Towards a national default option for low cost superannuation

Description
The purpose of this paper is to propose an approach to design a national default option to
maximize retirement savings in defined contribution superannuation, using a proportionate
shareholding approach (PSA) which minimizes total cost of investing for all investors.

Accounting Research Journal
Towards a national default option for low-cost superannuation
Wilson Sy
Article information:
To cite this document:
Wilson Sy, (2009),"Towards a national default option for low-cost superannuation", Accounting Research
J ournal, Vol. 22 Iss 1 pp. 46 - 67
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Towards a national default option
for low-cost superannuation
Wilson Sy
Australian Prudential Regulation Authority, Sydney, Australia
Abstract
Purpose – The purpose of this paper is to propose an approach to design a national default option to
maximize retirement savings in de?ned contribution superannuation, using a proportionate
shareholding approach (PSA) which minimizes total cost of investing for all investors.
Design/methodology/approach – Through analytic modelling, the author shows how transaction
costs in combination with size effects and agency incentives have limited the ability of professional
managers to use arbitrage and active investment to create a price-ef?cient market. Statistical models
show how investors would experience dif?culties in understanding fund performances due to inherent
noise in the data. The models suggest ?nancial intermediation has created an information asymmetry
which reduces the effectiveness of market competition to lower costs in superannuation.
Findings – The authors ?nd that the PSA is a collective optimal strategy and it is also an individual
optimal strategy, because of the presence of informational inef?ciency. Passive investing does not need
commercial indices. PSAis more passive and ?exible than standard indexing, and is fully-scalable and
available to all investors.
Research limitations/implications – Professional investment managers have not beaten the
market, not because the market is ef?cient, but because it is inef?cient due to a market failure to
recognise and resolve principal-agent con?icts of interest.
Practical implications – The proposed national default option has the potential to substantially
increase national savings through low-cost superannuation.
Originality/value – The paper provides a new rationale for passive investing based on the
hypothesis of market inef?ciency. It also provides the ?rst formal proof of the “Cost matters theorem.”
The proposed idea of a national default option will create a simple, understandable and cost-effective
alternative for all workers and will also provide a performance benchmark to encourage the
development of a more competitive and ef?cient superannuation market.
Keywords Pensions, Indexing, Savings, Investments
Paper type Research paper
1. Introduction
Many countries around the world face similar problems of funding the retirement of
ageing populations, now increasingly through de?ned contribution schemes. De?ned
contribution plans are pension schemes which are essentially self-funded through
The current issue and full text archive of this journal is available at
www.emeraldinsight.com/1030-9616.htm
The views and analysis expressed in this paper are those of the author and do not necessarily
re?ect those of Australian Prudential Regulation Authority (APRA) or other staff. The data
samples and methods used in this paper are selected for the speci?c research purposes of this
paper and may differ from those used in other APRA publications.
The author dedicates this paper to Jack Bogle, who has long championed tirelessly the basic
truths that inspire this paper. The author thanks, Jack Bogle, Keith Ambachtsheer, Anthony
Asher, Natalie Gallery, Charles Littrell, Katrina Ellis, Belinda Tracey and the participants of the
16th Australian Colloquium of Superannuation Researchers for helpful comments.
ARJ
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Accounting Research Journal
Vol. 22 No. 1, 2009
pp. 46-67
qEmerald Group Publishing Limited
1030-9616
DOI 10.1108/10309610910975324
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personal savings encouraged by tax incentives. It is important that the national
savings are maximized through ef?cient pension investments.
The recent trend towards individual investment choice in competitive markets for
pension products has given rise to a complex web of ?nancial intermediaries, including
trustees, consultants, fund managers, stockbrokers, ?nancial planners and other
service providers. The unbundling of investment services traditionally provided by the
pension trustees, and now often separately charged, has increased the overall cost of
the pension system (Coleman et al., 2006; Sy, 2008b) so that the growth of national
pension wealth is now sub-optimal.
It is arguable whether the increased ?exibility and complexity of market offerings
justify the increased cost to most members of the pension system. For the bulk of
workers typically before the age of 55 years, their ?nancial objectives are simply to
accumulate suf?cient wealth so that they may be in a ?nancial position to consider
more sophisticated alternatives through ?nancial advisors. It is imperative that some
simple, low-cost investment choices be understood by, and made available to, all
workers.
The observed inertia (Cameron and Gibbs, 2005), or lack of interest in making
investment choices is economically rational, given the low-account balances (less than
$100,000) of most workers. In the unlikely event that a worker can make a better
investment choice and beats the averages, the bene?ts of better gross returns would be
swamped by the additional costs involved in making a better decision, due to the lack
of scale, leading ultimately to lower net returns. Brown et al. (2002) have suggested the
establishment of a government-regulated universal default fund to cater for members
who are unable or unwilling to make informed choices. Similar expressions of the need
for national approaches for most workers have been made in other countries such as in
the UK (Turner et al., 2006) and Canada (Ambachtsheer, 2008). We will investigate a
particular non-coercive national default option in this paper.
Member investment choice, as distinct from “choice of funds” provided by
superannuation choice legislation in Australia, has led to a higher cost structure for the
pension industry without necessarily maximizing savings for most workers. Recent
regulatory effort by the Australian Securities and Investments Commission (2007) to
improve product disclosure serves to show that a complex cost structure has evolved in
the pension industry (Chant, 2008). Most investors are not able to understand and
quantify those costs directly due to inadequate information disclosure by their
superannuation funds (Gallery and Gallery, 2006).
Regardless of whether the trustee is a pro?t-making organisation or not, most of the
costs of a superannuation fund come from service providers that are pro?t-making
entities. A pro?t-driven market often fails to deliver simple, low-cost products, as
markets generally abhor simplicity. Simple products compete only on price, leading to
a process of commoditization which eliminates abnormal pro?ts. Hence, the expected
bene?t of market competition driving down the cost of superannuation may not have
been realised, or is being realised only very slowly. Competition is less effective when
there are complexities and there are few simple low-cost products which can serve as
benchmarks for comparison.
While individuals generally prefer more choice rather than less, too many choices
(Benartzi and Thaler, 2002; Brown et al., 2002) can ultimately lead to high costs and
lower savings for retirement. Everyone should be given the choice to decide on whether
Low-cost
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they need any choice at all (Fear and Pace, 2008), whereas many workers are forced
currently to face many choices. Governments have the responsibility to help
unsophisticated individuals (Thaler and Sunstein, 2003) and guide them through
default options (Gallery et al., 2004; Basu and Drew, 2006) to save in simple and effective
ways so that aggregate national savings are optimised.
The purpose of this paper is to propose an approach to create a national default option
for low-cost superannuation to increase total retirement savings. We are motivated by the
