--- title: Elm Wealth Research | Featured Insights (2) description: Featured Insights | Regular Elm Posts (2) --- [![Elm Partners](https://insights.elmwealth.com/hs-fs/hubfs/elm%20logo%20mini.png?width=100&height=100&name=elm%20logo%20mini.png "Elm Partners")](http://elmwealth.com/) Open main menu Close main menu # Elm Wealth Research Select Category Featured Insights How Elm Works In the News Investing 101 Risk and Return Tax Matters Uncategorized Posts about: ## Featured Insights (2) ![](https://insights.elmwealth.com/hubfs/Imported_Blog_Media/fabric-felicity-840x420.png) [Featured Insights](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights) ### [Measuring the Fabric of Felicity](https://insights.elmwealth.com/elm-wealth-research/measuring-the-fabric-of-felicity) Sep 17, 2018, 12:00:00 AM September 17, 2018 Featured Insights ## Measuring the Fabric of Felicity *By Victor Haghani and James White* [1](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-1-2223) We steer our financial course through life choosing how much to spend and how to invest what’s left, periodically updating our choices as circumstances evolve. This is the essence of financial planning: specifying in advance a desired spending and investment policy conditional on relevant aspects of our life, varying investment opportunities, and our preferences for the benefits derived from our wealth. It’s a pretty simple problem to put into words but finding an optimal solution has occupied some very bright minds for the past 60 years, since Harry Markowitz got the ball rolling with his brilliant but highly-stylized one-period, static portfolio selection paradigm.[2](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-2-2223) Since then, researchers have made tremendous progress in both specifying and solving increasingly realistic and relevant formulations of this problem, with the goal of maximizing expected lifetime utility at the core of most formulations. In fact, academic research in this area has been so rich that it’s given birth to an entire academic discipline with dedicated university courses and textbooks.[3](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-3-2223) And yet, even though the study of this problem has delivered novel and valuable insights, we haven’t seen meaningful adoption by the financial planning or wealth management industries. The issue may in part involve the “utility” formulation so common in academia. Utility is a measure of the happiness, or felicity,[4](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-4-2223) we derive from using our wealth, and researchers generally presume that an investor knows her own “utility function,” which is to say the relationship between her wealth and utility. Unfortunately, sparingly little has been written about how an individual should calibrate her personal utility function. Financial planners may also worry that these relatively stylized academic models of utility don’t adequately represent investor preferences in the real world and fear their clients will be confused by a process grounded in an unfamiliar abstract formalism. Are these reservations justified? We decided to create a short survey to address this question by testing whether a group of financially-sophisticated investors could comfortably communicate their preferences in a way which we could then translate into utility terms, and whether these preferences are reasonably consistent with core utility concepts. If the responses were affirmative, then we’d have reason to believe that putting the academic research findings into practice could have a big payoff. So, over the past six weeks, we gave the survey to a group of 31 friends and former colleagues from the finance industry.[5](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-5-2223) We make no claims that results from this selective sample are automatically applicable to broader groups, but we thought it would be a good starting point to see if there’s at least a kernel of people who would be comfortable thinking within this framework.[6](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-6-2223) Below is the survey.[7](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-7-2223) As we’ve done before in our research, we framed the survey questions using coin-flipping thought experiments.[8](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-8-2223) A utility-based framework is built into the structure of the questions, but you’ll see we don’t actually use the word “utility” anywhere in the questions, as we hoped to bring as few preconceptions into the survey as possible. We’d value getting more responses, so please [take the survey here](https://goo.gl/forms/vZ1MlmxUaQ0fG49v1) if you have a few minutes to give it some thought before reading on. 1. Imagine you are presented with a one-time investment opportunity where the outcome depends on whether a fair coin flip comes up heads or tails. The coin flip is uncorrelated with your existing portfolio of investments and consumption and bequest plans. If it comes up tails, you lose 10% of your wealth. Which of the following amounts of upside would make you feel most ambivalent between accepting or declining the coin flip? If you can’t decide between two options, tick both. In case it helps, the Sharpe Ratios of the different options are 0.05, 0.11, 0.18, 0.26 and 0.34 respectively. 