ArticlePDF Available

Abstract and Figures

This article presents a simple procedure for assessing the relative impact of luck and skill in determining investment performance. The procedure is then applied to the large-cap value managers. The results are consistent with earlier work that suggests that the great majority of the cross-sectional variation in fund performance is due to random noise.
Content may be subject to copyright.
First draft: June 2008
Current version: August 2008
Luck, Skill and Investment Performance
BRADFORD CORNELL
CALIFORNIA INSTITUTE OF TECHNOLOGY
PASADENA, CA 91125
626 564-2001
bcornell@hss.caltech.edu
I would like to thank Wayne Landsman, Steve Stubben and Elizabeth Tito for helpful
comments on earlier drafts of this paper. Of course, the errors remain my own.
Luck, Skill and Investment Performance
Abstract
This article presents a simple procedure for assessing the relative impact of luck
and skill in determining investment performance. The procedure is then applied to the
large cap value managers. The results are consistent with earlier work that suggests that
the great majority of the cross-sectional variation in fund performance is due to random
noise.
1. Introduction: The basic problem of skill versus luck
Successful investing, like most activities in life, is based on a combination of
skill and serendipity. Distinguishing between the two is critical for forward looking
decision making because skill is relatively permanent while serendipity, or luck, by
definition is not. An investment manger who is skillful this year presumably will be
skillful next year. An investment manager who was lucky this year is no more likely to
be lucky next year than any other manager.
The problem is that skill and luck are not independently observable. Instead all
that can be observed is their combined impact which is here called performance. The
central question, therefore, is to determine how much can be learned about skill by
observing performance. It turns out that there is a straightforward way to investigate that
question based on application of the bivariate normal distribution. Though the results
presented here are well known in statistics, they are not commonly applied in the context
of assessing portfolio managers. As shown, they can serve as the basis of a simple and
useful model for assessing the skill of competing fund managers. To illustrate how the
model works, I use the procedure to analyze the performance of large cap equity
managers tracked by Morningstar. It turns out, as one might expect given the volatility of
asset prices, that the relative performance of managers in any given year provides little
information about management skill.
2. A simple model for assessing luck and skill
To develop the model, assume that there exists a measure of performance, p, that
reflects the sum of skill, s, and luck, L. This formulation has a straightforward
- 1 -
interpretation in terms of portfolio management. In that context, p represents the
observed return on a specific portfolio, s represents the added expected return due to the
skill of the investment manager, and L represents the impact of idiosyncratic risk on the
portfolio’s return over the observed holding period.
More specifically, assume that both luck and skill are normally distributed in the
cross section and that
p = s + L , (1)
where s ~ n(E(s), sd(s)) and L ~ n(0, sd(L)). By definition, the mean of the luck
distribution is zero. Because p = L + s, p and s are distributed as bivariate normal with
mean vector [E(s), 0] and covariance matrix
sd(p) corr(p,s)*sd(p)*sd(s)
corr(p,s)*sd(p)*sd(s) sd(s)
Because luck cannot be correlated with skill, otherwise there would be a predictable
component of luck, it follows that
sd(p) = sd(s) + sd(L) and corr(p,s) = sd(s)/sd(p) . (2)
For the bivariate normal distribution, it is well known from the statistical literature1 that
E(s|p) = E(s) + corr(p,s)*sd(s)/sd(p)*[p – E(p)] . (3)
Substituting the relations from (2) into (3) gives,
E(s|p) = E(s) + [var(s)/var(p)]*[p-E(p)] . (4)
Because E(L) = 0, it follows that E(p) = E(s) so equation (4) can be written,
E(s|p) = E(s) + [var(s)/var(p)]*[p-E(s)] . (5)
1 See Mood (1974).
- 2 -
Equation (5) is the basic model. It states that when performance is observed in
excess of the mean, the assessment of skill is adjusted upward, but not all the way to the
observed level of performance, p. Instead, the assessment of s is adjusted upward by the
observed superior performance, p-E(s), times the ratio of the variance of s to the variance
of p. Therefore, the assessment of skill based on the observation of performance depends
critically on var(s)/var(p).
