Investment Style of Portfolio Management: Excel Applications




                              Stanley M. Atkinson*

     ...
2


       Investment Style of Portfolio Management: Excel Applications

                                        Abstract
...
3


Sharpe’s. Their main criticism is that the estimated style based on historical data is unstable

and the style is hard...
4


where Ri is the mutual fund return, Fn is the value of the nth factor, bin is the factor

sensitivities, and ei is the...
5


regression method is one with the restriction that the total allocation of assets is 100%. A

short discussion regardi...
6


squared, which is to be minimized. Given those constraints, the Solver function computes

the degrees of exposures to ...
7


III. Empirical Results

   The estimated factor loadings (e.g, exposures) from the quadratic programming in the

Solve...
8


IV. SENSITIVITY ANALYSES OF THE MODEL

A. Stability of asset allocation methodology

   Christopherson (1995) argues t...
9


This allocation pattern persists in the third sub-period: 76% in the Large Stock Index and

24% in the Small Stock Ind...
10


V. CONCLUSION

       We show that Sharpe's investment style model seems to produce fairly accurate and

consistent r...
11


APPENDIX

An Excel Illustration

   We illustrate the Excel model we employed in the previous section with the Magell...
12


REFERENCES

Benninga, S. (1997). Financial Modeling. The MIT Press, Cambridge.

Christopher, J. (1995). Equity style ...
13


Table 1. Style estimation based on the constrained regression vs. quadratic programming
on the three funds: the Short...
14


Table 2. Means and variances for stock and bond allocations and t-values for the
differences in means based on random...
15


ENDNOTES
1
 Sharpe (1992) uses twelve-class assets: Bills, Intermediate-term Government Bonds, Long-term Government
Bonds, Corporat...
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Investment Style of Portfolio Management: Excel Applications

