FTS Real Time System Project: Portfolio Diversification Note: this project requires use of Excel s Solver

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3 stock analysts use techniques involving historical data, fundamental analysis, and scenario analysis. Values based on the CAPM are provided by the FTS Real Time Client, though you can override them via the Parameters menu item. Given the portfolio weights, the expected return of the portfolio is: Finally, we need to describe risk. Modern portfolio theory uses the standard deviation of returns (also called volatility) as a measure of risk (or equivalently, the variance). To calculate the risk of a portfolio, you also need the covariances between stock returns. In notation, let ij denote the covariance between the returns of stocks i and j, so ii is the variance of the return of stock i. Then, given the portfolio weights, the variance of the portfolio return is: The volatility is the square root of this number. The FTS Real Time Client calculates all the covariances for you. The portfolio selection problem is to find weights that minimize the variance subject to some constraints. The first is that the sum of the weights equals 1; this simply means that you invest all the money you have allocated to stocks. The second is that the expected return from the portfolio equals your desired return. Beyond that, you can impose more conditions. For example, you may restrict short selling, either completely, which says w i 0. Or you may require that you will not invest more some amount in any one stock; this says w i 0.1. Common constraints for this case, which has 30 stocks, would be that you do not invest more than 15% in any one stock, and if there is no short selling, that you invest at least 2% in every stock. Project The objective of this project is to learn to apply modern portfolio theory by constructing portfolios, implementing the recommended portfolio, and tracking its performance over time. This is done by completing the following steps, and includes learning how to use Solver to calculate the portfolio weights. 1. From the RT Client, select Covariances and Returns in the Analytics area (at the bottom right); you will see your (raw) portfolio weights and the expected returns and covariances of all the stocks. In the Edit menu of the Analytics area, select Export to Excel. This will transfer the data into an Excel spreadsheet. You have two choices with the covariances and returns. If you select Covariances and Returns, it shows you either the default expected return or the value you entered under Parameters. If you select Covariances and Returns (CAPM) it shows you the expected return based on CAPM. 2. Decide on a target expected return. This could be a number you pick, e.g. 7%, or you could be guided by the past return of an index such as the S&P500, or the yield on Treasury bonds.

5 Example with the FTS 30 Stock Case We show you here a detailed example with the FTS 30 Stock case. We will allow short sales to show you how to adjust your spreadsheet appropriately. First, select Covariances and Returns from the analytics area: From the Edit menu circled above, select Export to Excel to get the data in a spreadsheet:

6 The stocks are in rows 7 to 36, the expected returns are in column C and the covariance matrix is in D7:AG36. The weights shown don t matter; we will be calculating new weights. The first thing we have to do is replace cells B2, B3 and B5 with formulas. B5 is the sum of the weights; this has to be 1, so enter the formula: =SUM(B7:B36)

7 B2 is the portfolio expected return;, so enter the formula =SUMPRODUCT(B7:B36,C7:C36) B3 is the portfolio variance; this formula is =SUMPRODUCT(B7:B36,MMULT(D7:AG36,B7:B36)), which is the Excel implementation of the double-summation formula for the portfolio variance. Finally, replace D3 with the formula =SQRT(B3) so the portfolio volatility is calculated correctly. What we have done so far is replace the values copied from the FTS Real Time Client into formulas where the values are calculated as a function of the portfolio weights. When we use Solver to find the portfolio weights, we will be asking it to find the lowest variance portfolio that matches our target return. The target return is in cell B4; initially, it is simply the expected benchmark return, which in the FTS 30 Stock Case is the S&P500 index. You can change that to a higher number if you want to try and beat the benchmark; we changed it to 0.09 for this example. Now, we have to think about constraints on the problem. The first constraint is that the sum of the weights is 1: B5 = 1. The second constraint is that the portfolio expected return equals our target return: B2 = B4 We could also decide to disallow short sales; that would be the constraint B7:B36 >= 0 but we will not impose that. Other constraints you could impose could be limits on allocations, e.g. no more than 10% in any one stock, or no more than 20% in any sector, and so on. Enter these into Solver and click `Solve

8 A 20% maximum and minimum on any stock would require adding the constraints \$B\$7:\$B\$36 <= 0.2 and \$B\$7:\$B\$36 >= For the base case, we obtained a solution:

10 We conclude with some comments: You will have to rebalance your portfolio periodically as prices change. You may decide to run Solver periodically and recalculate the weights; again, this can be costly so if you plan to do this, make sure you have enough resources available. To calculate the results required in Step 5 above, you will need to download your trading history from the Reports menu at the top: From the history window, generate the report End of Day Market Values and copy the selected information as shown and paste it into Excel:

11 Now, you can calculate what is needed; there is also a performance report that you can generate automatically.

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