# Conditional value at risk of portfolio

### Acknowledgements

First of all I would like to thank my god lord Ganesh for guiding me and bequeathing me the strength to successfully complete this myriad task.

I am ever so grateful to my supervisor Christine Gregory for the invaluable contribution, continuous support, guidance and encouragement she always warmly obliged during the course of this project. Also I would like to extend my heartfelt appreciation to Cormac Lucas, for his assistance with the software segment of this project.

Furthermore I would also like to thank The Mathematics School for all their support and assistance. Last but not least my humble appreciation goes to my family and friends who have supported me throughout the year, even during some hard times.

### Abstract

In the contemporary unstable financial climate reducing risk whilst maximizing profit has been the ultimate goal for almost every business. Portfolio optimization methods have been long used for this purpose. Optimal portfolios are normally computed using the portfolio risk measured in terms of its variance. Conditional Value at Risk (CVaR) is a risk metric used to confront this problem. This project uses non linear programming techniques to define and quantify the Conditional Value at Risk of a portfolio.

Existing methods of measuring Conditional Value at Risk and using Markowitz model to optimise a portfolio of stocks is detailed in this project. Furthermore this model was implemented using AMPL and the results obtained were analysed and discussed. The Variance-Covariance method and the Empirical approach were used to measure Value at Risk and Conditional Value at Risk.

### Overview guide to the chapters

Chapter1 - Introduction

This chapter explains the purpose of the research, the aims and objectives of the project.

Chapter 2- Portfolio Optimisation

This chapter in brief will go over portfolio optimisation.

Chapter 3-Value at Risk

This section will give an explanation of Value at Risk and Value at Risk methods.

Chapter 4- Conditional Value at Risk

This section will give an explanation of Conditional Value at Risk and Conditional Value at Risk methods.

Chapter 5: Markowitz Model

This chapter will take an in-depth look on Markowitz models and a brief introduction to the software AMPL.

Chapter 6: Design and Implementation

This chapter brings together all the topics talked about in the previous chapters. It will illustrate results from the Mean Variance Model, obtained using AMPL and the CVaR values obtained by the different types of methods mentioned before.

Chapter 7: Conclusions and Recommendations

In this chapter overall conclusions of the whole project will be discussed, referring back to the project's aims and objectives. Also further recommendations to improving this research will be talked about.

### Research objective

The objective of this project was to optimise a portfolio of stocks and calculate Conditional Value at Risk using different methods to obtain the smallest (minimum) Conditional Value at Risk. To optimise a portfolio of stocks an advanced computing software and widely known mathematics equations will be used. The data obtained from Datastream based upon stock prices will be implemented on the software AMPL. This will generate the minimum variance for the portfolio and will obtain weights for the stocks. These weights will then be used to obtain Conditional Value at Risk using commonly used methods.

### Research justification

The reason for choosing this subject was is because a lot of research has been done about Value at Risk. But as CVaR is now becoming a much more used risk measure than VaR only a limited number of research have been carried out.

In the past, research has been done to optimize Conditional Value at Risk, the study suggest that in past few years CVaR seems to have become somewhat much more incorporated in recent years (Uryasev). However this project will focus on optimising a portfolio of stocks using Markowitz Mean Variance method to obtain weights for the portfolio. CVaR will then be calculated using different methods. I believe it is crucial to explore this topic in depth as CVaR is a fairly new topic and much research needs to be conducted in this area of study.

### Chapter 2

### Portfolio Optimisation

### Introduction To Optimizing A Portfolio

This chapter gives an introduction to the subject of portfolio optimization. Portfolio optimization is generally defined as putting together a portfolio of assets e.g. stocks, in such a way that the portfolio provides the most expected return for a given level of risk or the lowest risk for the given expected return. A portfolio optimization problem can only be solved assuming that an initial portfolio of assets is available.