1. 11% 2. 12.5% 3. 14.5% 4. 17% 5. 20.5% 2. Same question as above, but now the downside if the coin comes up tails is you lose 20% of your wealth. Which upside would make you feel most ambivalent? Again, if you are finding two answers close, tick both. The Sharpe Ratios are 0.11, 0.25, 0.44, and 0.80 respectively. 1. 25% 2. 33% 3. 50% 4. 175% 5. I wouldn’t risk losing 20% of my wealth for any of these upsides. 3. Still flipping a fair coin. Now imagine that if it comes up heads, your wealth will increase 5-fold. What’s the most downside you’d accept to take that flip? Again, if your answer is between two options, please tick both. 1. 80% 2. 50% 3. 30% 4. 20% 5. 15% Before proceeding, we need to explain a little about how we picked the specific values in answers (a)-(e). The family of utility functions most commonly used in the academic literature is called Constant Relative Risk Aversion (CRRA) Utility.[9](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-9-2223) It’s hard to beat for simplicity, requiring just one parameter: the coefficient of risk aversion. One of the fundamental insights of the utility framework is that an individual’s utility curve defines her risk preferences, and in particular her level of risk aversion. Risk aversion comes from the asymmetry in utility between positive and negative changes in wealth. The parameter which describes the curvature of the utility function (the asymmetry) therefore also describes the level of risk aversion.[10](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-10-2223) Answers (a)-(e) each reflect one particular level (1, 2, 3, 4 and 5) of risk aversion. So, for example, an individual who displays CRRA Utility with a risk aversion level of 3 would choose answer (c) in all 3 questions. Two “ideal” individuals whose preferences are well represented by CRRA utility and who have the same coefficient of risk aversion would answer our survey in the same way, regardless of their wealth.[11](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-11-2223) And now to the survey results. In sum, we were encouraged by the responses. You can find the full response data [here](https://docs.google.com/spreadsheets/d/1dc_a5GwHaPhxHrvrTNe2AFiuKu76E8z1o5Mj_9mNtRw) and summarized in the chart below. Here’s what we take away from the survey data and follow-on conversations: - The style of calibration questions we used appears reasonable and comfortable to answer for our particular group, with not one respondent needing clarification before answering the questions. - The risk preferences of our respondents seem broadly consistent with CRRA utility. Respondents tended to answer the three questions in a consistent fashion. Most selected nearby letters or the same group of letters across all three questions. Had answers been selected at random, the average expected variation per respondent would have been 50% greater across the three questions. - The level of risk aversion evidenced by our respondents is strongly clustered around choice (c), equivalent to a risk aversion level of 3, as indicated in the bar chart above. We think Q2 is the mostly broadly relevant of the three questions, and the average value of answers to Q2 was 3.2, with not one person choosing (a) and only three choosing (e).[12](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-12-2223) The average value for Q3 was 3.0.[13](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-13-2223) We’d like to give a flavor for what’s implied by a risk-aversion level of 3. In line with the survey, an “ideal” investor with CRRA risk-aversion of 3 would never risk more than 30% of her wealth on an even-odds gamble regardless of the upside, and would need at least 50% of upside to risk 20% of her wealth. In an investing context under the most basic set of assumptions, such an investor would optimally invest 50% of her wealth if her only investment opportunity were a risky asset with an expected excess return of 4% and standard deviation of 16%.[14](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-14-2223) We look forward to writing a lot more about this topic in coming notes, using increasingly realistic sets of assumptions. Before concluding, we feel a few caveats are in order. The thinking of our respondent group may not be representative of the broader population. Seven have PhDs in finance, and four have published relevant research papers. Caution is also called for in light of the extensive body of behavioral finance research, much of it in the area of Prospect Theory, which holds that people systematically make choices inconsistent with the classical utility functions that we used in our survey.[15](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-15-2223) Furthermore, psychological studies suggest that most of us are not very good at knowing how we’ll feel in different future scenarios. However, there’s no escaping the type of questions posed in our survey if we want to make sound long-term financial decisions. An imperfect calibration of our preferences is likely better than no calibration at all. An approach that ignores or denies the premise of decreasing marginal utility of wealth will likely find itself trying to maximize expected future wealth, which can produce dangerous and nonsensical policy recommendations.