Names (codes) of the asset classes are as follows: S&P 500 (SPX), Russell 2000 (RTY), DJ Euro
Stoxx 50 (EUR), FTSE 100 (GBP), Topix (JPY), MSCI Latin America (MSCI-LA), MSCI Emerging
Markets Europe (MSCI-EME), MSCI AC Asia ex Japan (ASIA), JP Morgan Global Govt Bond Euro
(EUR-BND), JP Morgan Govt Bond US (USD-BND), ML US High Yield Master II (USD-HY), JP
Morgan EMBI Diversied (EM-BND), S&P GSCI (GSCI).
Results of the historical backtests are summarized in Table 6 and cumulative
performances represented in gure 2. The hierarchy in terms of average returns,
risk statistics, concentration and turnover statistics is very clear. The ERC portfolio
performs best based on Sharpe ratios and average returns. In terms of Sharpe ratios,
the 1/n portfolio is largely dominated
by MV and ERC. The dierence between
Notice that for the ERC portfolio, turnover and concentration statistics are here
superior to the ones of the previous example, which corroborates the intuition that
these statistics are increasing functions of heterogeneity in volatilities and correlation
A perceived lack of robustness or discomfort with empirical results have led investors
to become increasingly skeptical of traditional asset allocation methodologies that
incorporate expected returns. In this perspective, emphasis has been put on mini-
mum variance (i.e. the unique mean-variance ecient portfolio independent of return
expectations) and equally-weighted (1/n) portfolios. Despite their robustness, both
approaches have their own limitations; a lack of risk monitoring for 1/n portfolios
and dramatic asset concentration for minimum variance ones.
The dramatic drawdown of the 1/n portfolio in 2008 explains to a large extent this result.
the various components of the portfolio. This way, we try to maximize dispersion
of risks, applying some kind of 1/n lter in terms of risk. It constitutes a special
form of risk budgeting where the asset allocator is distributing the same risk budget
to each component, so that none is dominating (at least on an ex-ante basis). This
middle-ground positioning is particularly clear when one is looking at the hierarchy
of volatilities. We have derived closed-form solutions for special cases, such as when
a unique correlation coecient is shared by all assets. However, numerical opti-
mization is necessary in most cases due to the endogeneity of the solutions. All in
all, determining the ERC solution for a large portfolio might be a computationally-
intensive task, something to keep in mind when compared with the minimum vari-
ance and, even more, with the 1/n competitors. Empirical applications show that
equally-weighted portfolios are inferior in terms of performance and for any measure
of risk. Minimum variance portfolios might achieve higher Sharpe ratios due to lower
volatility but they can expose to higher drawdowns in the short run. They are also
always much more concentrated and appear largely less ecient in terms of portfolio
Empirical applications could be pursued in various ways. One of the most promis-
ing would consist in comparing the behavior of equally-weighted risk contributions
portfolios with other weighting methods for major stock indices. For instance, in
the case of the S&P 500 index, competing methodologies are already commercialized
such as capitalization-weighted, equally-weighted, fundamentally-weighted (Arnott
et al. ) and minimum-variance weighted (Clarke et al. ) portfolios. The
way ERC portfolios would compare with these approaches for this type of equity
indices remains an interesting open question.
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A.1 The MV portfolio with constant correlation
Let R = C
if i = j and
. We may write the covariance matrix as follows: Σ = σσ
. We have
(n − 1) ρ
− (n − 2) ρ − 1
With these expressions and by noting that tr (AB) = tr (BA), we may compute the
MV solution x = Σ
1 /1 Σ
. We have:
Let us consider the lower bound of C
This solution is exactly the solution of the ERC portfolio in the case of constant
correlation. This means that the ERC portfolio is similar to the MV portfolio when
the unique correlation is at its lowest possible value.
A.2 On the relationship between the optimization problem (7) and
the ERC portfolio
The Lagrangian function of the optimization problem (7) is:
f (y; λ, λ
(y; λ, λ
) = ∂
σ (y) − λ
) = 0
− c) = 0
is not dened for y
that the constraint
not be y = 0), then λ
that we face a well know optimization problem (minimizing a quadratic function
subject to lower convex bounds) which has a solution. We then deduce the ERC
portfolio by normalizing the solution y such that the sum of weights equals one.
Notice that the solution x may be found directly from the optimization problem (8)
by using a constant c = c − n ln (
A.3 On the relationship between σ
Let us start with the optimization problem (8) considered in the body part of the
0 ≤ x ≤ 1
We remark that if c
, we have σ (x (c
)) ≤ σ (x (c
because the constraint
− c ≥ 0
than with c
. We notice that if c =
, the optimization problem is exactly the MV problem, and x (−∞) is the MV
≤ −n ln n
portfolio. It comes that the solution for the general problem with c ∈ [−∞, −n ln n]
σ (x (−∞)) ≤ σ (x (c)) ≤ σ (x (−n ln n))
portfolio. It proves that the inequality holds:
A.4 Concentration and turnover statistics
The concentration of the portfolio is computed using the Herndahl and the Gini
indices. Let x
be the weights of the asset i for a given month t. The denition of
∈ [0, 1]
concentrated portfolio (i.e., where only one component is invested) and 1/n for a
portfolio with uniform weights. To scale the statistics onto [0, 1], we consider the
modied Herndahl index :
random variable on [0, 1] with distribution function F . Mathematically, the Lorenz
curve is :
L (x) =
θ dF (θ)
(x, L (x))
called the line of equality. If there is any inequality in size, then the
Lorenz curve falls below the line of equality. The total amount of inequality can be
summarized by the Gini index which is computed by the following formula:
G = 1 − 2
L (x) dx.
portfolio and 0 for the portfolio with uniform weights. In order to get a feeling of
diversication of risks, we also apply concentration statistics to risk contributions.
In the tables of results, we present the average values of these concentration statistics
for both the weights (denoted as ¯
(denoted as ¯
consecutive rebalancing dates with the following formula:
portfolio towards the 1/n target. However, apart in special circumstances, this eect
is of second order and we prefer to concentrate on modications of the portfolio
induced by active management decisions.In the tables of results, we indicate the
average values of T
across time. In general, we have preference for low values of H