### History On Portfolio Optimization

Portfolio optimization is a multiple objective process that uses Mean- Variance optimization. This mathematical technique was pioneered by Harry Markowitz in the early 1950s and was published in 1952 by the journal of Finance [1]. Implemented using Mean-Variance optimization, his theory of portfolio allocation to this day is the most successful applications of quantitative finance. The main aspect of this concept is asset diversification [2]. It can be said that one of the main objective of portfolio optimization is the process of selecting assets that complement one another on the basis of volatility and market movement. Mean-Variance optimization to achieve desired asset allocation was used by institutional investor and money managers for many years and recently because of interest to mainstream investors around the world due to the affordable access to global markets provided by the internet.

### Terminology

Before exploring the portfolio optimising model, some of the terminology that will be used in this project will be introduced.

### Portfolio Return

Assume there are assets with random rates of returns. The expected rates of returns are E [], E [], E [],...., E [].

Assuming we have a portfolio consisting of n assets, and is the weight of asset in the portfolio, such that

Hence the portfolio return is (2.1)

And the expected portfolio return is (2.2)

### Risk

Risk is usually defined as the variance of the portfolio,. The variance of a random variable is its second central moment, and its mathematical definition is

With the variance of individual assets given, the variance of the portfolio can be calculated using the covariance between asset and asset

Let and be the variance of asset and the portfolio, respectively. The variance of the portfolio can be calculated as shown below:

Alternatively, let be a vector representing the weights of each asset, be the current portfolio position, and be the variance covariance matrix. The variance of the portfolio can also be written as

### Optimising A Portfolio

When an investor is trying to optimize a portfolio they are actually trying to maximize performance to reduce loss.

### Maximum Expected Return Problem

For an investor who wants to achieve the maximum expected return, the following problem maximises the expected return of the portfolio

### Minimum Variance Portfolio

All financial portfolios are exposed to a risk of substantial financial loss. The model given in the above equation does not take into account the risk of a portfolio. Instead of maximising the returns of the portfolio, risk-averse investors may want to minimize the risk (variance) of the portfolio. The following model gives the portfolio with the minimum variance.

### Mean Variance Efficient Portfolio

Most investors would like to balance maximum returns and minimum variance portfolios in order to give a maximum return with a minimum risk. Hence the classical mean-variance portfolio optimization [3] can be formulated as shown below:

This model gives the optimum portfolio (i.e. minimum risk) for the minimum required wealth, H. That is, no other portfolios can give a similar or higher return with a lower risk (i.e. small variance is desired).

### Chapter 3

### Value At Risk (VaR)

### Value At Risk

In the mean variance portfolio optimisation model, the risk measure was the portfolio variance. The variance is a measure of the statistical dispersion that gives us the average of the squared distance of the possible values from the expected value. It treats both upside and downside payoff symmetrically. However, while most investors will be disturbed by a drop in prices, most investors will not mind an increase in price.

VaR stands for Value at Risk and is one of the most important risk measures used by financial institutions; it presents risk in terms of potential financial loss on the portfolio. Value at risk is defined as an estimate, with a given degree of confidence, of how much one can lose from one's portfolio over a given time horizon [4].

To put this into context, one might say that the VaR of their portfolio is £0.5 million at the 95% confidence level with the target horizon set to one week. This means that there is a 5 out of 100 chance that the portfolio will lose over £0.5 million within the target horizon under normal market conditions.

Value at risk is mathematically defined in [5] as:

Where is the confidence level, is a random variable, is Value At Risk and is Value At Risk at confidence level.

The calculation of VaR requires having the current prices of all assets in the portfolio, their volatility and the correlations between them. Volatility is important for the reason that if the underlying markets are volatile then investments of a given size are more likely to lose money than if markets were less volatile. Hence volatility here refers to the distribution of the return around the mean. Correlations between assets are also important.

To calculate Value at Risk we need two parameters: the portfolios holdings and historical market data. A method will then merge the market data and the portfolio holdings to give the daily returns on the portfolio.

### Calculating Value At Risk

There are three common methods that are used to compute Value at Risk [6] [7] [8], though there are numerous variations within each approach. The measure can be computed analytically by making assumptions about return distributions for market risks, and by using the variances in and covariance's across these risks. In this section, we will look at four methods of calculating Value at Risk:

- Empirical Approach
- Historical simulation
- Variance-Covariance
- Monte Carlo Simulation

### Empirical Approach

The basic concept of the empirical approach is to calculate VaR using the actual historical prices. This is done by obtaining a sample of historical values and calculating the change in portfolio values. VaR is then equal to the percentile of all the changes in portfolio values at the set confidence level over the given time horizon.