[16](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-16-2223) We’ve been encouraged by the results of this survey and are looking forward to taking the next steps towards applying the valuable and extensive body of academic research to the real world financial planning and investing challenges faced by us all. --- 1. Victor is the Founder and CIO of Elm Partners, and James is Elm’s CEO. **Past returns are not indicative of future performance.** This not is not an offer or solicitation to invest.   Thank you for the helpful comments and guidance of Chi-fu Huang, Jeff Rosenbluth, Andy Morton, Vlad Ragulin, Ben Miller, Ayman Hindy, Larry Hilibrand, Aron Landy and all our friends who generously shared their time with us in taking our survey. 2. Harry Markowitz, [Portfolio Selection](https://www.math.ust.hk/~maykwok/courses/ma362/07F/markowitz_JF.pdf). The Journal of Finance. pp 77–91 (Mar 1952). 3. The discipline is often called Lifetime Portfolio Choice and Consumption. Many consider the seminal paper Robert Merton’s [Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case](http://www.people.hbs.edu/rmerton/Lifetime%20Portfolio%20Selection.pdf), The Review of Economics and Statistics, Vol. 51, No. 3 (Aug. 1969). For a brief overview, see John Campbell, [Strategic Asset Allocation: Portfolio Choice for Long-Term Investors](http://www.nber.org/reporter/fall00/campbell.html), NBER, (2000).   For an in depth treatment, these books: John Campbell and Luis Viceira, [Strategic Asset Allocation](https://faculty.fuqua.duke.edu/~charvey/Teaching/BA453_2006/Campbell_Viceira.pdf) (2002), John Campbell, Financial Decisions and Markets (2017), Robert C. Merton, Continuous Time Finance (1992), John Cochrane, Asset Pricing (2005). 4. felicity: the quality or state of being happy. The “fabric of felicity” is attributed to utilitarian Jeremy Bentham: *“Nature has placed mankind under the governance of two sovereign masters, pain and pleasure. It is for them alone to point out what we ought to do, as well as to determine what we shall do…The principle of utility recognises this subjection, and assumes it for the foundation of that system, the object of which is to rear the **fabric of felicity** by the hands of reason and of law.”* Introduction to the Principles of Morals and Legislation, Chapter 1, (1789). 5. Our 31 respondents were a pretty special group, intentionally chosen to be highly financially sophisticated. The majority are investors with Elm Partners. All are reasonably affluent too, stating that their current wealth was significantly in excess of what would be needed to support their basic consumption needs. 6. Although our sample represents a small niche of investors as a proportion of all investors, it’s large in absolute terms, and influential. 7. Another approach would have been to ask people about their investment portfolios and their assumptions of expected return and risk of their portfolios in order to back out an implied degree of risk aversion from their portfolio choices. We felt that would be considerably more complex and blurred by other considerations that respondents might bring to their decision making, such as the degree of mean-reversion in equity markets, making it difficult to isolate the characteristics of their utility functions. 8. We continue to like posing the questions in terms of ambivalence or indifference between accepting or declining a gamble, as we discussed in this note: [How Much of a Good Thing is Best for You?](https://elmwealth.com/how-much-of-a-good-thing-is-best-for-you/) 9. *U(W) = (1 – W1-ƞ)/(ƞ – 1)* for *ƞ* ≠ 1, and *U(W) = ln(W)* for *ƞ = 1* , where *ƞ* is known as the coefficient of risk aversion. See this [Wiki article](https://en.wikipedia.org/wiki/Isoelastic_utility).   In addition to the simplicity, plausibility and tractability of CRRA Utility functions, a line of reasoning known as Portfolio Turnpike theory states that for long-term horizons, investor preferences are likely to converge to CRRA utility. See Mossin, [Optimal Multi-period Policies](http://web.math.ku.dk/~rolf/Mossin) (1968) or Cox and Huang, [A Continuous-Time Portfolio Turnpike Theorem](https://www.sciencedirect.com/science/article/pii/016518899290046H?via%3Dihub) (1992). 10. The degree of curvature of one’s utility function is also at the center of the spend-now versus save-and-spend-more-later decision. The greater the curvature, the greater the inducement required to defer consumption. We expect to delve deeper into the mechanics of this in upcoming notes. 11. Assuming their “subsistence” spending needs are a small fraction of total wealth. 12. A choice of (a) is equivalent to “log-utility”, the most commonly used level of risk-aversion amongst academics, and of interest to gamblers through the original statement of the Kelly Criterion. 13. We also expected to find that respondents would evidence a higher level of risk aversion for smaller gambles than larger ones, and indeed we found this to be the case. The average coefficient of risk aversion to Q1 (10% downside) was 4.0, while for Q2 (20% downside) it was 3.2. It is worth noting that for very small gambles, investors should logically be willing to accept a very small edge, as the price of risk should be proportional to variance, or bet size squared.   