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Portfolio construction:
seeking consistency between
strategic risk and active risk
Research paper
September 2014
- Document intended exclusively for professional clients in accordance with MIFID
#1
Strategic objective and active management
Alexis Flageollet
Senior Financial Engineer
Seeyond, Natixis Asset Management
[email protected]
Franck Nicolas
Head of Investment & Client Solutions
Natixis Asset Management
[email protected]
NATIXIS ASSET MANAGEMENT
The Seeyond and Investment
Solutions divisions in brief
&
Client
With €307 billion under management and 588 employees1, Natixis
Asset Management ranks among Europe's leading asset managers.
Natixis Asset Management offers its customers (institutional
investors, companies, private banks, distributors, and banking
networks) innovative, effective, customized solutions based on six
main areas of management expertise: Fixed Income; European
Equities; Investment and Client Solutions; Structured and Volatility
developed by Seeyond2; Global Emerging; and Responsible
Investment developed by Mirova3. Natixis Asset Management markets
these solutions through Natixis Global Asset Management's global
distribution platform, which offers access to the expertise of more
than twenty management companies in the United States, Asia, and
Europe.
In durably volatile markets in which trends have been depleted, the
teams of Seeyond, the volatility management and structured
products investment division of Natixis Asset Management,
implement an investment philosophy based on the conviction that it is
more efficient to exploit market volatility to generate value rather
than invest according to return forecasts that are sometimes
inaccurate and often unstable. They thus seek to use market
variability and dispersion to generate value and focus on risk
management to construct portfolios tailored to an increasingly
complex environment. With 32 employees, this division manages €15
billion in assets1.
The Investment and Client Solutions division relies on its global
allocation expertise to offer innovative, customized investment
solutions to institutional investors. These solutions use all
management techniques: benchmarked management, guaranteed
management, portfolio insurance, absolute performance, Liability
Driven Investment, etc. With 53 employees, this division manages
€36 billion in assets as well as €44 billion in asset allocation under
management delegation with other divisions of Natixis Asset
Management1.
1 - Source: Natixis Asset Management – 30/06/2014.
2 - Seeyond is a trademark of Natixis Asset Management.
3 - Mirova is a wholly owned subsidiary of Natixis Asset Management
2
Strategic objective and active management
CONTENTS
SUMMARY
1/// UTILITY FUNCTION IN THE INVESTMENT CHOICE THEORY
2/// DEFINITION OF THE PAIN THRESHOLD AND STRATEGIC
ALLOCATION
3/// BREAKDOWN OF THE RISK RELATED TO THE OVER/UNDEREXPOSURE OF THE ALLOCATION TO THE BENCHMARK
CONCLUSION
REFERENCES
APPENDIX: EXPRESSION OF THE GAMMA PARAMETER
3
Strategic objective and active management
SUMMARY
When investors purchase an actively managed multi-asset class
portfolio, they expose themselves to three types of risk:
•
•
•
The risk which derives from the strategic allocation the
investor has chosen (volatility);
The risk resulting from the tactical management
implemented by the allocation manager for this strategic
allocation (allocation tracking error);
And finally, the selection risk resulting from the different
investment vehicles, if they are actively managed (selection
tracking error).
Several questions emerge as a result:
•
•
•
How can we ensure that all the portfolio-related risks
actually correspond with the investor's expectations?
How can we account for risk aversion without resorting to
abstract and counterintuitive measures?
How can we spread the investor's maximum target risk
between these three components?
According to finance theory, investors choose from among several
solutions according to their investment horizon and risk aversion. The
classic approach consists of arbitrating between return expectations
and portfolio variance, once the investor has identified his theoretical
risk aversion. In this context, either the global risk of the portfolio is
considered, without distinguishing between the strategic risk and the
active management risk (tactics and selection), or the management
risk is used in isolation. Furthermore, theoretical risk aversion
(derived from utility functions) is not only difficult to measure but
also abstract and counterintuitive.
Is it therefore possible to construct a portfolio with overall
consistency between strategic risk, tactical risk, and selection risk
with regard to an investor's desired overall level of risk?
In this document, we specify the links between strategic risk, tactical
risk, and management risk within an asset class.
The objective of this approach is to construct a portfolio with
consistent management leeway with regard to the overall level of risk
that an investor is willing to take.
Our approach
The strategic allocation may be deduced by defining a maximum
statistical loss threshold for a given time horizon (the pain threshold)
directly dependent on the investor's implied risk aversion, and not the
reverse.
This approach has the advantage of being able to access a risk
measurement with concrete meaning for an investor, namely the
level of possible loss on a portfolio within a given time period.
Once the mix of asset classes is defined on the basis of the loss at a
horizon that an investor can take on, we can undertake a