### Historical Simulation

Historical Simulation uses historical data to simulate the shape of the distribution and VaR is calculated by looking at historical movements in market variables. Historical Simulation assumes that whatever changes were identified in the past can reflect the distribution of changes over a given time horizon. The changes that occurred in the rate of prices are then used to revalue the current portfolio; and hence a set of portfolio values relative to the changes are produced. Given sufficient amount of historical data, this method can give a realistic distribution as it will also take into account major market events such as stock market crashes. So the main aim of Historical Simulation method is to find a huge number of possible scenarios for the next day's portfolio value given today's value.

The primary advantage of Historical simulation is that it is easy to use and implement hence straightforward. The disadvantage of Historical Simulation is that because the simulation assumes the past represents a reasonable simulation of future events, the distribution is assumed to be stationary, the results are highly sensitive and there may be problems with collecting consistent historical data [9].

### Variance Covariance

The release of risk metrics [10] by JPMorgan has made the Variance- Covariance method of calculating value at risk popular. The variance covariance method uses information on the volatility and correlation of stocks to compute the VaR of a portfolio. The volatility of an asset is said to be a measure of the degree to which price fluctuations have occurred in the past and hence expected to occur in the future. Correlation is a measure of the degree to which the price of one asset is related to the price of another.

Variance-Covariance method seems to be one of the simplest and the fastest approach to calculating VaR. This assumes that a particular normal distribution for both the changes in market prices, rates and change in portfolio value. Once the distribution of the possible portfolio profits and losses are obtained, standard mathematical properties of normal distribution are then used to determine the loss that will be equalled or exceeded the X percent of time i.e. the value of risk.

The benefits of the Variance-Covariance model are the use of a more compacted and maintainable data set which are often being bought from third parties. Another benefit is the speed of calculation using optimized linear algebra libraries. Drawbacks include the assumption that the portfolio is composed of assets whose delta is linear. This means that a change in the value of the portfolio is linearly dependent on all the changes in the values of the assets, so that the portfolio return is also linearly dependent on all the asset returns. Another drawback is the assumption of a normal distribution of asset returns (i.e. market price returns).

### Monte Carlo Simulation

The basic concept of Monte Carlo approach is to simulate repeatedly a random process for variables based on specified distribution. Monte Carlo simulation is the generation of a distribution of returns and/or asset price paths by the use of random number [11]. Historical data drawn from normal distribution of the market prices and rates are then used to estimate the parameters of the distribution. This then builds a distribution of future scenarios and tells how likely the outcomes are based on how the range of estimates was created.

The Monte Carlo simulation method involves randomly generating scenarios based on parameters obtained from historical data. After generating these scenarios, we then proceed to calculate the profit and loss, followed by the value at risk in a similar fashion as described in the previous section about historical simulation.

### Distributions

In this section we briefly look at two types of distributions:

- Normal distribution
- Student t distribution

### Normal Distribution

Normal distribution is usually defined as the distribution of a random variable X for which the probability density function f is given in [12] by,

Where the parametersand are the mean and variance, respectively. The distribution is denoted by. If a random variable X has such a distribution, then this is denoted by ~ and the random variable may be referred to as a normal random variable.

### Students T Distribution

An alternative distribution used when normal distribution is not, is called the Student's t distribution. This distribution is important and used when the standard deviation is estimated from data. Due to the uncertainty in the estimate of the standard deviation the T distribution gives wider confidence intervals. The probability values depend on the number of degrees of freedom, which is the number associated with the estimate of the standard deviation. Tables of the T- distribution are widely available, but algorithms for direct computation are relatively lengthy.

