The impact of big winners on passive and active equity investment strategies Maxime Markovand Vladimir Markov

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The impact of big winners on passive and active equity

investment strategies

Maxime Markov∗and Vladimir Markov

Abstract

We investigate the impact of big winner stocks on the performance of active and

passive investment strategies using a combination of numerical and analytical tech-

niques. Our analysis is based on historical stock price data from 2006 to 2021 for a

large variety of global indexes. We show that the log-normal distribution provides a

reasonable ﬁt for total returns for the majority of world stock indexes but highlight the

limitations of this model. Using an analytical expression for a ﬁnite sum of log-normal

random variables, we show that the typical return of a concentrated portfolio is less

than that of an equally weighted index. This ﬁnding indicates that active managers

face a signiﬁcant risk of underperforming due to the potential for missing out on the

substantial returns generated by big winner stocks. Our results suggest that passive

investing strategies, that do not involve the selection of individual stocks, are likely to

be more eﬀective in achieving long-term ﬁnancial goals.

1 Introduction

One of the most signiﬁcant phenomena in the world of ﬁnance is the rise of passive investing.

Active investing strategies give portfolio managers discretion to select individual securities,

generally with the investment objective of outperforming a previously identiﬁed benchmark.

In contrast, passive strategies use rule-based investing to track an index, typically by holding

all its constituent assets or an automatically selected representative sample of those assets [1].

The share of passive investments is constantly increasing. For example, the rate of mutual

and exchange-traded funds in the United States rose from 3% in 1995 to 37% in 2017 [2].

Since then, the trend has only been upward, and passive investing is expected to overtake

active investing by 2026 [3]. The reason for this shift is the persistent underperformance

of active investing. Indeed, 99% of actively managed US equity funds sold in Europe have

failed to beat the S&P 500 index over the period of 10 years from 2006 to 2016, while

only two in every 100 global equity funds have outperformed the S&P Global 1200. The

situation is similar to active emerging market equity funds, 97% of which underperform [4, 5].

S&P regularly publishes S&P Indices Versus Active (SPIVA) research reports measuring the

performance gap between actively managed and index funds [6].

∗Corresponding Author: markov@theory.polytechnique.fr

arXiv:2210.09302v5 [q-fin.PM] 9 Oct 2023

Passive investing has several advantages over an active strategy. First, passive investing

products have lower fees relative to active mutual fund fees [7]. The Morningstar investment

research ﬁrm estimated that passive US fund investors saved $38 billion in fees in 2021

compared to what they would have paid to have their money in active funds. The higher

fee eﬀect is cumulative (this eﬀect is also called ”a tyranny of compounding costs”) and

represents a headwind for active investors. Second, active managers, like all humans, have

cognitive and emotional biases. In particular, the disposition eﬀect states that investors tend

to sell winning investments and hold on to losing investments. Third, the impact of missing

the market’s best days can be huge. For example, missing the ten largest days in S&P 500

leads to underperformance by 55%, and seven of the best ten days occurred within two weeks

of the ten worst days, which makes market timing challenging [8]. Another important factor

is the eﬀect of a few big winner stocks that can grow by a factor of 10 or more over a long

enough time-frame, which produces an outsized share of market returns. The last factor is

the objective of this study.

In this paper, we explore the impact of big winners on investment performance from

diﬀerent perspectives for a wide variety of global indices. First, we examine the distribution

of stock index returns using historical stock price data from 2006 to 2021 and quantify the

diﬀerence between average returns and typical returns (approximated by a mode or median)

for major stock indexes. We show that the log-normal distribution provides a reasonable ﬁt

for the total returns for most world stock indexes and highlight the limitations of this model.

We use an analytical expression for the sum of Nlog-normal random variables to quantify

the ratio of the typical mean to the true mean as a function of the number of stocks in a

portfolio. This shows how a typical small (concentrated) portfolio’s performance diﬀers from

that of an index portfolio.

Second, to better understand the mechanism of index returns, we ﬁt a geometric Brownian

motion (GBM) model to index constituents and extract index drift and volatility parameters.

We observe a diverse range of relations between drift and volatility, which can be used to

build a microscopic model of index returns, and quantify the eﬀect of big winners. We also

study a toy model with drift distributed according to a normal distribution and constant

volatility. The ratios of mean to median and mean to mode are given by an analytical

function of the parameters of the model.

