Analysis of Risk Measures in Portfolio Optimization for the Uganda Securities Exchange

For the most recent years, risk has become one of the essential parameters in portfolio optimization problems. Today most practitioners and researchers in portfolio optimization have used variance as a standard risk measure. This approach has been found subjective. The Markowitz (1952) mean-variance model considered variance as an adequate portfolio risk measure, and asset returns are multivariate normally distributed and that investors have a quadratic utility function which is subjective too. Other risk measures have been suggested to overcome the limitations of the mean-variance model. This paper analyzes which portfolio optimization models can better explain the optimal portfolio performance (high return, low risk) for the Uganda Security Exchange (USE). We compare Mean-Variance (MV), Mean Absolute Deviation (MAD), Robust Portfolios and Covariance Estimation Models (The Shrinked Mean-Variance (SMV) Models & Alternative Covariance Estimator (ACE) Models) and Mean-Conditional Value-at-Risk (Mean-CVaR) models in terms of the risk and performance. For the computed monthly returns and price data (February 2010 to January 2021) for USE selected stocks, we considered the results to show that Mean-CVaR and ACE portfolios have the highest performance ratio compared to other models. We find that VaR is the best risk measure for portfolio optimization for the USE since it has lower values across all models than other risk measures. It is vital to consider all the available risk measures for a regulator or practitioner to make a good decision since using one can be subjective; as seen in our results, different risk measures yield different results.


Background of the Study
A portfolio is a collection of financial instruments. Portfolio optimization is the process of selecting the best portfolio (asset distribution) by the investor from a list of portfolios that will give him or her the highest level of return given an investment objective. Risk measures are tools used to assess the performance of an investment basing on the exposure and degree at which the associated prices move (Volatility). The question is which risk measure to be selected for the appropriate portfolio investment management. Markowitz (1959) laid the groundwork for MPT defining an investor's portfolio selection problem regarding expected return and variance of return. He postulates that an investor should maximize expected portfolio return while minimizing portfolio variance of return. Since the introduction of Markowitz (1952) Mean-Variance (MV) model, variance has become the most common risk measure in portfolio optimization. However, this model relies strictly on the assumption that the returns of assets are multivariate normally distributed or the investor's utility function is quadratic (Hoe et al., 2010). The most recent literature has shown that the Markowitz framework and MV formulation based on these two assumptions seem not to hold in the real market. Markowitz argues that given estimates of the returns, volatilities, and correlations of a set of investments and constraints on investment choices (Nduku, n.d.). From this, it is possible to perform an optimization that results in the risk/return, or mean-variance efficient frontier, according to him, this frontier is efficient because every portfolio on this frontier is a portfolio that results in the greatest possible expected return for that level of risk. However, it is observed that in postwar US data, the slope of the mean-standard deviation frontier is much higher than reasonable risk aversion and consumption volatility estimates suggest. Brooks and Kat (2002) also show that hedge funds returns are not normally distributed.

Modern Portfolio Theory (MPT)
Therefore due to issues raised and limitations of Markowitz's (1952) Mean-Variance (MV) model, different measures of risk measures in portfolio optimization have been proposed, such as Mean Absolute Deviation (MAD) by Konno and Yamazaki (1991), Minimax (MM) attributed to Young (1998), Albuquerque (2009) Beta models and Silva et al. (2017) Beta-Conditional Value-at-Risk (CVaR) models. There have been several studies in Portfolio optimization using different risk measures by employing portfolio optimization models. Mayanja (2011) urges that to carry out portfolio optimization, one needs software, which must have inbuilt algorithms. This means that for one to think about portfolio optimization, one should first be assured of a portfolio that exists, then the next step would be which percentages in terms of allocation should be allocated to each portfolio. Such portfolios are available in Stock (Securities) Exchanges where they are traded; in these exchange markets, buyers and sellers hold securi-C. Birungi There exist portfolio optimization studies in the Uganda Security Exchange.