So, for an investor with coefficient of risk aversion of 3, say with $1mm of net worth, she should be indifferent to a fair coin flip of lose $10,000 versus gain $10,300, but we’ve observed that in practice most people demand a much bigger edge than that for small gambles. We are not overly disturbed by this tendency, as what really matters for the Lifetime Portfolio Choice and Consumption problem we are concerned with is risk aversion to more substantial risks. 14. As per the Merton rule: *µ / (ƞ σ2)* , where *µ* is the expected excess return over the risk-free rate, *σ* is the standard deviation of returns, and *ƞ* is the coefficient of risk aversion. See this [Wiki article](https://en.wikipedia.org/wiki/Merton%27s_portfolio_problem), or Merton’s original 1969 paper, [Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case](http://www.people.hbs.edu/rmerton/Lifetime%20Portfolio%20Selection.pdf) (page 253, equation 29′). 15. Behavioral finance mostly focuses on the question of what people actually do, not on questions of what people should do. The idea that behavioral heuristics, like those arising from classical utility functions, could help reduce inherent bias and produce more optimal outcomes is not inconsistent with behavioral finance generally. Other criticisms of the classical utility framework argue that people systematically display internally inconsistent preferences, for example as suggested by the [Allais paradox](https://en.wikipedia.org/wiki/Allais_paradox). 16. Even with technical constraints such as an arbitrary threshold on the probability of going broke. [James White](https://insights.elmwealth.com/elm-wealth-research/author/james-white) [Read More](https://insights.elmwealth.com/elm-wealth-research/measuring-the-fabric-of-felicity) ![](https://insights.elmwealth.com/hubfs/Imported_Blog_Media/hyundai-840x420.png) [Featured Insights](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights) ### [Is Vanguard More Rolls Royce, or Hyundai?](https://insights.elmwealth.com/elm-wealth-research/is-vanguard-more-rolls-royce-or-hyundai) Jul 17, 2018, 12:00:00 AM July 17, 2018 Featured Insights ## Is Vanguard More Rolls Royce, or Hyundai? *By Victor Haghani and James White* [1](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-1-2160) *“What we obtain too cheap, we esteem too lightly.”*   – Thomas Paine, 1776 A physician thinking about investing with Elm asked us: *“If I hurt my knee, I’m going to find the best orthopedic surgeon available and I’ll expect to happily pay top dollar. Why is hiring an investment manager any different?”* Great question. Alas, identifying the “best” investment managers isn’t so easy. Daniel Kahneman discusses this in “Thinking Fast and Slow”, asking the question *“When can you trust an experienced professional who claims to have an intuition?”* His answer is that the skill we are looking for should develop *“through prolonged practice”* in *“an environment that is sufficiently regular to be predictable…*\[and providing\] *immediate and unambiguous feedback.”* Many (if not most) crafts provide an excellent environment for developing unambiguous expertise, and it’s reasonable to expect to pay more for accessing that evident expertise. Not so with investing; it’s hard to imagine a domain more irregular, unpredictable and ambiguous. And not only is it challenging for investment managers to consistently develop such unambiguous skill, but for the consumer, it is very difficult to distinguish true skill from good luck with the amount of historical data typically available to investors.[2](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-2-2160) In our paper [What’s Past is Not Prologue](https://elmwealth.com/whats-past-is-not-prologue/), we suggested that even 20 years of historical performance data isn’t normally enough to distinguish with high confidence between skilled and average mutual fund managers. Investment products aren’t the only ones whose merits are difficult for consumers to assess. However, with most other products, better quality normally goes hand-in-hand with a higher price, and so we can use price as a filter to narrow down the choices when trying to find the best service or product. For example, at Elm we recently upgraded one of our laptops. We needed a higher-performance machine, and it was pretty easy to know how to identify one: we looked for higher-price laptops, and we were quickly able to narrow the options down to the few best. This heuristic doesn’t work with hiring investment managers though – partly because the difficulty of judging skill hinders price efficiency, and partly because the quality of the product and the price paid in fees are not ‘separable.’ When buying a car, the car’s quality and the money you pay for it are separable. A very wealthy person (or an extreme car enthusiast) might pay 10 times the price of a standard car for one which is only 10% better, and still be happy with their decision. Investing is different from most other consumer choices in that what you get and what you pay both come in the form of dollars, so they’re *not separable.* “What you get” is risk-adjusted excess return in dollars, and “what you pay” are the fees in dollars. No matter how wealthy a person is, she will never knowingly pay $2 to get $1 of extra wealth.