2 Empirical Data

In this study, we use 16 years of yearly data from January 1, 2006, to December 31, 2021, with

index constituents taken as of January 1, 2006. To avoid look-ahead bias, we take the index

constituents as of the start date. We group indexes according to their geographical location

into several groups, including the United States, Europe, Asia-Paciﬁc (APAC), Japan, and

BRIC (Brazil, India, and China) countries. We also study 10 sectors of S&P 500 GICS Level

1 indexes to better understand the role of heterogeneity within the S&P 500 index. The

composition of the groups is given below:

US indexes: S&P 500 (SPX ); NASDAQ Composite (CCMP); Russell 3000 index, which

is composed of 3,000 large US companies representing approximately 98% of market cap-

italization of the investable US equity market (RAY ); Russell 2000 index, which consists

of the smallest 2,000 companies in the Russell 3000 index representing approximately 8%

of the Russell 3000 index capitalization (RTY ); Russell 1000 index, which consists of the

largest 1,000 companies in the Russell 3000 index (RIY ); Russell 1000 Value, which consists

of Russel 1,000 companies with low price-to-book rations (RLV ); Russell 1000 Growth index

with high price-to-book ratio (RLG), and NASDAQ Biotechnology (NBI ).

S&P500 GICS Level 1 indexes: Consumer Discretionary (S5COND), Consumer Sta-

ples (S5CONS ), Energy (S5ENRS ), Financial (S5FINL), Health Care (S5HLTH ), Infor-

mation Technology (S5INFT ), Materials (S5MATR), Communication Services (S5TELS ),

Utilities (S5UTIL), and Industrials (S5INDU ).

European indexes (including the UK): Deutsche Boerse German Stock Index (DAX ),

French CAC 40 (CAC ), UK FTSE 100 (UKX ), Belgium BEL 20 (BEL20 ), Spain IBEX 35

(IBEX ), Danish OMX Copenhagen 20 (KFX ), Swedish OMX Stockholm 30 index (OMX ),

and Swiss Market Index (SMI ).

APAC indexes: Australia S&P ASX 200 Index (AS51 ).

Japanese indexes: Nikkei 225 (NKY ), Tokyo Price Index (TPX ).

BRIC indexes: Brazil Sao Paulo Stock Exchange Index (IBOV ), India NSE Nifty 40

Index (NIFTY ), MSCI India Index (MXIN ), Shanghai Stock Exchange Composite Index

(SHCOMP), and Shanghai Shenzhen CSI 300 Index (SHSZ300 ).

In this study, we consider the performance of equally weighted and ﬁxed portfolios or

indexes only. We neglect the eﬀect of portfolio and index weights and rebalancing and

concentrate on the impact that a few big-winner stocks have on a portfolio’s long-term

performance. In this framework, investors allocate capital randomly and in equal units. It

is a plausible model for uninformed investors.

3 Total Return Distribution

We start by considering the distribution of the total return, deﬁned as the ratio of the ﬁnal

price XTat time t=Tto the initial price X0:ρ=XT/X0. To have positive support for

ρ, we do not subtract one in the deﬁnition of total return ρ. All prices are adjusted for

dividends and splits. In Fig. 1, we show the total returns histogram for the CCMP (left

panel) and SPX (right panel) indexes. The histograms consist of the distribution body (blue

bins), as well as the left and right cumulative bins highlighted in red. The left cumulative

bin includes all beaten-down stocks satisfying condition ln(ρ)<−2 (approximately 86%

loss). The right cumulative bin aggregates the best-performing stocks, with a total return

of top 5% in the index distribution. We use twice the number of bins determined by the

Freedman-Diaconis [9] rule (see Table S1 and Figure S1 in Supplementary material 1).

Indexes can be divided into two groups: unimodal and bimodal. The ﬁrst group includes

indexes composed of stocks of well-established companies. The Belgium BEL20 and Swedish

OMX indexes are typical examples from the ﬁrst group. The left cumulative bin is small

and ﬁts well into the distribution body. In Section 5, we see that their distribution can

be approximated by log-normal. Indexes belonging to the second group have excessive left

cumulative bin, indicating a high number of depressed stocks that never recover. Examples

are tech heavy NASDAQ (CCMP), biotech NBI indexes, and RAY, RTY, and AS51 indexes.

1Link to the supplementary material

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TheimpactofbigwinnersonpassiveandactiveequityinvestmentstrategiesMaximeMarkov∗andVladimirMarkovAbstractWeinvestigatetheimpactofbigwinnerstocksontheperformanceofactiveandpassiveinvestmentstrategiesusingacombinationofnumericalandanalyticaltech-niques.Ouranalysisisbasedonhistoricalstockpricedatafrom200...

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