Notice that in both limiting cases, equation (5) makes intuitive sense. If var(L) is
much larger than var(s), then var(p) >> var(s) in which case E(s|p) goes to E(s). That is
reasonable because if performance is dominated by luck, then observation of performance
should play little role in the assessment of skill. On the over hand, if var(s) >> var(L)
then var(s) is approximately equal to var(p), which implies E(s|p) goes to p. That makes
sense because if luck has a relatively minor impact on performance, then observed
performance is a precise measure of skill.
Equation (5) has numerous applications in finance and is the basis for the
phenomenon referred to as regression toward the mean. Regression toward the mean
occurs because whenever the measure of performance, p, differs from the mean that
indicates two things. First, it indicates that the above average performance represents
above average skill. Second, it indicates that the above average performance reflects
good luck. In other words, above average performance is evidence of both good luck and
superior skill. However, whereas the skill element is permanent, the luck element is
transitory. Therefore, the expected performance next period reverts back toward the
mean from p because the luck variable has an expected value of zero. The greater var(L)
relative to var(s) the larger the regression back toward the mean.
- 3 -
What is here called luck can be interpreted as random measurement error in other
contexts. Consider, for instance, the problem of estimating beta. In that case, skill is
equivalent to the unobservable true beta and performance is the estimate of beta. For
betas, we know that the E(s) is 1.0. Therefore, equation (5) says that the best estimate of
beta is not the observed regression coefficient (measured by p), but E(s|p) which is a
weighted average of the estimated regression coefficient and the overall mean of 1.0.
Based on this property, Merrill Lynch developed a widely adopted weighted average
procedure for estimating beta. The task here, however, is not to estimate beta, but to
assess the contributions of luck and skill in determining investment performance. The
next section considers the application of equation (5) in that context.
3. Application of the model to mutual fund data
To illustrate the evaluation procedure, it is applied here to data on mutual fund
performance. Before turning to the data there is one important caveat. The model
attributes investment performance exclusively to the combination of skill and luck.
When the performance measure, p, is interpreted as the return on a portfolio, a third
element comes into play namely the risk level of the portfolio. There are two ways to
account for this. One is to perform the calculations in terms of risk adjusted returns, but
that introduces the problem of deciding how to adjust for risk, a problem that the finance
profession has not fully resolved after 40 years of research. The other approach is to
perform the analysis on a comparable cohort of investment funds. That is the approach
taken here.
- 4 -
The data are drawn from a comprehensive 2004 Morningstar database of mutual
fund performance.2 To assure that the data are relatively homogenous the sample is
limited to funds that invest primarily in the U.S. large capitalization value stocks. The
Morningstar database includes performance data on 1,034 large cap value funds during
2004. The first line of Table 1 presents cross-sectional summary statistics for the returns
on these funds which serves as the measure of performance, p. As shown in the table, the
mean return for the 2004 is 25.02% and the standard deviation across the 1,034 funds is
5.47%.
To apply equation (5) it is also necessary to estimate the standard deviation of s.
This is more difficult because s is not directly observable. There are two distinct
approaches for overcoming this difficulty. The first is to rely on judgment rather than
specific data. For example, an investor may conclude that the stock market is sufficiently
competitive that differences in skill among large cap value managers should lead to no
more than a 200 basis points differential from the mean for the vast majority of funds.
That judgment translates into a standard deviation of s on the order of 1.0%. If the
standard deviation of s is taken to be 1.0%, then the ratio of var(s) to var(p) of 0.033.
That ratio implies that the observation of annual performance should have virtually no
impact on assessment of the relative skills of the 1,034 large cap value managers. This
result is largely consistent with a large body of literature on mutual fund performance
beginning with the classic work of Jensen [1968] and continuing up through the work of
Nitzsche, Cuthbertson and O’Sullivan [2007].
2 The Morningstar historical data were graciously provided by Wilshire Associates.
- 5 -
An alternative approach is to use long-run return data to estimate the standard
deviation of s. If the fund return data were stationary and the sample period were
sufficiently long, then the cross-sectional standard deviation of s could be estimated with
little error. Unfortunately, neither is the case. With a maximum sample period of fifteen
years, luck still places a role in determining the cross-sectional distribution of returns.
This random noise results in an upward biased estimate of the standard deviation of s.
