Editorial Staff   Volume 14     SUMMER, 2015     Number 1 Managing Editor:    Luther Lawson, University of North Carolina‐ <1> Portfolio Performance Evaluation Benchmark: A Note Wilmington  Adam Y.C. Lei and Huihua Li Senior Editor:  Richard J. Cebula, Jacksonville University  <8> Delta Gamma Hedging and the Black-Scholes Partial Differential Co‐Editors (Economics):  Equation (PDE) Joshua Hall, West Virginia University  Sudhakar Raju Co‐Editors (Finance):  Bill Z. Yang, Georgia Southern University  <22> Black-Scholes Option Pricing: Implementing a Hands-On Assistant Editors:  Assignment Using Excel Gigi Alexander, Jacksonville University  Christi Wann Robert Houmes, Jacksonville University  Board of Editors (Economics):  <31> Market-Dependent Domestic Production Set Trisha L. Bezmen, Francis Marion  Henrik Egbert & Nadeem Naqvi Steven Caudill, Auburn University  Joy Clark, Auburn University at Montgomery  <41> Swansonomics: Using “Parks and Recreation” to Teach David Colander, Middlebury College  Economics Stephen Conroy, University of San Diego  L. Brooke Conaway and Christopher Clark Mike Daniels, Columbia State University  Paul Grimes, Mississippi State University  <69> Teaching the Law of Supply Using Karaoke John Marcis, Coastal Carolina University  Wayne Geerling and G. Dirk Mateer Kim Marie McGoldrick, University of Richmond  Franklin Mixon, Jr., Columbus State University  <79> Using Bloomberg Real-time Data and Analytics to Teach Usha Nair‐Reichert, Georgia Tech  Economics and Finance Inder Nijhawan, Fayetteville State University  Hossein S. Kazemi Carol Dole, Jacksonville University  James Payne, University of South Florida  < 94> Financial Performance Measures: A Review and Synthesis Polytechnic  H. Kent Baker and Prakash Deo Christopher Coombs, Louisiana State University ‐  Shreveport  <110> Gains from Specialization and Trade Revisited: “How” is Jason Beck, Armstrong Atlantic State University  Done and “Why” Must be Explained * Board of Editors (Finance):  X. Henry Wang, Bill Z. Yang and Binglin Li Robert Boylan, Jacksonville University  Kam (Johnny) Chan, Western Kentucky University  <120> Classroom Analysis of How to Value Shares of Public S. J. Chang, Illinois State University  Edward Graham, University of North Carolina at  Corporations: A Pedagogical Endeavor Including Earnings and Wilmington  Dividends Expectations Formation and Hybrid Modeling John Griffin, Old Dominion University  Richard J. Cebula , Luther Lawson , Ira S. Saltz, and John Buck Srinivas Nippani, Texas A&M University ‐  Commerce  Mario Reyes, University of Idaho  William H. Sackley, University of North Carolina at  Wilmington  Barry Wilbratte, University of St. Thomas  Bob Houmes, Jacksonville University  Academy of Economics and Finance Shankar Gargh, Holkar Science College, India  Shelton Weeks, Florida Gulf Coast University  Production Editor:  Doug Berg, Sam Houston State University JOURNAL OF ECONOMICS AND FINANCE EDUCATION • Volume 14 • Number 1 • Summer 2015 Portfolio Performance Evaluation Benchmark: A Note Adam Y.C. Lei and Huihua Li1 ABSTRACT Jones and Swaleheen (2014, this journal) examine the performance of an equity portfolio in a student managed investment fund and document the outperformance of the portfolio relative to the S&P 500 index on an absolute basis. We show that the apparent outperformance of the portfolio is due to using the index without its dividend component. Once we use the S&P 500 total return as the benchmark, the outperformance of the equity portfolio disappears. We explain why the S&P 500 total return should be used in this case, and propose and justify two alternative proxies for the S&P 500 total return. Introduction Jones and Swaleheen (2014, this journal) examine the actual performance of an equity portfolio in a student managed investment fund from January 2005 to April 2013, using the return of the S&P 500 index as the performance benchmark. While they find mixed evidence on the outperformance of the portfolio relative to the benchmark on a risk-adjusted basis (by the Sharpe ratio and the portfolio alpha), they find the outperformance of the portfolio on an absolute basis. Specifically, they report that the portfolio earns the cumulative return of 53.60% with an annualized return standard deviation of 13.41% from January 2005 to April 2013. The S&P 500 index, on the other hand, earns the cumulative return of 31.82% with