[3](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-3-2160) The uncertainty surrounding the “what you get” part of investment services can pave the way for us to unknowingly make a wealth-reducing exchange. In choosing a doctor, even if we fail to find the best doctor, an average doctor with the relevant medical qualifications is still pretty likely to do a good job. However, as we know from Sharpe’s “Arithmetic of Active Management,” the zero-sum nature of the investing game means the average investment manager is expected to deliver below average results. Despite all these obstacles, many investors and their advisors still persevere against the odds in the quest of identifying the “best” investment managers. Behavioral economists would explain this by citing our natural tendency to over-extrapolate from small samples of data, combined with our proclivity to be overconfident in our abilities and decision-making. Bill McNabb, former CEO of the Vanguard Group, explains that many of our consumer instincts about price and value lead us astray when it comes to choosing investment products: *“The whole cost argument from an investment perspective is counter-intuitive. If you think about your life in other areas, if you are out buying a car, you can buy a Rolls Royce and pay whatever Rolls Royces are going for today, or you can buy an inexpensive Hyundai. You are going to feel a difference in the car. Now whether it is worth it…only you as a buyer can make that decision. But, you are definitely going to feel the difference in quality. In investing, that equation doesn’t hold. And so, when you think about the average investor who is also a consumer, they are used to – the more I pay the higher the quality, the better the results I get. You come to investing and it is just the opposite.*[4](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-4-2160) *I think that it is really a hard behavior for people to unlearn.”*[5](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-5-2160) But to improve as investors, unlearn we must. --- 1. Victor is the Founder and CIO of Elm Partners, and James is Elm’s CEO. **Past returns are not indicative of future performance.** This not is not an offer or solicitation to invest. 2. This is not true for all track records and all situations, but it is true often enough. 3. Of course, in the context of investing, the equation is rarely so clear: “what you get” is, more formally, expected marginal utility in the context of your whole portfolio, and this is not directly observable. Historical net returns may be informative in this regard, but are still an imperfect proxy. 4. To wit, McNabb cited a recent Morningstar study which found that, “A fund’s annual fee is the most proven predictor of future fund returns.” [http://www.morningstar.co.uk/uk/news/149421/how-fund-fees-are-the-best-predictor-of-returns.aspx](http://www.morningstar.co.uk/uk/news/149421/how-fund-fees-are-the-best-predictor-of-returns.aspx) 5. Watch: [https://www.youtube.com/watch?v=SwkjqGd8NC4](https://www.youtube.com/watch?v=SwkjqGd8NC4), starting at 43:16. [James White](https://insights.elmwealth.com/elm-wealth-research/author/james-white) [Read More](https://insights.elmwealth.com/elm-wealth-research/is-vanguard-more-rolls-royce-or-hyundai) ![](https://insights.elmwealth.com/hubfs/Imported_Blog_Media/034-past-prologue-banner-1024x487.png) [Featured Insights](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights) ### [What’s Past is NOT Prologue](https://insights.elmwealth.com/elm-wealth-research/whats-past-is-not-prologue) Apr 11, 2017, 12:00:00 AM April 11, 2017 Featured Insights ## What’s Past is NOT Prologue *By James White, Jeff Rosenbluth and Victor Haghani* [1](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-1-429) Thank you to the 702 people who read and interacted with our note exploring when past returns are indicative of future returns. We received so much thought-provoking feedback that we decided a follow-up note was in order. We started by asking readers to guess how many flips they’d want to see in order to be able to discern, with 95% confidence, a fair coin from a coin biased 60% to land on heads.[2](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-2-429) Below is a histogram of the guesses. The median guess was 40 flips. While lower than the full-credit answer of 143, it does show that our readers appreciate it takes a really long time to identify an investment with this kind of risk/reward simply by track record. In Appendix I, we include the calculation used to arrive at 143.[3](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-3-429). Our readers are a pretty mathematical bunch, and we’re sure that if they took their time to calculate an answer, rather than giving a quick guess as we requested, most would have come close to the correct answer. But the point of the exercise was to illustrate how when we are thinking fast, we tend to put too much weight on small samples: a full 30% of respondents, the single largest bucket, thought 10 flips or less was sufficient. This built-in tendency to overweight small samples can easily lead us to ignore the dictum that *“past performance is not indicative of future results.”