Finance &Investments Analysis Homework help

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ECO366 Investments Analysis Problem Set 3 This problem set will require you to calculate and plot Efficient Frontier based on your data; the due date is Friday, November 5. Please, submit via Blackboard. Part 1. (a) Intro1 Harry Markowitz (1952, 1959) developed his portfolio-selection technique, which came to be called modern portfolio theory (MPT). Prior to Markowitz's work, security-selection models focused primarily on the returns generated by investment opportunities. Standard investment advice was to identify those securities that offered the best opportunities for gain with the least risk and then construct a portfolio from these. Following this advice, an investor might conclude that railroad stocks all offered good risk-reward characteristics and compile a portfolio entirely from these. The Markowitz theory retained the emphasis on return; but it elevated risk to a coequal level of importance, and the concept of portfolio risk was born. Whereas risk has been considered an important factor and variance an accepted way of measuring risk, Markowitz was the first to clearly and rigorously show how the variance of a portfolio can be reduced through the impact of diversification, he proposed that investors focus on selecting portfolios based on their overall risk-reward characteristics instead of merely compiling portfolios from securities that each individually have attractive risk-reward characteristics. A Markowitz portfolio model is one where no added diversification can lower the portfolio's risk for a given return expectation (alternately, no additional expected return can be gained without increasing the risk of the portfolio). The Markowitz Efficient Frontier is the set of all portfolios of which expected returns reach the maximum given a certain level of risk. The Markowitz model is based on several assumptions concerning the behavior of investors and financial markets: 1. A probability distribution of possible returns over some holding period can be estimated by investors. 2. Investors have single-period utility functions in which they maximize utility within the framework of diminishing marginal utility of wealth. 3. Variability about the possible values of return is used by investors to measure risk. 4. Investors care only about the means and variance of the returns of their portfolios over a particular period. 5. Expected return and risk as used by investors are measured by the first two moments of the probability distribution of returns-expected value and variance. 6. Return is desirable; risk is to be avoided1. 7. Financial markets are frictionless. Throughout this homework, investors are assumed to measure the level of return by computing the expected value of the distribution, using the probability distribution of expected returns for a portfolio. Risk is assumed to be measurable by the variability around the expected value of the probability distribution of returns. The most accepted measures of this variability are the variance and standard deviation. Given any set of risky assets and a set of weights that describe how the portfolio investment is split, the general formulas of expected return for n assets is: n E (rP ) wi E ri (X.1) i 1 where: n w = 1.0; n = the number of securities; wi = the proportion of the funds invested in security i; ri , rP = the return on ith security and portfolio p; and i 1 i By W.P Chen, H. Chung, K. Ho, T. Hsu, a chapter from a Handbook of Quantitative Finance and Risk Management. 