Personal Finance: Portfolio Optimization using Jensen's Performance Measure
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The document describes using Jensen's alpha and portfolio optimization to evaluate equity performance and allocate investments. It outlines Jensen's performance measure, defines relevant terms, presents sample stock price data, and describes using linear regression to calculate Jensen's alpha for different equities. The document then formulates an optimization model to maximize returns by allocating investments across 34 equities under various constraints, and solves the model using a linear programming solver.
![PERSONAL FINANCE: PORTFOLIO OPTIMIZATION USING JENSEN’s
PERFORMANCE MEASURE (that uses Least Squares Regression)
NOTE: The following project uses confidential data and plenty of independent research. The
data has been sanitized and therefore dummy variable names and data are used.
INTRODUCTION:
The project uses Linear Optimization and Jensen’s Performance Measure (Alpha) to evaluate the
performance of equities (and risk computation) compared to benchmarks (such as S&P 500 and
Treasury Bills) over time.
LITERATURE:
JENSEN’s PERFORMANCE MEASURE (ALPHA) [1]
Jensen measure (alpha) uses least squares regression to calculate risk-premium for an asset’s
return and market return. A positive alpha means that the asset performed better than the
benchmark (example: S&P 500, Treasury Bills) in risk-adjusted terms. If alpha is zero, the returns
between asset and benchmark.
Jensen’s formula: (ri – rf) = a + b (rm – rf)
Where
(ri – rf) is risk premium for asset i
rf is the risk free rate (For example treasury bills, in this project the average annual return of a
fixed deposit from a trustworthy bank is taken as risk free rate)
rm is the market return (in this project an index such as S&P 500 or SENSEX is taken as market
return)
rf is return on asset i
b is beta co-efficient
a is Jensen’s Performance Measure (alpha)](https://image.slidesharecdn.com/bana-7012-projectsarang-ananda-rao-160214031656/85/Personal-Finance-Portfolio-Optimization-using-Jensen-s-Performance-Measure-2-320.jpg)
![CONCLUSION:
Our optimal solution from the model after applying Linear Optimization is:
Equity Investment Average 3 Year Return
Expected Returns
(After one year)
% of
portfolio
Equity 1 $1,000.00 34.17% $1,341.65 1.00%
Equity 2 $0.00 42.75% $0.00 0.00%
Equity 3 $30,000.00 105.01% $61,504.21 30.00%
Equity 4 $10,000.00 79.17% $17,916.55 10.00%
Equity 5 $0.00 12.78% $0.00 0.00%
Equity 6 $0.00 18.31% $0.00 0.00%
Equity 7 $0.00 20.01% $0.00 0.00%
Equity 8 $30,000.00 199.52% $89,856.55 30.00%
Equity 9 $5,000.00 25.38% $6,269.01 5.00%
Equity 10 $0.00 14.36% $0.00 0.00%
Equity 11 $0.00 6.47% $0.00 0.00%
Equity 12 $8,000.00 70.41% $13,633.05 8.00%
Equity 13 $5,000.00 20.10% $6,005.02 5.00%
Equity 14 $0.00 6.77% $0.00 0.00%
Equity 15 $5,000.00 14.92% $5,746.10 5.00%
Equity 16 $0.00 37.28% $0.00 0.00%
Equity 17 $0.00 24.68% $0.00 0.00%
Equity 18 $0.00 37.21% $0.00 0.00%
Equity 19 $0.00 32.86% $0.00 0.00%
Equity 20 $5,000.00 36.88% $6,843.80 5.00%
Equity 21 $1,000.00 64.34% $1,643.43 1.00%
Equity 22 $0.00 63.82% $0.00 0.00%
Total Investment $100,000.00 Total Expected Returns $210,759.38
<=
$100,000.00
(Max. Investment)
As we can observe, we allocated funds worth $100,000 and after adopting Linear Optimization and using
Jensen’s Performance Measure, our Expected returns in one year is $210,759.38 which is a 111% increase.
