# Differences Between The Common Risk Measurement Finance Essay

Nowadays, investors are no longer focuses in investing in one type of investment only but rather on several type of investments all together such as in property, unit trust, shares, bonds and others. Investing in several type investments are called portfolio investment. Investing in portfolio investment could spread the risk of possible loss due to below expectations performance of one or a few of them. For investors to ensure gaining good returns and profits from their investment, they need to be able to measure the risk of all of their portfolio investments. "Risk is an asymmetric, relative, heteroskedastic, multidimensional concept that has to take into account asymptotic behavior of returns, inter-temporal dependence, risk-time aggregation, further sources of risk, and the impact of several economic phenomena that could influence an investor’s preferences." - Desirable Properties of an Ideal Risk Measure in Portfolio Theory (2004).

Historically, the most commonly used risk measure is the standard deviation (variance) of a portfolio’s return. In spite of its computation simplicity, variance is not a satisfactory measure due to its symmetry property and inability to consider the risk of low probability events. (Ekaterina N. Sereda, 2007). To overcome this, other approaches are considered, comparing the portfolio holdings produced by different risk measures, rather than the risk return trade-off. In this way we can see whether the risk measures used produce asset allocations that are essentially the same or very different and hopefully the best measurement could be found in measuring the risk in portfolio. (Peter Byrne and Stephen Lee, 2004)

## BACKGROUND OF STUDY

Many investors nowadays mistakenly base the success of their portfolios investments on returns alone. Few of them consider the risk that they took to achieve those returns. Since the 1960s, investors have known how to quantify and measure risk with the variability of returns, but no single measure actually looked at both risk and return together. Today, the most popular risk measurement on portfolio optimization is variance or mean-variance. However, there are other measurements out there that also could be useful to be use as a tool in measuring the risk of the investments. Measuring the risk of the portfolio could somehow help the investors makes decision whether to proceed with portfolio or not. Based on the risk measured, the investors can calculate the return of the portfolio and compare it with it risk to determine whether it worth buying or not.

This study is being conducted to compare the differences between the common use mean-variance in determining the risk of the investments than the other types of risk measurements such as Lower Partial Moment (LPM) and Mean Absolute Deviation (MAD).

## PROBLEM STATEMENT

Usually, the measure of investment risk used in portfolio approach is the variance. There are other approaches as well to measure the risk of the portfolio. However, these measures are less being used in the process compared to the traditional way of calculating risk. This study is being conducted to compare the main risk measurement used, Variance and other types of risk measurements used by investors to measure the risk on the portfolio they invest.

## RESEARCH QUESTION

The main research question of this study is:

Are there any differences of using other types of risk measurement such as Semi-Variance (SV), Lower Partial Moment (LPM), and Mean Absolute Deviation (MAD) rather than the common Mean-Variance (MV) in measurement of risk portfolio optimization?

## OBJECTIVE OF STUDY

To compare differences between the common risk measurement, Mean-Variance (MV) and other types of risk measurements or approaches which are Semi-Variance (SV), Lower Partial Moment (LPM), and Mean Absolute Deviation (MAD).

## SCOPE OF STUDY

The study will focus on determine risk of portfolio investment by analyzing data consists of share’s price of different companies in each industry in the stock market. The periods of this study that will be used are from year 2005-2010. The methods involved in determining the risk of portfolio investment are consists of different approaches which are Mean-Variance, Semi-Variance (SV), Lower Partial Moment (LPM), and Mean Absolute Deviation (MAD).

## LIMITATION OF STUDY

In the way to successfully complete this project paper, there are some constraints that have to be faced:

## Time horizon

This study is conducted in a limited extend of period where data analyze from 2005 to 2010 where supposedly the best way in determining the risk of portfolio allocation should be at least 10 years period.

## Data analyzing

\ The determining of portfolio risk using several types of risk measurements that are not commonly used by investors such as Semi-Variance (SV) and Lower Partial Moment (LPM) bring difficulties in determining the way of the risk measured which each methods are determine in different ways and approaches.