1 E = the expectation of the variable in the parentheses. The return computation is nothing more than finding the weighted average return of the securities included in the portfolio. The variance of a single security is the expected value of the sum of the squared deviations from the mean, and the standard deviation is the square root of the variance. The variance of a portfolio combination of securities is equal to the weighted average covariance2 of the returns on its individual securities: Var rp p2 wi w j Cov ri , rj n n (X.2) i 1 j 1 Covariance can also be expressed in terms of the correlation coefficient as follows: Cov ri , rj ij i j ij (X.3) where ij = correlation coefficient between the rates of return on security i, ri , and the rates of return on security j, r j , and i , and j represent standard deviations of ri and rj respectively. Therefore: Var rp wi w j ij i j n n (X.4) i 1 j 1 The concept of Markowitz efficient frontier Every possible asset combination can be plotted in risk-return space, and the collection of all such possible portfolios defines a region in this space. The line along the upper edge of this region is known as the efficient frontier. Combinations along this line represent portfolios (explicitly excluding the risk-free alternative) for which there is lowest risk for a given level of return. Conversely, for a given amount of risk, the portfolio lying on the efficient frontier represents the combination offering the best possible return. Mathematically the efficient frontier is the intersection of the set of portfolios with minimum variance and the set of portfolios with maximum return. Hence, to find it you could eitehr minimize level of risk given a certail level of return, or maximize level of return given certain level (budget) of risk. We will use the second approach. (b) Execution in Excel First, we need to calculate expected return, standard deviation, and covariance matrix. The expected return and standard deviation can be calculated by applying the Excel STDEV.S and AVERAGE functions to the historic monthly percentage returns data. The Variance-Covariance matrix can be obtained by using matrix multiplication in Excel: first, you need to find excess returns, or demeaned returns - to do so, for each series (columns) of returns you will create a new series (column) by subtracting mean (respective average that you have calculated) from each cell of the column. Hence, if you had five columns of raw returns, you will get new five columns of excess returns. Next, you highlight an n by n array (if you had 5 securities, it will be 5 by 5 square of cells) and type =MMULT(TRANSPOSE(Your_Exess_returns),(Your_Exess_returns)/(number_of_observations-1)) and press CTRL+Shift+ENTER. It should fill out your Var-Covar matrix space with numbers. Create a row of weights, randomly selected, by typing in a first cell of