Reference:
[1] Jenson’s Performance Measure p148-150, “101 Investment Tools for Buying Low and Selling High”, Jae
K. Shim and Jonathan Lansner, 2001](https://image.slidesharecdn.com/bana-7012-projectsarang-ananda-rao-160214031656/85/Personal-Finance-Portfolio-Optimization-using-Jensen-s-Performance-Measure-16-320.jpg)
Personal Finance: Portfolio Optimization using Jensen's Performance Measure
- 3. By adjustment, i.e. dividing each asset’s alpha by beta co-efficient (a/b), Jensen’s Alpha can be used to rank the performance of an asset relative to other assets. For the linear regression analysis, Y variable is (ri – rf) X variable is (rm – rf) DEFINITIONS: Table 1: Market Capitalization Definitions Market Cap: Lower Limit Upper Limit Market Cap (in crores) (in crores) $0.00 $4,999.00 Small Cap $5,000.00 $20,000.00 Mid Cap $20,001.00 $350,000.00 Large Cap Grades: Through independent research, equities are graded into A, B & C, which represent likelihood of purchase. For example, Grade A equities are equities that the investor is more likely to purchase than Grade B (and finally Grade C) Equities. Average 3 Year Return: The average return (mean of annual returns) of an equity in the last 3 years 2012-15 (Year on Year) Expected Returns: Total One Year expected returns for a particular equity
- 4. Data Preparation: A sample of 34 equities from 18 industries were shortlisted through independent research for the project. The 18 industries used in the project are: Table 2: Industries considered 1 Appliances 2 Automotive 3 Banking 4 Batteries 5 Conglomerate 6 Cookery 7 Courier 8 Energy 9 Fertilizers 10 FMCG 11 Food Processing 12 IT 13 Media 14 Metals 15 Personal Care 16 Pharmaceuticals 17 Textiles 18 Theatres Data for the equities such as Grades (calculated through independent research), Market Capitalization (Market Cap) and 5 year stock price data (first 2 weeks of November) were collected from reliable financial sources (such as Google Finance). ASSUMPTION: Since we are interested in long term risk measure for the equities, it is assumed that stock price does not fluctuate much in the first 2 weeks of November and therefore stock price during any day in first 2 weeks of November is considered for ease of data collection (since over a long term such as 5 years, fluctuation in stock price over 2 weeks is insignificant). In this project, the value of SENSEX (over a 5 year time period) is considered the market return (Rm) and the value of interest rate from fixed deposits through a national bank is considered as a risk free return (Rf).
- 5. Table 3: Sample of stock price over 5 year period (Refer worksheet “5 Year Data_Calculations” for complete data Equities - Stock Price during 2011-15 (Values adjusted to splits) 1st Week November 2015 2014 2013 2012 2011 SENSEX $26,265.24 $27,868.63 $21,196.81 $18,683.68 $17,562.61 Equity 21 $284.25 $298.50 $113.85 $83.95 $102.40 Equity 4 $87.80 $47.50 $16.75 $24.25 $27.90 Equity 12 $7,470.00 $6,521.70 $2,800.00 $1,709.60 $1,575.00 Equity 23 $170.00 $220.25 $148.50 $233.00 $226.50 Equity 3 $16,582.00 $12,671.00 $3,999.00 $2,390.00 $1,732.00 For computing the return of assets (Ri), a four year YoY (Year on Year) growth is calculated. The Y & X variables for linear regression are calculated using the above metrics (Ri, Rm, Rf). An average 3-year return is computed from the four year YoYs. Linear Regression Analysis: To calculate the Jensen’s Performance co-efficient for an equity, a linear regression is run with Y variable (ri – rf) X variable (rm – rf) Table 4: Regression Output for Equity 21 Regression Using Data Analysis Toolpak Equity Equity 21 X Y -14.0% -13.0% 23.3% 154.0% 5.3% 27.4% -1.8% -26.2% SUMMARY OUTPUT Regression Statistics Multiple R 0.912964 R Square 0.833504