management risk that is either a tactical allocation risk or an active
selection risk.
In order to determine the portfolio's overall relative risk, we use a
quadratic utility function within which the active management risk is
included in addition to the strategic risk.
The three pillars of a portfolio's overall risk are thus harmonized: the
strategic volatility, the tactical management tracking error, and the
selection tracking error on each of the asset classes in the strategic
mix.
4
Strategic objective and active management
INTRODUCTION
Is the average degree of active management introduced into a
portfolio based on the strategic allocation? At first glance, this is not
essential. Yet, an investor whose strategic allocation is defensive, and
therefore corresponds to a shorter horizon, should normally have
more passive management1. If an investor’s asset allocation is
conservative, it is because he does not wish to endure excessive
losses. So, there is no reason for overly aggressive active
management to put him in a situation of significant potential losses.
Conversely, a more dynamic investor, with more time ahead of him,
will have a riskier asset allocation and may therefore assume a
proportional active management risk.
Finance theory responds rather poorly to the risk that an investor can
endure. In particular, no connection is made between strategic risk,
tactical risk, and management risk within an asset class (or selection
risk). Here we propose an approach that reconciles the three
concepts by seeking overall consistency. This approach makes it
possible to construct a portfolio by defining management leeway
consistent with the overall level of risk that an investor is willing to
take.
1 In recent years, certain institutional investors have applied the opposite approach. They entrust more budget to very active
management when market visibility is reduced and when they rely more on manager performance than on the directional.
Conversely, when the directional becomes more attractive, their portfolios become more passive as the asset mix becomes
more aggressive.
However, we can no longer speak of just strategy here. The reduction of market risk in favor of manager risk in bearish
markets and vice versa is already a tactical act that can be carried out with unchanged overall active risk relative to a longterm strategic allocation. It is just the overall downside risk that becomes more sensitive to the manager in a bearish phase
and more sensitive to the market in a bullish phase on risk assets.
5
Strategic objective and active management
1
UTILITY FUNCTION IN THE INVESTMENT
CHOICE THEORY
The investment choice theory describes a portfolio selection process
in which the investor must choose among several solutions according
to his investment objective. Therefore, the first challenge of an
investment strategy is the proper identification of the objective.
Generally, a utility function helps economic agents rank their
preferences among several solutions in relation to the benefits that
they bring.
For a given portfolio, it is generally believed that the investor will
seek to maximize the utility that he will obtain from a portfolio's
accumulated wealth. Therefore, the utility function is often presented
with a positive first derivative and a negative second derivative. The
utility is:
• Increasing: a portfolio's utility increases with its wealth. This
element is crucial to justify that additional wealth will contribute
positively to the increase in utility. This is the concept of marginal
utility, which indicates that the utility gained is positively impacted by
additional wealth.
• Concave: the increase in utility will diminish with the portfolio's
value. It is the principle of satiety of an economic agent that
intervenes as the value of a portfolio appreciates.
In concrete terms, most utility functions used by practitioners are
quadratic.
This function takes the following form:
U ( X ) = X − λ. X 2
The absolute risk coefficient is also an increasing function of wealth
where:
A( X ) =
2.λ
1 − 2.λ . X
This utility function is particularly interesting because it gives access
to a mean-variance reasoning2 in which an investor will maximize his
expected wealth by basing his choice on the expected variance of
returns. Accordingly, the investor will attempt to maximize his utility
U as a function of two parameters: expected return, which acts
positively on utility, and variance of returns, which acts negatively on
U. The investment program then becomes:
• Maximize the expected return for a certain level of variance of
returns
• Minimize a portfolio's variance for a certain level of returns.
2 This also provides access to the concept of Value-at-Risk (VaR), which quantifies the maximum risk for a given period with a
certain confidence level. For example, an investment's one-year 99% VaR is defined as: VaR = µ −1.96.σ, while the 95% VaR
will be: VaR = µ −1.64.σ (in the Gaussian case).
Where: µ is a portfolio's return and σ is a portfolio's standard deviation.
6
Strategic objective and active management
If an investor decides to allocate between equities and bonds, his expected utility will become:
(
E (U ) = µ S .WS + µ B .WB − λ . σ S2 .WS2 + σ B2 .WB2 + 2.ρ .σ S .WS .σ B .WB
It is then possible to write the partial derivative of the expected
utility, which represents the change in utility generated by an
additional unit of equities or bonds in the asset allocation. The partial
derivatives with respect to equities and bonds are written
respectively:
(
(
∂E (U ) ∂WS = µ S − λ. σ S2 .WS + 2.ρ .σ S .σ B .WB

