The aggregate cost of equity underdiversification

• Author: Ulf Nielsson,Peter Raahauge,Jesper Rangvid,Bjarne Florentsen
• DOI: http://doi.org/10.1111/fire.12212
• Date: 01 November 2019
• Published date: 01 November 2019

DOI: 10.1111/fire.12212

ORIGINAL ARTICLE

The aggregate cost of equity underdiversification

Bjarne Florentsen1Ulf Nielsson1,2,3 Peter Raahauge1

Jesper Rangvid1,2,4

1Copenhagen Business School, Frederiksberg,

Denmark

2Danish Finance Institute, Frederiksberg,

Denmark

3PerCent,Frederiksberg, Denmark

4Reykjavík,Iceland

Correspondence

JesperRangvid, Copenhagen Business School,

SolbjergPl. 3, 2000 Frederiksberg, Denmark.

Email:jr.fi@cbs.dk

Abstract

We analyze equity diversification of all retail investors in a country

(Denmark). We find that underdiversification ispervasive. We calcu-

late the nationwide aggregate loss due to underdiversification and

express itin absolute and expected-return terms. The aggregate loss

is large. We find that investors with low education, low income, and

lowwealth are more likelyto underdiversify. Despite better diversifi-

cation, the larger fraction ofthe aggregate loss nevertheless adheres

to the top of the income/wealth distribution. Finally,our results indi-

cate that underdiversification arises because investors have limited

information about the benefits of diversification.

KEYWORDS

aggregate loss, investor characteristics, investor sophistication,

retail investors, stock holdings, underdiversification

JEL CLASSIFICATIONS

D14, G11, G15

1INTRODUCTION

Retail investors spread their equity investments over a few different stocks only. This equity underdiversification

was first described in Blume and Friend (1975) and has since then been reconfirmed in numerous studies, see, for

example, Barber and Odean (2000), Ivkovic, Sialm, and Weisbenner(2008), Kelly (1995), Polkovnichenko (2005), and

Goetzmann and Kumar (2008).1Equity underdiversification is surprising in light of the fact that idiosyncratic risks

can be reduced—without sacrificing expectedreturns—by holding well-diversified portfolios. A number of papers have

shown how standard deviationsof equity portfolios can be reduced by diversification and how many randomly selected

stocks are needed to eliminate idiosyncratic risks, starting with the by now famous Evans and Archer’s (1968) article,

and followed by Benjelloun (2010), Bloomfield, Leftwich, and Long (1977), Campbell, Lettau, Malkiel, and Xu (2001),

Domian, Louton, and Racine (2007), Elton and Gruber (1977), Statman (1987, 2004), and others. Suggested expla-

nations for the underdiversification puzzle include preferences for lottery stocks (Barberis & Huang, 2008), margin

1Retirementaccounts are also underdiversified, see, for example, Benartzi and Thaler (2001) and Agnew, Balduzzi, and Sunden (2003).

Financial Review.2019;54:833–856. wileyonlinelibrary.com/journal/fire c

2019 The Eastern Finance Association 833

834 FLORENTSENET AL.

and borrowing restrictions (Roche, Tompaidis,& Yang, 2013), solvency constraints (Liu, 2014), financial literacy (von

Gaudecker,2015), and other channels. But how large are the aggregate costs retail investors suffer from underdiversi-

fication? If these are minuscule, we may not be that concerned. On the other hand, if they are sizeable, it is even more

important that we continue our efforts to try to understand their underlying reasons.

The main contribution of this paper is that it calculates the aggregate cost of underdiversification of the total pop-

ulation of retail investors in a country,Denmark. We calculate the costs of underdiversification as the return foregone

due to underdiversification, that is, the return an investor foregoes by holding portfolios with idiosyncratic risk (that

is not compensated by expected return) instead of portfolios with systematic risk (that is compensated by expected

return). As an example, what is the expected return the investorforegoes by holding a portfolio with an idiosyncratic

level of risk of,say, 30%, instead of a portfolio with a systematic level of risk of 30%?

We find sizeable costs of underdiversification. In our baseline calculation, we find that retail investorscould poten-

tially increase their expectedreturn by up to three percentage points per year, without increasing risk, by shifting from

concentrated portfolios to diversified portfolios. We are able to calculate the aggregate cost of underdiversification

because we have access to data that contain the complete equity portfolios (outside retirement savings) of all retail

investors in Denmark. Wethereby differ from studies that examine subgroups of investors, such as online brokers and

respondents to surveys, as in Barber and Odean (2000), Blume and Friend (1975), Ivkovicet al. (2008), Kelly (1995),

Polkovnichenko(2005), Bodnaruk (2009), and Goetzmann and Kumar (2008).

We first describe the portfolios of our retail investors, concentrating on the number of different stocks entering

the portfolios. We find that the typical investor holds a veryconcentrated portfolio. Most investors, 66% of investors,

hold one stock only.We also find that as little as 1.8% of investors hold more than 10 stocks. This underdiversification

resembles that found in other studies, such as those mentioned above.

Weanalyze a sample of Danish stocks for which we have monthly data for the last 20 years. There are 91 such stocks.

Based on this sample of stocks, we calculate the level of risk (standard deviation of returns) of portfolios including

different numbers of stocks, that is, the relation between the number of stocks in a portfolio and its standard deviation,

in the tradition of Evans and Archer (1968) and the subsequent literature.2With one randomly chosen stock, we find

that the expectedstandard deviation is 38%. When increasing the number of stocks in the portfolio, standard deviation

of the portfolio naturally drops. We provide confidence bounds around the standard deviationbased on the empirical

distribution. We find that with around 50 stocks in the portfolio, the standard deviation of the portfolio reaches its

convergence point at around 15.5%. This is also the level of risk of the market portfolio, that is, the portfolio of all

91 stocks. As much as (38% – 15.5%)/38% ≈60% of risk can thus be eliminated by diversification. Our finding that it

takesaround 50 randomly chosen stocks to eliminate idiosyncratic risk is in line with results reported in Campbell et al.

(2001) for U.S. stocks.

We propose a simple wayto calculate the potential impact of diversification on expected returns. The model we use

is the capital-asset pricing model (CAPM) and its basic insight that investors get compensation for bearing systematic

risk, but not for bearing idiosyncratic and diversifiable risk. Instead of holding, for example,a one-stock portfolio with

a 38% standard deviation, the investor can hold a well-diversifiedportfolio (with risk equal to 15.5%) and lever up this

portfolio to 38% risk, that is, move up the capital market line and earn a risk premium. The historical risk premium on

the Danish stock market is around 5% (Dimson, Marsh, & Staunton, 2011). We use this as our benchmark estimate of

the expectedrisk premium (and provide robustness checks using alternative estimates for the expected risk premium).

We find that an investor,who moves up the capital market line from 15.5% to 38% risk, should expect an increase in

expected returns of slightly more than seven percentage points. This means that an investor foregoes at least seven

percentage points expected return by holding a one-stock portfolio, compared to holding a well-diversified portfolio.

We perform such calculations for portfolios with two stocks, three stocks, etc., up until 91 stocks.

2A by-product of our analysis, thus, is that we present results for the relation between portfolio standard deviation and number of stocks in a portfoliofora

non-U.S.market (Denmark, in our case).