weights array =NORMINV(RAND(), 0, 1). Drag it by the right lower corner to spread the formula across all the cells of the array except the last one (to make sure that they all equal to 1). For your last security weight, you apply =1-SUM(previous n-1 weights). To check for yourself that your weights sum up to 1, type in the cell next to your row of weights =SUM(your array of weights) and it always should be equal to 1. Type 1 in the cell right next to have a check for your optimization problem. Calculate expected return for the portfolio by typing =SUMPROD(weights, expected_returns_of_securities) Calculate standard deviation of your portfolio returns by typing High covariance indicates that an increase in one stock's return is likely to correspond to an increase in the other. A low covariance means the return rates are relatively independent and a negative covariance means that an increase in one stock's return is likely to correspond to a decrease in the other. 2 =sqrt(MMULT(weights, MMULT(Var-Covar_matrix, TRANSPOSE(weights)))), and press CTRL+Shift+ENTER. To install Solver, go to File, choose Options, choose Add-ins, choose Solver Add-in, and press GO. Check a box right next to Solver Add-in and press Ok. Now you should be able to find Solver in DATA (under Analysis). To change randomly selected weight you can press F9. Your values of expected return and standard deviation should change respectively. Save 20 pairs of them - these are your expected returns and measures or risk for 20 portfolios that are created with weights randomly selected from standard normal distribution. These portfolios are not necessarily optimal. But we will use level of their volatility (standard deviation) to find maximum possible return by changing the weights. Portfolios formed by these weights will be on the efficient frontier. To find these portfolios, use Solver (as specified in Excel practice sheet) and report the optimal values of returns. Finally, plot your optimal and randomly generated returns with respect to the standard deviation (volatility). If you use line graphs, your lines will not cross! The upper line is your Efficient Frontier. \fReturns 196301 196302 196303 196304 196305 196306 196307 196308 196309 196310 196311 196312 196401 196402 196403 196404 196405 196406 196407 196408 196409 196410 196411 196412 196501 196502 196503 196504 196505 196506 196507 196508 196509 196510 196511 196512 196601 196602 196603 196604 196605 196606 196607 196608 196609 196610 196611 196612 1 13.02 -3.25 4.98 4.8 3.39 3.77 0.85 3.8 -2.7 1.36 -3.11 -2.94 3.95 3.14 0.63 -1.7 -1.4 1.92 0.85 1.27 4.69 1.7 -0.16 -1.51 8.56 1.58 1.4 3.38 -0.6 -12.4 2.44 2.91 3.08 5.97 3.59 6.41 5.88 6.49 0.43 10.51 -12.3 -0.68 -2 -11.98 -1.88 -7.92 8.44 4.1 2 11.44 -3.49 -1.47 1.5 1.68 -1.17 0.24 2.15 0.26 -0.63 -4.32 -0.55 3.17 2.66 1.11 2.58 -1.17 0.63 3.88 -1.91 4.03 0.37 -0.41 -1.4 5.83 1.07 2.72 4.86 -2.32 -10.76 3.74 5.42 6.18 8.65 6.28 7.23 6.08 6.16 -0.85 4.68 -13.06 -0.52 -1.94 -10.59 -2.08 -4.77 3.68 2.83 3 9.39 -0.74 -0.15 1.88 2.72 -1.35 0.56 1.32 -1.09 1.24 -1.6 -0.9 4.11 2.26 2.38 -0.41 0.47 1.28 3.27 -1.78 5.91 2.65 0.72 -2.29 6.08 2.47 1.25 5.16 -1.55 -9.53 3.2 5.38 1.62 6.74 3.04 2.66 8.45 1.87 -1.7 5.78 -10.87 1.28 -2.41 -9.88 -1.73 -4.6 4.67 2.42 4 10.96 -0.83 1.24 3.61 4.12 -2.46 -0.02 2.29 -1.59 0.05 -1 -1.75 4.22 0.3 3.64 -1 0.45 0.77 2.64 -0.92 2.73 2.21 0.75 -2.62 7.22 3.04 2.33 4.07 -0.41 -8 2.29 3.8 3.94 5.63 8.09 3.87 9.58 4.64 -0.84 6.93 -10.76 1.54 -0.02 -11.07 -1.89 -1.16 2.46 0.85 Excess Returns 5 1 2 3 4 11.13 12.30737 10.21672 8.121895 9.517925 2.78 -3.962631 -4.713284 -2.008105 -2.272075 2.61 4.267369 -2.693284 -1.418105 -0.202075 2.78 4.087369 0.276716 0.611895 2.167925 7.92 2.677369 0.456716 1.451895 2.677925 -1.09 3.057369 -2.393284 -2.618105 -3.902075 -1.22 0.137369 -0.983284 -0.708105 -1.462075 4.73 3.087369 0.926716 0.051895 0.847925 -0.38 -3.412631 -0.963284 -2.358105 -3.032075 2.37 0.647369 -1.853284 -0.028105 -1.392075 -1.11 -3.822631 -5.543284 -2.868105 -2.442075 -0.97 -3.652631 -1.773284 -2.168105 -3.192075 4.87 3.237369 1.946716 2.841895 2.777925 4.49 2.427369 1.436716 0.991895 -1.142075 2.71 -0.082631 -0.113284 1.111895 2.197925 -0.9 -2.412631 1.356716 -1.678105 -2.442075 1.47 -2.112631 -2.393284 -0.798105 -0.992075 1.09 1.207369 -0.593284 0.011895 -0.672075 4.71 0.137369 2.656716 2.001895 1.197925 -0.49 0.557369 -3.133284 -3.048105 -2.362075 3.98 3.977369 2.806716 4.641895 1.287925 3.36 0.987369 -0.853284 1.381895 0.767925 0.52 -0.872631 -1.633284 -0.548105 -0.692075 -1.84 -2.222631 -2.623284 -3.558105 -4.062075 8.77 7.847369 4.606716 4.811895 5.777925 3.73 0.867369 -0.153284 1.201895 1.597925 3.25 0.687369 1.496716 -0.018105 0.887925 4.6 2.667369 3.636716 3.891895 2.627925 -1.27 -1.312631 -3.543284 -2.818105 -1.852075 -9.57 -13.11263 -11.98328 -10.7981 -9.442075 4.33 1.727369 2.516716 1.931895 0.847925 3.93 2.197369 4.196716 4.111895 2.357925 3.14 2.367369 4.956716 0.351895 2.497925 9.04 5.257369 7.426716 5.471895 4.187925 6.64 2.877369 5.056716 1.771895 6.647925 4.97 5.697369 6.006716 1.391895 2.427925 9.24 5.167369 4.856716 7.181895 8.137925 6.12 5.777369 4.936716 0.601895 3.197925 -1.1 -0.282631 -2.073284 -2.968105 -2.282075 3.89 9.797369 3.456716 4.511895 5.487925 -11.79 -13.01263 -14.28328 -12.1381 -12.20208 0.37 -1.392631 -1.743284 0.011895 0.097925 -1.03 -2.712631 -3.163284 -3.678105 -1.462075 -10.38 -12.69263 -11.81328 -11.1481 -12.51208 -1.24 -2.592631 -3.303284 -2.998105 -3.332075 -1.39 -8.632631 -5.993284 -5.868105 -2.602075 3.56 7.727369 2.456716 3.401895 1.017925 0.2 3.387369 1.606716 1.151895 -0.592075 196701 196702 196703 196704 196705 196706 196707 196708 196709 196710 196711 196712 196801 196802 196803 196804 196805 196806 196807 196808 196809 196810 196811 196812 196901 196902 196903 196904 196905 196906 196907 196908 196909 196910 196911 196912 197001 197002 197003 197004 197005 197006 197007 197008 197009 197010 197011 197012 197101 197102 20.42 7.48 7.2 6.92 0.36 15.79 9.32 -0.71 11.76 1.8 -1.1 13.13 3.14 -10.1 -1.52 21.57 10.75 -1.08 -4.9 4.54 7.16 -2.05 6.75 4.07 -0.53 -12.38 3.13 1.05 1.03 -14.64 -13.01 5.18 -3.01 13.98 -8.49 -8.18 -4.86 2.26 -6.51 -24.15 -10.19 -12.11 4.82 5.6 22.15 -11.13 -4.72 6.98 15.27 8 20.57 4.89 6.17 4.04 0.4 9.37 10.01 0.64 10.17 -3.98 3.79 10.35 2.57 -7.09 -0.01 17.49 12.58 0.1 -4.58 5.79 8.48 -0.03 7.7 2.34 -2.12 -10.73 2.47 -0.49 0.81 -12.92 -11.47 4.76 -5.02 12.63 -7.12 -7.45 -3.28 3.4 -5.24 -18.74 -10.17 -8.66 