### Chapter 4

### Conditional Value At Risk (CVaR) and Calculating Conditional Value At Risk (CVaR)

### Conditional Value At Risk

An alternative risk measure to VaR is CVaR. Whereas VaR asks the question "how bad can things get?" CVaR asks "if things do get bad, how much can we expect to lose?" [13]. CVaR is the conditional expectation of the loss above VaR for a given time horizon N, and confidence level X. This is illustrated by the graph below:

CVaR is said to be the average loss to match the loss that exceeds VaR. For continuous distributions CVaR is also known as Mean Excess Loss, Expected shortfall or Tail VaR. So CVaR is the expected loss exceeding VaR, for example it is the mean value of the worst losses. For instance at, CVaR is the average of the 1% worst losses. For discrete distribution CVaR is the weighted average of VaR, portfolios with a low CVaR also has a low VaR. So portfolios CVaR is the loss one expects to suffer, given that the loss is equal to or larger than its VaR.

Where is the confidence level, is a random variable, is Value At Risk and is Value At Risk at confidence level.

Looking at equation 4.1 you can see that CVaR is equal to the expectation of the random variable, where the random variable is equal to or greater than VaR, hence CVaRVaR.

### Why CVaR Is More Preferred Than VaR

Even Though VaR is a very popular risk measure, there are several reasons why CVaR is a much more preferred risk measure to VaR. VaR lacks sub-addictivity and convexity [15]; CVaR is a sub-additive measure of risk. Sub-addictivity means that diversification of a portfolio reduces CVaR but will not always reduce VaR. For example, the CVaR of a combination of two portfolios will be better than the sum of the CVaR of each portfolio, whereas the sum of the VaR of each portfolio may be worse than the VaR of the combination of the two portfolios [16]. VaR is a consistent risk measure only when it is based on standard deviation of a normal distribution. VaR is difficult to optimize when using scenarios [17]. Also VaR does not provide any information about the amount of loss exceeding VaR. Another defiency of VaR compared to CVaR is that minimization of VaR leads to an undesirable stretch of the tail of the distribution exceeding VaR. So for portfolios with skewed distributions minimization of VaR may result in a significant increase of high losses exceeding VaR. CVaR quantifies the losses exceeding VaR and behaves like an upper bound of VaR. So minimizing CVaR typically leads to a portfolio with a small VaR. Another benefit of CVaR is that CVaR methods require less data.

### CVaR Calculations

### Empirical Approach

Empirical Approach is an alternative to the parametric method. Empirical Approach is a non parametric method i.e. no assumption of the distribution type is required. The data speaks for itself when using the historical simulation method. Empirical Approach is a simple theoretical approach that requires relatively few assumptions about the statistical distributions of the underlying market factors.

Here we will look at the steps used to calculate CVaR.

- Step 1: Obtain sample of historical values, P.
- Step 2: Multiply the corresponding weights to the values, WP.
- Step 3: Calculate change in portfolio values.
- Step 4: VaR is equal to the percentile of all the changes in portfolio values at the set level.
- Step 5: Put the change in portfolio values in ascending order.
- Step 6: CVaR is the average of the sum of change in portfolio values which are less than VaR.

Consider a simple example. Assume you are calculating CVaR for a single asset with a confidence level of 99%.

VaR = Percentile of all the change in portfolio returns at 1%

= -77300The 99% one day VaR is equal to -77300.

CVaR is equal to the average of the sum of portfolio returns which are equal to or less than VaR.

CVaR = -80000/1 = -80000

Hence the 99% CVaR is -80000.

### Variance-Covariance Method

In this section we briefly describe how the Variance-Covariance method is used to calculate VaR. Following the calculation of VaR we will calculate CVaR. As indicated by its name in the Variance-Covariance method, VaR is calculated using the variance covariance matrix. Since the confidence level is obtained by assuming the distribution of returns follows the normal distribution, when using the Variance-Covariance method the assumption is that the returns of the assets are normally distributed. The steps I'm going to take to calculate CVaR using the Variance-Covariance method are as follows:

- Step 1: Compute the covariance matrix of my portfolio of assets ().
- Step 2: By using a column vector representing the weighting of each asset in your portfolio and the covariance matrix we can calculate the variance of the portfolio, variance.
- Step 3: Compute standard deviation. Standard deviation.
- Step 4: Now compute value at risk. , where is the corresponding percentile of the standard normal distribution.
- Step 5: To compute CVaR, I need to calculate the average of the VaR values beyond the specified level.