On the other hand, the 15-year sample is also impacted by survival bias. Whereas there
are annual data for 1,034 funds, the 15-year sample contains only 341 funds. Because the
funds that disappear from the sample are more likely to be underperformers, both the
mean 15-year return and the cross-sectional standard deviation are likely to be overstated.
Given that the calculation presented here is only illustrative, no attempt is made to adjust
for either of these offsetting effects on the estimated standard deviation of s.
The second line of Table 1 presents the cross-sectional summary statistics for the
341 funds in the 15-year sample. The mean annual return 10.03% and the standard
deviation is 1.57%. A standard deviation of 1.57% for s implies that the ratio of var(s) to
var(p) is 0.082. This indicates that approximately 92% of the cross-sectional variation in
annual performance is attributable to random chance.
3. Conclusion
The simple model presented here provides a useful, practical tool for assessing the
impact of skill and luck on portfolio performance. When the model is applied to a
sample of large cap value managers, the results indicate the most of the annual variation
in performance is due to luck, not skill. This finding is consistent with that reported in
- 6 -
- 7 -
other papers on mutual fund performance. Nonetheless, the model provides another way
of analyzing performance data.
The analysis also provides further support for the view that annual rankings of
fund performance provide almost no information regarding management skill. Potential
investors are better advised to consider the stated investment philosophies of competing
firms than to rely on such rankings. In any event, at best minors revisions of estimates of
skill such be based on annual performance data.
REFERENCES
Jensen, Michael C., 1968, The performance of mutual funds, Journal of Finance,
23: 2, 389-416.
Mood, Alexander, 1974, Introduction to the Theory of Statistics, Mc-Graw-Hill, New York.
Nitzsche, Dirk, Keith Cuthbetson and Niall O’Sullivan, 2007, Mutual fund performance,
SSRN working paper.
- 8 -
Table 1
Cross-sectional Statistics for Large Cap Value Funds
Annual data for period ended March 2004*
Number of funds Mean return Standard deviation
1,034 25.35% 5.47%
Fifteen-year data for period ended March 2004*
Number of funds Mean return Standard deviation
341 10.03% 1.57%
* All data from Morningstar
... It is true that studies by Coxwell, (2018) ;Fuels, (2010); Bruno and Martin (2009) ;Bradford, (2009) and others talked about business investment and locus of control but there was no relationship between business investments, locus of control and life during retirement mentioned in relation to interpersonal and intrapersonal conflict. This a serious gap to be covered by this investigation. ...
... The return on investment (ROI) is the ratio of money gained to the amount of funds invested. In case of passive investing (into shares and bonds), the ROI (or rate of return) includes a stream of income (dividends for shares and interest for bonds) as well as capital gains (appreciation in share or bond prices over time) (Bradford, 2009). The rate of return from a business investment is more than a function of the expected cash flows and capital appreciation. ...
Article
Full-text available
This study adopts Review Research Design with a qualitative advancement in data collection through knowledge and opinions gathering from various scholars, including personal experiences of the author to describe the variables under investigation tagged “Sustainable Business Investment In Active Years And Life During Retirement: A Case Of The Wise And The Foolish”. The study discovered from the various opinions reviewed that some employees do not inadequately visualize life during retirement from the active of years of services and do not invest in some of the sustainable businesses; they do not also believe in their internal locus of control as well as learn to take responsibilities of their failures in life. The study further discovered that many employees have not learned to be wise and draw moral lesson from the cited case of this study entitled: “The Parable of the Ten Virgins, also known as the Parable of the Wise and Foolish Virgins.”Among other things, it was recommended that employees should inadequately visualize life during retirement from the active of years of services, invest in sustainable businesses always, believe in their internal locus of control, learn to take responsibilities of their failures in life, learn to be wise with moral lesson from the case of “The Parable of the Ten Virgins, also known as the Parable of the Wise and Foolish Virgins.” and be regularly be trained and retrained on sustainable business investment while in active service to enjoy a happy life during retirement. Keywords: sustainability, sustainable business, sustainable business investment, active years, life during retirement, the wise and the foolish, internal and external locus, and control.