  1. 1. Investment Style of Portfolio Management: Excel Applications Stanley M. Atkinson* and Yoon K. Choi** October, 1999 ________________________ Department of Finance, College of Business Administration, University of Central Florida, PO Box 161400, Orlando, FL 32816. *Atkinson: (407) 823 – 2609, stan.atkinson@bus.ucf.edu ; FAX (407) 823 – 6676. **Choi: Corresponding author, (407) 823 – 5023, yoon.choi@bus.ucf.edu, FAX (407) 823 – 6676.
  2. 2. 2 Investment Style of Portfolio Management: Excel Applications Abstract Sharpe (1992) examined the style of a mutual fund using an asset class factor model. The purpose of this paper is to investigate the reliability of that model and to educate the readers in using the Excel’s solver function to conduct Sharpe’s style analysis. The results show that Sharpe’s investment style model is fairly accurate and consistent over time. The paper also shows that teaching with a spreadsheet can be not only fun but can be instrumental in motivating future financial managers. It is important to keep up with rapidly advancing technology in our profession of teaching and research. Naturally, the use of spreadsheets has been widely adopted in undergraduate and graduate curriculums. When appropriately used, it may enhance learning and teaching significantly. The purpose of the paper is to investigate Sharpe's investment style model of mutual funds in terms of asset allocation (style) using Excel’s solver function. Style analysis (or asset allocation) models provide a valuable tool for investors, plan sponsors, and consultants. Plan sponsors and consultants are interested in the manager’s observance of the investment objectives. Investors want to know the investment style so that they can obtain the effective mix of assets to their taste. Sharpe (1988, 1992) introduces an objective style model based on asset classes (or factors). Specifically, the mutual fund’s return is assumed to be a function of several factor exposures and firm-specific risks. The factor exposures or sensitivities determine the style or asset allocation of the fund. The impact of Sharpe's style analysis has been extensive with extensions and variations of his model following. Christohperson (1995) and DiBartolomeo and Witkoski (1997) point out a potential drawback of the style analysis based on return correlation, such as the
  3. 3. 3 Sharpe’s. Their main criticism is that the estimated style based on historical data is unstable and the style is hard to capture when a manager changes investment strategy. However, Trzcinka (1995) argues that the usefulness of Sharpe's model is an empirical question because economic models can be still useful even when economic data violate statistical assumption of the models. Also, Lobosco and DiBartolomeo (1997) derive an approximation of the confidence intervals of Sharpe style weights. In spite of some debates about the merits, the Sharpe style analysis has been popular and applied to real-world practices. Trzcinka (1995) further defends Sharpe's style model, stating that it is easy to implement and very objective, and the assumptions and data are clear to the analyst so that the results can be replicated. The following section reviews Sharpe’s asset class factor model and introduces Excel’s solver function as a solution method. Several mutual funds are analyzed here to show the rigor and flexibility of the Excel Solver function as a method of style analysis. We also provide a detailed instruction of using the Excel solver function for style analysis so that this technique can be used as a valuable teaching tool in advanced undergraduate or graduate courses. We finally summarize and conclude the paper. I. SHARPE'S ASSET ALLOCATION MODEL AND ITS ESTIMATION A. Style Analysis According to Sharpe, the asset allocation of a mutual fund is how the fund manager allocates his assets across a number of major asset classes. An asset class factor model is based on the following equation: Ri = bi1F1 + bi2F2 + ….+ binFn+ ei, (1)
  4. 4. 4 where Ri is the mutual fund return, Fn is the value of the nth factor, bin is the factor sensitivities, and ei is the unsystematic residual. The objective is to find the “best” set of asset class exposures (i.e., bi) that add up to 100% and lie between one and zero. More specifically, the best set of exposures is the one for which the variance of e i in Equation (1) is the least. Let us rearrange Equation (1) as follows: ei = Ri – [bi1F1 + bi2F2 + ….+ binFn] (2) Thus, the residual ei can be interpreted as the difference between the fund return (Ri ) and the return of a passive portfolio with the same style (bi1F1 + bi2F2 + ….+ binFn). The objective of the analysis is to choose the style (set of asset class exposures) that minimizes the variance of this difference (or the sum of the residual squared with all constraints). In a way, the technique is very similar to the least square estimation. However, the constraints on the exposure coefficients (i.e., bi) in quadratic programming make some of the simpler models based on regression analysis unsuitable. Sharpe suggests guidelines in choosing the asset class factors: such asset classes should be mutually exclusive, exhaustive and have returns that differ.1 By having exhaustive classes of assets, the asset class factors (F1 through Fn) as a whole can emulate the market portfolio as closely as possible. If there are missing assets in the equation that systematically affect the mutual fund’s return, then we will obtain wrong estimates of the sensitivity factors. B. Methodology in Excel We employ the constrained regression and the quadratic programming using the Excel Solver for asset allocation (i.e., fund style), and we compare the results. The constrained
