City Research Online City, University of London Institutional Repository Citation: Hayley, S. (2015). Cognitive error in the measurement of investment returns. (Unpublished Doctoral thesis, City University London) This is the accepted version of the paper. This version of the publication may differ from the final published version. Permanent repository link: https://openaccess.city.ac.uk/id/eprint/13172/ Link to published version: Copyright: City Research Online aims to make research outputs of City, University of London available to a wider audience. Copyright and Moral Rights remain with the author(s) and/or copyright holders. URLs from City Research Online may be freely distributed and linked to. Reuse: Copies of full items can be used for personal research or study, educational, or not-for-profit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way. City Research Online: http://openaccess.city.ac.uk/ [email protected] COGNITIVE ERROR IN THE MEASUREMENT OF INVESTMENT RETURNS Simon Hayley Thesis submitted for the award of PhD in Finance, Cass Business School, City University London, comprising research conducted in the Faculty of Finance, Cass Business School. April 2015 1 Table of Contents List of Tables and Figures………………………………………………………...page 3 Abstract…………………………………………………………………………….page 6 Summary and Motivation…………………………………………………………page 7 Chapter 1: Literature Review…………………………………………………...page 13 Chapter 2: Dollar Cost Averaging - The Role of Cognitive Error…………….page 44 Chapter 3: Dynamic Strategy Bias of IRR and Modified IRR – The Case of Value Averaging……………………….. page 72 Chapter 4: Measuring Investors’ Historical Returns: Hindsight Bias In Dollar- Weighted Returns ……………………………………………………….……..page 105 Chapter 5: Diversification Returns, Rebalancing Returns and Volatility Pumping……………………………..page 142 Conclusion……………………………………………………………………….page 169 2 List of Tables and Figures Chapter 2: Dollar Cost Averaging: the Role of Cognitive Error Table 2.1. Illustrative Comparison of Strategies as Share Prices Fall Table 2.2. Illustrative Comparison of Strategies as Share Prices Rise Table 2.3. Quantifying the Inefficiency of DCA (% of Initial Capital) Figure 2.1. Simple Model of DCA Strategy. Figure 2.2. Optimized Strategy Giving Identical Outturns to DCA Chapter 3: Dynamic Strategy Bias of IRR and Modified IRR – The Case of Value Averaging Table 3.1. Illustrative Comparison of VA and DCA – Declining Prices Table 3.2. Illustrative Comparison of VA and DCA – Rising Prices Table 3.3. Simulation Results: Performance Differentials Table 3.4. Measuring the Dynamic Efficiency of Value Averaging Figure 3.1. Simple Model of VA Strategy Figure 3.2. Optimized Strategy Which Generates Identical Outturns to VA Chapter 4: Measuring Investors’ Historical Returns: Hindsight Bias In Dollar-Weighted Returns Table 4.1. Investor Timing Effects Identified by Previous Studies Table 4.2. IRRs of Illustrative Two-Round Game Table 4.3. Illustrative Effects of Net Distributions on Return Weights Table 4.4. Impact of Distributions on the DW Return (NYSE/AMEX) Table 4.5. Decomposition of Timing and Hindsight Effects (NYSE/AMEX) Table 4.6. Decomposition on Alternative Return Assumptions Table 4.7. Decomposition with Alternative Corrections for Trend in Returns Table 4.8. Decomposition of Timing and Hindsight Effects (NASDAQ) 3 Table 4.9. NASDAQ Return Decomposition: Alternative Return Assumptions Figure 4.1. Illustrative Game Showing the Quit-Whilst-Ahead Effect Figure 4.2. Returns to Date and Net Distributions (NYSE/AMEX stocks) Figure 4.3. Timing and Hindsight Effects in DW Returns (NYSE/AMEX Stocks) Figure 4.4. Long-Term Equity Returns Figure 4.5. Distributions and Annual Returns (NASDAQ Stocks) Figure 4.6. Impact on DW Returns of Correlation between Distributions and Returns Chapter 5: Diversification Returns, Rebalancing Returns and Volatility Pumping Table 5.1. Expected AMs and GMs Derived Using E[GM]≈E[AM] - σ2/2 Figure 5.1. Rebalanced and Unrebalanced Portfolios – Varying Number of Assets Figure 5.2. GM vs. Variance for Rebalanced and Unrebalanced Portfolios (Varying Number of Assets) Figure 5.3. Rebalanced and Unrebalanced Portfolios – Varying Initial Portfolio Weights Figure 5.4. GM vs. Variance for Rebalanced and Unrebalanced Portfolios (Varying Initial Portfolio Weights) Figure 5.5. Volatility Pumping – Single Risky Asset Figure 5.6. Probability Density of the Terminal Wealth of Rebalanced and Unrebalanced Portfolios after 100 Years Figure 5.7. Proportion of Outcomes where Pr>Pu 4 Declaration I hereby grant powers of discretion to the University Librarian to allow this thesis to be copied in whole or in part without further reference to me. This permission covers only single copies made for study purposes, subject to normal conditions of acknowledgement. 