∂E (U ) ∂WB = µ B − λ. σ B2 .WB + 2.ρ .σ S .WS .σ B
)
)
(2.σ
(µ S − µ B )
2
S
.WS + 2.ρ .σ S .σ B .WB − 2.σ B2 .WB − 2.ρ .σ S .WS .σ B
)
Typically, here is the main problem with these approaches:
• This utility function does not take into account possible asymmetry
in the asset distribution. In reality, an investor will prefer an asset
with less asymmetry for an identical variance3.
• The risk aversion coefficient λ itself depends on the asset allocation.
This means that the coefficient can be known only when the
allocation is defined or, conversely, that the asset allocation can be
defined only when this coefficient is defined. However, the definition
of λ is precisely what is problematic.
The approach proposed below will therefore aim to break away from a
traditional approach in order to define a risk aversion level that will
deal with the strategic allocation, the level of active risk relative to
this strategic allocation tolerated by the coefficient, and the
distribution between core and satellite assets that can be supported
on a portfolio.
3 This is also a limit of logics in VaR in the context of Gaussian returns.
7
Where:
µS: expected return on
equities
µB: expected return on
bonds
WS: proportion of equities in
the allocation
If we equate these two derivatives, we get an implied risk aversion
that is a function of the expected returns, the standard deviations of
returns, the correlation between equities and bonds, and the asset
allocation of the investor.
λ=
)
WB: proportion of bonds in
the allocation
σS: standard deviation of the
return on equities
σB: standard deviation of the
return on bonds
ρ: correlation between
equities and bonds
λ: risk aversion coefficient
Strategic objective and active management
2
DEFINITION OF THE PAIN THRESHOLD
AND STRATEGIC ALLOCATION
In order to place ourselves in a portfolio management situation, we
define a management reference that can serve as a benchmark. In
concrete terms, this involves defining an investor's strategic
allocation from which he will deviate depending on his higher or lower
risk aversion (or higher or lower risk appetite).
Thus, consider B the benchmark's return and P the strategic
portfolio's return.
represents the relative performance, i.e.
Let N equal the number of assets making up the portfolio. ri and wi
are the return on asset i and its proportion in the portfolio P
respectively. Let
i equal the proportion of asset i in benchmark B.
i + ∆wi, which are the portfolio's active deviations
Consider wi =
relative to the benchmark.
N
We get:
N
∑w = ∑w
i =1
i
i =1
i
= 1 . So
N
∑ ∆w
i =1
i
=0
Consider:
H: an investment horizon4
R ≈ N (m× dt),∑×dt): the log-returns of N assets observed at a
relatively short frequency5.
m: the expected annualized log-returns vector
∑: the annualized variance/covariance matrix.
If we assume a unit initial wealth and consider PH and 1+ PH to be the
portfolio's return and the final wealth achieved at horizon6 H
respectively, we get:
 