5.6 6.45 16.54 -7.45 -2.12 7.32 13.78 7.47 19.27 5.58 4.95 3.54 -0.73 12.59 9.56 0.2 8.28 -1.68 -1.15 8.55 2.9 -5.44 -0.34 16.19 10.78 1.64 -3.73 4.86 6.93 1.4 6.5 0.69 -1.93 -9.05 3.56 1 1.01 -13.46 -10.84 4.95 -5.07 9.25 -6.15 -6.04 -3.88 4.09 -2.52 -14.96 -9.16 -7.3 5.11 7.14 15.64 -11.29 -2.09 10.44 16.31 6.1 20.61 4.51 7.53 4.04 -0.89 12.33 7.65 0.23 7.66 -1.63 -1.3 9.67 3.37 -6.95 -2.02 16.49 11.89 1.77 -1.51 4.1 5.94 0.65 6.59 -0.25 -1.31 -8.67 4.16 2.49 0.79 -12.59 -10.46 3.29 -2.21 7.94 -6.04 -8.22 -2.47 3.18 0.31 -12.98 -8.12 -8.28 6.46 6.77 11.59 -6.52 0.38 11.58 14.12 3.32 17.44 19.70737 19.34672 18.0019 19.16792 5.9 6.767369 3.666716 4.311895 3.067925 5.24 6.487369 4.946716 3.681895 6.087925 3.56 6.207369 2.816716 2.271895 2.597925 -0.71 -0.352631 -0.823284 -1.998105 -2.332075 10.05 15.07737 8.146716 11.3219 10.88792 11.85 8.607369 8.786716 8.291895 6.207925 2.09 -1.422631 -0.583284 -1.068105 -1.212075 4.86 11.04737 8.946716 7.011895 6.217925 -2.66 1.087369 -5.203284 -2.948105 -3.072075 -1.1 -1.812631 2.566716 -2.418105 -2.742075 10.13 12.41737 9.126716 7.281895 8.227925 6 2.427369 1.346716 1.631895 1.927925 -5.58 -10.81263 -8.313284 -6.708105 -8.392075 -0.08 -2.232631 -1.233284 -1.608105 -3.462075 15.77 20.85737 16.26672 14.9219 15.04792 11.74 10.03737 11.35672 9.511895 10.44792 0.19 -1.792631 -1.123284 0.371895 0.327925 0.81 -5.612631 -5.803284 -4.998105 -2.952075 4.83 3.827369 4.566716 3.591895 2.657925 6.32 6.447369 7.256716 5.661895 4.497925 1.66 -2.762631 -1.253284 0.131895 -0.792075 6.98 6.037369 6.476716 5.231895 5.147925 0.54 3.357369 1.116716 -0.578105 -1.692075 -1.34 -1.242631 -3.343284 -3.198105 -2.752075 -8.72 -13.09263 -11.95328 -10.3181 -10.11208 2.39 2.417369 1.246716 2.291895 2.717925 0.11 0.337369 -1.713284 -0.268105 1.047925 0.45 0.317369 -0.413284 -0.258105 -0.652075 -12.32 -15.35263 -14.14328 -14.7281 -14.03208 -7.92 -13.72263 -12.69328 -12.1081 -11.90208 2.77 4.467369 3.536716 3.681895 1.847925 -3.44 -3.722631 -6.243284 -6.338105 -3.652075 6.33 13.26737 11.40672 7.981895 6.497925 -7.39 -9.202631 -8.343284 -7.418105 -7.482075 -8.65 -8.892631 -8.673284 -7.308105 -9.662075 0.11 -5.572631 -4.503284 -5.148105 -3.912075 4.29 1.547369 2.176716 2.821895 1.737925 -1.29 -7.222631 -6.463284 -3.788105 -1.132075 -12.02 -24.86263 -19.96328 -16.2281 -14.42208 -10.45 -10.90263 -11.39328 -10.4281 -9.562075 -6.03 -12.82263 -9.883284 -8.568105 -9.722075 3.83 4.107369 4.376716 3.841895 5.017925 6.33 4.887369 5.226716 5.871895 5.327925 11.6 21.43737 15.31672 14.3719 10.14792 -4.99 -11.84263 -8.673284 -12.5581 -7.962075 0.21 -5.432631 -3.343284 -3.358105 -1.062075 10.74 6.267369 6.096716 9.171895 10.13792 15.93 14.55737 12.55672 15.0419 12.67792 4.06 7.287369 6.246716 4.831895 1.877925 197103 197104 197105 197106 197107 197108 197109 197110 197111 197112 197201 197202 197203 197204 197205 197206 197207 197208 197209 197210 197211 197212 197301 197302 197303 197304 197305 197306 197307 197308 197309 197310 197311 197312 197401 197402 197403 197404 197405 197406 197407 197408 197409 197410 197411 197412 197501 197502 197503 197504 8.34 0.78 -6.6 -0.42 -8.62 3.29 1.21 -7.86 -3.27 14.49 13.22 3.03 2.96 -1.23 -1.57 -4.21 -7.19 -1.62 -4.85 -3.45 -0.29 -1.77 -7.96 -14.07 -7.04 -11.85 -11.46 -5.13 18.94 -5.38 11.67 -2.93 -24.87 -4.02 10.24 -0.33 0.22 -4.71 -8.35 -3.42 -8.7 -9.49 -13.57 15.5 -6.33 -7.4 26.43 7.53 9.13 7.07 4.63 3.26 -5.1 -3.59 -8.06 4.38 2.09 -6.78 -5.79 14.24 13.38 5.48 0.1 0.19 -1.9 -4.67 -4.89 -2.64 -4.84 -3.28 3.94 -2.47 -4.99 -8.66 -4.03 -9.84 -10.8 -3.97 16.07 -6.99 10.87 0.38 -21.68 -4.82 14.29 -1.33 -0.92 -5.94 -7.55 -3.23 -5.62 -8.17 -8.34 7.13 -5.55 -8.14 29.54 5.42 10.72 6.32 7.86 2.24 -7.34 -2.29 -7.25 2.81 -0.12 -7.09 -3.76 10.74 11.86 4.16 -1.93 -1.07 -4.14 -3.32 -4.24 0.61 -4.81 -2.89 4.78 -3.35 -2.8 -8.51 -2.66 -6.7 -9.3 -4.38 12 -5.77 6.82 0.84 -18.94 -3.97 14.87 -0.23 0.5 -5.12 -8.11 -2.23 -5.13 -7.63 -6.77 6.99 -5.02 -7.27 28.13 3.15 9.09 4.23 6.49 2.74 -4.96 -1.24 -6.04 4.56 -0.63 -4.45 -4.39 11.78 10.84 4.22 -1.57 1.3 -2.26 -3.15 -3.34 -0.03 -3.74 -2.12 6 -2.61 -3.55 -8.13 -0.99 -6.08 -7.73 -3.07 13.57 -5.94 7.76 0.43 -16.71 -2.85 12.76 -0.11 1.56 -5.42 -6.79 -2.18 -4.71 -7.49 -6.93 7.79 -5.2 -7.5 27.78 3.47 7.92 2.82 5.13 7.627369 3.406716 6.591895 5.047925 1.86 0.067369 2.036716 0.971895 1.297925 -6.57 -7.312631 -6.323284 -8.608105 -6.402075 -2.16 -1.132631 -4.813284 -3.558105 -2.682075 -5.91 -9.332631 -9.283284 -8.518105 -7.482075 5.65 2.577369 3.156716 1.541895 3.117925 -2.38 0.497369 0.866716 -1.388105 -2.072075 -7.27 -8.572631 -8.003284 -8.358105 -5.892075 -5.6 -3.982631 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Var_Covariance St.Dev Variance Weights 1 2 3 4 5 0.712631 1.2232843137 1.268105 1.442075 1.591324 Step 1. Calculate expected return for ea 64.01896 52.80595 44.59696 40.71931 42.11764 Step 2. Calculate Variance-Covariance m To do so, highlight 5 by 5 array and type =MMULT(TRANSPOSE(data!G3:K614), ( where data!G3:K614 is your Excess Retu CTRL+Shift+Enter 52.805951248 47.391647297 39.698918183 36.669428525 38.146738527 44.59696 39.69892 36.01767 33.03757 34.64563 40.71931 36.66943 33.03757 32.13578 33.25013 42.11764 38.14674 34.64563 33.25013 37.11244 8.001185 6.8841591569 6.001472 5.668843 6.091998 64.01896 47.391647297 36.01767 32.13578 37.11244 -1.544621 0.7143427989 0.002292 Step 3. Calculate variance of each stock to check if your Var_Cov matrix is calcu 1.52424 0.303746 Step 4. Create random weights using w stocks into a risky portfolio. In first four =NORMINV(RAND(), 0, 1) which will ma normal. The last cell needs to be select weights add up to 1. Hence, type in =1Weights sum Constraint_1 1 1 Randomly selected Optimal E[r_portfolio] St. Dev E[r_portfolio] 2.4574337457 5.87446 1.54825434 1.7852222967 1.212921924 2.1756513622 1.6475911409 1.5453799312 2.229474622 2.2027503925 2.2580793754 2.5137960218 2.2244744383 1.3728492503 1.8611893195 1.9916535607 2.3362388838 2.7870297378 1.4160529548 1.2948782584 3.3614844127 3.8095123682 5.40 5.70 5.82 5.87 5.91 5.98 6.02 6.50 6.63 6.69 6.76 6.78 6.81 6.86 7.30 7.71 7.82 7.92 10.69 10.93 2.1796161665 2.3639041264 2.4284357261 2.4574337457 2.4748355058 2.5107514018 2.5256109158 2.7322486808 2.7825258455 2.8081029533 2.833767012 2.8395388707 2.8516736565 2.869138022 3.0259059534 3.1638470202 3.1997829769 3.2338598464 4.0568441596 4.1216846254 16 20 19 21 14 8 10 17 6 18 15 3 4 7 2 11 13 5 12 9 Step 5. To double check that your weigh up and compare to