Implied Returns
SEB Investment Management
House View Research
2014
Table of Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Sensitivity of the Markowitz Model to Input Parameters . . . . . . . . . . . 4
Complexity of Non-Binding Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 5
Implied Returns Without Inequalities in the Constraints . . . . . . . . . . . 5
Implied Returns with Inequalities in the Constraints . . . . . . . . . . . . . . 6
A Synthetic Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
A Real Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Uniqueness of the Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Litterature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Editorial
SEB Investment Management
Sveavågen 8,
SE-106 Stockholm
Authors:
Portfolio Manager, TAA: Peter Lorin Rasmussen
Phone: +46 70 767 69 36
E-mail: [email protected]
Analyst: Tore Davidsen
Phone: +45 33 28 14 25
E-mail: [email protected]
Portfolio Manager, Multi Management: Ruben Sharma
Phone: +46 8 763 69 73
E-mail: [email protected]
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This paper presents a numerical method to invert the standard Markowitz
portfolio optimization model with generic constraints. That is: a recipe of
how to find a set of implied returns from a given set of portfolio weights so
that the portfolio weights are “optimal” in a Markowitz model.
The original Markowitz model finds a set of “optimal” portfolio weights given a set of expected returns and covariances. A well-known problem of
the Markowitz model is its sensitivity to the input parameters. For example:
changing the expected returns only marginally can result in a set of very different “optimal” portfolio weights. This is naturally neither intuitive nor desirable, and as such a vast amount of literature suggests different solutions
to increase the robustness of the model. Generally, these suggestions focus
on either shrinkage of the input parameters or averaging of the output. For
a broad discussion on the literature, we refer to Meucci (2007).
This paper proposes a different model, namely the inverse Markowitz model. We state that this should be interpreted as a different model, because
in some sense it is not a portfolio optimizer. It merely provides you with a
way to conduct sanity checks on a given portfolio. That is: the model can tell
you what you are implying about the market through a given set of portfolio
weights. In the “model” the input becomes the portfolio weights and the
output becomes the expected returns. The results are therefore interpreted
in terms of the sensitive parameter (returns) instead of the robust (weights).
The example presented in this paper illustrates why this can sometimes be a
more desirable approach than that of the standard Markowitz model.
Inverting the Markowitz model is not a novel idea. However, most academic
studies have focused on the model where all constraints are binding and
the model therefore becomes solvable by Lagrange (closed form solution).
The novelty of this paper is the numerical approach by which the Markowitz
model can be inverted with generic constraints. A prominent example of a
constraint which invalidates the closed form solution is that all the portfolio
weights must be positive. Formally, the paper presents the inverse optimization of the Kuhn-Tucker problem. The approach present can even be used
for Mixed Integer Nonlinear Programming (MINLP) problems, and it is therefore extremely flexible.
In the following, we present a very simple example illustrating the sensitivity
of the Markowitz model with regard to the expected returns. We then present a short discussion on optimization problems with and without binding
constraints. Finally, we show by example how the model can be used.
Page 3
Introduction
First of all, we stress that the focus of this paper is solely on the sensitivity
with regard to the expected returns. It is, therefore, assumed that the higher
order moments: covariance, skewness and so forth are all known and fixed.
Note that the moments higher than the second only enters the Markowitz
model through an evaluation of the utility function. Since the vector of expected returns is the dominating parameter of the Markowitz model, this
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Page 4
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Page 5
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and
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the covariance matrix, 𝜇𝜇 is the vector of expected returns, and vector,
𝜇𝜇𝑝𝑝𝑝𝑝isis the
the
constraints,
thematrix,
efficient𝜇𝜇 frontier
can beof
estimated
using
Implied returns Without non-binding
is
the
covariance
is
the
vector
expected
returns,
and
𝜇𝜇
is
the
𝑝𝑝
required
requiredportfolio
portfolioreturn.
return.
simple linear algebra.