σ w2 


E (1 + P ) = exp H . mw +

2

 
H
and
σ (1 + P H ) = E (1 + P H ). exp( H .σ w2 ) − 1
4 In this case, we assume an investment horizon of more than one year. Therefore, the assumption of normal distribution is no
longer tenable given the geometrical nature of returns over the long term. A log-normal distribution is a more realistic
assumption.
5 We have dt = (1 / number of periods per year). If the portfolio's rebalancing frequency is fairly short and the assets are not
too volatile, the portfolio's log-return is Gaussian with mean value mw = w' m and σ2 = w'∑w because the log-returns of the
assets are approximately equal to the linear returns (Pt/Pt-1 -1).
2
If the approximation is not tenable, we must use the method of our choice to obtain mw and σ w in order to project the
portfolio's performance for the chosen horizon.
6 DE LA GRANDVILLE, O. - "The Long-Term Expected Rate of Return: Setting it Right", Financial Analysts Journal,
November/December 1998, pp. 75-80.
8
Strategic objective and active management
To identify an investor's risk profile, we choose to determine the
allocation that maximizes the portfolio's diversification under the
constraint of absolute loss. This allocation will have good capital
diversification and will respect a statistical performance threshold that
an investor does not wish to go below over a given time horizon7.
Once m and ∑ are defined, we seek the allocation that allows us to
expect to obtain, at the confidence threshold of X% (e.g. 95%), a
return at horizon H greater than or equal to an investor's pain
threshold SH.
This pain threshold will correspond to the statistical loss that an
investor will be willing to accept over a given time horizon and with a
given confidence level. It is therefore equivalent to the portfolio's
value at risk calculated at horizon H.
Let X% equal the confidence level and SH equal the investor's pain
threshold8; we can then write:
[
Pr 100% + P H ≤ 100% + S H
]
= 1− X %
We have9:
E (1 + P H ) − Q X % .σ (1 + P H ) = 1 + S H
⇔ E ( P H ) − Q X % .σ ( P H ) = S H
⇔ VaR X % ( P H ) = S H
Where:
QX % =
[
1 − exp N X % . H .σ w − 1 / 2.H .σ w2
[exp( H .σ
2
w
]
) −1
]
1/ 2
Where:
N x% the (1-X%)-quantile of the standard normal distribution (for
example, if X%=95%, then N 95% ≈-1.64).
Q X% the quantile of the normal distribution adjusted for the
investment horizon; it is positive because a normal log distribution
has exclusively positive values.
The strategic allocation w* is therefore determined using the
following optimization routine:
 w* = arg max H ( w) = 1 − w' w
w

