Portfolio construction using Markowitz model

Institutional investors, wealthy individuals, governments, pension's funds and many more have and will always try to beat the market and achieve superior returns. Carefully managing an investor's portfolio involves intelligently selecting assets and dynamically rebalancing their weights. This has created a controversy on whether the selection of these assets should only focus on domestic levels, or whether it would be more efficient if international assets were also considered. Accordingly, researchers have been questioning the efficiency of international portfolio diversification, whilst experimenting with the different aspect involved.

International portfolio diversifying has been the key to capture the growth of fast developing countries, reducing home bias investing, and hedging against currency risk. Many recent studies illustrate the importance of foreign markets to capture higher returns whilst lowering risk due to the effect of diversification. In return diversification is highly dependent on correlations between selected assets.

The principle of international portfolio diversification states that constructing a portfolio of assets based in different markets will effectively reduce portfolio risk and boost portfolio return due to lower correlation between financial markets relative to the correlation within assets in the same market. According to Solnik's finding in 1974, constructing an international portfolio that is well diversified could evade 90% of the portfolio volatility.

Nowadays, international portfolio diversification has become a very significant investment strategy especially for developing countries who encounter high country risk. Most researchers studied the benefits of international diversification to a particular investor based in a specific country. However, the main motivation behind this paper is to study the effectiveness of Markowitz model within international portfolio diversification under variable model constraints and scenarios.

The recent financial crisis urged the need for a model that can mitigate losses during global downturns. The questions that arise here is how efficient is the Markowitz model within the context of international diversification? Does the level of efficiency vary under variable model constraints and situations? Since International diversification is affected by the level of correlation between countries, which is not stable over time, periods of recession are exhibiting higher correlation between financial markets. This in turn will affect the allocation of weights to different country's assets, thus debating the solidity of the model.

Literature Review

Markowitz Portfolio Construction

As the number of financial assets is ever increasing in today's market, investors are exposed to great opportunities for improving their portfolio performance. The fact that these securities have different risk return characteristics enables investors to accordingly meet their goals and risk profiles. This in return confirms the need for a mechanism or selection technique that can select assets from different available options and their respective optimal proportions in order to create a portfolio. Harry Markowitz (1952), the father of modern portfolio theory suggests that there is a trade-off between return of an asset or a portfolio and the respective variance. His paper „Portfolio Selection? reveals that it is possible to compute the optimal weights of a portfolio of risky securities, given the corresponding historical returns and variance-covariance matrix of the securities in the portfolio through the use of a suitable model. The Markowitz portfolio selection model which is also called the mean- variance analysis creates an efficient frontier (figure 2.1) for each portfolio, which in return shows the maximum return given a defined level of risk (variance) and vice versa.

Asness (1996) argues that "a diversified portfolio provides more expected return per unit of risk." This highlights the importance of the Markowitz model in constructing well diversified portfolio where there is more value of an investment with lower risk.

The mean-variance analysis utilizes three major inputs of a portfolio of risky securities to allocate optimal weights to each of the assets in the portfolio. These three inputs are asset's return, asset's return variance and the correlation between the assets in the portfolio.

International Portfolio Diversification

Overview

The key assumption in the modern portfolio theory requires investors to hold a well diversified portfolio. In fact, the objective of the portfolio diversification is to minimize the volatility of return of a portfolio, specifically the systematic risk of a portfolio given portfolio expected return.

This implies that investors should diversify their portfolios by selecting assets from different markets, in different regions of the world, so that they are weakly correlated with the assets of the investor's home market. The benefit of international diversification of a portfolio is indirectly linked to market integration. Lower correlation between returns from different markets enables investor to enjoy lower risk through international portfolio diversification. (Arnold, G 2005).

Benefits of International Portfolio Diversification

The drawbacks to portfolio construction based on a single market are that securities tend to move in the same manner and are systematically influenced by similar factors. Grubel (1968) was the first to empirically experiment the benefits of international diversification, where he used monthly data of ten developed markets and the US. He was able to justify that gains made from international diversification are significant. Constructing portfolios using assets in different markets which are weakly co-integrated and are not controlled by similar factors ensures that these assets move independently from each other and thus enabling such portfolios to perform better than domestic ones.