- 6. Adjusted R Square 0.750256 Standard Error 0.410716 Observations 4 ANOVA df SS MS F Significance F Regression 1 1.688954 1.688954 10.0123 0.087036 Residual 2 0.337376 0.168688 Total 3 2.02633 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Uppe 95.0% Intercept 0.201779 0.211028 0.956171 0.439893 -0.7062 1.109761 -0.7062 1.1097 X 4.821131 1.523639 3.164222 0.087036 -1.73456 11.37682 -1.73456 11.376 R2 is a measure of fit of the line and has values between 0 and 1. Higher R2 means better fit of the line. Here, the value of R2 is 0.83, which is quite high, which means the curve has a better fit. For this analysis, we are only interested in the values of slopes and intercept (and not checking p-value for statistical significance or value of R2 ). The regression equation is (ri – rf) = a + b (rm – rf) In our case, (ri – rf) = 0.2018 + 4.8211 (rm – rf) The process is repeated for all the 34 equities and the values of Jensen’s Performance Measure as well as the Performance Ratio is calculated as shown below: Table 5: Jensen’s Performance Measure Calculation. Refer Excel worksheet “Regression” for complete analysis Y Performance Measure Ri - Rf Jensen's alpha Jensen's beta Jensen's Ratio Equities 2015 2014 2013 2012 Slope Y-Intercept alpha/beta Equity 21 -13.0% 154.0% 27.4% -26.2% 4.8211 0.2018 23.89308089 Equity 4 76.6% 175.4% -39.1% -21.3% 3.0756 0.3810 8.07337654 Equity 12 6.3% 124.7% 55.6% 0.3% 3.4533 0.3573 9.664072978 Equity 23 -31.0% 40.1% -44.5% -5.3% 1.7512 -0.1576 - 11.11264576 Equity 3 22.7% 208.7% 59.1% 29.8% 5.1956 0.6349 8.183522593
- 7. Using Jensen’s Performance Ratio, the equities are ranked from 1-34 (higher the ratio, lesser rank). A lesser rank equity has performed better than a higher ranked equity because it has yielded better returns compared to market (Rm) as well as risk-free returns (Rf). FINAL DATA: Table 6: Final Data (Refer “Equities” for complete dataset) OPTIMIZATION: We have a limited investment fund for investing in equities and we want to maximize returns from our investment. The equities are diverse (18 industries) and our best bet would be to diversify our investments into different sectors and not allocate all our funds into a single equity. This leads us to several constraints. In our portfolio optimization model, Objective Function: Maximize returns (over a one year time period) by investing in equities Decision Variables: Amount (Investment) to be allocated in each asset/equity Constraints: 1. Non-negativity constraint: All investments made in equities need to be non-negative 2. Equity constraint: No asset should be more than 30% of the portfolio 3. Total investment constraint: Total investment(sum of investments made in all equities) should be less than or equal to the funds available 4. Industry constraint: For diversifying the portfolio, investments in each industry have an upper and lower limit Table 7: Industry constraint Industry Lower Limit Upper Limit Appliances 1% 20% Automotive 1% 40% Banking 0% 30% Batteries 1% 40% Conglomerate 5% 30% Cookery 0% 20% Courier 0% 10%
- 8. Energy 5% 40% FMCG 5% 30% Food Processing 0% 20% IT 0% 40% Personal Care 0% 20% Pharmaceuticals 5% 30% Textiles 1% 25% Theatres 0% 10% 5. Market Cap Constraint: In order to diversify, equities should be allocated by all the 3 types of market cap subject to constraints: Table 8: Market Cap constraint Lower Limit Upper Limit Large Cap 10% 100% Medium Cap 0% 50% Small Cap 0% 40% 6. Jensen’s Alpha Constraint: Applies to only B & C Grade Equities: Choose Equities with positive Jensen's Alpha 7. Jensen’s Ratio Constraint: Applies to only B & C Grade Equities: Choose Equities with Jensen's Performance Rank<=25 Constraints 6 & 7 are computed by filtering data instead of using constraints in model because of limitation in number of constraints in Solver (limit of 200 variables) and scaling issues in Premium Solver(in spite of Automatic scaling). The objective function, decision variables and constraints are entered in the Solver parameters dialog box (as shown below). The solving method used is “Simplex LP” since the problem is a linear optimization problem. We need to check the box “Make unconstrained variables Non-Negative” to include the non- negativity constraint.