Say weportfolio
have
the
following
the
required
portfolio
return.problem:
required
return.
without
ere ι is a column vector of ones, same dimension as the weight vector, Ω
inequalities in
The
problem
Thesolution
solutiontoand
to
this
problem
is:
The
solution
tothis
this
problem
′ is:
he covariance matrix, 𝜇𝜇 is the vector of expected
returns,
𝜇𝜇
is
the
𝑤𝑤
Ω𝑤𝑤is:is:
𝑤𝑤to
=this
min
𝑝𝑝problem
The
solution
𝑤𝑤
the constraints
uired portfolio
return.
e solution to this problem is:
µµ(C
(Cµµ −−BB))++ιι((AA−−BBµµ ))
ww==ΩΩ−1−−11 µ (CµPPP − B ) + ι (22A − BµPPP )
AC
AC−−BB 2
= =𝜇𝜇𝑝𝑝Ω
𝑤𝑤 ′ 𝜇𝜇 w
AC − B
subject to:
𝑤𝑤 ′ 𝜄𝜄 = 1
µ (Cµ P − B ) + ι ( A −Where
Bµ P ) AA==µµ' Ω
Where
andCC==µµ' Ω
' Ω−−1−1µ
µ, , BB==µµ' Ω
' Ω−−1−ι11ι and
' Ω−−1−1µ
µ
1
w = Ω −1
Where
Where
,
and
A
=
µ
'
Ω
µ
B
=
µ
'
Ω
ι
C
=
µ
' Ω 1µ
2
AC − B
Now
the
returns
one
ofof
two
ways.
Either
directly
Nowvector
theimplied
implied
returns
can
be
deducted
inin
one
two
ways.
Either
directly
Now
the
implied
returns
canbe
bededucted
deducted
one
of
two
Either
directly
of ones,
same can
dimension
as theinweight
vector,
Ωways.
Where ι is a column
Now
the
implied
returns
can
be
deducted
in
one
of
two
ways.
Either
directly
−
1
isolate
in
the
equations
–
a
very
tedious
and
tricky
job
–
or
merely
µ
isolate
in
the
equations
–
a
very
tedious
and
tricky
job
–
or
merely
µ
matrix,
𝜇𝜇 isin the
of expected
is thejob – or merely opere A = µ ' Ω µ , B = µ ' Ω isι the
andcovariance
C = µ ' Ωisolate
µ
the vector
equations
– a veryreturns,
tediousand
and𝜇𝜇tricky
isolate µ in the equations – a very tedious and𝑝𝑝tricky job – or merely
optimize:
required portfoliooptimize:
return.
timize:
optimize:
w the implied returns can be deducted in one of two ways. Either directly
𝜇𝜇�𝐶𝐶𝜇𝜇
𝜇𝜇�𝐶𝐶𝜇𝜇𝑝𝑝𝑝𝑝−−𝐵𝐵�
𝐵𝐵�++𝜄𝜄�𝐴𝐴
𝜄𝜄�𝐴𝐴−−𝐵𝐵𝜇𝜇
𝐵𝐵𝜇𝜇𝑝𝑝𝑝𝑝��
this problem
ate µ in the equations – aThe
verysolution
tediousto and
tricky jobis: – or merely
−1
𝜇𝜇𝜇𝜇==min
�Ω−1
𝑝𝑝𝑝𝑝𝑤𝑤𝑖𝑖 �
min�Ω
𝜇𝜇�𝐶𝐶𝜇𝜇
−
𝐵𝐵�
+
𝜄𝜄�𝐴𝐴
−
𝐵𝐵𝜇𝜇𝑝𝑝 �−−𝑝𝑝𝑝𝑝𝑤𝑤
𝑖𝑖 �
𝑝𝑝
2
2
−1
𝜇𝜇𝜇𝜇 �Ω
𝐴𝐴𝐴𝐴
𝐴𝐴𝐴𝐴−−𝐵𝐵𝐵𝐵
imize:
− 𝑝𝑝𝑝𝑝𝑤𝑤𝑖𝑖 �
𝜇𝜇 = min
2
) − 𝐵𝐵
µ (Cµ 𝜇𝜇 − B ) + ι ( A − Bµ 𝐴𝐴𝐴𝐴
−1
−1
w = Ω −1
P
P
weights;
Where
𝑝𝑝𝑝𝑝𝑤𝑤
theinitial
initial
portfolio
weights;the
theweights
weightsfor
forwhich
whichwe
weseek
seektoto
Where
𝑝𝑝𝑝𝑝𝑤𝑤𝑖𝑖 are
𝜇𝜇�𝐶𝐶𝜇𝜇𝑝𝑝 − 𝐵𝐵� + 𝜄𝜄�𝐴𝐴 − 𝐵𝐵𝜇𝜇
𝑖𝑖 arethe
AC −portfolio
B2
𝑝𝑝 �
−1
are
the
initial
portfolio
weights;
the
weights
for
which
we
seek
to
Where
𝑝𝑝𝑝𝑝𝑤𝑤
−
�
𝜇𝜇 = min �Ω
Where
are
the
initial
portfolio
weights;
the
weights
for
which
we seek
𝑖𝑖
𝑖𝑖
find
the
implied
returns.
find
the
implied
returns.