* Our readers generally agreed that a 60/40 coin would represent an attractive investment opportunity. At a rate of one flip per year, this would equate to an investment with a Sharpe Ratio of 0.2, roughly comparable to most broad public markets.[4](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-4-429) In fact, a few suggested that 95% was too high a confidence test given the attractiveness of the opportunity. They remarked that they’d be happy to invest half their money on each coin, and not worry about figuring out which was which. This comment highlights the fact that in the two-coin problem we presented, you’ve got a 50% chance of picking the right coin without learning anything through flipping them. However, if we make the problem more realistic by asking how many flips are needed to discern a 60/40 biased coin from 3 fair coins, the number of flips required jumps to 220 (see Appendix I). So even if we had a lower confidence test but with more potential coins, as would be likely in the real world, it still takes a long time to figure out which is the good coin. Perhaps the most thought-provoking suggestion arising from our original note came from our friend Andy Morton. He proposed a more realistic setup of the problem wherein you can invest in 100 active fund managers, but only 15% are expected to generate a post-fee return of 1% a year in excess of their benchmark, while the other 85% are expected to lose 1% a year vs. their benchmark after expenses. We’ll assume that each has an annual risk vs. the benchmark of 10%, which makes the outperformance of the good funds vs. the bad funds similar in Sharpe Ratio to a 60/40 coin vs. a fair coin. If anything, this may be optimistic given S&P Dow Jones reports that only 1 in 10 active managers outperformed their benchmarks over relatively short horizons, while with our assumptions, just under 50% of funds would outperform their benchmark each year.[5](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-5-429) Let’s explore the ramifications of “chasing” returns with this setup. You start off by putting 1% of your portfolio into each fund, because you don’t yet have data to tell them apart. Your starting expected return is -0.7% p.a. versus the benchmark (*85% \* -1% + 15% \* 1%* ). Each year you move more of your portfolio to the funds that have been doing well, using a Bayesian update of the probability of each fund being one of the good ones, and that helps your expected return improve from the starting point (see Appendix II for more detail). Alas, even by the end of 5 years – a reasonable “lookback” window for real-world fund evaluation – the expected return versus the benchmark on your portfolio will still be -0.66%. Extending out to 10 years doesn’t help much either – you only improve to -0.6% expected return.[6](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-6-429) It’s generally believed that Warren Buffett-like investors are very rare – much rarer than 1 in 1,000 – but regardless let’s say all 15 of our “good” fund managers are like Buffet and produce an excess Sharpe Ratio of 0.45 (roughly Berkshire’s Sharpe Ratio of excess returns versus the S&P 500 since 1980). In this case, after 5 years we’d still only be breaking even across the whole portfolio, and would only have about 50% of our capital allocated to the 15 Warren Buffetts! We can clearly see that in the real world, where Buffetts are much rarer, 5 to 10 years of track record just doesn’t tell us that much when dealing with a portfolio composed mostly of mere mortals. As we discussed in our note from six months ago, [“What I learned from my daughter (about investing)”](https://elmwealth.com/what-i-learned-from-my-daughter-about-investing/), our attempt to build a portfolio of rare investment gems – a clutch of young Buffetts – also has to contend with the drag of the false positive. Even if we can identify a good from a not-so-good investment with 90% accuracy, if the good ones represent just 5% of the possible investments, the best we can expect is a portfolio comprised of 32% good ones and 68% not-so-good ones. The cognitive bias that makes it hard to see this result is called base-rate bias. In the real world, any kind of return-chasing strategy will also face additional headwinds, amongst them the fact that when an investment strategy was truly extraordinary in the past, its very discovery will diminish future performance (or even make future returns negative), as more capital is drawn towards it. This may be the the prime reason why investor returns (i.e. dollar-weighted returns) tend to be significantly below fund returns (i.e. time-weighted returns).[7](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-7-429) This doesn’t happen with coin flips. Another comment worth sharing, from the marketing capo of a well-established hedge fund group, was, *“I don’t know anyone who would look at a fund investment without looking at the past track record.”