## SUMMARY

Although there are not much researchers conducting this study before, this study is important as it could somehow be useful to some people especially the investors to measure and calculate the risk of the investments they are doing. They are no longer limited to only the traditional and common method, Mean-Variance (MV) but also other approaches such as LPM and MAD to be use in their investments risk measurement.

## CHAPTER 2

## LITERATURE REVIEW

## INTRODUCTION

Traditionally, the measure of risk used in portfolio optimisation models is the variance. However, alternative measures of risk have many theoretical and practical advantages and it is peculiar therefore that they are not used more frequently. This may be because of the difficulty in deciding which measure of risk is best and any attempt to compare different risk measures may be a futile exercise until a common risk measure can be identified (Peter Byrne and Stephen Lee, Different risk measures: different portfolio compositions?).

## PREVIOUS STUDY

Before this, there are not many studies being done to show the comparison of different approaches of risk measurement as the usage of Mean-Variance method in determine the risk of portfolio have been used for so long. Markowitz legendary work about portfolio optimization is accepted to be the pioneer of the modern portfolio theory. In the Markowitz model, risk is stated in terms of the predicted variance of portfolio return-a function that is quadratic in the decision variables. All other functions and constraints are assumed to be linear (Sharpe, 1971). However there are weakness of the MV approach which is that the underlying assumptions of multivariate normality or that investors have a quadratic utility function are not sustainable. This has led researchers to develop portfolio asset allocation models using other measures of risk that have many theoretical and practical advantages over MV.

Although the MV model is the most popular approach, it relies on the assumptions that returns are either normally distributed or that the investor’s utility function is quadratic. If either of these conditions hold, it can be shown that choosing among risky investments is compatible with the maximization of an investor’s expected utility (Tobin, 1958).Even so, the MV approach remains the most popular approach to the asset allocation problem. (Peter Byrne and Stephen Lee,2004). This may be because deciding which measure of risk is “best” is still unresolved (Stone, 1973). Reffering to Cheng and Wolverton (2001) for example, highlight the difficulty of comparing portfolios based on different risk criteria. They find that each approach produces results that minimise risk, but only in its own space. When the portfolio compositions of one risk measure are used to calculate the risk and return trade-off in another risk space the results are always “inferior” to the solutions produced inside that risk space. They argue that any attempt to find the portfolio model that offers the best risk return trade-off is likely to be useless until a common risk measure can be identified.

Based on (Ali Argun Karacabey,2006), Mean Absolute Deviation (MAD) model is said to be a viable alternative because (i) it does not require the covariance matrix of the returns, and (ii) MAD portfolios have fewer assets (Simaan, 1997). It is also argued that as the number of the assets decreases the transaction costs of the portfolio will decrease either. Furthermore, based on Sreenarayanapurath Madhavan Sunoj ,2004 on LPM, consider a portfolio with a random return X and assume individual has a target return t. An outcome larger than t is nonrisky and desirable, then individual faces only a one-sided risk called the downside risk that occurs when X falls short of t. Therefore LPM provides a measure that a specified minimum return (target return) may not be earned by a financial investment. Clearly, lower partial moments provide a summary statistics for the downside risk.

## SUMMARY

Understanding the previous studies done by other researchers is very important as it could provide guidelines and references for this study. Previous study gives clear contexts and information based on the portfolio risk measurements used in this study. The findings from the previous study will enhance the progress and completion of this study.

## CHAPTER 3

## METHODOLOGY AND DATA

## INTRODUCTION

This chapter will explain the methodology and data analyzing procedures to be use in this study including data collection, sampling frame, research design and others .The objective of this study is to investigate differences between the common risk measurement, Mean-Variance (MV) and other types of risk measurements or approaches which are Semi-Variance (SV), Lower Partial Moment (LPM), and Mean Absolute Deviation (MAD) which will analyze the data comprise of several company’s share price’s return in each stock price industry from year 2005-2010.

## DATA COLLECTION

The data that being used in this study were consists of company’s monthly share’s price which are taken from different companies in Malaysia. The share prices are taken from companies in each type of sector (13 sectors) in the stock market.