Consider a very simple example. Assume you are calculating CVaR for a single asset, where the potential values are normally distributed and assume the holding period is one day and the probability is 1%. The standard deviation (standard deviation of change in portfolio value) is calculated from a set of data to be $52500. The critical value for 1% is -2.32.

Value at Risk = -2.32 * Standard deviation of change in portfolio value

= -2.32 * 52500

= -121800

So the 99% one day VaR value is -121800.

The average of critical values of the normal distribution values beyond 99% is -2.6640.

Thus,

Conditional Value at Risk = -2.6640 * Standard deviation of change in portfolio value

= -2.6640 * 52500

= -139860

### Chapter 5

### Markowitz Model

### Brief History On Markowitz

Modern Portfolio Theory (MPT) was introduced by Harry Markowitz with his paper on "Portfolio Selection" which appeared in the 1952 Journal of Finance. This work of his earned him a share of 1990 Nobel Prize in Economics. A basic principle in economics is that due to the shortage of resources, all economic decisions are made in the face of trade-offs [18]. Markowitz identified that the trade-off facing the investor was risk versus expected return. His theory brought out optimal portfolio construction, asset allocation and benefits of investment diversification [19]. Most of the efficient frontier represents well diversified portfolios. This is because diversification is a powerful means of achieving risk reduction; therefore mean-variance analysis gives precise mathematical meaning to the saying "Don't put all your eggs in one basket".

Mean Variance portfolio optimisers respond to the uncertainty of an investment by selecting portfolios that maximise profit subject to achieving a specified level of calculated risk, or equivalently minimize variance subject to achieving a specified level in expected gain.

### Mean Variance Models

Two mean variance models are described below [20, 21]. The first mean variance model minimizes the risk subject to a desired level of return. This MV model determines the efficient portfolios given investment opportunity and a specified confidence level.

Where:

defines the expected rate of return of asset

the level of return for the portfolio

the covariance between asset and asset

the fraction of the portfolio value invested in asset

The objective function minimizes the covariance term, which minimizes the risk of the portfolio. Constraint (1) specifies the return expected from the portfolio; this represents the return of the efficient portfolio. Constraint (2) assures that the whole budget is invested. By specifying a level of expected return on the portfolio, the above quadratic model computes the corresponding minimum risk of the portfolio.

The second mean variance model given below minimizes the risk and maximizes the return.

Where is a parameter By varying the parameter between zero and one, the efficient frontier is computed. Constraint (1) assures that the whole budget is invested.

Mean variance models are expected to be a very powerful tool for portfolio optimizers. This is to efficiently allocate their wealth to different investment alternatives by reducing overall portfolio risk and/or achieving the maximum anticipated profit.

As I have been focusing more on risk (variance) and I will be calculating Conditional value at risk for a given set of asset, I am therefore going to use the first model, without constraint 1. This is because it will minimize risk, theoretically giving me the portfolio with the lowest CVaR value.

### Assumptions Of Mean Variance Models

As with any model, it is crucial to understand the assumptions of mean-variance analysis to use it effectively. Firstly mean variance analysis is based on a single period model. So at the start of the period the investor will allocate his wealth among a choice of asset classes, assigning a non-negative weight to each asset. During the period, each asset operates a random rate of return so that at the end of the period, his wealth has been changed by the weighted average of the returns. We have to clearly note that in selecting asset weights, the investor faces a set of linear constraints, one of which is that the weights must sum to one.

### Computational Study

There is a number of portfolio optimisation software available for a quantitative analyst intent on using the latest technologies as an aid by no means exhaustive. In this project I am going to be using AMPL to compute the weights for my portfolio and then on excel I will be calculating the CVaR of the portfolio.

### AMPL

AMPL stands for A Mathematic Programming Language. AMPL is said to be comprehensive and powerful algebraic modelling language. AMPL aims to introduce a linear programming model tool. AMPL is a programming language in which optimisation problems may be specified. It solves these problems using solvers such as CPLEX, FortMP, MINOS, IPOPT and SNOPT and so on to obtain the solutions. AMPL is a high level language describing mathematical programs. It allows a mathematical programming model to be specified independently of the data used for a specific instance of the model [22]. A functional diagram of how AMPL is used is shown in Figure 5.1 below.