Boucrelle and Le Fur (1996) examined the benefits of international diversification through the level of correlations and return volatilities of different markets. Using correlation matrices of different countries and constructing the corresponding efficient frontier of international portfolio, they were able to show how it is more favourable to include assets of different countries in a portfolio than investing in one single market.

Having stocks from different markets in the portfolio reduces the systematic risk of the portfolio where different countries in the world undergo economic cycle at different points in time. This in turn implies that including assets from different markets can lower the risks of the portfolio substantially.

The benefits of international diversification have been discussed extensively in literature. A recent study by Levy and Zerman(1998) illustrates how U.S investors specializing in bond investment could have enhanced their portfolio returns over the period 1960-1980 by 5% a year if they have invested internationally rather than limiting their portfolio to domestic assets. This further confirms that investing in international markets allows investors to grasp better opportunities and to be involved in the growth of other countries.

The Home Bias Effect

Home bias is the phenomenon whereby investors tend to hold a greater proportion of domestic equities compared to foreign equity in their portfolio despite the merits from international diversification.

Grubel(1968) was the first to empirically show that portfolios diversified to a single country's financial assets underperform portfolios that are diversified into different markets? assets. Securities in a domestic portfolio will still face the same sources of systematic risk; no matter how hedged away is the unsystematic risk. This implies that the volatility of a domestic portfolio will always be higher than an internationally diversified portfolio.

Although many countries have recently lifted restrictions on international trading, it is still obvious that investors have a greater proportion of their portfolio invested in domestic assets. This is referred to as the "Home Bias" effect. So far, there has not been a reasonable explanation for this phenomenon.

The Role of Correlation

Equity trading is considered to be a risky investment; however international diversification reduces the risks associated with such an investment. International diversification can be defined as an investment approach that aims to combine securities with returns that are weakly correlated to reduce the total risk of the portfolio. (David S, Krause, 2006) Therefore, the effectiveness of international diversification depends heavily on the level of correlation among international markets. This implies that understanding the correlation dynamics between domestic and foreign equity will always be crucial when considering diversifying internationally.

It is noteworthy to mention that the perfect environment for international diversification would be as low correlation as possible between financial markets. The law of international diversification states that the lower the correlation level between financial markets, the higher the benefits achieved from investing in foreign equity. This inverse relationship between correlation of two securities and the respective gains from combining them in one portfolio explains the fact that internationally diversified portfolios have lower volatility of returns relative to domestically diversified portfolios. This highlights the importance of correlation in determining the risk of a portfolio.

Longin and Solnik (1995) have empirically computed the changing nature of correlation among international equity returns. They were able to show that correlation between financial markets increase as the volatility of the markets increase. Therefore, international diversified portfolios performed poor during periods of high market volatility which imply that the benefits to international diversification are influenced by correlation levels. Despite the fact that correlations level was variable in the analysis of Solnik's papers, correlation was never equal to one.

The benefits of international diversification are apparent, and risks can be mitigated by adding up securities with less than unity correlations. International diversification strategies are most needed in times of high market volatility. However, correlation is not steady in different market conditions and it tends to increase in periods of high volatility. This in return shows that benefits of foreign investments diminish when markets are unstable.

International Asset Allocation

Evidence supporting the considerable gains of international diversification is well documented in the literature; however the strategy of how to achieve its goal in real world is ambiguous. As it is mentioned before international diversification should allow investors to hold portfolios that are less risky relative to domestic portfolios. Nevertheless, issues like the number and the class of assets to include in the portfolio and the constraints to be imposed on the investments is not broadly set for efficient internationally diversified portfolio construction. Therefore, it is valuable to understand what asset allocation and security selection problems to be considered when an investor or a portfolio manger wants to diversify his portfolio into international equity. The question of how to diversify optimally in reality can be answered by "asset allocation".