... We first start with our most important issue, ignored in the agency literature; namely, to use a robust measure of skill. The measure of skill, that is robust and tenable in asset pricing theory is from Ambarish and Seigel (1996), as opposed to say Cornell (2009), and defined as below. ...
Preprint
Full-text available
Arrow-Debreu (AD) create an abstract world with profit maximizing firms, utility maximizing consumers and a fixed number of commodities to establish a competitive equilibrium for the prices of these commodities. While foundational from an academic perspective of asset pricing, AD securities do not and will never exist as there is no natural issuer and "states"/"commodities" are hard to specify. We present a more realistic normative, general equilibrium model, where investors invest to achieve multiple stochastic investment goals like retirement, saving for a child's education, health etc., but we treat the utility function as "implicit" because we use the actual objectives of investors. Additionally, the creation and issuance of unique and innovative retirement and education bonds by Brazil in 2023, to good and sustained demand, offers a chance to view these securities as "Goals-based Arrow Debreu securities"(GADs). We show that the existence of just two of these securities, under certain realistic market conditions and objective functions of investors, along with the absolute risk-free asset, can be used to determine the price of all assets and provide effective asset allocation recommendations for multiple stochastic investment goals. Given the duality of equilibrium pricing equations, we term this a Janus Equilibrium as there are no free parameters. We demonstrate reasonable prices for securities, given live correlation and volatility assumptions about GADs, and, in turn, the wages that must be earned by investors/consumers/workers to hold such portfolios. Moreover, we can also specify a generic (implicit) production and labor allocation function, such that the firms generating such desired cash flows, paying appropriate wages to the investors/consumers/workers who allocate their time to different entities, could be treated as profit maximizers and their securities and wages paid priced in the same general equilibrium. Hence, this is a much more realistic model than the AD General Equilibrium model as it is based on positive observations, but with implicit utility, production and labor allocation functions.
... Interestingly, even the Modigliani-Modigliani (1997) measure completely ignores time, and skill, and the multiperiod nature of investing. As Muralidhar (2015) notes, even the treatment of skill by Cornell (2009) and Grinold and Kahn (1999) is less than optimal or robust. ...
Preprint
In this note, we specifically examine the aspect of "implicit time" in the Sharpe Ratio that has not been addressed in the literature. We demonstrate that there is a critical element embedded in the Sharpe Ratio that provides further clarity on what the Sharpe Ratio actually is. We introduce two "time" concepts-one the length of track record (t) and the second, dimensionality (T). We show T is the time for the asset under consideration to grow beyond its "SELF NOISE", with a given probability. We describe the Sharpe Ratio, which has been used statically (with Square Root adjustments), as fundamentally a dynamic process. As an aside if the variance instead of the standard deviation had been used in the Sharpe Ratio, a truly static and dimensionless index would have followed. JEL Classification: G11, G12
... Chapter 4 -RAP and Importance of Skill to resolve the issue. While Cornell (2009) tries to address this question of luck versus skill, Ambarish and Seigel (1996) frame the question in a truly dynamic world and derive useful equations which can inform investors about the skill of a manager, but also about the relevance of the Sharpe Ratio and how it might be adapted in a dynamic multi-period world. ...
