0.7099 Low Kurtosis, platykurtic distribution

Skewness -0.1429 Negative means long left tail

Range 0.0680

Minimum -0.0350

Maximum 0.0330

Sum 0.2376

Count 314

Exhibit 2. VaR historical method - Characteristics of the distr ibution

Characteristics of the distribution

Bin Frequency Cumulative %

-3.50% 0 0.00%

-3.00% 2 0.64%

-2.50% 4 1.92%

-2.00% 5 3.51%

-1.50% 16 8.63%

-1.00% 20 15.02%

-0.50% 31 24.92%

0.00% 55 42.49%

0.50% 82 68.69%

1.00% 47 83.71%

1.50% 23 91.05%

2.00% 13 95.21%

2.50% 10 98.40%

3.00% 3 99.36%

3.50% 2 100.00%

More 0 100.00%

Exhibit 3. DAX historical method - Histogram

Histogram

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The chart below the actual distribution for DAX index from January 1, 2014 to March 31,

2015. It summarize all considerations made above, as negative skewness (long left tail), low

kurtosis (thin tails) and bulk of the distribution between -1% to +1%.

The final part of the historical method adopted in this project involves the estimate of the

conditional VaR (cVaR), also called expected shortfall. Conditional VaR measures the risk of

extreme losses and it may be considered as an extension of VaR. It tells investors how much

they could lose in the left tail of the distribution. CVar is calculated through the weighted

average of the VaR estimate and the expected losses beyond VaR (Kidd, 2012). A cVaR

estimate cannot be lower than a VaR estimate. The graph below shows the relationship

between VaR and cVar.

Graph 1. DAX historical method - Actual distribution

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Coming back at DAX analysis, investing €10.000.000, conditional on being in the left tail,

the expected loss is €194.463 with a 90% level of confidence, €236.119 with a 95% level of

confidence and €304.907 with a 99% level of confidence. The expected loss in percentage

terms is shown below (exhibit 4).

The next section will focus on variance-covariance approach to measure the Value at Risk for

DAX index.

Graph 2. DAX historical method - cVaR in terms of the probabil ity density function

CVaR Investing €10.000.000, conditional on being in the left tail, the expected loss is:

Conditional VaR10% -1.94% 194,463.45

Conditional VaR5% -2.36% 236,119.25

Conditional VaR1% -3.05% 304,907.91

Exhibit 4. DAX historical method. Expected shortfal l

Expected Shortfall ( conditional VaR)

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5.1 DAX analysis – Variance-Covariance method

Variance-Covariance is another method used to measure the Value at Risk. It is also called

delta-normal approach. The idea behind this method is that the portfolio exposures are linear

and the risk factors are jointly normally distributed. As the portfolio return is a linear

combination of normal variables, hence it is normally distributed itself. Therefore, utilizing

covariance matrix and weight vector allows to calculate the portfolio volatility (Li Fan, Li,

Zhou, Jin, Liu, 2012). It is a very simple method as it need only two factors to estimate: an

expected return and a standard deviation, which enable to draw a normal distribution curve.

Examining the DAX returns, the daily average of the returns is very close to zero (0.08%)

and the standard deviation is 1.08%. Hence, it is reasonable to affirm that the daily returns for

DAX are normally distributed. Through the Excel function “NORM.INV” which returns the

inverse of the normal cumulative for 0.08% mean and 1.08% standard deviation, on one hand

we get the following results: VaR 10% = -1.31%, VaR 5% = -1.70%, VaR 1% = -2.44%. On

the other hand, assuming the distribution has a mean of 0 and a standard deviation of 1, we

get the following results: VaR 10% = -1.38%, VaR 5% = -1.78%, VaR 1% = -2.51%.

Finally, using the formula exp(-(Za%^2)/2)/(a*(2*p)^0.5) we are able to compute the

conditional VaR as shown in the exhibit below.

VaR 10% (130,873.12)

VaR 5% (170,126.17)

VaR 1% (243,758.36)

VaR 10% (138,465.50)

VaR 5% (177,718.55)

VaR 1% (251,350.73)

Z-Value

1.282 CVaR 10% 1.75%

1.645 CVaR 5% 2.06%

2.326 CVaR 1% 2.66%

Exhibit 5. DAX Variance-Covariance method

Expected Shortfall Investing €10.000.000

175,378.15

Using Excel formula Norm.Inv.