Most of the studies conducted on portfolio optimization in Uganda Securities
Exchange have concentrated on the tests for the stock performance of the models, generating more risk measures like volatility, Sharpe Ratio (SR), Risk Parity (RP), Expected Shortfall (ES) or CVaR which they used to assess stock performance (Baganzi et al., 2017). These studies conducted were on how the portfolio performs, but not on the investigation of which portfolio optimization models can better explain the portfolio performance in terms of return and risk for the

The Problem Statement
One way of selecting optimal portfolios at USE is through market surveillance and speculation. However, USE is not mature enough to facilitate investors' investment choices, leading to the low utility of the same to the investors. Litera- ture shows that the approach of market surveillance and speculation is insufficient and subjective. One way of facilitating optimal investment portfolios in the market is by ensuring that the models used in portfolio optimization are as close to the actual market scenario as possible. Therefore, we want to answer the central question: Can we get an alternative method of selecting optimal portfolios at USE? To answer this question, we compared traditional portfolio optimization models to select optimal portfolios with the proposed risk models. We used variance, Covariance, Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) as risk measures to find out which model is efficient (high return, low risk) for USE.

Main Objective
This study's main objective is to establish the best risk measure for Portfolio Optimization for the USE.

Specific Objectives
The specific objectives of this research are: 1) To develop optimal portfolios using the traditional portfolio optimization models and the proposed risk models at USE.
2) To analyze the performance of the portfolios in terms of returns at USE.
3) To analyze the risk associated with each portfolio using different risk measures at USE.

Significance of the Study
The study will: 1) To provide an alternative method of selecting optimal portfolios at USE.
2) Establish the best risk measure for Portfolio Optimization for the USE.
3) Develop portfolio optimization models to select an optimal portfolio for investment at Uganda Securities Market and other Ugandan financial institutions with interest in portfolio investment. 4) Add on the foundation and further research portfolio optimization on Securities Exchanges especially developing markets like USE.
We structured the rest of this paper as follows. The next section 2, the mathematics discusses the mathematical models, mainly traditional portfolio optimization models and the proposed risk models. Section 3 methods discusses data used, parameter estimation, portfolio compositions. In the later section 4, we discuss the formed portfolios' computational and performance results by employing the five optimization models mentioned using the proposed data on section 3 of the Uganda Securities Exchange.

The Mathematical Formulation for Optimal Portfolio Formation
The data collected was used in testing and analysis of different risk measures in Mean-CVaR (7) models to compare the portfolio performance in terms of returns and risk associated with each optimal portfolio. The data include computed monthly returns and prices for the period from February 2010 to January 2021. We employed different portfolio optimization models to form these optimal portfolios.

Mean-Variance (MV) model
As proposed by Markowitz (1952), the MV model seeks to minimize Variance ( ) V R at a given level of expected returns ( ) E R shown in Equation (1). Equation (1), together with its constraints, is the famous Markowitz's Mean-Variance Model, also commonly known as the Modern Portfolio Theory (MPT) model (Markowitz, 1952). Markowitz laid the groundwork for the modern portfolio theory. The MV model's objective is to find the weight of assets that will minimize the portfolio variance at a level of the required rate of return. This model is a quadratic programming model (Hoe et al., 2010). We formulate the mathematical model as follows: Subject to:

5)
, is the covariance between assets j and k.
Parameters, N is the number of assets, T is time, jt y is return of asset j at time t, j y is the mean return of asset j, kt y is the return of asset k at time t, k y is the mean return of asset k, j w is the portfolio allocation for asset j, j u is the maximum budget share that can be invested in assets j. k w is the portfolio allocation for asset k. Then p µ and 2 p σ will be the desired level of the expected return on the portfolio and ts variance, respectively. Konno and Yamazaki (1991) proposed the Mean Absolute Deviation (MAD) model as a risk measure to overcome the Mean-Variance (MV) model's weakness. The MAD model assumes that the standard deviation is a satisfactory portfolio risk measure. MAD employs a mean's absolute deviation for measuring risk instead of the variance. Literature shows that if returns are normally distributed, both MAD, MV and MM yield the same results. Basing on (Konno & Yamazaki, 1991), MAD is mathematically formulated as follows:

Mean-Absolute Deviation (MAD) Model
, ρ is a parameter representing the minimal rate of return required by an investor. This constraint means portfolio expected return exceeds some minimum (ρW).