- 10. Understanding Sensitivity Report: Table 9: Decision Variable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $F$10 Equity 1 Investment 1000 0 1.341654303 0.362477516 1E+30 $F$11 Equity 2 Investment 0 - 0.364171905 1.427482628 0.364171905 1E+30 $F$12 Equity 3 Investment 30000 0 2.050140486 1E+30 0.258485953 $F$13 Equity 4 Investment 10000 0 1.791654533 0.258485953 0.087522713 $F$14 Equity 5 Investment 0 -0.66387386 1.127780672 0.66387386 1E+30 $F$15 Equity 6 Investment 0 - 0.608528372 1.18312616 0.608528372 1E+30 $F$16 Equity 7 Investment 0 0 1.200120467 0.504011352 1E+30 $F$17 Equity 8 Investment 30000 0 2.995218483 1E+30 1.291086664 $F$18 Equity 9 Investment 5000 0 1.253801048 0.450330772 1E+30 $F$19 Equity 10 Investment 0 0 1.143638805 0.560493014 0.078954092 $F$20 Equity 11 Investment 0 - 0.078954092 1.064684713 0.078954092 1E+30 $F$21 Equity 12 Investment 8000 0 1.704131819 0.087522713 0.060699628 $F$22 Equity 13 Investment 5000 0 1.201004732 0.503127087 1E+30 $F$23 Equity 14 Investment 0 - 0.081486472 1.067733049 0.081486472 1E+30 $F$24 Equity 15 Investment 5000 0 1.149219521 0.554912299 0.081486472 $F$25 Equity 16 Investment 0 0 1.372798676 0.331333144 1E+30 $F$26 Equity 17 Investment 0 - 0.125296976 1.246783057 0.125296976 1E+30 $F$27 Equity 18 Investment 0 0 1.372080033 0.332051787 0.125296976 $F$28 Equity 19 Investment 0 0 1.328606867 0.375524952 1E+30 $F$29 Equity 20 Investment 5000 0 1.368759266 0.335372553 1E+30 $F$30 Equity 21 Investment 1000 0 1.643432191 0.060699628 1E+30 $F$31 Equity 22 Investment 0 0 1.638216929 0.065914891 1E+30 The reduced cost tells us how much the objective co-efficient needs to be reduced in order for a non- negative variable that is zero in the optimal solution to become positive. Since Equity 1 is already present in the model, the reduced cost is zero. However, the reduced cost for Equity 2 is 0.3642 i.e. the objective function of Equity 2 has to increase from 1.42 to 1.78 if investment is to be made in Equity 2 (i.e final value to be positive).
- 11. Table 10: Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $F$34 Total Investment Investment $100,000.00 1.704131819 100000 2000 8000 $U$10 Appliances % of Portfolio 0.01 0 0.2 1E+30 0.19 $U$11 Automotive % of Portfolio 0.4 8752.271324 0.4 0.08 0.02 $U$12 Banking % of Portfolio 0 0 0.3 1E+30 0.3 $U$13 Batteries % of Portfolio 0.3 0 0.4 1E+30 0.1 $U$14 Conglomerate % of Portfolio 0.05 0 0.3 1E+30 0.25 $U$15 Cookery % of Portfolio 0 0 0.2 1E+30 0.2 $U$16 Courier % of Portfolio 0.08 0 0.1 1E+30 0.02 $U$17 Energy % of Portfolio 0.05 0 0.4 1E+30 0.35 $U$18 FMCG % of Portfolio 0.05 0 0.3 1E+30 0.25 $U$19 Food Processing % of Portfolio 0 0 0.2 1E+30 0.2 $U$20 IT % of Portfolio 0 0 0.4 1E+30 0.4 $U$21 Personal Care % of Portfolio 0 0 0.2 1E+30 0.2 $U$22 Pharmaceuticals % of Portfolio 0.05 0 0.3 1E+30 0.25 $U$23 Textiles % of Portfolio 0.01 0 0.25 1E+30 0.24 $U$24 