* We think the reason for this, and perhaps the central problem with investing in active fund managers, is that there is virtually no other way to develop a forward-looking expectation other than to look at the past track record. Unfortunately, as we’ve illustrated with our two-coin example, there is very limited statistical power in the typical historical return dataset which is often limited to 3-5 years due to issues around manager turnover. Perhaps when a fund picker says, *“We only will look at funds that have at least a 3-year track record,”* he is thinking that if he observes the funds on a weekly basis, that gives him a lot more data points to draw upon to reach a good level of confidence. Unfortunately, it doesn’t help because when we observe the managers on a more frequent basis, it’s requiring us to discern a coin that is closer to 50/50, which takes exactly the extra flips you get by sampling more frequently. Bob Merton pointed this out in a 1980 paper titled, “On Estimating the Expected Return on the Market,” *“…nothing is gained in terms of accuracy of the expected return estimate by choosing finer observation intervals for the returns.”* [8](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2#easy-footnote-bottom-8-429) One more remark we think worth sharing was, *“An investment in a fund is always about the manager.”* That at least recognizes the limited value of past returns, especially in isolation and over typical, relatively short horizons. The problem is it’s very hard to separate our qualitative evaluation of a manager’s character from our knowledge of their track record – most managers with an apparently sterling, but often short-term, track record also present as highly confident and competent people, in a way which might not be the case with a different frame. Once you accept the very low discriminating power of past returns in most practical investing contexts, what are you to do? Of course, you’ll want to look at all the information at your disposal regarding any investment, and track record should be part of the due diligence of any investment. If it’s particularly long and distinguished, it might have some impact on your forecast, when used together with other factors. It just shouldn’t be the primary or isolated driver of your return forecast. And if you’re ever unsure whether you’re being unduly influenced by a good track record over a normal (relatively short) time horizon, ask yourself whether you’d still make the investment if the return had been half as bad as it was good. If you answer yes, you’re probably giving about the right weight to track record in your decision process. All this discussion on the low value of historical returns in most investment contexts may leave you feeling either depressed or liberated, depending on your perspective and occupation. If you’re an investor, it’s really not so bad: just stick to investments where you don’t need to rely on the track record to make your decision. That leaves almost all of the direct, cost-efficient, investible universe open to you– bonds, equities, real estate and any strategy where you can produce a reasonable forward-looking return estimate without relying on past returns. --- ### Appendix I: Flipping a biased coin and one or more fair coins We would like to calculate how many flips of 2 coins, one biased and one fair we need to be 95% confident that the coin with more heads is the biased one. Above, we assumed that the biased coin has probability of heads of 0.6, here we will be a bit more general and represent this probability by *p* . Since each coin has a binomial distributions and the coins are assumed independent, the joint probability mass function of the two coins after *n* flips is the product of two binomial probability mass functions. Denote by *Q(n,k,j)* the probability of *k* heads for the biased coin and *j* heads for the fair coin after *n* flips, thus   Q(k,j,n) = (nk)  pk  (1 – p)n – k  (nj)12j12n – j This allows us to calculate the probability that the coin with more heads is biased as   1 2n Σ j \< k k ≤ n ( n k ) ( n j )  pk  (1 – p)n – k Choosing *n* to be 143 and *p = 0.6* , we obtain 95.01%. If we have more than one fair coin, the analysis is very similar. We calculate the joint probability mass function for the biased coin and the fair coins simply by multiplying together the individual probability mass functions. Then we sum over all of the cases where the biased coin has more heads than the maximum heads of any fair coin. All though this is a closed form solution, the summation has way too many terms to handle by hand. Therefore, we enlist the aid of a computer to calculate it. --- ### Appendix II: Bayesian Capital Allocation for 100 Funds In our 100 fund example from above, we used a “return-chasing” capital-allocation rule based on Bayesian learning, and here we explain the logic behind that in more detail. Bayes’ Theorem (sometimes also called Bayes’ Rule) provides a powerful tool for updating our beliefs about the world as new information is received. In the 100 funds example, for each fund we would have a level of belief about whether that fund is in the Good group or the Bad group. Before we have any data, our belief that a given fund is good would be at 15% for each fund. As we have incremental data about fund performance, we can use Bayes Theorem to update our level of belief for each fund, in a way that’s rigorous and consistent with the information-value contained by the new data. Specifically, we update our level of belief by the *relative* likelihood of seeing the new data if our hypothesis is true versus if it’s false. To give a simple example: if we get new data in, and that particular data is no more likely when our