Asset allocation is considered to be principal determinant of portfolio risk and return. Figure 2.2 shows the importance of asset allocation in determining the performance of a portfolio. The process of asset allocation can be summarized by the following steps:

  1. Define goals and time horizon
  2. Assess your risk tolerance
  3. Identify asset mix of current portfolio
  4. Create target portfolio (asset model)
  5. Specific investment selection
  6. Review and rebalance portfolio

Asset allocation must be consistent with the goal of an investment, time horizon and investment constraints. The traditional approach of asset allocation rely on portfolio selection models, however there are some alternative practical guides in the literature suggested by Svensen(2005) and Siegel(2008). Whether an investor wants to minimize portfolio risk for a required rate of return or alternatively maximize portfolio returns for an acceptable risk level, asset allocation will be the key factor to achieve investors? goals. Different assets operating in different markets will have various risk-return characteristics; therefore combing these assets together will determine the risk and return of a portfolio. For example, Indian equity is riskier than American Equity; however an American investor holding a domestic portfolio could still lower the risk of his portfolio by investing in Indian Equity. However, the key to successful asset allocation is to constantly rebalance the weights of the portfolio according to changes in market dynamics. As mentioned previously, the level of correlation between markets is not stable over time and economies are in cyclic conditions, hence dynamic rebalancing and constant reviewing of goals and investment constraints is essential for thriving asset allocation.

A problem with mutual funds and managers is they that shift asset classes within their funds as they alter the asset allocation of a portfolio. This in returns tends to suboptimal allocation of weights to the assets in the portfolio. Therefore, it is argued that for efficient asset allocation, managers should review investment goals and constraints on ongoing basis. (P. Brinson, D. Singer, and L. Beebower, 1991)

According to a study by Odier and Solnik (1993) where they investigated the minimum variance frontier of global equity markets, a rational asset allocation strategy provided a return more than 1.5 times the return provide by a US domestic portfolio given the same level of risk. For that reason asset allocation is crucial in achieving the benefits of international diversification. Never the less, an investor has to consider different aspects involved with asset allocation. This includes issues like the number of assets, classes of assets, investable markets and investment constraints. (Refer appendix C for further details on the determinants of portfolio performance).

Portfolio Performance

Risk adjusted portfolio performance measures are essential tools in raking the investment opportunities available to an investor. By analyzing the risk-return profiles and computing respecting premium of each portfolio, performance measures enable investor to select the optimal portfolio relative to their risk tolerance. In this study, three main risk adjusted performance measures will be used which are Sharpe ratio, Treynor ratio and Jensen's alpha. In addition, the capital asset pricing model (CAPM) will be used to validate returns generated by the model.

The Sharpe ratio is the reward to volatity ratio. It is defined as follows:

Sharpe ratio can be presented graphically as in the figure below. This ratio measures the gradient of the line from the risk free rapture to the combined return and risk of the portfolio; the steeper the capital assetline of a portfolio, the higher the Sharpe ratio of the portfolio. (R.Korajcyzk, 1999). In the figure below, line B has higher Sharpe ratio, hence this portfolio offers a better investment opportunity.

The Treynor measure is similar to the Sharpe ratio; however the denominator has systematic risk only without specific risk. Its equation is as follows

Bp represents the systematic risk of a portfolio represented by Beta. Detailed calculations of beta and the risk adjusted performance measures will be shown in later chapters.

Jensen's alpha measures the deviation of a portfolio from the securities market line. It represents the average return of a portfolio over and above the predicted return of the portfolio, given portfolio's beta and the average market return. Jensen's measures is calculated as:

Where;

rp= Expected portfolio return

rf = Risk free rate

Bp= Portfolio's Beta

rm= Expected portfolio return

The CAPM gives a precise prediction of the relationship between the risk of a portfolio and its expected return. It also provides a benchmark rate of return to evaluate other investments. (Bodie, Kane and Marcus, 2009) In this study, the CAPM will be used to validate the results of portfolio returns generated by the Matlab model. If the returns from CAPM are similar or close to the ones from the Markowitz model constructed on Matlab, this indicates the reliability of our model. In all, CAPM provides an equilibrium rate of return of a portfolio using the following equation:

Where,

rf = Risk free rate

Ba= Portfolio's Beta

rm= Expected market return

Data Collection

Countries, Indices and Study Period Selection

It is important to have a good geographical spread that covers major economic zones in the world and major commodities. Morgan Stanley Capital International standard country indices are a good example to have a consistency among data collected. In addition, the collected data should be in one currency as currency risk implications are not included in this study and only absolute returns will be needed. These MSCI indices can be downloaded from DataStream.