Theatres % of Portfolio 0 0 0.1 1E+30 0.1 $U$10 Appliances % of Portfolio 0.01 - 36247.75165 0.01 0.08 0.01 $U$11 Automotive % of Portfolio 0.4 0 0.01 0.39 1E+30 $U$12 Banking % of Portfolio 0 -50401.1352 0 0.08 0 $U$13 Batteries % of Portfolio 0.3 0 0.01 0.29 1E+30 $U$14 Conglomerate % of Portfolio 0.05 - 45033.07718 0.05 0.08 0.02 $U$15 Cookery % of Portfolio 0 - 56049.30143 0 0.08 0 $U$16 Courier % of Portfolio 0.08 0 0 0.08 1E+30 $U$17 Energy % of Portfolio 0.05 - 50312.70873 0.05 0.08 0.02 $U$18 FMCG % of Portfolio 0.05 - 55491.22986 0.05 0.08 0.02 $U$19 Food Processing % of Portfolio 0 - 33133.31438 0 0.08 0 $U$20 IT % of Portfolio 0 - 33205.17865 0 0.08 0 $U$21 Personal Care % of Portfolio 0 - 37552.49519 0 0.08 0 $U$22 Pharmaceuticals % of Portfolio 0.05 -33537.2553 0.05 0.08 0.02 $U$23 Textiles % of Portfolio 0.01 - 6069.962836 0.01 0.08 0.01
- 12. $U$24 Theatres % of Portfolio 0 - 6591.489051 0 0.08 0 $I$10 Equity 1 % of portfolio 0.01 0 0.3 1E+30 0.29 $I$11 Equity 2 % of portfolio 0 0 0.3 1E+30 0.3 $I$12 Equity 3 % of portfolio 0.3 25848.59532 0.3 0.1 0.2 $I$13 Equity 4 % of portfolio 0.1 0 0.3 1E+30 0.2 $I$14 Equity 5 % of portfolio 0 0 0.3 1E+30 0.3 $I$15 Equity 6 % of portfolio 0 0 0.3 1E+30 0.3 $I$16 Equity 7 % of portfolio 0 0 0.3 1E+30 0.3 $I$17 Equity 8 % of portfolio 0.3 129108.6664 0.3 0.08 0.02 $I$18 Equity 9 % of portfolio 0.05 0 0.3 1E+30 0.25 $I$19 Equity 10 % of portfolio 0 0 0.3 1E+30 0.3 $I$20 Equity 11 % of portfolio 0 0 0.3 1E+30 0.3 $I$21 Equity 12 % of portfolio 0.08 0 0.3 1E+30 0.22 $I$22 Equity 13 % of portfolio 0.05 0 0.3 1E+30 0.25 $I$23 Equity 14 % of portfolio 0 0 0.3 1E+30 0.3 $I$24 Equity 15 % of portfolio 0.05 0 0.3 1E+30 0.25 $I$25 Equity 16 % of portfolio 0 0 0.3 1E+30 0.3 $I$26 Equity 17 % of portfolio 0 0 0.3 1E+30 0.3 $I$27 Equity 18 % of portfolio 0 0 0.3 1E+30 0.3 $I$28 Equity 19 % of portfolio 0 0 0.3 1E+30 0.3 $I$29 Equity 20 % of portfolio 0.05 0 0.3 1E+30 0.25 $I$30 Equity 21 % of portfolio 0.01 0 0.3 1E+30 0.29 $I$31 Equity 22 % of portfolio 0 0 0.3 1E+30 0.3 $U$32 Large Cap % of portfolio 0.55 0 1 1E+30 0.45 $U$33 Mid Cap % of portfolio 0.15 0 0.5 1E+30 0.35 $U$34 Small Cap % of portfolio 0.3 0 0.4 1E+30 0.1 $U$32 Large Cap % of portfolio 0.55 0 0.1 0.45 1E+30 $U$33 Mid Cap % of portfolio 0.15 0 0 0.15 1E+30 $U$34 Small Cap % of portfolio 0.3 0 0 0.3 1E+30 The shadow price tells how much the value of the objective function will change as the right hand side of the constraint is increased by 1. For example, if we consider the total investment constraint, the RHS of constraint is $100,000. If we increase the total funds by $1, our objective function will increase by 1.704 i.e. our returns will increase by $1.704. However, the allowable increase for the RHS is $2000 for our optimal solution to be the same. If we increase the total funds by $2001, then we need to resolve the model to get the optimal solution.