needtohelpworkers who are mostlyinthe wealthaccumulationphase andhave little or no
other assets apart from their superannuation assets and perhaps their homes.
This situation would be the case for the bulk of the working population under the age
of 55.
The rationale for proposing a low-cost default option for Australian superannuation
is based on two original insights discussed in this paper. The ?rst insight, discussed in
the next section, is that the market is inef?cient which entices investors to try to exploit
the inef?ciencies. However, we show that the inef?ciencies are not being fully exploited
by professional investment managers for the bene?t of investors due to transaction
costs, fund size impediments and adverse agency incentives. The situation is not easily
recognised by investors because information asymmetry from noisy statistical data on
investment performance prevent investors from distinguishing good managers from
bad ones, thus perpetuating an inef?cient market with high costs.
The second insight, discussed in Section 3, is that the rationale for passive
management is to lower costs (Bogle, 2005b; Sharpe, 1991) in an inef?cient market,
rather than due to the common belief that professional managers cannot beat an
ef?cient market (Malkiel, 2005). Since cost is the quintessence of this insight, it is
worthwhile to convert the “Cost matters hypothesis” (Bogle, 2005b) to a “Cost matters
theorem,” by formally proving for the ?rst time that the asset-weighted average return
of all investors is equal to the market return minus transaction costs.
If loweringcost is what reallymatters, thenthe formal proof indicates that we canlower
costs even further and improve upon existing approaches to passive management. In
Section 4, we propose a new truly passive approach that extends the capitalisation
weighted indexing approach to a simpler and more ?exible approach of proportionate
shareholding. In Section 5, we suggest ways to implement the proportionate shareholding
approach (PSA) in a national default option for most workers to have low-cost
superannuation.
In Section 6, we summarise our arguments for government intervention and discuss
the potential bene?ts for doing so.
2. Inef?cient markets and ?nancial intermediation
In this section, we argue that investment markets are inef?cient and this explains why
active investment strategies have been widely adopted. However, exploiting market
inef?ciencies through delegated professional investment managers is unlikely to
succeed, but not because managers have no skills and therefore cannot beat the market.
Rather, the presence of transaction costs, competition, fund size impediment, adverse
agency incentive and information asymmetry all combine to discourage exploitation
and to maintain market inef?ciencies.
Market bubbles, scams, scandals, Ponzi schemes and ?nancial crises clearly show
investment markets are not price ef?cient, where the relationship between cost and
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bene?t of acquiring a ?nancial product is often highly unstable. Many long-term
empirical studies formally support this observation of price inef?ciency (Shiller, 1981,
2005; Summers, 1986). The capital needed and the risk involved in arbitrage can limit
its ability to exploit the price anomalies for a long period of time (Shleifer and Vishny,
1997).
If markets are price-inef?cient, then the concept of information ef?ciency, where
prices fully re?ecting all available information (Fama, 1970, 1991), does not seem so
relevant. Grossman and Stiglitz (1980) have also shown that the cost of information
gathering means ?nancial markets cannot be information-ef?cient.
The market portfolio as de?ned, for example, by the Standard and Poors (S&P) 500
index is sub-optimal (Arnott et al., 2005; Hsu, 2006) in the sense that it is not
mean-variance ef?cient, a criterion from modern portfolio theory (Markowitz, 1952;
Sharpe, 1964). That is, the broad market is inef?ciently priced relative to risk, which is
determined by the variances and co-variances of the returns of market securities.
Cost ef?ciency is the most common notion of ef?ciency in economics (Coase, 1998)
and we believe it is the most important notion. The superannuation market would be
cost ef?cient if the total cost in investing for all investors is the minimum, where the
total cost includes transaction costs, advisor fees, consultant fees, manager fees,
operating costs, taxes and any other costs which are deducted from investment returns.
In this paper, we suggest a way to substantially improve the cost ef?ciency of the
superannuation market.
Clearly, the market being inef?cient in so many ways can be exploited and many
individual investors have done this systematically and consistently over long periods
of time. Many, such as Madoff, have done this illegally and others, such as Buffett,
have done this legally and skilfully. The fact that some successful investors often
adopt similar approaches (Buffett, 1984 also appendix to Graham (1973)) indicates that
we are not simply being “fooled by randomness” (Taleb, 2005).
Marketing of well-documented investment successes by investment ?rms has
enticed many individual investors and superannuation funds to ?nd talented active
managers to beat the market on their behalf. Unfortunately, long-term empirical
studies (Bogle, 2005a; Malkiel, 2005) have shown this strategy of delegation to improve
investment performance has mostly been unsuccessful.
We argue that professional investment managers or institutions have not beaten the
market, because it is not their main objective as agencies to beat the market. Rather,
their main objective is to maximize business pro?ts and is not necessarily aligned with
that of their investors when they operate in the market with transaction costs,
competition, fund-size impediments and existing fee structures. We suggest this
principal-agent con?ict of interest can explain a number of observations of the
investment market.
Current attempts to explain many observed anomalies or market inef?ciencies are
based on theories of irrationality (Shiller, 2005) or bounded rationality (Lo, 2005) of
individual behaviour. However, it is self-evident that the behaviour of ?nancial
markets is determined largely by institutional intermediaries, and not by individuals.
For example, most pension assets in Australia are managed by ?nancial
intermediaries, not by individuals (Sy, 2008a).
The process of ?nancial intermediation introduces additional layers between
individual investment decisions and transactions in the markets. In many cases,
Low-cost
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individual investment objectives have no direct impact on the markets. Hence, overall
market behaviour is not determined by the aggregate action of optimizing individuals,
contrary to the assumptions of conventional equilibrium ?nance and economic
theories.
We explain observed market inef?ciencies by the rational behaviour of ?nancial
intermediaries, where we argue that the principal-agent con?ict of interest is resolved
in favour of professional investment managers optimizing their own welfare ahead of
optimizing the welfare of their clients. We suggest that professional investment
managers rationally do not attempt to beat the market because of information
asymmetry, fund-size impediment and asymmetric business incentives.
2.1 Information asymmetry
Inherent market volatility, or noise, requires many years of data for superior or inferior
investment performance to emerge as being statistically signi?cant. This leads to a
sort of market failure due to information asymmetry where investors and investment
advisers overly rely on short-term performance data to distinguish good or bad
superannuation funds, since there is inadequate disclosure of other information by
superannuation funds (Gallery and Gallery, 2006).
If a good long-term investment strategy involves potentially a few years of below
average performance, then the strategy may be rejected because of unacceptable
business risk to the investment manager. This leads to a strong incentive for a
manager with skills to limit risk-taking, since markets can remain irrationally priced
longer than the manager can remain solvent. The informational inef?ciency and
short-term focus lead to price-inef?ciencies, which professional managers do not
exploit for sound business reasons.
To illustrate the information problem from statistical sampling in superannuation
fund performance tables, let