H
H
 s.c. VaR X % ( P ) = S

s.c. ∑ w = 1, wi > 0

7 The chosen diversification measure is the Herfindahl index: H(w) =1− w'w
8 If SH = 0% and X%= 95%, then we seek the allocation that generates a return greater than zero in 95% of cases. If SH =
−5% and X%= 95%, then we seek the allocation that gives a loss less than or equal to -5% in only 5% of cases.
9 ALBRECHT, T. - "The Mean -Variance Framework and Long Horizons", Financial Analysts Journal, vol. 54, no. 4, July/August
1998, pp. 44-49.
9
Strategic objective and active management
The implied risk intolerance parameter is defined as:
q=
E ( P1 )
σ ( P1 )
[1]
Where:
E(P1) and
σ(P1): respectively the expected return and the portfolio's
volatility.
As we will see later, a portfolio's target TE is a decreasing function of
the risk intolerance parameter.
It is therefore by defining a maximum statistical loss threshold at a
given time horizon, which depends directly on the investor's implied
risk aversion, that the strategic allocation will be deduced and not the
reverse. This has the advantage of being able to access a risk
measurement with concrete meaning, since nothing is more telling for
an investor than the level of possible loss on a portfolio within a
certain time period.
Naturally, this reasoning is not free from a logic consistent with the
CAPM, in which the expected return and the volatility of the assets
are positively related. However, if we want to introduce volatility
taking into account a symmetry of the distribution for example (with
a Cornish-Fisher correction10), this is also possible.
For example, let's choose an allocation based on three asset classes:
equities, bonds, and an "alternative" asset class. We determine the
following long-term return/risk profiles (annualized performance of
returns in log):
Equities
Bonds
Alternative
Annualized return
8.0%
5.0%
6.0%
Annual volatility
15.0%
5.0%
8.0%
With the following correlation matrix:
Equities
Bonds
Alternative
Equities
1.0
0.2
0.5
Bonds
Alternative
1.0
0.5
1.0
Later we will look at the sensitivity of various portfolio characteristics
as a function of the pain threshold and at the considered investment
horizon. We have chosen 3, 5 and 8 years.
10 LEE, Y. S. & LIN, T. K. - "Higher-Order Cornish Fisher Expansion", Applied Statistics, 1992, vol. 41, pp. 233-240.
10
Strategic objective and active management
The figure here shows the volatility of
portfolios obtained according to the above
optimization routine on the basis of the
investor's pain threshold and investment
horizons. The more the investor is willing to
consider a high potential loss, the higher the
portfolio's volatility11.
The longer the investment horizon, the
greater the portfolio's volatility may be. The
share of equities (bonds) therefore decreases
(increases) as long as the pain threshold
increases and the investment horizon
shortens (see figure on the right).
The risk intolerance coefficient [1] is
positively related to the pain threshold (see
graph).
We note that the shorter the investment
horizon, the more sensitive the risk
intolerance is to the pain threshold.
We also see a sensitivity of the pain threshold
to the selected quantile (95% and 97.5%).
The more one wishes to identify the maximum
possible loss with a high probability, the lower
this minimum will be because it will
encompass a greater share of the possible
distribution (while reserving, once again, the
possibility
of
exploring
higher-order
moments12).
11 The confidence level of the VaR is equal to 97.5% here.
12 KRITZMAN, M. - "About Higher Moments", Financial Analysts Journal, vol. 50, no. 5, 1994, pp. 10-17.
11
Strategic objective and active management