an annualized return standard deviation of 15.58% over the same period. The higher cumulative return and the lower return standard deviation of the portfolio relative to those of the S&P 500 index, if true, would indeed suggest that the portfolio outperforms its benchmark in absolute terms. In this note we show that the equity portfolio examined in Jones and Swaleheen (2014) does not outperform its benchmark in absolute terms. The detailed information provided in Jones and Swaleheen (2014) allows us to trace the documented outperformance to the use of the performance benchmark without its dividend component. Specifically, although “dividends and interest (on cash in the fund) is included in the returns” for the portfolio in Jones and Swaleheen (2014, p. 60), we verify that the return of the S&P 500 index used as the benchmark is without dividends. Once we use the appropriate S&P 500 total return, which includes the return from dividends, as the performance benchmark, the outperformance of the equity portfolio disappears: The S&P 500 total return index earns the cumulative return of 56.90% with an annualized return standard deviation of 15.57% from January 2005 to April 2013. The equity portfolio therefore has a lower cumulative return and a lower return standard deviation than those of the appropriate benchmark over the entire period. Given the availability of the data on the S&P 500 total return index, we also propose and examine two alternative proxies for the S&P 500 total return: The total return of the SPDR S&P 500 ETF Trust (symbol: SPY) and the total return of the iShares Core S&P 500 ETF (symbol: IVV). Both SPY and IVV attempt to replicate/track the performance of the S&P 500 index and have high levels of liquidity. Our analyses suggest that the total returns of SPY and IVV are reasonable proxies for the S&P 500 total return: Their monthly returns and return variance are statistically identical to those of the S&P 500 total return from 1 Adam Y.C. Lei, CFA, Dillard College of Business Administration, Midwestern State University, 3410 Taft Blvd, Wichita Falls, TX 76308, [email protected]. Huihua Li, CFA, Herberger Business School, St. Cloud State University, 720 4th Ave S, St. Cloud, MN 56301, [email protected]. We thank Bill Yang and an anonymous referee for helpful comments and suggestions. Errors, if any, are our own. 1 JOURNAL OF ECONOMICS AND FINANCE EDUCATION • Volume 14 • Number 1 • Summer 2015 January 2005 to April 2013 and for longer periods. We also show that the effect of ignoring the dividends in the benchmark return lies largely on benchmark return level instead of its return variance. Assuming the return variance as a risk measure, this finding suggests that potential biases of using a benchmark without its dividend component likely come from the underestimation of the benchmark return rather than the under- or overestimation of the benchmark risk. The rest of this note is organized as follows: In the following section we review the relevant standards in portfolio performance reporting and evaluation, and explain why the total return of the performance benchmark should be used. We provide the details on the S&P 500 total return index and introduce the two alternatives next. We then illustrate our data and methods, followed by our empirical results. We conclude in the last section. The appendix provides the institutional details on the SPDR S&P 500 ETF Trust and the iShares Core S&P 500 ETF. Portfolio Performance Reporting and Evaluation Under SEC Rule 205-1 (promulgated under the Investment Advisers Act of 1940), both the investment performance of an investment company and the investment record of an appropriate index of securities prices should be defined on a total return basis, for which the total return includes both capital gains (losses) and cash distributions. In the widely followed Global Investment Performance Standards of the CFA Institute (2010), Provision 2.A.1 on the requirements of calculation methodology specifies that “TOTAL RETURN MUST be used” (capital letters as in the original provision). Provision 5.A.1.e on the requirements of presentation and reporting specifies that “the TOTAL RETURN for the BENCHMARK for each annual period” “MUST be presented in each COMPLIANT PRESENTATION.” In addition, “the BENCHMARK MUST reflect the investment mandate, objective, or strategy of the COMPOSITE.” In this case the