S
t
be the capital value of a random sample of portfolios of
active investors at time t and S
t
is the corresponding capital value of the portfolios of
passive investors. Starting with equal initial capital values

S
0
¼ S
0
:
x
t
¼

S
t
S
t
ð1Þ
is a stochastic variable, which we assume, for illustrative simplicity, to be a Gaussian
variable diffusing under Brownian motion. This simplifying assumption is for the sake
of convenience of structuring the argument and for making rough conservative
estimates, rather than for quantitative accuracy. Non-normal stochastic variables
(Taleb, 2005) will further strengthen the conclusion we will draw. It is well-known that
the stochastic variable evolves in time according to:
z
t
¼ lnðx
t
Þ ¼
lnðx
0
Þ 2
_
mþ
1
2
s
2
_
t
s
??
t
p : ð2Þ
The variable is a unit normal variate, where the standard deviation s quanti?es the
uncertainty introduced by random sampling and the drift parameter m quanti?es the
non-random effect of active investment costs (Miller, 2007; French, 2008), which is
average rate of under-performance of unskilled active versus passive investors. Active
versus passive investing is dicussed in the next section.
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The probability that the asset weighted average return of the random sample of
portfolios is greater than the passive return is given by the probability that x
t
. 1 or
z
t
. 0, which has the value N(z
t
), given by the cumulative normal probability function.
Note the asymptotic property that N(z
t
) !0 as s !0 or t !1, that is, larger
samples and longer time periods lead to more certain results. Also the longer the
timeframe of return comparison, the less the error from randomsampling will matter in
demonstrating the stochastic dominance of the market return.
In Table I, we provide numerical examples of a few typical cases, where the drift
parameter for costs is 2 or 3 per cent and the dispersion of one-year returns of the
random sample has a standard deviation of 5 or 7 per cent. It can be seen that the
smaller the sample (the larger the standard deviation) and the lower the transaction
cost, the higher the probability that the random sample will be seen to beat the passive
investors.
Hence, sampling errors in empirical studies can potentially give a misleading, but
statistically inconclusive, impression that the group of active investment managers has
beaten the passive managers, at least in the short-term. However, in the longer term,
say 30 years or more, the compound effect of costs overwhelms any statistical error
from random sampling and the probability is very small that any group of active
investors will be seen to beat passive investors. These observations and explanations
are supported by empirical studies (Bogle, 1999).
In summary, we have shown that over short periods, say less than ?ve years or less,
groups of unskilled active investors could by chance beat passive investors due to
errors from random sampling. It could take up to 30 years, given our parametric
assumptions, for systematic effects to overwhelm statistical noise. Hence, it is dif?cult
to distinguish a good investment manager from a bad one based on short-term
statistical comparisons alone.
In the absence of more reliable information to make investment choices, the market
suffers from information asymmetry (Gallery and Gallery, 2006), leading to a failure to
distinguish good from bad products, for long-term performance. It is rational for
professional managers to chase only short-term opportunities to add value and not to
take unnecessary business risk in attempting a feat which is intrinsically dif?cult and
potentially unpro?table.
Cost ¼ 2 Cost ¼ 3
Return dispersion
(per cent)
Return dispersion
(per cent)
Years 5 7 5 7
1 33.5 37.4 26.6 32.1
3 23.1 28.9 14.0 21.1
5 17.1 23.7 8.1 15.0
10 8.9 15.5 2.4 7.1
15 5.0 10.7 0.8 3.6
20 2.9 7.6 0.3 1.9
30 1.0 3.9 0.0 0.6
50 0.1 1.2 0.0 0.1
Table I.
Probability (per cent) of
the average return of the
random sample
exceeding the passive
return
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2.2 Size and cost
It is intrinsically dif?cult for a large fund to outperform its smaller peers. Performance
comparison tables normally do not take into account fund size, potentially misleading
investors on how the average fund performs. For example, in a market with one
$10 billion fund and ten $1 billion funds, a 1 per cent under-performance by the large
fund could be balanced by a 1 per cent out-performance by each of the small funds,
suggesting a misleading conclusion that the average manager has beaten the market.
In the next section, we show that taking into account fund size, through
asset-weighted averages, is essential in understanding investment performance of fund
managers. In particular, we will prove that for a given period, if R is the market return
and

R is the asset-weighted average return of all investors, then:

R ¼ R 2t; ð3Þ
where t is the percentage transaction cost paid by active investors for trading. Since,
this relationship is so important and critical to the ?nancial services industry (Bogle,
2008), it deserves a careful, rigorous and formal treatment. For now, we illustrate how
the cost in this equation combined with a fund-size effect creates a daunting task for
active investment managers to better market returns.
In any given period, there are winners whose returns beat the market return and
there are losers whose returns are beaten by the market return. If we de?ne fund
capital amounts and average active excess returns net of costs of the group of winners
and the group of losers by C
W
, a
W
, C
L
, a
L
, respectively, then we have a mathematical
identity for capital changes:
CðR 2tÞ ¼ C
W
ðR þa
W
Þ þC
L
ðR þa
L
Þ ð4Þ
on account of equation (3). On noting that C ¼ C
W
þC
L
, we can simplify equation (4)
and write:
a
W
¼ 2t
C
C
W
2a
L
C
L
C
W
: ð5Þ
This result shows that due to the conservation of capital within the market and
presence of transaction costs, for there to be any winners at all (a
W
. 0), we need to
have losers, with negative excess return given by:
2a
L
. t
C
C
L
: ð6Þ
The ratio on the right of equation (6) is greater than one and a
L
, 0, by de?nition. For
there to be any winners, the losers have to bear all the transactions costs (including
those of the winners), say typically 2 per cent per annum, with their own below-market
returns. As winning capital increases, the losses sustained by losing capital have to
increase rapidly. For example, if two thirds of the active managers perform marginally
above the market, and if costs are 2 per cent for the period, then the losers have to
perform on average 6 per cent below the market, from equation (6). As the pool of loser
capital dwindles, the task of beating the market by the winners eventually becomes
impossible.
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This provides a rigorous explanation for the observation by Bogle (1999, p. 257) that
in professional investing “nothing fails like success”. As a fund manager succeeds,
more and more capital ?ows to the fund until the combination of fund size and
transaction costs eventually causes it to fail. If performance league tables of fund
managers were accurate, everyone will switch their money to top-performing funds,
leading to increases in funds’ size to such levels that they will eventually fail. Picking
winners from well-publicized performance league tables is unlikely to be a sound
long-term investment strategy, because of the costs in switching to chase yesterday’s
winners, whose winning streak is unlikely to persist.
A well-known historical example is the Fidelity Magellan Fund, which had a great
investment performance record in its earlier years, but grew to an enormous size of
more than $100 billion (Gross, 2005). Poor performances in the 1990s forced it to close
to new investors in 1997 and today the Magellan Fund is only about half of its peak
size.
2.3 Asymmetric incentive
Even before an active fund manager reaches a “too large not to fail” status, it is often
already hampered by a business asymmetric incentive, leading to a defensive strategy
called “closet indexing”. Empirical studies exist (Chevalier and Ellison, 1997) to
suggest that the level of risk-taking by mutual funds changes in response to incentives.
We will supplement such studies with an analytical model, which assumes that it is
rational for a large fund manager to have as its primary goal to stay in business by
retaining funds under management. It is only a secondary goal to increase funds under
management. This creates an asymmetric incentive, which is stronger for larger fund
managers.
If a manager performs poorly in the sense of returning less than the market,
investors will withdraw funds from the manager, leading to a threat to the primary
goal of retaining funds. On the other hand, if a manager performs well in the sense of
returning more than the market, investors will make additional contributions leading
to achievement of the secondary goal. Funds under management from existing
investors are generally much greater than their future contributions or new fund ?ows.
The asymmetric incentive leads to risk aversion on the part of the manager, relative to
benchmark performance and the phenomenon of “closet indexing”. This problem
occurs whether the manager has real skills or not, and it is more pronounced for large
fund managers.
To illustrate potential investment manager behaviour arising from asymmetric
incentives, we consider a simple model shown in Figure 1, where the reward for
performance relative to the benchmark is a linear function of the net active return. The
reward function r(x) is de?ned by:
rðxÞ ¼ b
þ
x x . 0
¼ b
2
x x , 0
ð7Þ
where the gradients are different but are both positive. A reward symmetry ratio can
be de?ned by r ¼ b
þ
=b
2
, where r , 1 and the smaller the ratio, the less symmetric
the reward incentive.
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The payoff P for an investment manager faced with risk-taking under such an
incentive structure is given from equation (7) by:
P ¼
_
1
21
rðxÞf ðxÞdx ¼
_
0
21
b
2
xf ðxÞdx þ
_
1
0
b
þ
xf ðxÞdx ð8Þ
where f(x) is a probability density function of the outcome from active risk-taking by
the investment manager. As our general conclusions will be robust against the details
of f(x), we assume for simplicity a shifted normal distribution:
f ðxÞ ¼
1
??????
2p
p
s
exp 2
1
2
x 2e
s
_ _
2
_ _
ð9Þ
where e is the expected net active return (due to skills or con?dence) of the manager
and s is the volatility or deviation from expectation from risk-taking relative to the
market benchmark. It is straightforward to evaluate the integral in equation (8) given
equation (9) to ?nd the ratio of the payoff to the risk taken as:
P
s
¼ b
þ
kNðkÞ þb
2
kNð2kÞ þðb
þ
2b
2
Þ
exp
_
2
1
2
k
2
_
??????
2p
p ð10Þ
where N(k) is the unit cumulative probability function and we have introduced k ¼ e=s,
which is the ratio of expected net return to the risk taken.
If the manager has negative expectations such that e # 0, an expected excess net
return which is not positive, the ?rst two terms in equation (10) are non-positive (k # 0)
Figure 1.
Model of active return and
asymmetric reward
Net active return
4.0 3.0 2.0 1.0 0.0 –1.0 –2.0
P
r
o
b
a
b
i
l
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y