## SOURCES OF DATA

All these data and information were being collected from:

DataStream

Bloomberg

## SAMPLING FRAME

The sampling frame covers the share prices and returns of 57 companies in all 13 types of sectors in Malaysia for the period of 6 years which are from year 2005-2010.

## DATA ANALYSIS AND TREATMENT

## Risk Calculation

## Mean-Variance (MV)

The variance is the most commonly used risk measure in portfolio optimization models. It is a measure of the dispersion of a set of data points around their mean value. Variance is a mathematical expectation of the average squared deviations from the mean.

Formula (Investopedia):

C:\Users\kRuiSer\Desktop\123.png

Where:

n = the total number of observations

rt = the observed value

average = the mean or target value of the data set

## Semi-Variance (SV)

A measure of the dispersion of all observations that fall below the mean or target value of a data set. It is an average of the squared deviations of values that are less than the mean.

Formula (Investopedia):

Where:

n = the total number of observations below the mean

rt = the observed value

average = the mean or target value of the data set

## Lower Partial Moment (LPM)

A measure of downside risk computed as the average of the squared deviations below a target return. This measure of downside risk is more general than semi- variance which is computed as the average of the squared deviations below the mean return.

Formula (S. Gorard, 2004):

Where:

T = number of data

Rt = data element

= the target value of the data

## Mean Absolute Deviation (MAD)

The mean deviation (also called the mean absolute deviation) is the mean of the absolute deviations of a set of data about the data's mean.

Formula (S. Gorard, 2004):

Where:

T = number of data

Rt = data element

E(R) = the mean value of the data

## Portfolio Performance

Portfolios are developed employing the Mean-Variance (MV), Mean Absolute Deviation (MAD), Lower Partial Moment (LPM) and Semi-Variance (SV) models in order to compare the portfolio performances of different optimal portfolios. The portfolio performance is calculated using the reward/return per risk equation:

Portfolio Performance = mean return / risk

## SUMMARY

This research will be done in accordance to the objective of this study as to investigate the differences between the common risk measurement, Mean-Variance (MV) and other types of risk measurements or approaches which are Semi-Variance (SV), Lower Partial Moment (LPM), and Mean Absolute Deviation (MAD).After calculate all the risk based on all the risk measurements, the performance of each methods will be measure by observing the trends and calculate the portfolio performance.

## CHAPTER 4

## FINDINGS AND ANALYSIS

## 4.0 INTRODUCTION

This chapter will explain the result gain from this study. This study is being conducted to compare the differences between the common use mean-variance in determining the risk of the investments than the other types of risk measurements such as Lower Partial Moment (LPM) and Mean Absolute Deviation (MAD) which analyze the data comprise of several company’s share price’s return in each stock industry (13 industry) from year 2005-2010.

## 4.1 RISK CALCULATION

## 4.1.1 Mean-Variance

2005

2006

2007

2008

2009

2010

Mean

-1.67%

2.40%

3.41%

-4.06%

2.13%

2.56%

Variance (Risk %)

0.097%

0.137%

0.400%

0.320%

0.100%

0.120%

## 4.1.2 Semi-Variance

2005

2006

2007

2008

2009

2010

Mean

-1.67%

2.40%

3.41%

-4.06%

2.13%

2.56%

Semi-Variance (Risk index)

0.067%

0.220%

0.910%

0.590%

0.192%

0.240%

## 4.1.3 Lower Partial Moment (LPM)

*Target return = 0%

2005

2006

2007

2008

2009

2010

Mean

-1.67%

2.40%

3.41%

-4.06%

2.13%

2.56%

LPM

(Risk index)

0.150%

0.400%

0.580%

0.594%

0.394%

0.200%

## 4.1.4 Mean Absolute Deviation (MAD)

2005

2006

2007

2008

2009

2010

Mean

-1.67%

2.40%

3.41%

-4.06%

2.13%

2.56%

MAD

(Risk index)

2.36%

2.42%

4.96%

3.93%

2.32%

2.63%

Year

Risk (%)

The graph above shows the value of risk for all for methods (MV, SV, LPM and MAD) for year 2005 until 2010. The risk values for all methods shows almost the value except MAD where the risk values are much higher than the rest methods because the method measures the average absolute deviation of observations from their forecasts without the squared mean. The SV, MV and MAD methods shows almost the same pattern of risk value however the LPM shows a bit different because the measure is based on downside risk computed as the average of the squared deviations below a target return where the target value for this study is 0.