hypothesis is true versus when it’s false, then our level of belief stays the same. On the other hand, if we get in new data and that data is highly likely if our hypothesis is true, but highly unlikely if it’s false, then our level of belief will go up substantially. Bayes’ Theorem: For a hypothesis *H* with prior probability *P(H)* and new data *D* with *P(D) ≠ 0* then,   P(H|D) = P(D|H)P(H) P(D) Using the mechanics of Bayes’ Theorem and assuming that capital is allocated proportionally to our level of belief that a fund is in the Good group, we can derive a closed-form solution for the mathematical expectation of capital that would be allocated to the Good funds at each point in time as more fund data becomes known. We can also see in closed form the mathematical expectation of the updated probability that a Good fund is good and a Bad Fund is bad, at each point in time: As we can see from the above chart, even with this fairly sophisticated update strategy, relatively short periods of time don’t meaningfully impact our probability estimate that a Good fund is Good, or that a Bad fund is Bad. --- 1. James is Elm Partner’s CEO, and Victor is the Founder and CIO. **Past returns are not indicative of future performance.** This not is not an offer or solicitation to invest. 2. Example inspired by: “Good and bad properties of the Kelly criterion,” MacLean, Thorp, Ziemba, 2010, page 8. 3. Another version of the problem, posed and solved by Costas Kaplanis, lets the observer stop the flipping whenever desired. In this version of the problem, the solution involves using Bayes’ Theorem to update the probabilities regarding the identities of each coin. The result is that you usually need significantly less than 143 flips to be 95% confident you’ve identified the correct coin, but there is still a significant probability that you will need well in excess of 143 to get to that level of confidence. Stay tuned for more from Costas on this topic. 4. We did have some respondents who suggested that they only invest in opportunities with a Sharpe Ratio of 1 or higher. In our experience, investments that appear to have such high Sharpe Ratios are closed to outsiders, implicitly bear significant, hidden negative tail risk, are ephemeral or, infrequently, are fraudulent. For a more academic treatment, see Harvey, Liu, Zhu, *“…and the Cross-Section of Expected Returns,”* (2015), on [SSRN](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2249314). 5. See SPIVA Persistence Scorecard [here](https://us.spindices.com/documents/spiva/spiva-us-mid-year-2016.pdf) for S&P Dow Jones report. 6. We can make the example more realistic by introducing some mean-reversion into the picture in order to capture the effect that when an investment strategy truly was extraordinary in the past, its very discovery will tend to diminish future performance (or even make future returns negative), as more capital is drawn towards it.   This doesn’t happen with coin flips. The result is that a moderate amount of this effect means that even if the good managers are generating 4% more return than the not-so-good managers, even after 50 years the expected return of your portfolio still won’t be positive.   To give this result some context, this amount of excess return is similar in risk-adjusted terms to Warren Buffett’s past 40 year track record relative to the S&P 500. The amount of mean-reversion we introduced is such that each fund is expected over the coming year to make back or give up 20% of the return it lost or made relative to the benchmark and to its underlying expected return over the past five years.   So, if a fund outperformed by 10% over the past five years, it would be expected to do 2% worse than its normal expected return over the next year. Of course, a strategy set up to explicitly take advantage of the mean reversion would do better, but that is the topic for another note. 7. See our Elm paper on return-chasing [here](https://elmwealth.com/return-chasing-can-be-hazardous-to-your-wealth/). Also, see Morningstar’s “Mind the Gap” notes and Dalbar’s annual investor behaviour reports. 8. Merton, Robert, C., “On Estimating the Expected Return on the Market: An Exploratory Investigation,” Journal of Financial Economics 8 (1980) 323-361. See Appendix 1, pages 355-357. To see why you can’t get around this inconvenient truth, note that in continuous time, the number of years of observation, using the annual Sharpe Ratio (*SR* ) as an input is *2 \* (1.645 / SR)2* , where 1.645 is the 95% cumulative probability level in a Normal distribution. If we sample more frequently, *f* times per year, the required number of periods goes to *2 \* (1.645 / (SR√f))2 = 2f \* (1.645 / SR)2* .   So, we’ll need to observe *f* times as many periods as when we look annually, which is exactly how many more periods we get to observe by breaking the year into *f* intervals. Note this exact result depends on a strong assumption about the distribution returns are being drawn from. [James White](https://insights.elmwealth.com/elm-wealth-research/author/james-white) [Read More](https://insights.elmwealth.com/elm-wealth-research/whats-past-is-not-prologue) [1](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights) [2](https://insights.elmwealth.com/elm-wealth-research/tag/featured-insights/page/2)