Morningstar Magazine - August/September 2017 - 43

Strategies

The Inefficient Frontier
Risk and reward are not necessarily
linked.

QUANT U

Paul D. Kaplan

How often is it said, "If you want more return, you
have to take more risk"?
In this edition of Quant U, I illustrate why for most
portfolios, this is not the case.
Efficient and Inefficient Portfolios
Arguably, the most important development in
finance was Harry Markowitz's mean-variance
model for portfolio construction, commonly called
modern portfolio theory, or MPT. Markowitz
published his theory in a paper in 1952, which he
later expanded into a book in 1959.1 In 1990,
Markowitz shared the Nobel Memorial Prize in
Economic Sciences for his groundbreaking work.

Markowitz's main insight was that investors care
about both risk and expected return. Before
Markowitz, no one explicitly took risk into account
when constructing a portfolio. Markowitz made
risk central to the analysis. His other great
insight was that up to a point, risk can be reduced
through diversification. A portfolio that has the
lowest possible risk for a given level of expected
return is said to be efficient.
This notion of efficient portfolios led to the idea
of the efficient frontier. By plotting each level
of expected return against the corresponding
lowest level of risk possible with long-only
portfolios, we have a positively sloped curve that
shows a trade-off between risk and expected
return (E X H IB IT 1 ).

It is important to remember that the portfolios
represented on the efficient frontier are,
well, efficient. Therefore, it is only if an investor
is holding an efficient portfolio that achieving
a higher level of expected return necessarily
means increasing the level of risk by moving to a
riskier efficient portfolio. In other words, the
trade-off between risk and expected return is only
among efficient portfolios.
Getting MPT Wrong: The SRRI
A common misunderstanding of MPT is that the
trade-off between risk and expected returns
holds for all portfolios, and not just efficient ones.
This is unfortunate because by definition, nearly
all portfolios are inefficient. This is especially
true for single-asset-class portfolios, such as most
funds. Generally, only a well-constructed
multiasset-class portfolio (using multiple funds
or a fund that holds multiple asset classes)
approaches efficiency.

This misunderstanding of MPT is embodied in the
Synthetic Risk and Reward Indicator, or SRRI.
Fund regulators in the European Union require the
SRRI to appear on the Key Investor Information
Document, or KIID, which must be presented to
potential investors for each fund they are
considering. In most cases, the SRRI is based
on a five-year standard deviation that is mapped
into an integer between one and seven using
a fixed set of breakpoints. EXHIB IT 2 shows how
it is displayed on the KIID.
As E X HIB IT 2 shows, the SRRI claims to not
only indicate the level of risk, but also to indicate
"typically lower rewards" or "typically higher

rewards." Not only does this fly in the face of MPT,
properly understood, it flies in the face of reality.
EXHIB IT 3 is a scatter of the five-year standard
deviation compared to the five-year compound
annual return on the oldest share class of all
European funds that are denominated in euros and
that have five years of monthly returns over
the period June 2012 to May 2017. As you can
see, there is no clear relationship between risk and
return beyond there being greater variation
of returns among the riskier funds. Hence, the
SRRI, which carries the weight of regulatory
authority, may inadvertently nudge investors up
the risk spectrum of risk in their search for
higher returns. This is surely not the regulators'
intention, but it is a potential consequence
of misunderstood theory.

The Inefficient Frontier
Returning to EXHIB IT 1 , we see that not only is
there an efficient frontier, which gives the
trade-off between risk and expected return, but
also an inefficient frontier, which shows the
riskiest portfolio for each level of expected return.
In other words, it shows for each level of expected
return what the worst long-only portfolio is.
Therefore, all long-only portfolios fall between the
efficient and inefficient frontiers.

There are a few notable points about the
inefficient frontier:
1 The inefficient frontier has the same end points

as the efficient frontier. Consequently, these
two frontiers wrap all possible long-only portfolios.
2 The inefficient frontier consists of:
a The securities or asset classes with the worst
combinations of risk and expected return.
b Portfolios of pairs of consecutive securities or asset
classes that are the most inefficient. Hence,
just as the efficient frontier represents the
most diversified portfolios, the inefficient frontier
consists of the most concentrated portfolios.
The existence of the inefficient frontier should
serve as a reminder that not all portfolios
are efficient, and thus, there is no trade-off

1 The paper is "Portfolio Selection," Journal of Finance, Vol. 7, No. 1 (1952), pp. 77-91. The book is Portfolio Selection, Oxford: Basil Blackwell, 1959, reprinted in 1991.

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