2)
1 N j j w W = ≤ ∑ the total allocations to the portfolio do not exceed the budget (W).
, total allocations or portions or fraction of capital allocated in selected assets equals 1, meaning all the money must be invested.
Parameters, N is the number of assets, T to be used later is the time, j Y , is a random variable representing return per period for asset j, j w , is portfolio allocation to asset j, j u , is the maximum budget share that can be invested in asset j, W ρ is the minimum level of return, W is the total allocation. Konno and Yamazaki (1991) assume that the expected value can be approximated by the average over Here, jt y is the realization of random variable j Y during period j. Substituting Equation (3) into Equation (2) we have Equation (4) ( ) ( ) From Equation (4) (5) and now the optimization problem in (2) can be written as, Subject to the same constraints in Equation (2). By simplifying Equation (5) further that is jt j , total allocations, portions of capital allocated in selected assets should be 1, meaning all the money must be invested.
Equation (6), together with its constraints, becomes our linear optimization problem. Hoe et al. (2010) show that there is no need to calculate the covariance matrix for this linear problem. Furthermore, it is a linear program and Equation (6) penalizes both negative and positive deviations. Literature shows that investors prefer higher positive deviations and avoid lower negative deviations in portfolio return (Hoe et al., 2010).

Mean-CVaR Models
Most literature shows that traditional optimization models fail to provide efficient portfolios, especially when financial assets' returns are highly volatile, Silva  Since Equation (7) is an optimization problem, the author proposes minimizing CVaR α and VaR α are not equivalent. They, therefore, consider the fol-  Würtz et al. (2015) proposed Robust Portfolios and Covariance Estimation Models to compute the mean and covariance matrix of the set of financial assets to achieve better stability properties compared to traditional minimum variance portfolios. We use two different approaches implemented by Würtz et al. (2015) that is robust mean and covariance estimators, and the shrinkage estimator. Würtz et al. (2015) considered a convex combination of the empirical estimator with some suitable chosen target. According to the Authors, a mixing parameter was selected to maximize the expected accuracy of the shrinked estimator; this was done by using an analytic estimate of the shrinkage intensity. Unlike the computational cost required in the MV model (1), Shrinked Mean-Variance Models increases and in terms of boundness, shrinkage estimate is always positive definite and well-conditioned, which is advantageous in terms of convergence. Würtz et al. (2015) provide an alternative to estimate covariance from an R's recommended packages, such as MASS, which has inbuilt functions to generate optimal portfolios. We apply this method on the 11 years' historical price data (132 months from 2010-02-26 to 2021-01-26) for nine stocks listed on the LSI USE indexes to compare the performance of this model with other models.

Data Collection
This study uses secondary data (computed monthly returns and prices for the selected stocks from February 2010 to January 2021) from USE Local Share Index (LSI), which tracks only the USE's local companies. This period was not randomly selected; we chose this period because the Ugandan economy reported solid economic growth, especially from 2016 to 2019, estimated at 6.3%, the expansion of services drove this. We considered nine stocks, Uganda Clays Ltd Airways, Jubilee Holdings Ltd., Equity Bank Ltd., Kenya Commercial Bank Ltd., Nation Media Group, Centum, UCHUMI) among others. Mathematically the nine stocks out of seventeen stocks by then we consider will be an excellent sample to represent the whole stocks listed at USE for this analysis to study the portfolio composition and risk measures. Information on the trading dates, opening price, closing price, Stock names, low/high prices, the volume traded was collected from the USE website. Using a data collection sheet, we only considered stock names, trading dates, and closing prices since we are interested in calculating expected values, standard deviation, and correlations of stock returns to calculate expected returns and volatilities of these stocks. The monthly returns were computed using Equation (9) from February 2010 to January 2021.