3
BREAKDOWN OF THE RISK RELATED TO
THE OVER/UNDER-EXPOSURE OF THE
ALLOCATION TO THE BENCHMARK
Once the mix of asset classes is defined on the basis of the loss at a
horizon that an investor can take on, a management risk can be
undertaken. This risk can be:
• a tactical allocation active risk (by opportunistic deviation between
asset classes through market timing),
• or an active selection risk (by selecting on each of the asset classes
more or less active investment strategies that will add to the
portfolio's active risk).
In order to model the active risk contained in a portfolio of
performance P, we consider the following linear regression equation:
P = α + β .B + e
[2]
where:
B: the benchmark return from the previous strategic allocation
exercise cov(B,e) = 0
Consider TE2 (e) =σ2 (e).
: contribution of active management not correlated with benchmark.
We get: P − B =
By considering
: + (β −1).B + e
and β = 1 +γ, we get:
~
P = α + γ .B + e
[3]
The term γ is the βeta of the difference in performance (P-B) with the
benchmark. A positive value indicates an overexposure (positive
directional on the benchmark), while a negative value indicates an
underexposure (negative directional on the benchmark).
The relative expected performance is expressed as:
~
E ( P ) = α + γ .E ( B )
[4]
Clearly, if the benchmark's performance is
overexposure implies a positive outperformance.
positive, then an
Of course, there's no such thing as a free lunch. Therefore, if no
active risk not correlated with the benchmark (tracking error, TE(e))
is taken, no additional uncorrelated performance (alpha, ) can be
recorded. If TE2 (e) = 0 =>
= 0.
12
Strategic objective and active management
That is why we consider13:
α = k .TE (e)
[5]
We thus express the portfolio's active risk in the form of tracking
error (TE) and get:
TE 2 ( P) = γ 2 .σ 2 ( B) + TE 2 (e)
[6]
Therefore, the overall volatility becomes:
σ 2 ( P) = β 2σ 2 ( B) + TE 2 (e)
⇔
σ 2 ( P) = σ 2 ( B) + 2γσ 2 ( B) + γ 2σ 2 ( B) + TE 2 (e)
⇔
σ 2 ( P ) = σ 2 ( B ) + 2γσ 2 ( B) + TE 2 ( P)
[7]
Lastly, we saw earlier that the over/underexposure coefficient,
namely the portfolio's βeta to the benchmark, was expressed as:
βP =
cov(P, B)
= 1+ γ
σ 2 ( B)
In the appendix, we show that:
N
γ = ∑ ∆wi β i
[8]
i =1
In a way, Gamma14 (γ)
measures the aggressiveness
of the investment policy
from a tactical point of view.
This parameter is important
because it will make it
possible to tie the tactical
leeway to the TE.
13 For the quantified examples, we will use a level of 0.3, which can correspond to a target information ratio for an active
management.
14 See Appendix for another expression of Gamma
13
Strategic objective and active management
4
DETERMINATION
RELATIVE RISK
OF
THE
OVERALL
We therefore now take a quadratic utility function15, within which we
can include the active risk management in addition to the strategic
risk.
λ
~ λ
.σ 2 ( P) = E ( B) + E ( P ) − . (1 + γ ) 2 .σ 2 ( B) + TE 2 (e)
2
2
λ
λ
~
U = E ( B) − .(1 + 2.γ ).σ 2 ( B) + E ( P ) − .TE 2 ( P)
24244443 144
2 444
1444
42
3
U = E ( P) −
[
Strategic risk
]
Active risk
.σ (B)
.σ(B) + TE(e)
From [1], we consider E(B) =
) = γ.
and we have: E(
We get:
U = q .(1 + γ ).σ ( B ) −
λ
.(1 + γ ) .σ 2 ( B ) + k .TE (e) −
2
2
λ
2
.TE 2 (e)
By seeking to maximize the investor's utility within this framework,
the first-order optimality conditions become:
• On the one hand:
∂U
2
= 0 ⇔ q .(1 + γ ) − λ .(1 + γ ) .σ ( B) = 0
∂σ ( B)
⇔ λ=
q .(1 + γ )
(1 + γ ) .σ ( B)
2
=
q
(1 + γ ).σ ( B)
[9]
• On the other hand:
∂U
=0
∂TE (e)
⇔
⇔ TE * (e) =
*
⇔ TE (e) =
k − λ.TE (e) = 0
k
λ
k
.(1 + γ ).σ ( B )
q
[10]
By substituting [10] in [4], we get:
k