Advantages of AMPL are that the marketing and support can be decentralized and also a new AMPL company can be expanded gradually. One particular advantage of AMPL is the similarity of its syntax to the mathematical notation of optimization problems. This hence allows for a very concise and readable definition of problems in the domain of optimization. Disadvantages include not much are known about the user base and development of new features can be hard to coordinate.

AMPL is fairly new software with a lot of advantages to solving optimization problems as mentioned above. Hence I will be using AMPL to apply Markowitz model for my portfolio of returns to obtain the optimal weights which will then be used to calculate CVaR.

### Chapter 6

### Design and Implementation

The goal of this project is to implement Markowitz first model described in Section 5.2, and to obtain the optimal weights for my portfolio. Then on Microsoft Excel I will be computing CVaR using the two different methods described in Section 4.3. The following sections describe how I calculated CVaR for the portfolio.

### Data Source

Historical prices of the assets are needed to calculate the rate of return and covariance matrix. Historical prices of assets from the FTSE 100 can be found online on sites such as yahoo! Finance and Google finance. However I want data for the last ten years on a weekly basis so that I can perform a more comprehensive test and neither websites were able to provide that amount of data. Therefore I decided to obtain the historical data from the DataStream provided in Brunel university library.

DataStream is a comprehensive database of company which provides financial and economic data. It contains a very precise updated data. The data comes from some of the biggest companies in the sector and also data from government agencies, IMF and OECD. Coverage of data goes back 20-30 years, depending on the data that you require and is available across a variety of different categories. The category I have chosen is equity indices which mean stock prices for the past ten years. From the data I got from DataStream I have randomly selected 50 stocks. The data of these 50 stocks were collected over a time period of 10 years which span from 20/12/1999 to 21/12/09.

### Portfolio Data

Figure 6.1 is a screenshot of half of my data which I got from FTSE 100. This portfolio consists of 103 stocks and its weekly prices over 10 years which means 523 weeks. As it can be seen from above, the stock prices for some stocks are not there for the earlier time periods, meaning they had not yet been included in the FTSE100, so therefore I will be excluding them from my data.

I have randomly chosen 50 stocks from the data above and using that for the rest of my research.

So the portfolio data I'm using for my project is a portfolio which consists of 50 stocks and its prices for 523 weeks. I have calculated the log returns for the 50 stocks as shown in Figure 6.2 below.

Since my portfolio consists of stock prices on a weekly basis, log return was calculated using the formulae:

The log return portfolio will be used for the rest of this project.

### Model Implementation

I will be using the first Markowitz model described in Section 5.2. This model has been implemented in AMPL to give the maximum weights for the portfolio.

The model that was implemented is shown below in Figure 6.3.

Line 1 'set stocks' refers to my sets, which in this case is my 50 stocks. Line 2 and 3 are my parameters. Where 'param returns' refers to the returns on stocks and 'param covar' holds that dimension of stocks going across and down. Line 4 and 5 are my variables. Line 6 is my objective and the last two lines are my constraints.

The corresponding data that was implemented is shown below in figure 6.4.

In my data file, as mentioned above the first line is the number of stocks I have. Line 'param returns' is the total of the returns of each stock and 'param covar' is the covariance matrix I had calculated using the log returns.

The solution that was given in AMPL is shown in figure 6.5

The solution file shows the optimal portfolio weights to obtain the optimal solution. The objective of Markowitz model was to generate the minimum variance portfolio; which is given in figure 6.3 as 0.0004152547393164389. It can be seen that the weights are only given for 20 stocks out of the 50 stocks from my portfolio. This is because only 20 stocks were given a weight greater than 0. Also when adding the weights up they add to 0.99999937, which can be rounded to 1.

### Calculating CVaR With Empirical Approach

The first step is to have all the historical values of your portfolio and multiply each week's rate of return by their corresponding weights which I computed on AMPL. Therefore as I only got weights for 20 stocks, I will only be using those 20 stocks as the other stocks weights are equal to zero.