Book
Full-text available
The book is meant to provide a new perspective on asset pricing, asset allocation and risk adjusted performance – the three facets of investment theory and investment finance - by focusing on a positive approach to normative theory. Current theory starts with the assumption that “investors maximize the expected utility of wealth”, when in reality this assumption can be faulted on at least four observations based on reality. Instead, as demonstrated in the book with examples from multiple pension fund Investment Policy Statements: “investors maximize (net) relative risk-adjusted returns for multiple stochastic goals through the use of (hopefully) skillful agents”. As a result, current theory is, as noted by GSB Distinguished Prof. Paul Pfleiderer, is a nest of “chameleons”. Incorporating multiple stochastic goals (like retirement, retirement health, saving for a child’s college etc.) is a major step up relative to current theory, which typically assumes just one deterministic (as in CAPM) or one stochastic goal (say peers, background risk etc.). Moreover, the theory on agency is limited and does not incorporate the fact that investors want to delegate to skillful agents. So, the book is meant to fill a key void, by using positive observations to lay out a normative theory with numerical simulations (and hence will not have substantial empirical evidence). As a result, the recommendations of this theory are easily applied by practitioners too and could lead to better investment results. Some of the recommendations for example, for new financial instruments in Chapter 3 (including one proposed with Nobel Laureate Robert C. Merton), have been implemented in Brazil in 2023, and other countries are in the process of adapting/adopting the same. In summary, while we seek to alter the course of theory by incorporating a “relative” lens as opposed to the current absolute lens (as in physics), the broader purpose is to alter practice to improve investment results for a range of stochastic goals. GOAL: The goal of the book is to use this new approach to change both theory and practice as incorrect theory or the incorrect application of theory has led to repeated financial crises. The most common and widespread crises being the global retirement crisis and one could even attribute the collapse of Silicon Valley Bank to the focus on absolute wealth as in current theory as opposed to relative wealth, which is the correct measure in practice and as noted over the last 20 years by Nobel Laureate Robert C. Merton. Additionally, the research that forms the basis of parts of the book has already led to the implementation of financial innovations in Brazil for retirement and education financing and is being copied around the world. The target of the book is both academics and practitioners (e.g., CFAs, MBAs; CIOs of investment pools; regulators; policy makers).
... Our most important issue, ignored in the agency literature is a robust measure of skill, even though Cornell and Roll (2005), following in the wake of Brennan (1993), attempt to create a delegated agent model of asset pricing, but they also ignore goals. The measure of skill, that is robust and tenable in asset pricing theory, is from Ambarish and Seigel (1996), as opposed to say Cornell (2009), and defined as below. ...
Preprint
Full-text available
Arrow-Debreu (AD) securities, while foundational from an academic perspective of asset pricing, do not and will never exist as there is no natural issuer and "states" are hard to specify. Moreover, because of the spanning argument, to price all securities, would require an inordinate number of AD securities. In reality, investors invest to achieve investment goals like retirement, saving for a child's education, health etc. The creation and issuance of unique and innovative retirement and education bonds by Brazil in 2023 offers a chance to view these securities as "Goals-based Arrow Debreu securities"(GADs). We show that the existence of just two of these securities, under certain realistic market conditions, along with the absolute risk-free asset, can be used to determine the price of all assets. More GADs just require more equilibrium conditions, thereby addressing the John Cochrane "free parameter problem" in typical asset pricing models.
... This contradicts our expectation and might suggest that followers are attracted by a trader's past performance but show concerns about a trader with good trading results during the current year. According to Hartzmark (1991) and Cornell (2009), investment performance is affected by a combination of luck and skill. While luck is serendipity, skill is relatively permanent. ...
Article
Full-text available
Online social trading offers an opportunity for less-experienced individuals or firms to follow top traders by mimicking their behavior, but little is known about the determinants of leadership that shape such relationships. To study this, we build on signaling theory using fixed-effect panel least squares estimations to analyze 250 top traders in a network of around 1,100 traders; we examine their trader credentials, the volume of trades, performance, and risk signals. Contrary to our initial expectations, findings show that trader credentials are more important than performance, volume, or risk signals, but there are significant differences between virtual and real money traders. This study proposes a network signaling theory approach by linking it to herd behavior and the disposition effect. Our findings can have practical implications not only for top traders, followers, and social trading platform managers but also for policy-makers and regulators of such investment instruments.
Article
We consider the problem of identifying skilled funds among a large number of candidates under the linear factor pricing models containing both observable and latent market factors. Motivated by the existence of non-strong potential factors and diversity of error distribution types of the linear factor pricing models, we develop a distribution-free multiple testing procedure to solve this problem. The proposed procedure is established based on the statistical tool of symmetrized data aggregation, which makes it robust to the strength of potential factors and distribution type of the error terms. We then establish the asymptotic validity of the proposed procedure in terms of both the false discovery rate and true discovery proportion under some mild regularity conditions. Furthermore, we demonstrate the advantages of the proposed procedure over some existing methods through extensive Monte Carlo experiments. In an empirical application, we illustrate the practical utility of the proposed procedure in the context of selecting skilled funds, which clearly has much more satisfactory performance than its main competitors.