Using Excel formula Norm.S.Inv.

exp(-(Za%^2)/2)/(a*(2*p)^0.5)

Standard Normal

-1.38%

-1.78%

-2.51%

VaR 10%

VaR 5%

VaR 1%

-1.31%

-1.70%

-2.44%

DAX analysis - Variance Covariance Method

206,130.02

266,338.90

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5.2 DAX analysis – Monte-Carlo simulations

Monte-Carlo approach was designed in the 1940s by J. Von Neumann, S. Ulam and N.

Metropolis. This method utilizes arbitrary samples from known populations of simulated data

in order to define a statistic’s behaviour. The idea behind Monte-Carlo method is that

generally inference system characterizes the distribution of returns by supposing any standard

joint distribution like the joint-normal distribution and designating a covariance matrix and

mean vector. In other words, such an approach envisages to build a model for future stock

price returns and perform multiple hypothetical trials via the model.

In order to perform the Monte-Carlo simulation for DAX, the first step involved running

10000 trials of daily returns. We implement geometric Brownian motion to generate returns.

It means that trend and volatility terms are proportional to the current value of S:

ΔS = µSΔt+σSΔz

The returns are generated in the following way:

LN (DAX t/DAX t-1)= Mean + standard deviation * Zt where Zt is the random variable

either shock or event not controlled.

After generating returns, we implement the analysis of simulated data. From the simulations

of 10000 return we get: VaR 10% = 1.32%, VaR 5% = -1.71%, VaR 1% = -2.45%.

The exhibit below shows also the VaR in EUR terms.

mean stand.dev

Analysis of simulated data 0.0008 0.0108

Sample size 10,000

VaR 10% Size 1,000

VaR 5% Size 500

VaR 1% Size 100

VaR 10% -1.31% (131,441.94)

VaR 5% -1.67% (167,178.09)

VaR 1% -2.43% (243,268.22)

Exhibit 6: DAX analysis - Monte-Carlo method

Geometric Brownian motion

DAX analysis - MonteCarlo simulation

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Equally interesting the results stem from the characteristics of the distribution. Specifically,

the skewness became positive, meaning right tail longer than left and therefore higher

probability to get large positive values. Kurtosis sharply decreased, hence thin tails

(platykurtic).

Finally we implement the Histogram for Monte-Carlo simulations and show the simulated

distribution.

Characteristics of distributionMean 0.07%

Standard Error 0.0001

Median 0.0007

Mode #N/A

Standard Deviation 1.09%

Sample Variance 0.0001

Kurtosis 0.0214

Skewness 0.0161

Range 0.0841

Minimum -0.0413

Maximum 0.0428

Sum 7.2581

Count 10000

Exhibit 7: DAX analysis - Monte-Carlo method. Characteristic of the distribution

HistogramBIN Frequency Cumulative %

-3.50% 5 0.05%

-3.00% 12 0.17%

-2.50% 67 0.84%

-2.00% 184 2.68%

-1.50% 472 7.40%

-1.00% 852 15.92%

-0.50% 1358 29.50%

0.00% 1790 47.40%0.50% 1758 64.98%

1.00% 1523 80.21%

1.50% 1049 90.70%

2.00% 544 96.14%

2.50% 261 98.75%

3.00% 90 99.65%

3.50% 25 99.90%

More 10 100.00%

Exhibit 8: DAX analysis - Monte-Carlo method. Histogram

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6 Advantages and limitations of the three methods

VaR can be utilized to define the event risk in all areas of risk management such as market,

credit, operational or insurance risk. It may be estimated via several methods, each one with

own benefits and caveats.

For purpose of our DAX’s VaR measure, we used three commonly used approaches:

historical method; variance-covariance; Monte-Carlo simulations.

Historical method. The strength of this approach is its simplicity. In fact, it does not need to

do statistical assumptions beyond stationary of the distributions of returns and the VaR relies

on actual price movements, assuming that history will repeat itself. Although its significant

benefits, it carries on any weaknesses:

Past is not guaranteed will repeat itself in the future. All three approaches used

historical data, but the historical method is much more reliant on them than variance-

covariance and Monte-Carlo simulations because VaR is totally calculated from

historical prices variations.