5 Abstract This thesis identifies and quantifies the impact of cognitive errors in certain aspects of investor decision-making. One error is that investors are unaware that the Internal Rate of Return (IRR) is a biased indicator of expected terminal wealth for any dynamic strategy where the amount invested is systematically related to the returns made to date. This error leads investors to use Value Averaging (VA). This thesis demonstrates that this is an inefficient strategy, since alternative strategies can generate identical outturns with lower initial capital. Investors also wrongly assume that the lower average purchase cost which is achieved by Dollar Cost Averaging (DCA) results in higher expected returns. DCA is a similarly inefficient strategy. Investors also adopt strategies such as Volatility Pumping, which appears to benefit from high asset volatility and large rebalancing trades. This thesis demonstrates that any increase in the expected geometric mean associated with rebalancing is likely to be due to reduced volatility drag, and that simpler strategies involving lower transactions costs are likely to be more profitable. Academic papers in highly-ranked journals similarly misinterpret the reduction in volatility drag achieved by rebalanced portfolios, mistakenly claiming that it results from the rebalancing trades “buying low and selling high”. The previously unidentified bias in the IRR has also affected an increasing number of academic studies, leading to misleadingly low estimates of the equity risk premium and exaggerated estimates of the losses resulting from bad investment timing. This thesis also derives a method for decomposing the differential between the GM return and the IRR into (i) the effects of this retrospective bias, and (ii) genuine effects of investor timing. Using this method I find that the low IRR on US equities is almost entirely due to this bias, and so should not lead us to revise down our estimates of the equity risk premium. This method has wider applications in fields where IRRs are used (e.g. mutual fund performance and project evaluation). In identifying these errors this thesis makes a contribution: (i) to the academic literature by correcting previous misleading results and improving research methods; (ii) to investment practitioners by identifying avoidable errors in investor decision-making. It also makes a contribution to the field of behavioural finance by altering the range of investor behaviour which should be seen as resulting from cognitive error rather than the pursuit of different objectives. 6 Chapter 1 Summary and Motivation This chapter summarises the key findings of this thesis, and sets out the relationship between these results and existing financial research within this field. This thesis identifies and quantifies the impact of cognitive errors in certain aspects of investor decision-making. These errors relate primarily to the behaviour of retail investors, although as will be discussed below, institutional investors and academics are not immune from these errors. They are: (1) Investors wrongly assume that the lower average purchase cost which is achieved by Dollar Cost Averaging (DCA) results in higher expected returns. Far from improving returns, DCA is a demonstrably inefficient strategy (chapter 2). (2) Investors are unaware that the Internal Rate of Return (IRR) is a biased indicator of expected terminal wealth for any dynamic strategy where the amount invested is systematically related to the returns made to date. Specifically, this error leads investors to follow Value Averaging (VA), a strategy which generates attractive IRRs. Chapter 3 demonstrates that these high IRRs are misleading, and that VA is an inefficient strategy, since alternative strategies can generate identical outturns with lower initial capital. (3) Investors are misled by the maths of rebalanced portfolios, leading them to adopt strategies such as Volatility Pumping which aim to increase the scale of rebalancing trades. I demonstrate instead that any increase in the expected geometric mean associated with rebalancing is likely to be due to reduced 7 volatility drag, and simpler strategies involving lower transactions costs are likely to be more profitable. This is covered in chapter 5. Academic studies in the highest ranked journals have also been affected by errors (2) and (3). The bias in the IRR has not previously been identified, so the difference between historical geometric mean (GM) and IRR figures has wrongly been attributed to bad timing by investors. This has led