Article
This article aims to examine the complexity of lead investor's signaling by information disclosure in equity crowdfunding. We use 40 samples from a Chinese equity crowdfunding source to explore why lead investor influences followers' decision making and how lead investor signals effectively by using fuzzy‐set qualitative comparative analysis (fsQCA) approach. Our findings enrich the literature on the lead investors and syndicate by testing the complexity of signaling in equity crowdfunding, shed new light on investigating the decision‐making model of individuals at a disadvantage of information asymmetry, and offer insights into signaling strategically online.
Chapter
Kapitel 3 fasst den aktuellen Stand der wissenschaftlichen Forschung über die Märkte und ihre Teilnehmer zusammen. Hierbei werden sowohl das kontrovers diskutierte Spannungsfeld zwischen der Möglichkeit eines aktiven oder passiven Anlagemanagements auf Basis der Markteffizienzdebatte aufgearbeitet als auch die bisher gängisten in der Theorie und Praxis vorgebrachten Erfolgsstrategien beleuchtet. Bezogen auf das Verhalten der Marktakteuere werden anschließend die wichtigsten empirischen Erkenntnisse zusammengetragen.
Article
We evaluate the academic research on mutual fund performance in the US and UK concentrating particularly on the literature published over the last 20 years where innovation and data advances have been most marked. The evidence suggests that ex-post, there are around 2-5% of top performing UK and US equity mutual funds which genuinely outperform their benchmarks whereas around 20-40% of funds have genuinely poor. Key drivers of relative performance are, load fees, expenses and turnover. There is little evidence of successful market timing. Evidence on picking winners suggests past winner funds persist, particularly when rebalancing is frequent (i.e., less than one year) - but transactions costs and fund fees imply that economic gains to investors from actively switching into winner funds may be marginal. However, recent research using more sophisticated sorting rules (e.g., Bayesian approaches) indicate possible large gains from picking winners, when rebalancing monthly. The evidence also clearly supports the view that past loser funds remain losers. Broadly speaking results for bond mutual funds are similar to those for equity mutual funds but hedge funds show better ex-post and ex-ante risk adjusted performance than do mutual funds. Sensible advice for most investors would be to hold low cost index funds and avoid holding past "active" loser funds. Only very sophisticated investors should pursue an active investment strategy of trying to pick winners - and then with much caution.
Article
In this paper I derive a risk-adjusted measure of portfolio performance (now known as Jensen's Alpha) that estimates how much a manager's forecasting ability contributes to the fund's returns. The measure is based on the theory of the pricing of capital assets by Sharpe (1964), Lintner (1965a) and Treynor (Undated). I apply the measure to estimate the predictive ability of 115 mutual fund managers in the period 1945-1964 - that is their ability to earn returns which are higher than those we would expect given the level of risk of each of the portfolios. The foundations of the model and the properties of the performance measure suggested here are discussed in Section II. The evidence on mutual fund performance indicates not only that these 115 mutual funds were on average not able to predict security prices well enough to outperform a buy-the-market-and-hold policy, but also that there is very little evidence that any individual fund was able to do significantly better than that which we expected from mere random chance. It is also important to note that these conclusions hold even when we measure the fund returns gross of management expenses (that is assume their bookkeeping, research, and other expenses except brokerage commissions were obtained free). Thus on average the funds apparently were not quite successful enough in their trading activities to recoup even their brokerage expenses.
Article
Using a comprehensive data set on (surviving and non-surviving) UK equity mutual funds, we use a cross-section bootstrap methodology to distinguish between 'skill' and 'luck' for individual funds. This methodology allows for non-normality in the idiosyncratic risk of the funds -- a major issue when considering those funds which appear to be either very good or very bad performers, since these are the funds which investors are primarily interested in identifying. Our study points to the existence of stock picking ability among a relatively small number of top performing UK equity mutual funds (i.e. performance which is not solely due to good luck). At the negative end of the performance scale, our analysis strongly rejects the hypothesis that most poor performing funds are merely unlucky. Most of these funds demonstrate 'bad skill'. Recursive estimation and Kalman 'smoothed' coefficients indicate temporal stability in the ex-post performance alpha's of winner and loser portfolios. We also find performance persistence amongst loser but not amongst winner funds.