composite is defined as “an aggregation of one or more portfolios managed according to a similar investment mandate, objective, or strategy.” Following those standards, the performance evaluation benchmark for a portfolio not only should be appropriate to the specific portfolio (see, e.g., Reily and Brown, 2012, p.992 for the characteristics of an appropriate benchmark), but also should include returns from both capital gains (losses) and cash distributions. In the case of an equity portfolio, the return of the benchmark should include both capital gains (losses) and dividends. The rationale for using the total return of the benchmark is intuitive: As the return of a portfolio being evaluated commonly includes both capital gains (losses) and cash distributions, the benchmark return should also include both components to provide a fair evaluation. In addition, Constable (2011) shows that cumulative returns that reflect only capital gains (losses) but not dividends are often misleading. Zewig (2011), nonetheless, reports several instances in which the S&P 500 index return without dividends is incorrectly used as the performance benchmark in practice. S&P 500 Total Return Index and Alternatives The S&P 500 index (Bloomberg symbol: SPX) is a market value-weighted stock index of 500 large-capitalization firms in the U.S., provided by S&P Dow Jones Indices LLC. As of June 30, 2004, the market capitalization of the constituents ranges from $3.77 billion to $560.34 billion, with a median (mean) of $18.11 ($36.81) billion (S&P Dow Jones Indices LLC, 2014). This index is widely circulated and often cited as a portfolio performance evaluation benchmark because of its large coverage of the total U.S. market value. However, the S&P 500 index is a price index and not a total return index: A return calculated from the change of the index level reflects only the return of the index constituents from capital gains (losses) but not from dividends. The S&P 500 total return index (Bloomberg symbol: SPXT), on the other hand, is the total return version of the S&P 500 index also provided by S&P Dow Jones Indices LLC. A return calculated from the change of the total return index level reflects both the return of the index constituents from capital gains (losses) and the return from dividends. However, unlike the S&P 500 index for which the index level is widely available, the availability of the data on the S&P 500 total return index is more limited. For 2 JOURNAL OF ECONOMICS AND FINANCE EDUCATION • Volume 14 • Number 1 • Summer 2015 instance, although the Center for Research in Security Prices (CRSP) stock database includes the S&P 500 index data, the S&P 500 total return index data from CRSP are available only to its indexes subscribers. Because of the limited availability of the data on the S&P 500 total return index, we propose two alternative proxies for the S&P 500 total return: The total return of the SPDR S&P 500 ETF Trust (symbol: SPY) and the total return of the iShares Core S&P 500 ETF (symbol: IVV).2 SPY starts its trading in 1993 and IVV in 2000. Both SPY and IVV have high levels of liquidity (e.g., average daily dollar trading volume in 2013 more than $19 billion and $703 million, respectively), and both of them attempt to replicate/track the performance of the S&P 500 index, despite using different strategies. We summarize their institutional details in the appendix. The suitability of those alternatives is of particular interest to portfolio managers who use the Bloomberg terminal to evaluate their portfolios: Although the Bloomberg terminal provides the data on the S&P 500 total return index, only the S&P 500 index but not the S&P 500 total return index is available as a performance benchmark in its automated portfolio evaluation functions without an additional subscription charge. Portfolio managers, however, can specify a security as the performance benchmark in those functions (see, e.g., Lei and Li, 2012). Data and Methods We obtain the data on the S&P 500 index and the S&P 500 total return index from the Bloomberg terminal, and the data on the SPDR S&P 500 ETF Trust and the iShares Core S&P 500 ETF from the Center for Research in Security Prices (CRSP) database. As in Jones and Swaleheen (2014), the entire sample period is from January 2005 to April 2013. Each annual period starts in May and ends in April of the following year. Unlike Jones and