Graph 3: DAX analysis - Monte-Carlo method. Simulated distribution

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Trends in the data. Taking a sample of 314 daily observations from January 2, 2014 to

March 31, 2015 it is feasible that the sample has been affected by trend of either

increasing or decreasing volatility which could lead to underestimate or overestimate

the Value at Risk.

New Assets. The historical method cannot be utilized to estimate VaR’s new assets

simply because there are not historical data available

Variance-Covariance. As the historical approach, the main advantage of variance-

covariance approach is its simplicity. Moreover, it requires only two parameters to compute,

mean and deviation standard. However, the variance-covariance method presents at least

three main drawbacks:

Wrong distributional assumption. If conditional returns are not normally distributed,

the actual VaR will be higher than calculated VaR.

Parameters errors. Variance-covariance matrix, input of the approach, is a collection

of estimates some of which have large error terms, leading to a mistake in the

computed VaR.

Non-stationary variables. Inputs of the model (variance and covariance) across assets

tend to change over the time, resulting in wrong calculated VaR.

Monte-Carlo simulations. The major advantage of the Monte-Carlo method is the freedom to

choose alternate distributions for the variables. Moreover, these distributions can be adjusted

by subjective judgements. The main pitfalls of this approach are:

It makes inherent sampling variability due to randomization as different casual

numbers will lead to different results.

It requires to make assumptions on the stochastic process.

Time-consuming.

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7. Summary of findings

All three models provide similar results. The relevant point is that the historical approach

looks to overestimate the VaR compared to the other two methods. Moreover the differences

grow up by increasing the level of confidence.

Nevertheless, the analysis proves the daily expected loss should not exceed €291.015

investing €10.000.000 in worst-case scenario or in percentage terms investors are 99% sure

their daily return will be higher than -2.91%.

The exhibit below shows the comparison of results across the three methods used.

8. Conclusions

Banks and other financial institutions have increased the usage of VaR as tool to risk

management. There are three main approach adopted to compute the VaR: the historical

method, the variance-covariance and the Monte-Carlo simulations. Each of them has own

benefits and drawbacks and each of them requires a number of assumptions. Moreover, there

is not a method better than others, and the best solution is to compute VaR using all three

methods and make comparison across them.

As the DAX analysis unveiled, the VaR analysis is very useful tool to manage risk because it

allows to allocate a cushion which could be necessary to meet regulatory requirements.

SUMMARY DAX ANALYSIS

HISTORICAL -1.34% (134,283) -1.87% (187,292) -2.91% (291,015)

VARIANCE-COVARIANCE -1.31% (130,873) -1.70% (170,126) -2.44% (243,758)

MONTE-CARLO -1.31% (131,441) -1.67% (167,178) -2.43% (243,268)

Exhibit 9 . Summary of VaR for DAX

VaR 10% VaR 5% VaR 1%

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References

Benninga Simon, Wiener Zvi, Value at Risk. Mathematica in Education and Research, 1998.

Culp Christopher, Miller Merton, Neves Andrea, Value at Risk: uses and abuses. Journal of

Applied Corporate Finance, 2004.

Duffie Darrell, Jun Pan, An overview of Value at Risk, 1997.

Garp Philippe, Financial Risk Manager Handbook. John Wiley & Sons, 2009.

JPMorgan/Reuters, RiskMetrics-Technical Document. Morgan Guaranty Trust Company of

New York, 1996.

Hao Li, Xiao Fan, Yu Li, Yue Zhou, Ze Jhin, Zhao Liu, Approaches to VaR. Stanford

University, 2012.

Kidd Deborah, Value at Risk and Conditional Value at Risk: a comparison. CFA Institute,

2012.

Zvi Bodie, Alex Kane, Alan J. Marcus, Essentials of Investment. McGraw-Hill, 2003.

Web Sources

Investopedia : http://www.investopedia.com/articles/04/092904.asp

DAX: http://www.dax-indices.com/EN/index.aspx?pageID=1