d
e
n
s
i
t
y

o
f

a
c
t
i
v
e

r
e
t
u
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n

a
n
d

r
e
w
a
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d

f
u
n
c
t
i
o
n
0.5
0.4
0.3
0.2
0.1
0.0
–0.1
–0.2
–0.3
–0.4
–0.5
Expected net active return
Gain from out performance
Loss from under performance
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and the last term is negative ðb
þ
, b
2
Þ. Hence, the manager’s optimal strategy is to
take no risk relative to the benchmark to avoid a negative payoff.
Even if the manager has positive expectations in the sense that e . 0, it may still be
rational for the manager not to take risk against the benchmark because of reward
asymmetry. Using the symmetry ratio r ¼ b
þ
=b
2
and the mathematical identity
Nð2kÞ ¼ 1 2NðkÞ, we can write equation (10) also as:
P
b
2
s
¼ k þðr 21Þ kNðkÞ þ
exp
_
2
1
2
k
2
_
??????
2p
p
_ _
ð11Þ
The ?rst term on the right hand side is positive k . 0 since e . 0 on assuming positive
expectation. The second term is negative since r , 1 due to asymmetric incentive.
Whether the manager should take risk s . 0 to beat the benchmark depends on
whether e is large enough to make the right hand side of equation (11) positive. Let us
assume the manager takes risk ef?ciently so that s ¼ t þ e, where t is the cost of
active management. For given symmetry ratio r and cost t, we can solve for e which
makes the right hand side of equation (11) is greater than zero and obtain the excess net
and gross return hurdles which make active investment a rational proposition.
A sample of the results is shown in Table II, where we see that high cost t . 2 per cent
and low-reward symmetry r , 0.4 would make active management unattractive.
When a fund is newly established and asset size is low, the reward symmetry ratio
is closer to one because the reward of fund ?ow due to investment performance is
more symmetrical. As a fund becomes well established with a larger asset size, the
symmetry ratio would become substantially smaller and the risk of money out?ow
due to under-performance is not adequately compensated by the reward of money
in?ow from out-performance. Hence, as asset size becomes large, asymmetric incentive
increases and the symmetry ratio falls, it is logical to reduce cost, target a lower excess
return and take a lower level of risk.
The tendency to “hug benchmarks” has been widely observed in the industry,
leading to the phenomenon called “closet indexing”, where active management fees
have been charged for largely passive management (French, 2008). Since the portfolios
of many active investment managers are de facto substantially passive relative to their
benchmarks, the cost of residual active management is very high (Miller, 2007). Indeed,
the market may already be de facto substantially passively managed, but paying
substantially higher active fees. The attempt to “separate alpha” by pension funds
where active fees are paid for only active management, and lower passive fees are paid
for passive management, has led to the rapid growth of hedge funds in recent years.
A similar explanation can also be advanced for the emergence of boutique fund
managers in Australia and elsewhere. An investment manager who has established
Cost in per cent
Reward symmetry ratio 1 2 3
0.2 2.8 5.5 8.2
0.4 1.6 3.2 4.7
0.6 1.3 2.5 3.8
0.8 1.1 2.2 3.3
Table II.
Excess gross return
hurdle to overcome cost
and asymmetric incentive
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a record of good returns for several years with limited investor capital attracts substantial
fund in?ows. The funds under management become larger until performance falters, due
to asset size and cost effects discussed above. Realising the impossibility of the task, the
manager resigns fromthe large-size fundwhichpaidgoodbonuses until thenandsets upa
boutique investment ?rmwith a much smaller amount of investor capital to invest, but is
now compensated by a large equity stake.
3. The cost matters theorem
Cost has been central to our arguments about inef?ciencies in markets and in
particular, equation (3) has been used extensively in the above discussion. Verbal
statements which are more or less equivalent to equation (3) have been made by others
such as Sharpe (1991, p. 7): “the return on the average actively managed dollar will
equal the return on the average passively managed dollar” before costs, and Bogle
(2008, p. 98): “Gross return in the ?nancial markets, minus the costs of the system
equals the net return actually delivered to investors.”
The question of cost is “so important that it should be hard to think about anything
else” (Bogle, 2008, p. 95). However, verbal statements on cost lack the precision, clarity
and certainty of formal proofs, which are particularly important in clarifying the
assumptions made. The cost question deserves a more thorough treatment. Moreover,
important theorems in mathematics are often proved in several ways, because different
methods of proof provide different insights. We show from our proof below that it is
almost free of assumptions according to “relentless rules of humble arithmetic” (Bogle,
2005b, p. 22 and that important insights gained in the process about the nature of
passive investing will be applied in new ways.
There is no precise universal de?nition for the market, which could include any
traded securities. We will show later that by itself this is an important insight. To make
our discussion simpler and more concrete, we restrict ourselves initially to stocks, as
the generalisation to include other securities is straightforward.
We de?ne a stock market as consisting of a couple of objects: a set of listed company
stocks and the set of all investors or traders who deal in those stocks; so depending on
different sets of selected company stocks, there are different stock markets. After
de?ning a particular stock market, we might subsequently refer to it as the stock
market for convenience. The notional portfolio which includes all stocks of the market
is de?ned as the market portfolio. The market portfolio is owned collectively by all
investors.
Consider a stock market with N listed companies, the market capitalisation C
t
at
time t is de?ned as:
C
t
¼

N
i¼1
n
i
p
it
ð12Þ
where n
i
and p
it
are, respectively, the number of outstanding shares in i-th stock and its
stock price at time t.
Ignoring dividends for the moment for simplicity, the market return R
t
for one
period starting at time t is de?ned by:
R
t
¼
C
tþ1
2C
t
C
t
: ð13Þ
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We show ?rstly that the market return de?ned in equation (13) is the capitalisation
weighted average return of individual stocks. The capital weight w
it
of i-th stock in the
market portfolio at time t is de?ned by:
w
it
¼
n
i
p
it
C
t
ð14Þ
and its return r
it
for one period starting at time t is de?ned by:
r
it
¼
p
iðtþ1Þ
2p
it
p
it
: ð15Þ
Combining equations (12) and (13) then using de?nitions (14) and (15), we ?nd:
R
t
¼

N
i¼1
n
i
p
it
C
t
p
iðtþ1Þ
p
it
21
_ _
¼

N
i¼1
w
it
r
it
: ð16Þ
The market return is by de?nition the return of the market portfolio; it is also equal to
the capital weighted average return of individual stocks. Note that the market return
de?ned here has nothing to do with any conception of market equilibrium or the
market return of the capital asset pricing model (CAPM). The CAPM market portfolio
(Sharpe, 1964; Lintner, 1965) is an abstractly de?ned concept based on optimising
behaviour of individual investors in a costless world. There is no reason to suppose our
market return is mean-variance ef?cient, as Hsu (2006) has shown that capital
weighted portfolios are mean-variance sub-optimal.
We provide two formal proofs below: the longer proof which is perhaps more intuitive,
requires the notion of stock returns, whereas the shorter one does not. Consider the stock
market having a total of Minvestors. Let the j-th investor own u
ijt
number of the i-th stock
at time t. The value of the investor’s portfolio v
jt
at time t is de?ned by:
v
jt
¼

N
i¼1
u
ijt
p
it
ð17Þ
which represents a fraction f
jt
of the total market given by:
f
jt
¼
v
jt
C
t
: ð18Þ
Since the sum of all investor holdings of a particular stock must equal to the number of
outstanding shares in that stock, we have:
n
i
¼

M
j¼1
u
ijt
: ð19Þ
Equations (17) and (19) together imply the obvious mathematical identity that the sumof
all investor capital

C
t
equals the total market capital:
C
t
¼

C
t
¼

M
j¼1
v
jt
; ð20Þ
where the right hand equality is a de?nition of total investor capital.
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The j-th investor’s return r
jt
for the period starting at time t can be shown to be
given by:
r
jt
¼
v
jðtþ1Þ
2v
jt
v
jt
¼

N
i¼1
u
ijt
p
it
v
jt
r
it
ð21Þ
The coef?cient ðu
ijt
p
it
=v
jt
Þ ahead of the stock return r
it
is the capital weight of the stock
relative to the investor’s total portfolio value. For now, we can think of prices changing
due to trading at the end of the period, which normally requires the existence at least
some active investors. In general, for an active investor, these capital weights will not
be equal to the corresponding capital weights of the market portfolio. The individual
investor returns will be different from the market return given by equation (16).
However, the asset weighted average return

R
t
of all investors for the period
starting at time t is given by:

R
t
¼

M
j¼1
f
jt
r
jt
ð22Þ
where the quantities under the summation are de?ned by equations (18) and (21).
Substituting the de?nitions (18) and (21) in equation (22) and interchanging
summations, we ?nd:

R
t
¼

N
i¼1

M
j¼1
u
ijt
p
it
C
t
r
it
¼

N
i¼1
n
i
p
it
C
t
r
it
¼

N
i¼1
w
it
r
it
ð23Þ
where we have used the identity (19).
Comparing equations (16) and (23), we have proved the asset weighted average
return

R
t
of all investors (without needing the concept of active or passive investing) is
exactly equal to the market return R
t
of the market portfolio:

R
t
¼ R
t
, assuming no
trading between investors over the period. Again this result is a mathematical identity
and is valid independent of assumptions about stock price volatility, investor risk
preferences and market ef?ciency. Note that the result is valid not only for the whole
market (containing all stocks), but also it is valid for any subset of stocks which we
might like to de?ne as “the market”.
For a shorter proof, we note that in a frictionless market, without costs, market
capital is conserved despite changes in shareholdings among investors in the sense
that from equation (20):
C
tþ1
2C
t
¼

M
j¼1
ðv
jðtþ1Þ
2v
jt
Þ; ð24Þ
which provides another way of proving the equality: R
t
¼

R
t
. To see this, we divide
both sides of equation (24) by to obtain from equations (18) and (21):
C
tþ1
2C
t
C
t
¼

M
j¼1
v
jt
C
t
v
jðtþ1Þ
2v
jt
v
jt
¼

M
j¼1
f
jt
r
jt
: ð25Þ
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This equation leads directly to R
t
¼

R
t
on applying equations (13) and (22). Essentially,
the statement is a direct consequence of the law of conservation of capital. Trading or
active management does not create capital but merely re-distributes it.
The conservation of capital holds only if there is no trading or if the trading is
costless over the period. Otherwise, if investors traded stocks inter-period or over the
period, then for the market as a whole, capital gains and losses for market timing or
stock selection cancel, leaving investors in aggregate with a net loss from brokerages,
other trading costs and taxes from realising capital gains. The asset weighted average
return

R
t
of all investors for the period t is given by:

R
t
¼ R
t
2t
t
: ð26Þ
The market return is R
t
and t
t
denotes the impact of transaction costs and capital gain
taxes, leading to a loss of capital of t
t
C
t
over the period. Note this, is the actual total net
market return for all investors for when price changes from trading only occur between
periods. The result also holds when there is inter-period trading (see the Appendix). All
quantities in equation (26) would vary with time. But since we expect t
i
. 0 in any
period, we would conclude

R
t
, R
t
for all time periods. The total capital gain (and
dividend income) of all investors must be less than the capital gain (and dividend
income) of the total market, when costs are included.
Our result about cost is more than a hypothesis, as in Bogle’s “Cost matters
hypothesis,” it is a mathematical certainty and hence it should be called a “Cost matters
theorem” in accordance with the “relentless rules of humble arithmetic” (Bogle, 2005b,
p. 22; Sharpe, 1991).
4. Proportionate shareholding
There are two key insights from our proof of the “Cost matters theorem”. First, it does
not matter what set of securities we choose to de?ne as the market, the asset-weighted
average return of all investors cannot exceed the market return. Second, the collective
optimal return for all investors is achieved when trading and transaction costs are
therefore minimized. We show here a fair and simple way for all investors to
collectively own all the securities of the market, with everyone getting a fair share of
the market return, which is also the collective maximum.
Consider an investor who holds a constant fraction p of the outstanding shares of
every stock, then the number of the i-th stock held is pn
i
and the portfolio value v
t
of
the investor at time t is given by:
v
t
¼