## 4.2 PORTFOLIO PERFORMANCE

Portfolios are developed employing the Mean-Variance (MV), Mean Absolute Deviation (MAD), Lower Partial Moment (LPM) and Semi-Variance (SV) models in order to compare the portfolio performances of different optimal portfolios. The data consists of monthly returns of 56 stocks included in the Bursa Malaysia with all 13 industries from 2005 until 2010. The portfolio performance is calculated using the reward/return per risk equation:

Portfolio Performance = mean return / risk

MV

SV

LPM

MAD

Mean

0.0083

0.0083

0.0083

0.0083

Risk

0.0026

0.0040

0.0035

0.0396

Performance

3.1923

2.0750

2.3714

0.2096

Summary statistics of the optimal portfolios generated

As shown in table above, the mean returns of all risk measurement are the same with all four models. The most risky portfolio is the SV model (0.0040) while the less risky portfolio is MV model (0.0026). The MV model (3.1923) shows the highest performance whereas the MAD model (0.2096) gives the lowest performance.

## 4.3 SUMMARY

This chapter explains the result for this study using the methods that had been state in the previous chapter. From the result, the pattern of portfolio risk values from 2005 to 2010 for methods MV, SV and MAD are the same while LPM shows different pattern. The portfolio values for MV, SV and LPM shows almost the same while MAD a bit higher than the rest methods. Based on the portfolio performance, the most risky portfolio is the SV model while the less risky portfolio is MV model. The MV model shows the highest performance whereas the MAD model gives the lowest performance.

## CHAPTER 5

## CONCLUSION AND RECOMMENDATIONS

## CONCLUSION

The main purpose of this study is to determine the differences of several types of portfolio risk measurement. It is mainly to compare those measurements and discover whether there are differences between using the common Mean-Variance (MV) and other types of risk measurements or approaches which are Semi-Variance (SV), Lower Partial Moment (LPM), and Mean Absolute Deviation (MAD). The study focuses on determine the risk of portfolio investment by analyzing data consists of share’s price of different companies in each industry in the stock market. The periods of this being used were from year 2005-2010. The methods involved in determining the risks of portfolio investment are consist of different approaches which are Mean-Variance, Semi-Variance (SV), Lower Partial Moment (LPM), and Mean Absolute Deviation (MAD). To compare the differences of each method, several methods and tools had been used. From the result, the pattern of portfolio risk values from 2005 to 2010 for methods MV, SV and MAD are the same while LPM shows different pattern. The portfolio values for MV, SV and LPM shows almost the same while MAD a bit higher than the rest methods. Based on the portfolio performance, the most risky portfolio is the SV model while the less risky portfolio is MV model. The MV model shows the highest performance whereas the MAD model gives the lowest performance.

This paper discusses the theory of risk measures and compares the portfolio optimization models with different risk measures. The overall result shows that MM model outperforms the other models. As shown by the result of this study, the MV model does not perform as well as other models. As such, the MV model is a better choice for portfolio optimization compared to the other models for it ranks highest in terms of performance. The results thus explain why the Mean-Variance method is still the most popular choice for investors in determining portfolio risk and optimization. However, other methods have their own advantages compare to MV such as SV and LPM, these models are appropriate for investors who have a strong downside risk aversion where it evaluate the return or data that are lower than the mean value or target value.

## 5.1 RECOMMENDATION

There are some recommendations for future researches that wishes to continue this study further more. They should include more alternative risk measures that can be use to compare and determine the best risk measurement. This is because there are other methods of measurement that can be tested such as Minimax and conditional value at risk in portfolio optimization and this might give more better and interesting result. Other than that, future researches can also use other methods and tools in analyzing and comparing these methods in order to gain better and more accurate results. Furthermore, other stock markets can also be use to wider the results of this study as well as using longer duration period such as 10 years period.

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