  5. 5. 5 regression method is one with the restriction that the total allocation of assets is 100%. A short discussion regarding the constrained regression in Excel may be helpful here. Since the regression function is not flexible enough to allow for some constraints, we need to modify the regression equation with the constraints, bi1 + bi2 + ….+ bin = 1, before applying the regression function. By applying the constraint to equation (1), we obtain Ri = bi1F1 + bi2F2 + ….+ (1 - bi1 - bi2 -….- bin-1 )Fn + ei. (3) Rearranging the parameters, equation (3) can be written as Ri - Fn = bi1(F1 - Fn ) + bi2 (F2 - Fn.) + . . . . + bin-1 (Fn-1 - Fn ) + ei. (4) That is, the variables in the equation are expressed in the excess returns net of F n . Therefore, the number of the independent variables is reduced to n-1 for the regression inputs of the regression function. Furthermore, we need to impose the intercept in the regression to be zero and add this constraint in the regression function. A major drawback is the inability of the regression function to add the constraints of positive factor sensitivities. That is, a negative sensitivity is possible, which is inconsistent with a positive holding in a portfolio. Meanwhile, quadratic programming is, as described, used to solve non-linear programming problems with the constraint that the total asset allocation is 100% and all the factor sensitivities are non-negative and lie between zero and one. Specifically, in the Excel Solver, the investment style problem is to obtain a set of exposure coefficients that minimizes the sum of the residual squared, Σei2 with the constraints that each of the factor sensitivities (or exposures), bi, is constrained to lie between zero and one, and the factor sensitivities should add up to one (e.g., bi1 + bi2 + ….+ bin = 1). For input data for the Solver function, we need to write an objective function in a cell: Σei2 – the sum of the residual
  6. 6. 6 squared, which is to be minimized. Given those constraints, the Solver function computes the degrees of exposures to each asset class (bis) that minimize the sum of the residual squared. 2 II. Data We obtain the asset classes from various sources, including the BARRA investment data (available in the Internet, www.barra.com). We concentrate on the domestic asset world for the purpose of demonstrating the excel models, employing the S&P 500/BARRA Growth Index, the S&P 500/BARRA Value Index, the S&P MidCap 400/BARRA Growth Index, the S&P MidCap 400/BARRA Value Index, the S&P SmallCap 600/BARRA Growth Index, the S&P SmallCap 600/BARRA Value Index, the IBBS Corporate Bond Index, the IBBS Government Bond Index, and finally the IBBS Treasury Bond Index. Since the small cap index data series did not start until January 1994, we use three- year monthly index returns from January 1994 through December 1996 for our analysis. IBBS above stands for the data series from the Ibbotson and Sinquefield’s yearbook. To be consistent with the requirements for the asset classes, we pick three domestic stock and bond mutual funds: Fidelity Short-term Bond, Fidelity Magellan Fund, and Fidelity Balanced Fund for an illustration. We obtain the mutual fund data from the Morningstar CD-ROM database.
  7. 7. 7 III. Empirical Results The estimated factor loadings (e.g, exposures) from the quadratic programming in the Solver function are shown in Table 1 for the three mutual funds: Fidelity Short-term Bond, Fidelity Magellan, and Fidelity Balance Fund. Table 1 also shows the results from the constrained regression analysis. The Fidelity Short-term Bond Fund has 3.6% in Mid-cap growth stocks, 13.8% in the Corporate bonds, 5.8% in the Government Bond, and 76.8% Treasury Bills as respective factor loadings (i.e., exposures). The Magellan Fund has factor loadings of 16.7% in the Large Growth Stock (S&P500), 16% in the Large Value Stock, 47.8% in the Mid-cap Growth Stock, 10.1% in the Mid-cap Value Stock, and 9.3% in the Government Bond Index, respectively. This reflects its major holdings in stocks, particularly in the mid-cap growth-oriented stocks. However, about 10% of the Fund’s holdings seem to be exposed to government bonds as well. Finally, the Balance Fund has factor loadings of 8.3% in the S&P500 Growth Stock, 8.5% in the S&P500 Value Stock, 2.5% in the Mid-cap Growth Stock, 1.4% in the Small Growth, 14.6% in the Small Value Stock, 33.8% in the Corporate Bond, 4.1% in the Government Bond, and 26.7% in the Treasury Bills. Thus the loadings reflect the investment style of balancing between stocks and bonds evenly. Meanwhile, the constrained regression results on the three mutual funds show in Table 1 that some exposure estimates have negative signs, which is not a desirable property of good estimation methodology. Therefore, the above quadratic programming is superior to the constrained regression because of its capability of restricting the factor loadings (i.e., exposures) estimates to be positive.3
