Jensen's alpha and the market‐timing puzzle

• Author: Marco Wilkens,Martin Rohleder,Sebastian Bunnenberg,Hendrik Scholz
• Published date: 01 April 2019
• DOI: http://doi.org/10.1002/rfe.1033
• Date: 01 April 2019

ORIGINAL ARTICLE

Jensen's alpha and the market‐timing puzzle

Sebastian Bunnenberg

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Martin Rohleder

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Hendrik Scholz

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Marco Wilkens

2

1

Finance, ESB Business School,

Reutlingen University, Reutlingen,

Germany

2

Finance and Banking, University of

Augsburg, Augsburg, Germany

3

Finance and Banking, Friedrich-

Alexander-University Erlangen-Nürnberg,

Nürnberg, Germany

Correspondence

Sebastian Bunnenberg, Finance, ESB

Business School of Reutlingen University,

Alteburgstraße 150, 72762 Reutlingen,

Germany.

Email: sebastian.bunnenberg@reutlingen-

university.de

Abstract

Theory predicts that market‐timing activities bias Jensen's alpha (JA). However,

empirical studies have failed to find consistent evidence of this bias. We tackle

this puzzle in a nested model analysis and show that the bias contains an exoge-

nous market component that is unrelated to market‐timing skill. In a comprehen-

sive empirical analysis of US mutual funds, we find that the timing‐induced bias

in JA is mainly driven by this market component, which is uncorrelated with

measured timing activities. Measures of total performance that allow for timing

activities are virtually identical to JA, even if timing activities are present in the

evaluated fund. Hence, we conclude that JA is a sufficient measure of total per-

formance.

JEL CLASSIFICATION

G11, G23

KEYWORDS

market‐timing, mutual fund performance, stock selection, total performance

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INTRODUCTION

Jensen's (1968) alpha (JA) and its multifactor variants are still among the most widely used approaches to measure the eco-

nomic value that portfolio managers add for their clients. However, criticizing his own model, Jensen (1972) shows that JA

is biased downward when applied to funds that successfully engage in market‐timing activities: A manager who perfectly

times the market may have a statistically significant, negative JA. Treynor and Mazuy (1966) (TM) and Henriksson and

Merton (1981) (HM) suggest performance models that allow the separation of funds’selection activities from their timing

activities. While these models are still standard approaches in the literature,

1

the impact of market‐timing on JA is hardly

ever discussed. Moreover, the empirical studies of this subject find no evidence of a significant market‐timing bias, leaving

a disparity between theoretical and empirical research. We address this puzzle and provide consistent evidence why the tim-

ing‐induced bias in JA is usually irrelevant. Thus, our key contribution is the demonstration that JA is an adequate empiri-

cal measure of total performance for mutual funds.

To briefly illustrate how timing activities bias JA, Figure 1 shows scatterplots of portfolio excess returns over market

excess returns in a stylized stochastic simulation. Assuming a single‐factor model, we draw 36 market excess returns from a

normal distribution with a mean of 0.5% and a volatility of 4.536%.

2

The portfolio excess returns follow either a TM market‐

timing strategy—that is, exposure to market risk depends linearly on the market excess return—or an HM market‐timing

strategy, which implies that the portfolio manager chooses between two levels of market risk, depending on the sign of the

market risk premium. In both plots, a linear regression of portfolio excess returns against market excess returns as depicted

has a positive intercept, that is, JA. In our study, we analyze if JA unbiasedly reflects the joint economic value that results

from selection activities and timing activities present in portfolios that exhibit such timing qualities.

Figure 1 illustrates that the presence of market‐timing activities violates the assumption of a linear relation between port-

folio and market excess returns, which is implicit within JA. Grant (1977) relates the impact of this violation to parameters

Received: 1 February 2018

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Accepted: 8 March 2018

DOI: 10.1002/rfe.1033

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© 2018 The University of New Orleans wileyonlinelibrary.com/journal/rfe Rev Financ Econ. 2019;37:234–255.

of the distribution of the market excess return. Admati and Ross (1985) and Dybvig and Ross (1985) show that, under con-

ditions of market equilibrium, the sign of JA is unpredictable in the presence of successful market‐timing activities. Grin-

blatt and Titman (1989) attribute the timing‐induced bias in JA to a false estimation of a fund's exposure to market risk

during the evaluation period, which results from the varying exposure to market risk implied by timing activities. However,

none of these studies addresses the problem empirically.

Grinblatt and Titman (1994) provide empirical evidence by comparing the measured performance of JA with that of the

TM model for a small sample of 279 mutual funds. They find that the choice of model has little impact on the results.

However, they do not incorporate the HM model into their considerations. Coles, Daniel, and Nardari (2006) compare JA

with TM and HM total performance using bootstrapped mutual fund returns. They, too, conclude that the choice of model

has no significant impact on measured total performance. However, neither study analyzes why the theoretic ally predicted

market‐timing bias is not observed in the empirical results, leaving this an unresolved puzzle.

We contribute a solution of the market‐timing puzzle in mutual fund performance in three ways. First, using measures

of timing performance and total performance based on the TM and HM models, we take a nested model perspective on the

issue, similar to Aragon and Ferson (2006) and Chen, Ferson, and Peters (2010). We show that the market‐timing bias is

determined by two components: the extent of market‐timing activity, which is a fund‐specific component, and the market

conditions experienced by the fund, which are an exogenous component beyond the control of the fund manager.

TM model

(a)

(b) HM model

–0.05 0.00 0.05 0.10 0.15

Excess return of the portfolio

–0.10 –0.05 0.00 0.05 0.10 0.15

Excess return of the market

Portfolio excess return

TM regression

JA regression

–0.05 0.00 0.05 0.10 0.15

Excess return of the portfolio

–0.10 –0.05 0.00 0.05 0.10 0.15

Excess return of the market

Portfolio excess return

HM regression

JA regression

FIGURE 1 Stylized illustrations of the bias in JA as a result of timing activities according to the TM and HM models. For these

illustrations, we sample 36 market excess returns from an identical and independent normal distribution with an expected value of 0.5% and a

standard deviation of 4.536%. Using these sampled market excess returns, we generate two portfolios which follow market‐timing activities as

imposed by the TM and HM models. (a) The portfolio excess returns follow a TM market‐timing strategy: The exposure to market risk depends

linearly on the market excess return, resulting in a quadratic relation between portfolio and market excess returns. (b) The portfolio excess returns

follow a HM market‐timing strategy: We vary between a low level and a high level of exposure to market risk with the sign of the market

excess return. In this case, the relation between portfolio and market excess return is kinked for a market excess return of 0 [Colour figure can

be viewed at wileyonlinelibrary.com]

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