Applying Markowitz Portfolio Model in Management Science - Math Problem Example

Solution 1 The given problem can be written in the following manner in the matrix where, elements of the matrix are the costs of transportation per motor: Plants (1) (2) (3) (4) Supply A 120 130 41 59.5 500 B 61 40 100 110 700 C 102.5 90 122 42 800 Demand 400 900 200 500 2000 Since the total demand of motors is equal to the total supply of the motors, so it is a balanced transportation problem which requires the transportation cost to be minimized. The problem can be expressed in the following model: Minimize Z = 120*x11 + 130*x12 + 41*x13 + 59.5*x14 + 61*x21 + 40*x22 + 100*x23 + 110*x24 + 102.5*x31 + 90*x32 + 122*x33 + 42*x34 Subject to x11 + x12 + x13 + x14 = 500; x21 + x22 + x23 + x24 = 700; x31 + x32 + x33 + x34 = 800; x11 + x21 + x31 = 400; x12 + x22 + x32 = 900; x13 + x23 + x33 = 200; x14 + x24 + x34 = 500; XIJ >= 0; where, XIJ’s are the number of motors transported. Step 1: Finding Initial of Basic Feasible Solution: To find the initial feasible solution Vogel’s Approximation Method (VAM) is used, which, gives the following matrix where, the numbers inside the boxes are the number of motors to be transported from the harbor to the assembly plant. Plants (1) (2) (3) (4) Supply D1 D2 D3 D4 A 120 130 41 59.5 500 18.5 60.5 - - B 61 40 100 110 700 21 21 21 21 C 102.5 90 122 42 800 48 48 48 12.5 Demand 400 900 200 500 2000 D1 41.5 50 59 17.5 D2 41.5 50 - 17.5 D3 41.5 50 - 68 D4 41.5 50 - - Since the number of allocations is equal to the (no. of rows + no. of columns - 1 = 6), hence we move to the next step to test the optimality of the solution. Step 2: Optimality Test: The optimality is tested by calculating the opportunity costs of unallocated cells using the formula: OC = Cij – (Ui + Vj), where Cij’s are the costs in each cells and Ui’s & Vj’s are the difference between the lowest cost and the second lowest cost in each row and columns. Plants (1) (2) (3) (4) Supply Ui A 120 0 130 22.5 41 59.5 500 U1=17.5 B 61 8.5 40 100 126.5 110 118 700 U2=-50 C 102.5 90 122 98.5 42 800 U3=0 Demand 400 900 200 500 2000 Vi V1=102.5 V2=90 V3=23.5 V4=42 Since Opportunity Costs (OC) in each unallocated cell is non-negative, hence the initial feasible solution is the optimal solution. Step 3: Total Minimum Cost: Total minimum cost = Σ (xij * Cij) = 133,350 Step 4: Interpretation of Results: The outcome of the above transportation problem suggests that in order to minimize the transportation costs the management must follow the following transportation routes and patterns: Number of motors From To 200 Amsterdam Liege 500 Amsterdam Tilburg 700 Antwerp Nancy 400 Le Havre Leipzig 200 Le Havre Nancy 200 Le Havre Tilburg Solution 2 Markowitz Portfolio Model suggests how the risk of a portfolio can be minimized by including the assets of different features such as risk and return. These basic features of the assets are used to model the portfolio which gives optimum return at the minimum risk. Markowitz Model requires investigating the relationship between risk and return for an individual asset. The Capital Asset Pricing Model, in essence, predicts the relationship between the risk of an asset and its expected return. This relationship is useful in two folds; first it produces a benchmark for evaluating various investments, second, it helps us to make an informed guess about the return that can be expected from an asset that has not yet been traded in the market. The risks and returns of individual assets are then combined to form the portfolio. So, if there are two assets with 1 and 2 as their expected returns and an investor has invested in these two assets in w1 and w2 weights respectively, then the expected return of the portfolio for that investor would be: w11 + w22 a) The risks and expected returns of each business sector is given in the following table: Water Brewers Telecoms Property Mean Return 9.685 10.97 9.24 9.765 Standard Deviation 1.729812 4.732406 3.276414 4.922739 Sharpe Ratio (return per unit of risk) 5.598874 2.31806 2.820156 1.983652 From the above table we can interpret that the expected return for Brewers sector is the highest while for the Telecoms it’s the lowest. The standard deviation which represents the total risk (Systematic and unsystematic risks) associated with an asset is lowest for the Water sector while it’s highest for the Property sector. By just looking at these figures we cannot interpret which sector is the best industry which can give the optimum return with minimum return. So, here Sharpe ratio comes handy which, represents the expected return per unit of the risk. The Sharpe ratio is highest for the Water sector while lowest for the Property sector. This signifies that although the expected return is highest for the Brewer sector but it’s not advisable to invest the money in the sector because the risk is also high. b) Since the investor in the case is risk-minimiser, so he/she should select either an asset or a portfolio which can provide an optimum return at the minimum risk. If the investor wants to invest its money in a single sector then he/she must invest that amount in Water sector which provides the optimum return at minimum risk among all the sectors. However, the investor also has the options of diversifying the risk and investing the amount in various sectors rather than just investing in one sector. For building up a two sector portfolio the investor must select those two sectors which move in the opposite directions. To measure their movement correlation coefficients can be used which is given in the following table: Sectors Correlation Water, Brewers -0.033 Water, Telecom 0.123 Water, Property -0.339 Brewer, Telecom 0.698 Brewer, Property -0.230 Telecom, Property 0.059 The following possible portfolios can be formed which are given in the table with the expected return and standard deviation of the portfolios: Portfolio Weights Return SD Water, Brewer 90%, 10% 9.8135 1.612061 Water, Telecom 80%, 20% 9.596 1.602423 Water, Property 80%, 20% 9.701 1.400321 Brewer, Telecom 10%, 90% 9.413 3.296696 Brewer, Property 50%, 50% 10.3675 2.995507 Telecom, Property 70%, 30% 9.3975 2.800697 Out of all these portfolios, the one with water and property sector has the least standard deviation, hence the investor should assign his/her amount in this portfolio which is formed by investing 80% of the amount in the Water sector and remaining 20% in the Property sector. If the investor wish to further increase the portfolio then it can be done by taking three or four sectors at a time and then assigning weights to each of the sectors and the one with the least standard deviation can be selected. c) In order to achieve a return of 10%, the investor has two options either investor can go for a single sector investment or many sectors investment. If investor wants to invest in a single sector then Brewers sector provides a return of slightly higher than 10% but not exactly 10% and the other concern is its high risk. So, investing in a single sector doesn’t seem to be a feasible option. Hence the investor is left with the combination of two sectors. Although in that option as well there will be few cases where it will not be possible to get a 10% return by combining the two sectors’ returns, reason being for the combination to get 10% at least one of the sector must have the minimum 10% return and the other less than 10%. Following can be the cases: Portfolio Return Return 1 Return 2 Weight 1 Weight 2 Portfolio SD Water, Brewer 10 9.685 10.97 0.754864 0.245136187 1.71761864 Brewer, Telecom 10 10.97 9.24 0.439306 0.560693642 3.60985263 Brewer, Property 10 10.97 9.765 0.195021 0.804979253 3.85606655 Since the returns of the all the three portfolios are 10% so, the investor must select the portfolio with the least risk or standard deviation. Out of these three options, the option of Water and Brewer is t with the least standard deviation; hence, the investor must choose this portfolio. Solution 3 Read More