a constraint 1 (this w in an optimuization problem) Step 6. Based on your randomly selecte expected return for your portfolio by =S B2:F2) and pressing Enter, and standard =SQRT(MMULT(B13:F13, MMULT((B4:F and pressing CTRL+Shift+Enter; These a Use F9 button to change your weights. E[r_portfolio and St. Dev. will change. R each time you press F9. Repeat this 20 Step 7. Install Solver. In Solver set obje returns ($A$26 if first field and check "M values of your weights: $B$13:$F$13; S Weights sum up to 1, or $F$19=$F$20 a of risk, meaning we want our risk equal $B$26=$B$28. Dont forget to uncheck t button. In this particular case Solver wi return for teh level of risk=11.08365; Yo exercise for all left over 19 cases. Just c last constraint = to $B$29 (Press OK), th forget to report the optimal values! Step 8. Now you have 20 data points of sellected portfolo returns. Sort these n and plot them against your St. Dev. You increasing lines that do not cross. The u 0.4389013071 11.08 4.1653380475 1 Step 8. Now you have 20 data points of sellected portfolo returns. Sort these n and plot them against your St. Dev. You increasing lines that do not cross. The u frontier! Efficient Frontier 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 Randomly selected weights Optimally select lculate expected return for each stock lculate Variance-Covariance matrix fro these five stocks. highlight 5 by 5 array and type TRANSPOSE(data!G3:K614), (data!G3:K614)/611), a!G3:K614 is your Excess Returns in a data sheet. Press ft+Enter lculate variance of each stock's returns and st. deviation f your Var_Cov matrix is calculated correctly eate random weights using which you combine your o a risky portfolio. In first four cells type V(RAND(), 0, 1) which will make draws from standard he last cell needs to be selected to make sure your dd up to 1. Hence, type in =1-SUM(B13:E13) double check that your weights add up to 1, sum them mpare to a constraint 1 (this will also be your constraint muization problem) sed on your randomly selected weights, calculate return for your portfolio by =SUMPRODUCT(B13:F13, d pressing Enter, and standard deviation by MULT(B13:F13, MMULT((B4:F8), TRANSPOSE(B13:F13)))) ng CTRL+Shift+Enter; These are Not Efficient portfolios! tton to change your weights. Each time your olio and St. Dev. will change. Report these two values you press F9. Repeat this 20 times. stall Solver. In Solver set objective: maximize expected A$26 if first field and check "Max" box); By changing your weights: $B$13:$F$13; Subject to constraints: 1). um up to 1, or $F$19=$F$20 and 2) we use all "budget" eaning we want our risk equal to some set number: $28. Dont forget to uncheck teh Non-Negative Variables this particular case Solver will calculate maximum teh level of risk=11.08365; You will have to repeat the or all left over 19 cases. Just click on Solver and CHANGE raint = to $B$29 (Press OK), then $B$30 and so on. Dont eport the optimal values! ow you have 20 data points of optimal, and randomly portfolo returns. Sort these numbers based on St. Dev. hem against your St. Dev. You chould obtain two lines that do not cross. The upper one is your efficient Efficient Frontier ted weights Optimally selected weights