TE ( P ) = γ .σ ( B ) +  .(1 + γ ).σ ( B ) 
q

2
2
2
2

k2
2
⇔ TE 2 ( P) = γ 2 + 2 .(1 + γ ) .σ 2 ( B)
q


15 COLLINS, RA. & GBUR, E.E. - "Quadratic Utility and Linear Mean-Variance: A Pedagogic Note", Review of Agricultural
Economics, Vol. 13, No. 2 July, 1991, pp. 289-291.
14
Strategic objective and active management
The tactical allocation tracking error is thus defined with:
k2
2
TE ( P) = γ + 2 .(1 + γ ) .σ ( B )
q
2
[11]
Formula [11] shows that the target TE is proportional to the
benchmark's volatility; it is a decreasing function of the risk
intolerance and positively tied to the manager's ability to deliver noncorrelated performance in case of the portfolio's outperformance.
Just like the previous figures, the graph here
illustrates the dependence of the TE with the
pain threshold. The TE depends positively on the
pain threshold and the investment horizon. Thus,
the more risk-averse an investor is, the higher
his pain threshold is (right movement on the Xaxis) and the lower the active risk
(TE) will be.
As previously seen, the active risk increases with
the investment horizon. The figures on the right
show that the tactical leeway also changes the
active risk more strongly the longer the
investment horizon is16.
A defensive positioning (βeta<1) hardly changes
the TE, while a more aggressive positioning
(βeta>1) increases the TE17. The active risk is
increased because, in a way, the decision to
increase the portfolio's risk brings about a
positive view of the risk asset.
16 We tested the following as leeway: a 20% increase (decrease) in equities funded by a 15% decrease (increase) in bonds and
5% in alternative.
15
Strategic objective and active management
We also realize that:
• The figure here shows that the lower the pain threshold (less riskaverse investor), the greater the share of satellite.
• Formula 14 below shows that the greater the
information
ratio
of
the
satellite
active
management, the higher the TE.
Actually, if we opt for a combination18 of replication
management with a low tracking error (Core) and
management with higher added value (Satellite),
for each asset class, the percentage to be invested
in Satellite is itself a function of an aversion
parameter λ (see equation 10).
It is calculated by assuming a zero correlation between core and
satellite:
IRS .TES − IRC .TEC + λ .TEC2
Satellite =
λ.TE S2 + λ.TEC2
[12]
We have set the information ratios to 0.3, and we have selected the
following tracking error parameters for each management:
Core
Satellite
Equities
3.0%
8.0%
Bonds
1.5%
4.0%
Alternative
4.0%
12.0%
Lastly, before concluding, we summarize our method with the
following example: let's assume that an investor has an investment
horizon of five years and a pain threshold of -5%; given our
assumptions of returns/volatilities and correlations, we get a
confidence level of 97.5%:
• A strategic allocation of 46% equities, 24% bonds, and 30%
alternative
• The target TE is 3.7%
• The satellite share is 37% for equities, 62% for bonds, and 28% for
alternative
• Considering ±10% of tactical leeway on equities and bonds, the
"tactical" TE varies between 3.4% and 4.4%.
18 AMENC, N.; MALAISE, Ph. & MARTELLINI, L. – “Revisiting Core-Satellite Investing”, The Journal of Portfolio Management,
Fall 2004, pp.64-75
16
Strategic objective and active management
CONCLUSION
We have seen that there is a possibility of harmonizing the three
pillars of a portfolio's overall risk: the strategic volatility, the tactical
management tracking error, and the selection tracking error on each
of the asset classes in the strategic mix.
A simple quadratic utility function can be used starting from the
minimum performance that an investor is prepared to tolerate within
a given time period. The deduced implied risk aversion parameter
makes it possible to:
• set the investor's strategic allocation,
• position the active risk due to the tactical allocation,
• and therefore know the distribution of this selection risk among
more or less active management strategies.
17
Strategic objective and active management
REFERENCES
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n° 4, pp. 422-436, May 2008
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September 2003, pp. 1-15.
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Finance, 2006, p. 45.
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LOBOSCO, A. & DIBARTOLOMEO, D. - "Approximating the Confidence Intervals for Sharpe Style
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LUCAS, A. & KLAASSEN, P. - "Extreme Returns, Downside Risk, and Optimal Asset Allocation",
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SORTINO, F. & FORSEY, H. - "On the Use and Misuse of Downside Risk", Journal of Portfolio
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Quarterly Journal of Economics, 1991vol.106, pp. 1039–1061.
18
Strategic objective and active management
APPENDIX: EXPRESSION OF THE GAMMA
PARAMETER
Let N equal the number of assets making up the portfolio. ri and wi
are the return on asset i and its proportion in the portfolio P
respectively. Let
i equal the proportion of asset i in benchmark B.
We start from the expression of the portfolio's βeta with regard to the
benchmark:
βP =
cov(P, B)
σ 2 ( B)
N
βP =
⇔
cov(∑ wi ri , B)
i =1
2
σ ( B)
N
i =1
cov(ri , B)
σ 2 ( B)
cov(ri , B) N
cov(ri , B)
+ ∑ ∆wi
2
σ ( B)
σ 2 ( B)
i =1
β P = ∑ wi
⇔
N
= ∑ wi
i =1
N
βP =
⇔
cov(∑ wi ri , B)
i =1
2
σ ( B)
N
β P = 1 + ∑ ∆wi
⇔
i
i =1
N
+ ∑ ∆wi
i =1
cov(ri , B)
σ 2 ( B)
cov(ri , B )
σ 2 ( B)
Consider:
N
γ = ∑ ∆wi β i
i =1
N
It is relatively easy to reinstate the constraint
∑ ∆w
i =1
i
= 0 as
follows.
Consider the Nth asset of the portfolio to be the one that has the
smallest βeta relative to the benchmark and the first asset to have
the greatest, i.e. βmax = β1 ≥ β2 ≥ …≥βN = βmin.
Therefore, [8] becomes:
N −1
γ = ∑ ∆wi (β i − β N )
i =1
with
(β i − β N ) ≥ 0
Obviously, the overexposure (∆wi > 0) of assets with high βeta
increases and therefore the portfolio's overexposure to the
benchmark and the tracking error.
19
Strategic objective and active management
It is also important to note that the beta of the difference in
performance (P-B) can be expressed both as a linear combination of
βetas of the portfolio's assets or βetas of differences in performance
of assets toward the benchmark.
N
~
N
γ~ = ∑ ∆wi β i = ∑ ∆wi
i =1
⇔
i =1
 cov(ri , B) cov(B, B) 
−

2
σ 2 ( B) 
 σ ( B)
N
γ~ = ∑ ∆wi 
i =1
⇔
cov(ri − B, B)
σ 2 ( B)

N

γ~ = γ − 1 ∑ ∆wi  = γ
 i =1

20
Strategic objective and active management
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Strategic objective and active management
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