- 13. Table 11: Binding Constraints Cell Name Cell Value Formula Status $F$34 Total Investment Investment $100,000.00 $F$34<=$F$36 Binding $U$10 Appliances % of Portfolio 1.0% $U$10<=$W$10 Not Binding $U$11 Automotive % of Portfolio 40.0% $U$11<=$W$11 Binding $U$12 Banking % of Portfolio 0.0% $U$12<=$W$12 Not Binding $U$13 Batteries % of Portfolio 30.0% $U$13<=$W$13 Not Binding $U$14 Conglomerate % of Portfolio 5.0% $U$14<=$W$14 Not Binding $U$15 Cookery % of Portfolio 0.0% $U$15<=$W$15 Not Binding $U$16 Courier % of Portfolio 8.0% $U$16<=$W$16 Not Binding $U$17 Energy % of Portfolio 5.0% $U$17<=$W$17 Not Binding $U$18 FMCG % of Portfolio 5.0% $U$18<=$W$18 Not Binding $U$19 Food Processing % of Portfolio 0.0% $U$19<=$W$19 Not Binding $U$20 IT % of Portfolio 0.0% $U$20<=$W$20 Not Binding $U$21 Personal Care % of Portfolio 0.0% $U$21<=$W$21 Not Binding $U$22 Pharmaceuticals % of Portfolio 5.0% $U$22<=$W$22 Not Binding $U$23 Textiles % of Portfolio 1.0% $U$23<=$W$23 Not Binding $U$24 Theatres % of Portfolio 0.0% $U$24<=$W$24 Not Binding $U$10 Appliances % of Portfolio 1.0% $U$10>=$V$10 Binding $U$11 Automotive % of Portfolio 40.0% $U$11>=$V$11 Not Binding $U$12 Banking % of Portfolio 0.0% $U$12>=$V$12 Binding $U$13 Batteries % of Portfolio 30.0% $U$13>=$V$13 Not Binding $U$14 Conglomerate % of Portfolio 5.0% $U$14>=$V$14 Binding $U$15 Cookery % of Portfolio 0.0% $U$15>=$V$15 Binding $U$16 Courier % of Portfolio 8.0% $U$16>=$V$16 Not Binding $U$17 Energy % of Portfolio 5.0% $U$17>=$V$17 Binding
- 14. $U$18 FMCG % of Portfolio 5.0% $U$18>=$V$18 Binding $U$19 Food Processing % of Portfolio 0.0% $U$19>=$V$19 Binding $U$20 IT % of Portfolio 0.0% $U$20>=$V$20 Binding $U$21 Personal Care % of Portfolio 0.0% $U$21>=$V$21 Binding $U$22 Pharmaceuticals % of Portfolio 5.0% $U$22>=$V$22 Binding $U$23 Textiles % of Portfolio 1.0% $U$23>=$V$23 Binding $U$24 Theatres % of Portfolio 0.0% $U$24>=$V$24 Binding $I$10 Equity 1 % of portfolio 1.00% $I$10<=$K$10 Not Binding $I$11 Equity 2 % of portfolio 0.00% $I$11<=$K$11 Not Binding $I$12 Equity 3 % of portfolio 30.00% $I$12<=$K$12 Binding $I$13 Equity 4 % of portfolio 10.00% $I$13<=$K$13 Not Binding $I$14 Equity 5 % of portfolio 0.00% $I$14<=$K$14 Not Binding $I$15 Equity 6 % of portfolio 0.00% $I$15<=$K$15 Not Binding $I$16 Equity 7 % of portfolio 0.00% $I$16<=$K$16 Not Binding $I$17 Equity 8 % of portfolio 30.00% $I$17<=$K$17 Binding $I$18 Equity 9 % of portfolio 5.00% $I$18<=$K$18 Not Binding $I$19 Equity 10 % of portfolio 0.00% $I$19<=$K$19 Not Binding $I$20 Equity 11 % of portfolio 0.00% $I$20<=$K$20 Not Binding $I$21 Equity 12 % of portfolio 8.00% $I$21<=$K$21 Not Binding $I$22 Equity 13 % of portfolio 5.00% $I$22<=$K$22 Not Binding $I$23 Equity 14 % of portfolio 0.00% $I$23<=$K$23 Not Binding $I$24 Equity 15 % of portfolio 5.00% $I$24<=$K$24 Not Binding $I$25 Equity 16 % of portfolio 0.00% $I$25<=$K$25 Not Binding $I$26 Equity 17 % of portfolio 0.00% $I$26<=$K$26 Not Binding $I$27 Equity 18 % of portfolio 0.00% $I$27<=$K$27 Not Binding $I$28 Equity 19 % of portfolio 0.00% $I$28<=$K$28 Not Binding $I$29 Equity 20 % of portfolio 5.00% $I$29<=$K$29 Not Binding