Swaleheen (2014), however, we define the Sharpe ratio of a performance benchmark as the average monthly excess return of the benchmark relative to the one-month T-bill rate, divided by the monthly return standard deviation of the benchmark. Using the monthly excess return preserves the information available at the monthly frequency, and the return matches the interval for which the return standard deviation is calculated. We obtain the one-month T-bill rates from the Fama/French research factors provided by Kenneth R. French. Results Table 1 shows the returns and return standard deviations (annualized) of the different performance benchmarks from January 2005 to April 2013. We reproduce the returns and return standard deviations of the equity portfolio (i.e., the student managed portfolio, SMP hereafter) and the index examined in Jones and Swaleheen (2014) from their Table 1. From Table 1, it is clear that the index return used as the performance benchmark in Jones and Swaleheen (2014) is the return of the S&P 500 index without dividends, as their reported index returns and return standard deviations match exactly the returns and return standard deviations of SPX. The larger cumulative return (53.60%) and the lower annualized return standard deviation (13.41%) of SMP relative to those of SPX (31.82% and 15.58%, respectively) lead to their conclusion of the SMP outperformance on an absolute basis. On the other hand, once we use the appropriate S&P 500 total return (SPXT) as the performance benchmark, SMP no longer outperforms the benchmark in absolute terms. The cumulative return and the annualized return standard deviation of SPXT from January 2005 to April 2013 is 56.90% and 15.57%, respectively. SMP therefore has a lower cumulative return and a lower return standard deviation than those of the appropriate benchmark over the entire period. We also notice that within the annual periods, SMP has a higher return than the benchmark only when the benchmark performs poorly (column SMP–SPXT shows the return differences between the two). For instance, SMP has a return higher than the S&P 500 total return from 2007 to 2008 (-1.92% vs. -4.68%), from 2008 to 2009 (-28.91% vs. -35.31%), and from 2011 to 2012 (9.30% vs. 4.76%). This finding is consistent with SMP having lower return and lower risk than the benchmark overall. 2 Vangard S&P 500 ETF (symbol: VOO) also attempts to track the performance of the S&P 500 index. Due to its shorter history (i.e., launched on September 7, 2010 and began trading on September 9, 2010), we do not consider it here. 3 JOURNAL OF ECONOMICS AND FINANCE EDUCATION • Volume 14 • Number 1 • Summer 2015 Table 1: Returns and Return Standard Deviations of Different Performance Benchmarks from January 2005 to April 2013 Jones and Swaleheen (2014) Total Returns SMP Year SMP Std SPXT Std SPY Std IVV Std –SPXT 2005-2006 5.80 8.56 15.42 6.20 15.61 6.90 15.51 6.67 -9.62 2006-2007 12.66 7.89 15.24 7.13 14.86 7.19 14.90 7.06 -2.58 2007-2008 -1.92 12.55 -4.68 11.92 -4.96 11.60 -4.81 11.97 2.76 2008-2009 -28.91 23.07 -35.31 27.93 -35.10 27.85 -35.14 27.74 6.40 2009-2010 35.47 10.99 38.84 11.65 38.63 11.86 38.68 11.83 -3.37 2010-2011 15.35 13.12 17.22 18.09 17.12 17.98 17.12 18.06 -1.87 2011-2012 9.30 13.74 4.76 16.67 4.67 16.67 4.58 16.41 4.54 2012-2013 11.03 9.94 16.89 10.23 16.70 10.21 16.85 10.25 -5.86 Cumulative 53.60 13.41 56.90 15.57 56.18 15.54 56.50 15.52 -3.30 Sharpe 0.092 0.091 0.091 Jones and Swaleheen (2014) Returns without Dividends Year Index Std SPX Std SPY.X Std IVV.X Std 2005-2006 13.29 6.17 13.29 6.17 13.58 7.13 13.74 6.86 2006-2007 13.11 7.22 13.11 7.22 12.79 7.18 12.83 7.04 2007-2008 -6.53 11.92 -6.53 11.92 -6.76 11.54 -6.65 11.89 2008-2009 -37.01 27.92 -37.01 27.92 -36.77 27.62 -36.74 27.52 2009-2010 35.96 11.53 35.96 11.53 35.91 11.92 35.98 11.86 2010-2011 14.91 18.16 14.91 18.16 14.83 17.80 14.84 17.87 2011-2012 2.52 16.69 2.52 16.69 2.52 16.91 2.44 16.64 2012-2013 14.28 10.31 14.28 10.31 14.16 9.94 14.38 9.99 Cumulative 31.82 15.58 31.82 15.58 32.11 15.51 32.60 15.48 Sharpe 0.053 0.053 0.054 Notes: Both returns and return standard deviations (annualized) are shown in percentage (%). Column SMP for the returns of the student managed portfolio, column Index for the returns of the S&P 500 index, and the associated return standard deviations are reproduced from Jones and Swaleheen (2014) Table 1. Table 1 also shows that