N
t¼1
pn
i
p
it
¼ pC
t
: ð27Þ
Clearly, by holding a constant fraction p of the outstanding shares of every stock, the
investor holds the same fraction p ¼ v
t
=C
t
of total market capital. The return of such
a portfolio r
t
for the period starting at time t is equal to the market return R
t
, since
from equations (13) and (27):
r
t
¼
pC
tþ1
2pC
t
pC
t
¼ R
t
: ð28Þ
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This is the PSA to obtaining exactly the market return, which can also include
dividend income over the period. It is important to note that the strategy consists
simply of holding the same proportion p of the outstanding shares of every stock in the
market.
This strategy is independent of individual company prices or capitalisations. To
mention prices, capitalisation and portfolio weights is to make a very simple idea
unnecessarily complicated in the ?rst instance. We only need to know the total number
of outstanding shares for each stock to easily construct such a portfolio. Once this
simple, but profound idea is grasped, we can go beyond capitalisation weighted
indexing to a truly passive and highly ?exible approach to collective share ownership.
As an example, consider how we construct a two-company share portfolio which
does not involve trading and rebalancing as their prices ?uctuate. Assume two
companies: BRKA and ABC. BRKA has 1.55 million shares with a capitalisation of
$281 billion and ABC has 161 million shares with a capitalisation of $6.2 billion on a
given day. Under the PSA, a portfolio with a 10 per cent constant proportion of the
market would hold 155,000 million BRKA and 16.1 million ABC shares. The initial cost
of acquiring such a portfolio obviously depends on individual stock prices, which may
vary over the period of construction. In the case of quick construction, we pay
approximately $28.1 billion for BRKA and $620 million for ABC. The same fractional
ownership of companies implies the same fractions of shares and therefore the same
fractions of capital of the companies.
There is no subsequent maintenance cost of the portfolio, which has no need to
rebalance due to price ?uctuations. Portfolio adjustments are needed rarely, only when
the numbers of outstanding shares change due to corporate actions such as new
issuances or buy-backs. Obviously, for a given set of stocks, our approach is equivalent
to a capitalisation weighted approach, for which we have provided a formal proof in
the previous section (equation (16)). However, the emphasis on capital weighting is an
unnecessary distraction from the basic ownership concept because weighting is a
portfolio construction concept due to starting from a ?xed set of different stocks.
Passive investing does not require the use of commercial indices.
We emphasize our approach is categorically not indexing, because indexing refers
now to any rule-based approach to investing, which may be active or passive. The
recently created research af?liates fundamental index (Arnott et al., 2005) is active
because regular trading is required to rebalance the portfolio and the portfolio is not
fully scalable. Even some capitalisation weighted index funds can be quite active in
portfolio rebalancing, if performance is simulated using only a subset of stocks in the
index.
Our PSA portfolio is truly passive, because there is no turnover from rebalancing
due to price ?uctuations and there is no possibility of deviation from the market return.
Moreover, we have a highly ?exible concept of the market due to our abandonment of
the index concept, which is unnecessarily restrictive in ?xing the set of stocks.
An index generally refers to a ?xed de?nition for portfolio construction, such as the
Dow-Jones industrial average (DJIA) created in 1896 or S&P indices: S&P90 created in
1923 and S&P500 created in 1957. These indices are useful for a variety of purposes
including investment comparison and ?nancial research and we expect they will
continue to perform their useful roles. However, they have inherent limitations for
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purely investment purposes due largely to pre-de?ned universes of stocks. Such
indices are not necessary for passive management.
The selection of stocks to include in an index is based on criteria adopted by the
indexer, which might not coincide with those of investors. For example, the set of
different stocks (too few or too many), the types of stocks (e.g. for ethical investment) or
stock weight adjustments (for liquidity reasons) might not suit some investors. Hence,
if we adopt a particular index, we are forced to invest in ways which are dictated by
some outside agency that pays no heed to our particular investment objectives.
Using PSA, we could create a fully passive and highly diversi?ed portfolio which
excludes certain stocks that may be regarded as unethical (or socially irresponsible) or
redundant to our approach, such as exchange traded funds or some listed investment
companies. There is also no need to set arbitrary numerical upper or lower limits to the
set of stocks the portfolio can hold, causing unnecessary trading when stocks enter or
exit the portfolio with a ?xed number of included stocks. The portfolio structure can
vary with the number and selection of stocks over time, depending, for example, on
cash ?ows or corporate action, but at all times maintaining the principle of passivity.
Let us consider a simple example to differentiate our approach from standard
cap-weighted indexing. Suppose a large-cap stock with about 3 per cent weight of an
equity index went bankrupt, having its share price written down to near zero.
This would induce an ASX/S&P 200 indexer to sell down fractions of holdings of 199
stocks and buy a new stock, incurring transaction costs. For a $100 billion fund with
0.1 per cent brokerage, the cost would be about $3 million. In our approach, as the
precise number of different stocks in our portfolio is unimportant, we need to do
nothing in this case, thus saving $3 million in transaction costs.
The PSA portfolio is totally passive and fully scalable from a relatively small asset
size all the way to the whole market (scalable in the full range: 0 # p # 1) and it is
available to all investors, at least through a collective arrangement if individual asset
sizes are small. Note that unless the relative stakes are the same, with a constant
proportion for every stock, the approach will not be fully scalable (e.g. DJIA, which is
not cap-weighted) and the return will not equal the market return as de?ned by
equation (13). The PSA portfolio has an identical structure to the market portfolio.
In summary, PSA is a passive investment method which emphasises collective
ownership of companies. The PSA is ?exible in terms of the number or selection of
securities so that its portfolio structure cannot be easily anticipated or replicated. Some
residual uncertainty is important in practice as it prevents short-term gaming by the
rest of the market which could otherwise take advantage of the trading activities of
what could be a portfolio of enormous asset size. The concept of index tracking error is
also inappropriate for PSA, as we are unconcerned by short-term differences in
performance against any particular benchmark index. Performance appraisal of our
PSA portfolio should be based on comparisons against the performance of aggregate
portfolios of similar asset sizes.
5. A national default option
We have proved that the totally passive PSA is the collective optimal strategy for all
investors, because it minimizes the total cost in an inef?cient market. Even though the
strategy is not necessarily optimal for every individual investor, it is optimal for most
individuals who need to use directly or indirectly (through collective investments)
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professional investment managers as ?nancial intermediaries. We discuss how PSA
can be used to create a national default option for low-cost superannuation and the
rationale for government intervention to use this product to stimulate greater
competition in the superannuation industry.
In Section 2, how an inef?cient market has been perpetuated was explained. This
market failure may have led many individual investors who have suf?cient resources to
withdrawfromthe market and to do it themselves inself-managedsuperannuation, which
is the fastest growing sector of the industry. A recent study (Sy, 2008a) suggests the
superior performance of self-managed small APRA funds is partly due to most small
funds having much simpler and more cost-ef?cient structures. However, the majority of
workers under 55 years of age lack suf?cient assets to economically run their own
portfolios, and they are mostly unable to escape the high cost of institutional
superannuation.
What matters most for future national ?nances is to have the maximum collective
retirement savings accumulated over time and to have a relatively narrow spread of
individual savings (Burtless, 2003) to minimize the number of individuals in the lower tail
of distribution requiring public assistance. We suggest that a national default option
based on PSA is the optimal solution to achieve those objectives for future national
?nances.
The current approach to default options offered by superannuation funds may be
sub-optimal (Gallery et al., 2004), because different default options are offered by different
funds and there is generally only one default option per fund for all demographics within
that fund. Basu and Drew (2006) suggest that there needs to be much heavier tilting
towards growth assets, particularly for young workers, to maximize wealth accumulation
over the long-term. It may be important to take into account a worker’s age and wealth
level to design a more appropriate default option for that worker.
We suggest the creation of a retirement growth account (RGA), which is a national
growth fund managed on PSA principles, providing a low-cost, high-growth product,