  8. 8. 8 IV. SENSITIVITY ANALYSES OF THE MODEL A. Stability of asset allocation methodology Christopherson (1995) argues that the asset allocation (or effective-mix) methodology is not trustworthy due to its vulnerability of returns toward noisy data and style dynamics. However, Trzcinka (1995) finds some problems in Christopherson's conclusion. Christopherson shows that correlations can be spurious, and provides an example where Shapre's style model fails with the Russell database. However, the sample size is too small to generalize Christopherson's results. Also, the sample of managers in the Russell database is not a random sample of all managers. In order to shed lights on the issue, we examine asset allocation over time to ascertain the stability of the asset allocation methodology over time. We do that by extending the asset allocation methodology over January 1981 to December 1995. We estimate asset allocation for the three 5-year sub-periods: 1981 – 1985; 1986 – 1990; and 1991 – 1995. If we observe stable asset allocation over the sample periods, the asset allocation estimation methodology can be trustworthy to some extent. Since many BARRA market indices were not available in early periods, for this investigation, we employ the Ibbotson and Sinquefield’s market indices instead.. As an illustration, we examine Fidelity’s Magellan fund using six market indices as asset classes: Large Stock Index, Small Stock Index, Long-term Corporate Bond Index, Long-term Government Bond Index, Intermediate Government Bond Index, and Treasury Bill Index. In the first sub-period (1981 – 1985), the result shows that the optimal asset allocation was 32% in the Large Stock Index, 63% in the Small Stock Index, and 5% in the Long-term Government Bond Index. However, the asset allocation in large stocks is much greater in the second sub-period: 72% in the Large Stock Index and 28% in the Small Stock Index.
  9. 9. 9 This allocation pattern persists in the third sub-period: 76% in the Large Stock Index and 24% in the Small Stock Index. This result is fairly consistent with the investment style known for the Magellan fund - a major large equity stock fund. In sum, the quadratic programming methodology used in the Solver function seems to produce the optimal asset allocation (investment style) fairly consistently over time. B. Comparison with actual asset holdings In order to examine the accuracy of the style model, we compare the asset allocation derived from the model with the actual holdings for the same period. We randomly picked 60 domestic balanced funds for the period of 1994 - 1996 and obtained the 3-year average stock and bond holdings in percentages. Table 1 shows a summary of the actual and predicted asset allocations between stock and bond holdings. Since our model used four stock indexes and three bond indexes including treasury bills, we added the estimated allocation to generate stock and bond asset holdings as a whole.4 The model predicts the actual asset allocation remarkably well. The actual average stock and bond holdings were 55.56% and 43.71%, respectively, while the predicted allocation was 55.32% and 44.65%, respectively. The difference in asset allocation between the model estimates and the actual allocation was virtually zero. The t values for the mean differences between the actual and predicted allocation were very insignificant at all conventional levels.
  10. 10. 10 V. CONCLUSION We show that Sharpe's investment style model seems to produce fairly accurate and consistent results over time. The model provides an objective and reliable method to evaluate the mutual (or pension) fund manager's asset allocation. The required information for the analysis is the historical portfolio returns without actual asset holdings. The analysis can be extended to international portfolio management by employing foreign asset classes. We also have demonstrated how to use the Solver function in Excel for style analysis (Sharpe's asset allocation). This will provide a valuable tool for advanced undergraduate and graduate students, and investment professionals as well. We can enhance the quality of learning significantly by familiarizing our students with this indispensable tool. Furthermore, teaching with the spreadsheet applications can also be fun and instrumental in motivating and training our future high-tech investment wizards.