using the total return of the SPDR S&P 500 ETF Trust (SPY) and the total return of the iShares Core S&P 500 ETF (IVV) as alternative proxies of the S&P 500 total return produces similar results. In contrast, the results from using the return of the SPDR S&P 500 ETF Trust without dividends (SPY.X) and the return of the iShares Core S&P 500 ETF without dividends (IVV.X) are similar to the results using the return of SPX. Over the entire period from January 2005 to April 2013, the S&P 500 total return is 25.08% higher than the S&P 500 return without dividends, and the S&P 500 total return has a higher Sharpe ratio of 0.092 than the 0.053 of the S&P 500 return without dividends. The annualized return standard deviations, nonetheless, seem to be similar regardless of whether the return is measured by total return or return without dividends. For instance, the annualized return standard deviation from January 2005 to April 2013 is 15.57% for SPXT and 15.58% for SPX. We formally test whether the total return and the return without dividends are same at the monthly interval, and whether they have the same variance next. Table 2 shows the p-values on testing the monthly return differences between the different performance benchmarks from January 2005 to April 2013. The results suggest that the S&P 500 total return (SPXT), the total return of the SPDR S&P 500 ETF Trust (SPY), and the total return of the iShares Core S&P 500 ETF (IVV) are statistically the same over the entire period. For instance, the p-value for rejecting the null hypothesis that the monthly return difference between SPXT and SPY is zero is 0.801. In addition, the S&P 500 return without dividends (SPX), the return of the SPDR S&P 500 ETF Trust without dividends (SPY.X), and the return of the iShares Core S&P 500 ETF without dividends (IVV.X) are also statistically the same. The difference between the total return and the return without dividends, however, is significantly different from zero in all cases. For instance, the p-value for rejecting the null hypothesis that the monthly return difference between SPXT and SPX is zero is smaller than 0.000. 4 JOURNAL OF ECONOMICS AND FINANCE EDUCATION • Volume 14 • Number 1 • Summer 2015 Table 2: P-values on Testing the Monthly Return Differences between Different Performance Benchmarks from January 2005 to April 2013 SPXT SPY IVV SPX SPY.X SPY 0.801 IVV 0.859 0.896 SPX 0.000 0.000 0.000 SPY.X 0.000 0.000 0.000 0.976 IVV.X 0.000 0.000 0.000 0.894 0.811 Table 3 shows the p-values on testing the equality of return variances between the different performance benchmarks from January 2005 to April 2013. The results suggest that the S&P 500 total return (SPXT), the total return of the SPDR S&P 500 ETF Trust (SPY), the total return of the iShares Core S&P 500 ETF (IVV), the S&P 500 return without dividends (SPX), the return of the SPDR S&P 500 ETF Trust without dividends (SPY.X), and the return of the iShares Core S&P 500 ETF without dividends (IVV.X) all have the same variance over the entire period. For instance, the p-value for rejecting the null hypothesis that SPXT and SPY (SPXT and SPX) have the same return variance is 0.989 (0.991). Assuming the return variance as a risk measure, the results from Table 2 and Table 3 collectively suggest that potential biases of using a benchmark without its dividend component likely come from the underestimation of the benchmark return rather than the under- or overestimation of the benchmark risk.3 Table 3: P-values on Testing the Equality of Return Variances between Different Performance Benchmarks from January 2005 to April 2013 SPXT SPY IVV SPX SPY.X SPY 0.989 IVV 0.976 0.986 SPX 0.991 0.981 0.970 SPY.X 0.973 0.984 0.997 0.964 IVV.X 0.956 0.967 0.981 0.948 0.984 Conclusion We reexamine the outperformance of an equity portfolio documented in Jones and Swaleheen (2014) relative to the S&P 500 index on an absolute basis. We show that the apparent outperformance of the portfolio is due to using the S&P 500 return without its dividend component as the performance benchmark. Once we use the S&P 500 total return, the outperformance of the equity portfolio disappears. We explain why the S&P 500 total return should be used in this case and propose two alternative proxies for the S&P 500 total return: The total return of the SPDR S&P 500 ETF Trust (symbol: SPY) and the total return of the iShares Core S&P 500 ETF (symbol: IVV). We show that the total returns of those two ETFs are reasonable proxies for the S&P 500 total return. We also show that ignoring the dividends in the benchmark return affects largely the benchmark return level instead of its return variance. Our results collectively highlight the importance of using the appropriate benchmark return on portfolio performance evaluation. References BlackRock, Inc. 2013. 