universally available to all individuals for superannuation. This pension product will
run in a similar way and complement the existing retirement savings account (RSA)
guaranteed by the government and operated by authorised service providers such as
banks. Combinations of RGA and RSA could provide ?exible alternatives in addition
to current default options offered to workers by their employers.
For example, a simple version of a national default option would be for employers to
direct the contributions from the super guarantee levy on behalf of their employees to
both the RSA and RGA. The contribution for each employee could be split with (115 –
age) per cent directed to a RGA and the balance to a RSA, based on the investment
life-cycle assumption. Other rule-based variations in asset allocation are also possible.
The RGA concept can be extended to include other asset classes, such as international
shares and ?xed interest securities.
Such a national default option provides a simple, very low-cost product. Retail index
funds in Australia have typical management expense ratios of around 0.4 per cent of
assets per annum (Vanguard, 2008). With higher passivity of PSA, increased stability
and greater economies of scale of the national default option, fees could be reduced
by more than half. The approach could potentially save all investors collectively up to
$20 billion per year (depending on the uptake), on total superannuation assets
exceeding $1 trillion.
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Another important bene?t of the proposed national default option scheme is that it
serves as an actual performance benchmark (net of fees and taxes) against with which
other offerings could be compared to promote a competitive market. More complex and
sophisticated products will continue to exist as ?nancial intermediaries continue to
seek abnormal pro?ts through innovative products. However, the existence of a
benchmark, such as the national default option, will encourage market forces to operate
more competitively and ef?ciently.
It is commonly argued that active investing increases liquidity helping the price
discovery process and therefore improves price-ef?ciency of the market, whereas
passive investing does not contribute to this process and it is therefore a “free rider”
obtaining an unpaid-for bene?t. The suggestion that more passive investing would
lead to greater price-inef?ciency has little theoretical or empirical support. The
empirical evidence is just the opposite: market bubbles tend to occur with very high
levels of turnover, and subsequent busts prove over-trading caused price-inef?ciency.
Theoretically, both information-inef?ciency and price-inef?ciency could rationally
induce the “herd” instinct, as evident in chartists’ momentum trading, which could lead
to market instabilities (Chiarella et al., 2008).
Price volatility and liquidity are important risk factors for speculators, particularly
those who are highly leveraged. The risk factors are of little concern to the long-term
investor who is seeking the rewards of business earnings and dividends over a long
period of time. We could hardly express the matter better than Keynes (1936, p. 155):
Of the maxims of orthodox ?nance none, surely, is more anti-social than the fetish of liquidity,
the doctrine that it is a positive virtue on the part of investment institutions to concentrate
their resources upon the holding of “liquid” securities. It forgets that there is no such thing as
liquidity of investment for the community as a whole. The social objective of skilled
investment should be to defeat the dark forces of time and ignorance which envelop our
future.
Instead, better long-term investment returns can only be obtained ultimately from
greater business pro?ts, through better business operations and investments.
Shareholders can increase their chance of real wealth creation through their demand
for better corporate governance (Bogle, 2005a). We note in passing that the stable
shareholdings implied by the PSA can be used to facilitate the creation of a registry for
the ef?cient transfer of voting rights to a new class of proxy capitalists, leading to a
new form of representative capitalism.
6. Summary
In this paper, we have argued that market inef?ciencies have not been exploited
effectively due to the process of ?nancial intermediation. Professional investment
managers have inherent con?icts of interest between short-term business pro?ts and
long-term performance of client portfolios. The information asymmetry that exists due
to the low-information content of short-term performance statistics makes the
identi?cation of superior investment managers dif?cult for most investors (Sy and Liu,
2009). This informational inef?ciency leads to a relative absence of vigorous
competition in the market for the services of professional investment managers and
perpetuates the structural inef?ciency of the market.
We have argued that the rationale for passive investing for individuals is not that
the market is ef?cient (Malkiel, 2005), but because the market is inef?cient. Nor is the
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rationale based on the supposition that individual active investors are likely losers to
professional investment managers who are likely winners due to better information
and greater skills (Thorley, 2002). While individuals can potentially delegate their
investments by hiring active managers, we suggest ?nancial intermediation fails to
exploit the market inef?ciencies because of information asymmetry and adverse
manager incentives. Instead, our rationale for passive investing for individuals rests on
maximizing net return though minimizing costs (Bogle, 2008) in a structurally
inef?cient market.
By converting Bogle’s (2005b) “Cost matters hypothesis” to a “Cost matters
theorem” through a formal mathematical proof, we show that minimizing the total cost
of investing leads to a collective optimal investment strategy for all investors. As
passive investing does not require the use of commercial indices, we propose the PSA
which minimizes cost and is totally passive, fully scalable and available to all
investors. In contrast to approaches based on commercial indices, the PSA is more
passive and more ?exible, thus enabling transaction costs to be reduced to a minimum.
In relation to pension savings, creation of a RGA fund using the PSA principles
would provide all workers with the opportunity to gain exposure to growth assets at
minimum costs. The RGA, together with the low-risk RSA, could provide the basic
elements of a national default option scheme. Like the RSA, the RGA can be operated
by selected authorised private-sector service providers under prescribed mandates by
the government.
Such a national default option scheme based on passive investing strategy of the
PSA can save the superannuation system more than 1 per cent of total assets per year,
which amounts to around $10 billion or more per year. Even if only half of total
national assets are managed in the manner proposed, the impact on ultimate national
savings over the long-term is substantial. Clearly, the cost reduction is achieved
through removal of layers of ?nancial intermediation, which functions largely to
re-distribute wealth, rather than create it. The global ?nancial crisis serves as a
powerful and painful reminder of this fact.
The envisaged national default option will provide a simple and easy-to-understand
alternative, in addition to all other alternatives, available to all individuals, particularly
those in the early stages of wealth accumulation for retirement. It would be particularly
useful for employers who do not want to nominate a default fund or who prefer to use a
ready-made choice for their employees. It provides a choice for people who do not want
to make choices. Importantly, the national default option will provide a performance
benchmark to promote a competitive and ef?cient superannuation industry.
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Appendix
We provide a proof of equation (26) when there is inter-period trading. The relationship (26) is
obviously correct when trading occurs only between periods, when some stocks are assumed to
be traded and portfolios are adjusted at the end of one period and held ?xed until the end of the
next period.
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To show that the result is true also when there is inter-period trading, we assume that for a
given period t in equation (26) there are n sub-periods, which need not be regularly spaced, when
there is no trading inter-period and trading only occurs between sub-periods, then equation (26)
holds (including dividends) for each sub-period:
R
i
¼ R
i
2t
i
ði ¼ 1; 2; . . . ; nÞ ðA1Þ
The net return of all investors R
t
for the period t is de?ned by:
R
t
¼

n
i¼1
ð1 þR
t
Þ 21 ðA2Þ
Substituting equation (A2) into equation (A1) and using a similar de?nition as equation (A2) for
R
t
, we have proved equation (26) holds with inter-period trading, except that the transaction cost
t
t
has to be determined by costs incurred between sub-periods:
t
t
¼

n
i¼1
ð1 þR
i
Þ 2

n
i¼1
ð1 þR
i
2t
i
Þ ðA3Þ
To leading orders, assuming small sub-period transaction costs t
i
ði ¼ 1; 2; . . . ; nÞ the total
transaction cost t
t
of the period t is given by:
t
t
¼

n
i¼1
t
i
1 2

n
j¼1; j–i
R
j
_ _
ðA4Þ
Corresponding author
Wilson Sy can be contacted at: [email protected]
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This article has been cited by:
1. Wilson Sy. 2011. Redesigning Choice and Competition in Australian Superannuation. Rotman
International Journal of Pension Management 4, 52-61. [CrossRef]
2. Wilson Sy, Kevin Liu. 2010. Improving the Cost Efficiency of Australian Pension Management. Rotman
International Journal of Pension Management 3, 38-47. [CrossRef]
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