Parameters in the Mean-CVaR and Robust Portfolios and Covariance Estimation Models
Different R packages under library (fPortfolio) were used to estimate most parameters for the computed monthly returns data for each of the selected stocks using Equation (9) while performing the optimization of portfolios. The (1 − α) is the confidence level.

Data Analysis
We used the 11 years' historical price data (132 months from 2010-02-26 to 2021-01-26) for nine stocks listed on the LSI USE indexes from the USE website (https://www.use.or.ug/). We computed the returns for each stock at the monthly price for the monthly prices of the selected stocks at the monthly price, which we used for analysis. For analysis purposes, we assigned zeros(0 prices) where the stock was not traded, for example, the first two months of NIC stock and the first 35 months of UMEM stock. The sample of the first 12 months of 132 months for the selected stock prices are as shown on Figure 1. We then computed the monthly returns for each of the selected stocks using Equation (9) where, i denote the stock number, t denotes the period in months, , 1 i t P + denote the stock i price at month 1 t + , , i t P denotes the stock i price at month t and , 1 i t r + denote the stock i return at month 1 t + . We used Equation (9)

Portfolio Compositions
Basing on the mathematical framework of MV (1), MAD (2), Mean-CVaR (7) models and Robust Portfolios and Covariance Estimation Models in subsection, (2.4), Portfolios were developed. Weights were assigned on the selected stocks using different techniques, for example, equal weights feasible portfolio with "LongOnly" constraints and others we consider optimal portfolio allocation using   We then allocated these weights in percentages of each stock based on the five optimization models to develop five optimal portfolios. The portfolio compositions are shown on Table 1.
The results in Table 1 show portfolios generated by the five portfolio optimization models and their compositions. Due to differences on the weight of stocks, results from portfolio compositions normally differ. Difference in weight may be probably due to the non-normality displayed by data (Byrne & Lee, 2004). For example, different stocks have different performance over time. We used different optimal asset allocations techniques to attach weights on all assets.
In section 4, we use Variance (Sigma), Covariance, Value at Risk (VaR) and Conditional Value at Risk (CVaR) as risk measures for all the five optimal portfolios to compare which model is efficient (high return, low risk) for USE.

Risk Measure Performances
In terms of risk measure for each portfolio discussed in subsection (1.2), we illustrated the portfolio compositions and how risk measure would be the best based on the lowest values. Therefore, we use this reference (taking the smallest value for each risk measure of each portfolio as the risk measure value), as summarized in Table 2. Table 2 presents the results of the investment portfolios and the risk measures. We analyzed only the Variance, Covariance, VaR and the CVaR. The reason for this is that both risk measures or measures of dispersion have the same unit of measurement. The portfolios generated using traditional portfolio optimization models i.e. Markowitz's based models (MAD (6), MV (1) and SMV (2.4.1)) except ACE (2.4.2) with covariance 4.0759 (higher covariance than the Mean-CVaR model) showed lower values in terms of risks. This result is consistent with Silva et al. (2017). The values of the Covariance, Variance, VaR were lower for MAD (6), MV (1) and SMV (2.4.1) models while CVaR has higher values across all portfolios or models except in Mean-CVaR (7) model, this is because the primary objective of Mean-CVaR model is to minimize the CVaR as a risk measure. When we considered all the assets (9) for the Mean-CVaR portfolio to make sure all assets are represented, there was a reduction in the risk (lower values of the risk measures), that is when we reduced assets from 9 to 7 assets on the Mean-CVaR portfolio the values for risk measures Covariance, VaR and the CVaR increased. This result shows that a portfolio with more assets performs better in terms of risk. From