- 15. $I$30 Equity 21 % of portfolio 1.00% $I$30<=$K$30 Not Binding $I$31 Equity 22 % of portfolio 0.00% $I$31<=$K$31 Not Binding $U$32 Large Cap % of portfolio 55.0% $U$32<=$W$32 Not Binding $U$33 Mid Cap % of portfolio 15.0% $U$33<=$W$33 Not Binding $U$34 Small Cap % of portfolio 30.0% $U$34<=$W$34 Not Binding $U$32 Large Cap % of portfolio 55.0% $U$32>=$V$32 Not Binding $U$33 Mid Cap % of portfolio 15.0% $U$33>=$V$33 Not Binding $U$34 Small Cap % of portfolio 30.0% $U$34>=$V$34 Not Binding When the LHS of constraint is equal to RHS, the constraint in binding. In our example in the total investment constraint, we utilize all the allocated funds. Therefore, the LHS of the constraint is equal to the RHS of the constraint and the constraint is binding. In the Small Cap % of portfolio constraint, we require at max 40% of our portfolio to be small cap. However, our allocation is only 30% which means LHS is not equal to RHS and therefore the constraint is not binding. Table 12: Slack for constraints Slack is the difference between the LHS and RHS of constraint. For example, in Appliances % of portfolio constraint, we have at max 20% as the RHS of the constraint. However, our allocation in Appliances in only 1%, which means that there is a slack of 19%.
- 16. CONCLUSION: Our optimal solution from the model after applying Linear Optimization is: Equity Investment Average 3 Year Return Expected Returns (After one year) % of portfolio Equity 1 $1,000.00 34.17% $1,341.65 1.00% Equity 2 $0.00 42.75% $0.00 0.00% Equity 3 $30,000.00 105.01% $61,504.21 30.00% Equity 4 $10,000.00 79.17% $17,916.55 10.00% Equity 5 $0.00 12.78% $0.00 0.00% Equity 6 $0.00 18.31% $0.00 0.00% Equity 7 $0.00 20.01% $0.00 0.00% Equity 8 $30,000.00 199.52% $89,856.55 30.00% Equity 9 $5,000.00 25.38% $6,269.01 5.00% Equity 10 $0.00 14.36% $0.00 0.00% Equity 11 $0.00 6.47% $0.00 0.00% Equity 12 $8,000.00 70.41% $13,633.05 8.00% Equity 13 $5,000.00 20.10% $6,005.02 5.00% Equity 14 $0.00 6.77% $0.00 0.00% Equity 15 $5,000.00 14.92% $5,746.10 5.00% Equity 16 $0.00 37.28% $0.00 0.00% Equity 17 $0.00 24.68% $0.00 0.00% Equity 18 $0.00 37.21% $0.00 0.00% Equity 19 $0.00 32.86% $0.00 0.00% Equity 20 $5,000.00 36.88% $6,843.80 5.00% Equity 21 $1,000.00 64.34% $1,643.43 1.00% Equity 22 $0.00 63.82% $0.00 0.00% Total Investment $100,000.00 Total Expected Returns $210,759.38 <= $100,000.00 (Max. Investment) As we can observe, we allocated funds worth $100,000 and after adopting Linear Optimization and using Jensen’s Performance Measure, our Expected returns in one year is $210,759.38 which is a 111% increase. Reference: [1] Jenson’s Performance Measure p148-150, “101 Investment Tools for Buying Low and Selling High”, Jae K. Shim and Jonathan Lansner, 2001