Identifying the Investor and the Benchmark Market

In order to examine the nature of international diversification relative to investors based in different market maturities. This will enable testing our models efficiency from different market's perspective.

The fact that investors within two countries have different risk tolerances means that different investors will have variable tendencies to diversify into international equity.

The benchmark markets selected are based on the location of the investor. For instance, a UK based investors will have the FTSE return benchmarked against his portfolio return to test how good an international diversified portfolio is performing.

Riskless Security Selection

Government bond rates are used over as a benchmark return of a risk free investment. These government bonds can be downloaded from Bloomberg.

Methodology

To assess the benefits of international diversification, the risk-return profile of different portfolio compositions should be analyzed.The Markowitz portfolio construction model is used to compute the optimal weights of different portfolio compositions.

Constructing the International Portfolio

In today's financial market place, a well constructed portfolio is crucial to an investor's or manager's success. Different kinds of investors need to know the asset allocation that best satisfies their investment goals and risk tolerance. By following the Markowitz systematic approach, investors can construct portfolio aligned to their goals. The steps of the procedure will be discussed as follows.

Index Return

Logarithmic returns are computed for each index using the following Formula: = ( - )

Where Pi t is the value of the index at time t and Pi t-1 is the previous period's value (previous month in this case)

To remove the negativity in returns and enable the computation of the geometric mean return of an index over a specific period of time. The following is done:

1+

Geometric mean return formula is used to compute the return of the index over a particular period. This is computed as follows: = = /-

Where n is the number of period returns.

Index Volatility (Risk)

The index volatility is computed using the standard deviation formula:

= - ( - =)

Where ri is the index average return

Portfolio Return

According to the Markowitz convention, the return of a portfolio is equal to the weighted average return of its constituent assets. This is computed according to the following formula: = =

Where wi is the weight of asset (index) i in the portfolio, is the geometric mean return of asset (index) i, and N is the number of assets in the portfolio. It should also be noted that there is a constraint in the weights as follows: wi=1

This return formula is however expressed in matrix notation as this is required to perform the portfolio variance computation which involves variance-covariance matrix. Hence portfolio return is equal to: r p = W'*R

Where W is the vector of the weights of each asset in the portfolio and R is the vector of returns

W= w1w2... = 1 2...

Portfolio Variance:

The variance of a portfolio with different assets is equal to the weighted average covariance of returns of its individual assets. It can be computed using the equation below.

Where wi and wj are the allocated weighted to asset (index) i and j respectively. Cov (ri, rj) is the covariance which measures the extent of the correlation between asset's returns. In this case there are two assets in the portfolio i and j .For these two assets the covariance can be calculated using the following formula:

, = , * * , = ()- ()- = (()- )= (()- )=

Where, is the correlation between asset i and asset j, , is the covariance between asset i and j, ri t is the index return at time t, and ri is the index average return of asset i, σi is the standard deviation of asset iand σj is the standard deviation of asset j.

A positive correlation between two assets indicates that these assets move in the same direction and vice versa.

Efficient Frontier

By computing all the previous measures and solving the two quadratic problems: portfolio return and variance, and the optimal weights allocated to different portfolio assets can be determined. The two equations are solved in order to find the optimal weights and the respective efficient frontier of the portfolio selected. The frontier will graphically illustrate the optimal solution which is the minimum variance portfolio.

Testing International Diversification

In this section, we will discuss the nature of the tests associated with investigating the efficiency of the model in constructing a well diversified international portfolio. These tests will have different investment constraints, different allocation of weights to domestic and international equity and are performed in different study periods to incorporate various business cycles of the economy.

As mentioned earlier, indices will be treated as equity of different countries. To make this assumption viable and make index return in the range of equity return, index returns of all country's is multiplied by two. This will make return results more consistent with returns of equity observed in real life.

The tests are designed in a way to verify as many aspects as possible in the context of diversifying internationally under the Markowitz portfolio construction

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