Portfolio Performances
In this subsection, we analyze the five models' portfolio performances discussed in (3.3) for the five models. We compare the performance of the traditional portfolio optimization models and the proposed risk models at USE. For the portfolios generated using Markowitz's based models, we considered variance or covariance as risk measures that is, by taking the smallest value of variance or covariance of each portfolio as the risk measure value) while for the mean-CVaR portfolio model, we considered the lowest value of conditional Value at Risk as the risk measure in this portfolio. For the portfolio expected return, we considered the highest fraction value of mean in each portfolio as the Expected Return of that particular portfolio, as shown in Table 3. Using this criterion, we

Conclusion
This study's main intention was to establish the best risk measure for portfolio optimization for the Uganda Securities Exchange (USE) and an alternative method of selecting optimal portfolios adopted in the USE. This involved: 1) Developing optimal portfolios; 2) Analyzing the performance of the portfolios in terms of returns; and 3) Analyzing the risk associated with each portfolio using different risk measures.
We used the USE secondary data (computed monthly returns and prices for the selected stocks from February 2010 to January 2021) from USE Local Share Index (LSI). We considered a sample of nine stocks from the USE Local Share Index. We chose nine stocks out of seventeen. Mathematically the nine stocks out of seventeen stocks listed at USE by then will be an excellent sample to represent the whole stocks listed at USE for this analysis to study the portfolio composition and risk measures.
There are concerns about model appropriateness for portfolio optimization. Hoe et al. (2010) find that some traditional portfolio optimization models are appropriate for investors who have a substantial downside risk aversion but not Journal of Financial Risk Management all of them. There was a general weakness of market surveillance and speculation method of selecting optimal portfolios in the literature, especially in developing markets. This study compared traditional portfolio optimization models to determine optimal portfolios with the proposed risk models. We used variance, Covariance, Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) as risk measures to find out which model is efficient (high return, low risk) for USE.
We analyzed different portfolio optimization models by comparing the risk measures (comparing risk measure estimates) in all portfolios. We developed portfolios, measured their performances in terms of the return and the risk associated with each portfolio. We used five portfolio optimization models, MV (1) We then compared the expected return and risk of the best performing models. In this case, we concluded that the Mean-CVaR (7) and ACE (2.4.2) in terms of both the expected return and risk. We find that ACE (2.4.2) performs better than the Mean-CVaR (7), with a performance ratio of 1.097318 and 1.783111, respectively. In terms of risk, ACE (2.4.2) was less risky than Mean-CVaR (7) though the difference was not too big. After considering all the factors, we conclude that the traditional portfolio optimization models generated an optimal portfolio with a higher return. The proposed risk models develop portfolios with less risk.
We also find that it is not about a specific risk measure; what matters is allocating weights to assets in a portfolio and the optimization method. Portfolios with more assets are less risky, i.e. lower risk measures' estimates were observed in portfolios with more stocks than those with fewer assets. However, we would like to quote the "decision on which risk measure to use should depend on the regulator or practitioner's intended use". Regulators or Practitioners should focus on the strengths and weaknesses of each if they are to adopt it. It is crucial to consider all the available risk measures for a regulator or practitioner to make a good decision since using one can be subjective; as seen in our results, "different risk measures yield different results". 2) We lacked enough tools to run different optimizations, especially on the models. The different types of portfolio optimization models and different R packages under the library (fPortf olio) we used in optimal portfolio formation,

Recommendations
were not efficient computationally. To overcome the computational inefficiency, further research on portfolio optimization at USE and other financial markets while comparing different portfolio optimization models for many assets should consider using other softwares that can perform the optimization. Other methods like Eigen decomposition-based methods can be used. The Eigen decomposition-based methods are recommended to obtain high-quality bounds on the optimal portfolios' solutions of traditional portfolio optimization problems and the proposed risk models with and without cardinality constraints.