2013 Propectus iShares Core S&P 500 ETF. Jersey City, NJ. CFA Institute. 2010. Global Investment Performance Standards. Charlottesville, VA. 3 Our results are statistically the same using the monthly (total) returns of the SPDR S&P 500 ETF Trust since its inception (i.e., from February 1991) and the monthly (total) returns of the iShares Core S&P 500 ETF since its inception (i.e., from June 2000) to December 2013. 5 JOURNAL OF ECONOMICS AND FINANCE EDUCATION • Volume 14 • Number 1 • Summer 2015 Constable, Simon. 2011. “Price Charts Can Mislead.” Wall Street Journal, December 5, available at http://online.wsj.com/news/articles/SB10001424052970204443404577054332688164006. Johnston, Michael. 2012. Closer Look At S&P 500 ETF Options. Available at http://etfdb.com/equity-etfs/closer-look-at-sp-500-options. Jones, Travis L., and Mushfiq Swaleheen. 2014. “Student Managed Investment Fund Performance: A Look at Equity Portfolio Data.” Journal of Economics and Financial Education 13: 58-63. Lei, Adam Y.C., and Huihua Li. 2012. “Using Bloomberg Terminals in a Security Analysis and Portfolio Management Course.” Journal of Economics and Finance Education 11: 72-92. NASDAQ Stock Market. 2014. Exchange Traded Funds by Net Assets. Available at http://www.nasdaq.com/investing/etfs/etf-list.aspx?selectedtype=size. PDR Services, LLC. 2014. SPDR S&P 500 ETF Trust Prospectus Dated January 22, 2014. New York, NY. Reilly, Frank K., and Keith C. Brown. 2012. Investment Analysis & Portfolio Management. Mason, OH. S&P Dow Jones Indices LLC. 2014. S&P 500® - S&P Dow Jones Indices. Available at http://us.spindices.com/indices/equity/sp-500. Zweig, Jason. 2011. “Here’s One Way to Beat the Market.” Wall Street Journal, June 4, available at http://online.wsj.com/news/articles/SB10001424052702304563104576363892725584866. APPENDIX: SPDR S&P 500 ETF TRUST (SPY) AND ISHARES CORE S&P 500 ETF (IVV) SPDR S&P 500 ETF Trust (SPY) The SPDR S&P 500 ETF Trust (symbol: SPY) is a unit investment trust that “seeks to provide investment results that, before expenses, correspond generally to the price and yield performance of the S&P 500 Index” (PDR Services, LLC, 2014, p. 1). Commonly referred to as the first ETF in the U.S., this trust was launched by State Street Global Advisors on January 22, 1993 and began trading on January 29, 1993. As a unit investment trust (and unlike other more recent ETFs), this trust has a stated termination date of January 22, 2118. More importantly, this trust has a high level of liquidity that renders it a viable investment alternative and performance benchmark: Its average daily (dollar) trading volume is 121.54 million shares ($19.88 billion) in 2013. As of June 30, 2014, this trust has the net assets of $168.46 billion and a net expense ratio of 0.0945%. iShares Core S&P 500 ETF (IVV) The iShares Core S&P 500 ETF (symbol: IVV) “seeks to track the investment results of the S&P 500 (the underlying index)” (BlackRock, Inc., 2013, p. S-2). This ETF was launched by Barclays Global Investors on May 15, 2000 and began trading on May 19, 2000. BlackRock, Inc. later acquired Barclays Global Investors and the associated iShares ETFs in December 2009. This EFT also has a relatively high level of liquidity: Its average daily (dollar) trading volume is 4.25 million shares ($703.99 million) in 2013. As of June 30, 2014, this ETF has the net assets of $57.76 billion and an expense ratio of 0.07%. In July 2014, it is the second largest ETF in the U.S., preceded only by the SPDR S&P 500 ETF Trust (NASDAQ Stock Market, 2014). 