The next step is to find the change in portfolio value. This is done by calculating the total portfolio return. To calculate VaR I computed the percentile of the 522 weeks change in portfolio at the desired level. Here I have used 95% and 99%. Then I have arranged the change in portfolios values in ascending order. CVaR is equal to. Hence to calculate CVaR I computed the average of all the change in portfolio values which are equal to or less than the VaR value. So looking at VaR at 95%; The VaR value is -0.0333101. So I have taken the average of all the numbers from -0.1057354 to -0.0333161. This is because the number after -0.0333161 is -0.0331959, which is greater than the VaR value at 95%. The CVaR value at 99% is calculated the same way.

### Calculating CVaR Using Variance-Covariance Method

The first step is to compute the variance covariance matrix of my 50 stocks. This is done by installing the variance covariance add-in into Microsoft excel. Then by matrix multiplication of transpose weights and the covariance matrix I get:

As seen from above, the weights I computed using Markowitz model on AMPL is used for their corresponding stocks.

To calculate variance I multiplied the transpose of weights matrix by the variance covariance matrix I computed before using the covariance add-in on excel. Then I multiplied the solution matrix of that again by the weights matrix to get the variance. Standard deviation is hence the square root of the variance.

To calculate CVaR I need the confidence level with their corresponding critical values of normal distribution. I have calculated CVaR for 95% and 99%; respectively the confidence level goes up by 0.25% for 95% and 0.05% for 99%. The smaller the difference the more precise and better the CVaR value. To calculate CVaR at 95%, the confidence level confidence level increases in the following order: 95%, 95.25%, 95.5%, 95.75%, 96%, 96.25%....99.75%. The critical value of normal distribution was calculated using the following formula in excel: normsinv(1-). Then the critical value for 95% which is -1.64485, multiplied by the standard deviation 0.020378835 gives you the VaR value at 95%. The VaR value at 95% is -0.033520201. Then I computed the VaR value for the rest of the percentages mentioned above. Finally to calculate CVaR I calculated the average of the sum of the VaR values I calculated as seen above. So CVaR equals:

The CVaR for 99% was calculated similarly.

### Conclusion and financial analysis

The main aim of the analysis was to calculate CVaR of the portfolio. Two methods for calculating CVaR were applied and four CVaR values were found.

By applying the Empirical Approach, we found at 95% level VaR was -0.0333101 and at 99% level VaR was -0.0592752. At 95% level CVaR was-0.0507345 and at 99% level CVaR was -0.0756934.

By applying the Variance-Covariance method, we found that at 95% level VaR was -0.033520201 and at 99% level VaR was -0.04740826. At 95% level CVaR was -0.04117 and at 99% level CVaR was -0.05358.

The graph visibly shows that the minimum CVaR value is -0.0756 which was given by the Empirical Approach at 99% and the maximum CVaR value of -0.0411 which was given by the Variance-Covariance method. The values on the graph represent the percentage of loss of the portfolio. For example the value -0.0756 represents a 7.56% loss of the portfolio, the value -0.0535 represents a 5.35% loss of the portfolio and so on. So the minimum CVaR value is -0.0411 which represents a 4.11% loss of the portfolio. Hence we can conclude that the smallest CVaR for this portfolio is a 4.11% loss which was given by the Variance-Covariance method at 95%.

### Chapter 7

### Conclusion

In this project I have looked at optimising a portfolio using Markowitz Mean Variance model and measured CVaR by two different commonly used methods. Markowitz model and the two methods Variance Covariance and Empirical approach make the main scope of the project.

The first part of this project was to obtain the minimum variance portfolio using Markowitz model. The implementation of Markowitz code in AMPL gave me the weights for stocks which gave the minimum variance portfolio. Analysis showed that only 20 stocks were given for the optimal portfolio.

The second and main part of this project was to calculate CVaR using two methods: Empirical approach and Variance-Covariance. The Empirical approach at 99% confidence level gave the worst VaR, which gave the maximum loss for the portfolio. However as shown in data analysis the Variance Covariance method at 95% confidence level gave the minimum CVaR for the portfolio.

