Portfolio Optimization and Portfolio Performance Measurement - Case Study Example

Portfolio Optimization and Portfolio Performance Measurement Portfolio Optimization and Portfolio Performance Measurement Introduction The modern works on portfolio selection concentrated on the Markowitz portfolio selection theory. However, it should be noted that same work have been done with other authors and contributors with great convenient pedagogical format and in details (Paul, 2008). Nonetheless, Markowitz’s approach usually commences with the investor’s rational position that helps the investor to seek optimal combination of return and risk on the investment. In other words, Markowitz’s approach notes that the higher the acceptable risk levels the greater the required return expectation (International Symposium… at al. 2008). However, it is worth noting that this approach is not that startling, but its highly complicated than the all times known approaches that were pegged on the philosophy that an investor usually expected the highest possible return without pegging the possibility on the risk borne levels. Markowitz approach defines the expected risks and returns in relation to a statistician approach. According to his definition, return expectation is a mean value based on the theoretical probability distribution and risk is defined as the variance of this theoretical probability distribution. However, later evaluation realized that it is convenient to define risk by substituting variance with its substituent standard deviation (Paul, 2008). Applying these definitions including their mathematics approach and properties, Markowitz among other contributors drew a series of conclusions. Nonetheless, the most remarkable conclusion among other conclusions was that an individual portfolio with high risky securities might equally be ‘efficient’ since the same may also provide minimum risk for any return expectation. This may be so because very simple individually risky securities may be characters with anticipated fluctuations that may offset another thereby providing a composite investment that is safer than some individual returns with low risk securities. Thus, efficient portfolios are generally diverse in character. Moreover, conclusion on interests based on Markowitz principles can easily show that the efficient frontiers are often obtainable by holding different assets combinations selected from a specific population. In other words, the best risk and return combination can be described from a curve where risks increase with returns; however, the same relationship or combination is only true with a diminishing rate. Mathematically, an efficient frontier is expected to convex in nature. Additionally, Markowitz among other contributors of the portfolio optimization concluded that asset selections must for determining investment optimization must be selected from a population assets that are entirely risk free (International Symposium… at al. 2008). Hence, for all other points to be on the left of tangency, then the combination of portfolio and safe assets presented by the tangency points must offer holding that is more efficient than any portfolios of purely risky assets. The understanding of tangency points in determination of frontiers in the portfolio optimization is sometimes described as separation theorem since it suggests that at any given point in the determination of portfolio optimization, any efficient portfolio is separable into two portions including portfolios consisting of only risky assets (Paul, 2008). The other portion includes the net holding portion that consists of only perfectly safe assets (there are either cash or highly liquid assets). Therefore, cautious investors must hold the same risk assets combination and regard them as more sanguine; however, such holdings will only consist of smaller risk adverse proportions on investor’s portfolio. In this case, the balance will comprise of perfectly safe assets holdings. Portfolios usually have different effects and are of different effects to the investors and shareholders. Notably, some portfolios are usually of public interest specifically for public scrutiny and they include mutual fund portfolios (International Symposium… at al. 2008). On the other hand, the portfolio separation theorem has since had certain consequences including application for the proper mutual fund’s administration. Particularly, the portfolio separation theorem is usually applied to mean that when the mutual fund performance is in the same axis, then there should be a cluster below and along the straight line. This is because the straight line defines the best possible performance when return attained is combined with risk borne for any investment management (International Symposium… at al. 2008). In this case, it can be assumed that the mutual fund is subject to the management of competent professional; thus, the investment portfolio will be fairly close to achieving optimal possible performance. However, it is worth noting that there are high possibilities that human factors are likely to prevent the full achievement of the optimal possible performance (Paul, 2008). Moreover, it should be noted that this approach creates a complex mutual fund’s performance standards than the simple period return over criterion. The criterion herein usually demands less due to its low legitimacy described by low return rates. This situation can only be achieved when there is low volatility degree equivalent to return. With the above understanding of the optimal portfolio, this essay aims at discussing different aspects of the optimal portfolio concepts using data from stock markets. Mean, Variance, Efficient, and Portfolio Frontier These calculations will be conducted from an American investor with a six stock index portfolio. This American’s portfolio comprises of six stock indexes including (Jarvis, 2003; Pg. 135): 1. The United States’ (S&P500), 2. Switzerland’s (Swiss Market Index, SMI), 3. United Kingdom’s (FTSE100), 4. Hong Kong’s (Hang Seng Index, HSI), 5. Singapore’s (Straits Times Index, STI) 6. Korea (Korea Composite Stock Price Index, KOSPI) (Paul, 2008) The above data will be considered on a monthly basis with effect from January 1990 to December 2006. The same data will be used to determining the optimal portfolio allocations. Using the Markowitz portfolio, the data will be divided into three parts; the first part will involve the data for calculating efficient frontiers. Secondly, choosing the optimal risky portfolio that can only be obtained from capital allocation line (this can only be determined by establishing the point at which the efficient frontier intersects the calculated tangent). Finally, the optimal complete portfolio is to be used to allocate funds between risk free assets and risky portfolio. Establishing Efficient Frontier: First, standard deviation, expected return, and covariance matrix must be calculated and the table below shows the same calculations. Table 1: The six stock indexes’ performance Establishing covariance matrix Table 2: showing calculated of covariance matrix the six stock indexes Using the above data, the result for portfolio variance, standard deviation, and means are as recorded in the table 3 below. Note: Determining Efficient Frontier The efficiency frontier is usually the curve that indicates all the efficient portfolios within which the return and risks lie (Paul, 2008). It is usually considered as the portfolio that maximizes the return within a given risk amount (also regarded as the standard deviation). For example, investors usually invest on efficient portfolios with a desired amount of risk; otherwise, such an investor will go nuts if they do not have the maximum possible expected return. Therefore, calculation of the efficient frontier will require maximization of risk and this translated to the determination of the standard deviation. In this case, the variance or the standard deviation squared is taken as the objective function (International Symposium… at al. 2008). However, before determining efficient frontier, it is usually essential to determine the benchmark for calculating efficient portfolio. The same is obtained by multiplying the border by covariance matrix. This result to the portfolio variance as indicated in the table above. For the above data, the standard deviation is 17.11 percent. The portfolio efficient frontier Moreover, restricted and unrestricted efficient frontier portfolios can also be calculated as indicated in the table below: Riskless Asset, Portfolio Frontier and Tangent Portfolio References Top of Form Bottom of Form Top of Form Bottom of Form Top of Form INTERNATIONAL SYMPOSIUM ON INTELLIGENT INTERACTIVE MULTIMEDIA SYSTEMS AND SERVICES, & TSIHRINTZIS, G. A. (2008). New directions in intelligent interactive multimedia. Berlin, Springer. JARVIS, D. S. L. (2003). International business risk: a handbook for the Asia-Pacific region. Cambridge [u.a.], Cambridge Univ. Press. PAUL, J. (2008). Business environment: Text and cases. New Delhi: Tata McGraw-Hill Publishing. Bottom of Form Read More