6 JOURNAL OF ECONOMICS AND FINANCE EDUCATION • Volume 14 • Number 1 • Summer 2015 Further Differences between SPY and IVV Although both the SPDR S&P 500 ETF Trust and the iShares Core S&P 500 ETF are listed on NYSE Arca and attempt to replicate/track the performance of the S&P 500 index, there are still fundamental differences between the two that could affect their returns. First, the SPDR S&P 500 ETF Trust effectively uses a full replication strategy in order to replicate the performance of the S&P 500 index.4 The iShares Core S&P 500 ETF, on the other hand, uses a representative sampling strategy.5 Second, while the SPDR S&P 500 ETF Trust does not hold or trade futures or swaps, the iShares Core S&P 500 ETF could invest in futures and swaps. In addition, the iShares Core S&P 500 ETF could lend securities up to one-third of its total assets. Finally, dividends received by the SPDR S&P 500 ETF Trust from its securities holdings are held as cash in a non-interest bearing account until paid as quarterly dividends to its shareholders. The iShares Core S&P 500 ETF, in contrast, has no such requirement of holding the received dividends in cash until quarterly payout. Johnston (2012) suggests that holding the received dividends in cash may lower the return of the SPDR S&P 500 ETF Trust in up markets but increase it in down markets, relative to that of the iShares Core S&P 500 ETF. 4 Specifically, “the Trust seeks to achieve its investment objective by holding a portfolio of the common stocks that are included in the Index (the “Portfolio”), with the weight of each stock in the Portfolio substantially corresponding to the weight of such stock in the Index.” “At any time, the Portfolio will consist of as many of the Index Securities as is practicable” (PDR Services, LLC, 2014, p. 2). 5 Specifically, this strategy “involves investing in a representative sample of securities that collectively has an investment profile similar to the Underlying Index. The securities selected are expected to have, in the aggregate, investment characteristics (based on factors such as market capitalization and industry weightings), fundamental characteristics (such as return variability and yield) and liquidity measures similar to those of the Underlying Index. The Fund may or may not hold all of the securities in the Underlying Index” (BlackRock, Inc., 2013, p. S-2). 7 JOURNAL OF ECONOMICS AND FINANCE EDUCATION ∙ Volume 14∙ Number 1 ∙ Summer 2015 Delta Gamma Hedging and the Black-Scholes Partial Differential Equation (PDE) Sudhakar Raju1 ABSTRACT The objective of this paper is to examine the notion of delta- gamma hedging using simple stylized examples. Even though the delta-gamma hedging concept is among the most challenging concepts in derivatives theory, standard textbook exposition of delta-gamma hedging usually does not proceed beyond a perfunctory mathematical presentation. Issues such as contrasting call delta hedging with put delta hedging, gamma properties of call versus put delta hedges, etc., are usually not treated in sufficient detail. This paper examines these issues and then places them within the context of a fundamental result in derivatives theory - the Black-Scholes partial differential equation. Many of these concepts are presented using Excel and a simple diagrammatic framework that reinforces the underlying mathematical intuition Introduction The notion of delta hedging is a fundamental idea in derivatives portfolio management. The simplest notion of delta hedging refers to a strategy whereby the risk of a long or short stock position is offset by taking an offsetting option position in the underlying stock. The nature and extent of the option position is dictated by the underlying sensitivity of the option’s value to a movement in the underlying stock price (i.e. option delta). Since the delta of an option is a local first order measure, delta hedging protects portfolios only against small movements in the underlying stock price. For larger movements in the underlying price, effective risk management requires the use of both first order and second order hedging or delta-gamma hedging. In some cases, a third order approximation (delta-gamma-speed hedging) may also be required. The objective of this paper is to examine the notion of delta-gamma hedging using simple stylized examples and to illustrate these concepts using Excel. Even though the delta-gamma hedging concept is among the most challenging concepts in derivatives portfolio management, standard textbook exposition of delta-gamma hedging usually does not proceed beyond a perfunctory mathematical presentation of delta hedging with calls. See Chance and Brooks (2010), Hull (2008), Kolb and Overdahl (2007), Chance (2003), Jarrow and Turnbull (2000). 1 Professor of Finance, Rockhurst University, Kansas City, MO 64110, [email protected] 8