There can be a number of reasons for these results. Firstly Empirical Approach is non parametric and does not require any distributional assumption. While both methods estimate VaR using historical data, Empirical Approach is much more reliant on them than Variance-Covariance for the simple reason that the Value at Risk is computed entirely from historical price changes. This method also saves us the trouble and related problems about distributions of returns but it implicitly assumes that asset returns in the future will have the same distribution as they have in the past [23]. In a market where assets are volatile and structural shifts occur at regular intervals, this assumption is difficult to sustain.

Therefore an advantage of Variance-Covariance is that you are able to consider possible portfolio returns that could happen. Variance-Covariance is easy as it only needs two factors: average return and standard deviation. Drawbacks of Variance-Covariance are that it assumes returns are symmetric and normally distributed [24]. Another drawback can be that even if the standardized return distribution assumption holds, the VaR can still be wrong if the variance and covariance used to calculate it are incorrect. To the extent that these numbers are estimated using historical data, there is a standard error associated with each of the estimates. In other words the variance covariance matrix that is input to the VaR measure is a collection of estimates, some of which may have very larger error terms. A final critique that can be levelled against the variance-covariance method to calculate VaR is that it is designed for portfolios where there is a linear relationship between risk and portfolio positions. Consequently, it can break down when the portfolio includes options, since the payoff on an option is not linear.

### Critical appraisal

The aim of the project was to measure CVaR for a portfolio and the objective of this project was to carry out a test between the different ways of measuring CVaR. This task was successfully completed and showed that the Empirical method was the better model which gave the minimum CVaR.

One problem of this project was that the topic was new to me as I had not done any financial modules in my degree so far. But as I had chosen modules like 'Corporate investment' and 'Strategic financial management' during this year, these modules covered topics on Markowitz model and portfolio risk, which helped me to understand more on this project.

After doing this project I have grasped a firm foundation on portfolio risk, in particular Value at Risk and Conditional Value at Risk. Looking back at my draft plan, I feel that I have achieved most of my aims and objectives and made good use of my Gantt chart as a tool to manage my time.

### Statement of Initiative

I decided to select this project because it sounded interesting. The moment I was given the title, I was researching and trying to understand the topic. My supervisor without a doubt helped me and gave me direction, but this project does have an independent piece of work. I took the initiative to go beyond any suggestion my supervisor made to ensure that I got information from as many sources as possible. An example of this is I would go beyond books recommended by my supervisor and research into more journals to help me understand my project more. I loaned books from the library and printed off a lot of research that I had gathered. I also familiarised myself with AMPL, which was tricky at first. I researched Markowitz Mean Variance model for AMPL. Coding the data file for 50 stocks was tricky at first but with a lot of tries and a bit of assistance I managed to get it done successfully.

### Recommendation

I feel that Conditional Value at Risk and portfolio optimisation is a very interesting topic. I definitely recommend this as a fascinating subject. Portfolio optimisation covers many topics and its relation to reducing risk whilst maximizing profit being the ultimate goal for almost every business makes it quite motivating. I would suggest Conditional Value at Risk and portfolio optimisation as a final year project because it is understandable and can be related to other topics and Conditional Value at Risk is a fairly new topic. In addition, there is a lot of information available for optimisation.

Due to time, there are a few more interesting topics that I would have liked to explore. The main theorem I would have liked to have explored is computing CVaR for a portfolio in AMPL. This is because as mentioned in the start CVaR is a fairly new topic and AMPL is a growing software for optimizing portfolio, hence I feel more accurate and easily obtainable results can be generated, which will help industries in the future. There are also other factors that can be looked at in the future such as measuring CVaR for non normal distributions and also for non linear portfolios such as bonds.

If I had two or three more months to work, I would have continued the exploration of calculating CVaR. I would have researched more into Markowitz model for obtaining portfolios greatest return with minimum CVaR and would have optimised CVaR in AMPL. I would have also explored more on Markowitz efficient frontier graphs. As a suggestion, a title for project next year could be 'Optimising CVaR in AMPL'.

Overall I was able to finish the programming and write up in the time given.

### References

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