  11. 11. 11 APPENDIX An Excel Illustration We illustrate the Excel model we employed in the previous section with the Magellan fund. First, we need to have input return data on mutual funds and all asset class factors. Refer to Appendix 1, 2, and 3 for the details. We collected 3-year monthly returns for the Fidelity Magellan fund and nine asset classes. Then we compute the residual from Equation (2) and the sum of the residual squared (e.g., the number in cell M76 in Appendix 1), which the Solver program eventually minimizes. In calculating the residual, we use numbers in cells, B76 to J76, as default parameters for factor loading parameters. At this point, any arbitrary numbers are used in these cells as long as the sum equals one (e.g., write an excel formula in cell K76 as shown). Then we start the Solver function by choosing the option, Solver, under the TOOLS menu in the Excel command bar.5 On the input screen, as shown in Appendix 2, we provide the cell reference for the target cell (e.g., M76) and check MIN for minimization. Then we type in B76:J76 for changing cells. The numbers in these cells (B76 through J76) are the choice variables (i.e., asset exposures) for the program. Finally we need to specify all constraints needed for the Solver function. There are two constraints: (1) the sum of the factor loadings is equal to 1 (e.g., K76 = 1); (2) and each factor loading lies between zero and one (e.g., B76:J76 <= 1 and B76:J76 >= 0). After we click the button “add”, we begin to type in these constraints in the appropriate boxes. When we click the button "Solve", the Solver searches for the optimal solutions for the allocation coefficients that minimize the target cell. The output for the Magellan is in cell range B78:J78: 16.7% for the S&P/Barra Growth, 16.0% for the S&P/Barra Value, 47.8% for the Mid-Cap 400/Barra Growth, 10.1% for the Mid-cap 400 Value, and 9.3% for the Government Bond.
  12. 12. 12 REFERENCES Benninga, S. (1997). Financial Modeling. The MIT Press, Cambridge. Christopher, J. (1995). Equity style classifications. J Portfolio Management, 32-43. DiBartolomeo, D. and E. Witkowski. (1997). Mutual Fund Misclassification: Evidence Based On Style Analysis. Financial Analyst Journal 53, 32-43. Lobosco, A. and D. DiBartolomeo. (1997). Approximately The Confidence Intervals For Sharpe Style Weights. Financial Analyst Journal 53, 80-85. Sharpe, W. (1988). Determining a fund’s effective asset mix. Investment Management Review, 59-69. Sharpe, W. (1992).Asset allocation: management style and performance measurement. J Portfolio Management, 7-19. Trzcinka, C. (1995). Equity Style Classifications: Comment. Journal of Portfolio Management, 21, 44-46.
  13. 13. 13 Table 1. Style estimation based on the constrained regression vs. quadratic programming on the three funds: the Short-term Bond, the Magellan, and the Balance fund. _______________________________________________________________________ Short-term Bond Magellan Balance _____________ ____________ __________ Asset class C Q C Q C Q ________________________________________________________________________ Large growth -3.1% 0% 16.8% 16.7% 8.2% 8.3% Large value 0.7% 0% 11.6% 16.0% 15.2% 8.5% Mid growth 18.9% 3.6% 48.3% 47.8% 6.9% 2.5% Mid value -6.9% 0% 15.6% 10.1% -15.3% 0% Small growth -7.8% 0% 0.4% 0% - 0.4% 1.4% Small value -2.2% 0% 0.6% 0% 18.3% 14.6% Corporate bond 36.6% 13.8% 0.6% 0% 49.4% 33.8% Government bond -10.7% 5.8% 14.4% 9.4% -7.7% 4.2% Treasury bond 74.5% 76.8% -8.4% 0% 25.4% 26.7% _________________________________________________________________________ Total 100% 100% 100% 100% 100% 100% _________________________________________________________________________ All equity indexes are based on the S&P/BARRA data and all bond indexes are from the IBBS index series data. C denotes the constrained regression, while Q denotes the quadratic programming.
  14. 14. 14 Table 2. Means and variances for stock and bond allocations and t-values for the differences in means based on randomly chosen 60 balanced funds for the period of 1994 - 1996. The number in the parenthesis is t-value. Mean Allocation (%) ________________________________________________________ Actual Model Difference ________________________________________________________ Stocks 53.36 55.32 -1.76 (-0.819) Bonds 43.71 44.65 -0.94 (-0.478) ________________________________________________________
  15. 15. 15 ENDNOTES
  16. 16. 1 Sharpe (1992) uses twelve-class assets: Bills, Intermediate-term Government Bonds, Long-term Government Bonds, Corporate Bonds, Mortgage-Related Securities, Large-Cap Value Stocks, Large-Cap Growth Stocks, Medium-Cap Stocks, Small-Cap Stocks, Non-U.S. Bonds, European Stocks, and Japanese Stocks. 2 Refer to the Appendix for detailed instructions on using the Excel function. 3 Sharpe notes that in computing R2 as a measure of data fit, the quadratic programming may produce slightly different values than those from regression analysis because the residual returns may be correlated with the asset classes (factors). But he claims that such differences are insignificant in practice. 4 The reason we group various allocations into an aggregate stock or bond allocation is that we obtain only the aggregate allocation information from the Morningstar database. 5 Refer to Benninga (1997) for more applications of the Solver functions.

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