𝜇𝜇
𝐴𝐴𝐴𝐴 − 𝐵𝐵2
thethe
implied
returns.
tofind
find
implied
returns.
Where A = µ ' Ω −1µ , B = µ ' Ω −1ι and C = µ ' Ω −1µ
Either
way
results
inin the
Eitherfor
way
results
the same
same solution.
solution. ItIt should
should be
be noted
noted that
that the
the
which
we seek
ere 𝑝𝑝𝑝𝑝𝑤𝑤𝑖𝑖 are the initial portfolio weights; the weights
Either
way
results
in in
theto
same
solution.
It should
be noted
that the
optimiEither
way
results
the
same
solution.
It
should
be
noted
that
optimization
isis
optimization problem
problem presented
presented above
above isis aa simple
simple NLP-problem,
NLP-problem, which
whichthe
d the implied returns.
Now the implied zation
returns
can beproblem
deducted
in one
of above
two
Either
directly
problem
presented
above
is aways.
simple
NLP-problem,
whichwhich
is easily
optimization
presented
is
a
simple
NLP-problem,
is
easily
solvable
––even
without
predefined
derivatives.
easily
solvable
even
without
predefined
derivatives.
isolate µ in thesolvable
equations
–
a
very
tedious
and
tricky
job
–
or
merely
–
even
without
predefined
derivatives.
easily solvable – even without predefined derivatives.
her way results in the sameoptimize:
solution. It should be noted that the
The
semi
solution
TheNLP-problem,
semiclosed
closedform
form
solution
presentedabove
aboveisisiningeneral
generalnot
notapplicable
applicableinin
Implied
returns
Implied
returns
imization problem presented
above
is a simple
which
is presented
The
semi
closed
form
solution
presented
above
is
in
general
not
applicable
Implied
returns
practise,
isis that
practise, as
as itit isis only
only valid
valid with
with binding
binding constraints.
constraints. The
The problem
problem
that ain
a
with
with
ily solvable – even without
predefined derivatives.
𝜇𝜇�𝐶𝐶𝜇𝜇
−
𝐵𝐵�
+
𝜄𝜄�𝐴𝐴
−
𝐵𝐵𝜇𝜇
�
practise,
as
it
is
only
valid
with
binding
constraints.
The
problem
is
that a
𝑝𝑝
𝑝𝑝
−1constraint
with
natural
such
as
positive
portfolio
weights
is
a
non-binding
natural
constraint
such
as
positive
portfolio
weights
is
a
non-binding
− 𝑝𝑝𝑝𝑝𝑤𝑤𝑖𝑖 �
𝜇𝜇 = min �Ω
2 positive
The
semiconstraint
closed form
solution
presented
aboveweights
is in general
applicable
𝜇𝜇natural
such
portfolio
is a not
non-binding
𝐴𝐴𝐴𝐴
−
𝐵𝐵as
Implied returns with
e semi closed form solution presented above is in
general
notasapplicable
in with binding constraints. The problem is that a
in
practise,
it
is
only
valid
inequalities in the
Implied
ImpliedReturns.docm
Returns.docm
ctise, as it is only valid with Where
binding𝑝𝑝𝑝𝑝𝑤𝑤
constraints.
The constraint
problem
that aasthe
natural
such
positive
coninitial
portfolio is
weights;
weightsportfolio
for whichweights
we seekistoa non-binding
𝑖𝑖 are the
Implied
Returns.docm
constraints
constraint.
invalidates the closed form solution and makes the inverted
ural constraint such asinequalities
positive
weights
is aThis
non-binding
in returns.
straint.
This
invalidates
the
closed
form
solution
and
makes
the
inverted
find theportfolio
implied
constraint.problem
This invalidates
the closed form solution and makes the inverted
inequalities
in Markowitz
rather complex.
the
constraints
Markowitz
problem
rathercomplex.
complex.
3(9)
Markowitz
problem
rather
theEither
constraints
Implied Returns.docm
way results in the same
solution. It should be noted that the
To invert
invert the
the Markowitz
Markowitz model,
with
non-binding
wewe
define
thethe
To
with
non-bindingconstraints,
constraints,
define
optimization problem
presented
above is model,
a simple
NLP-problem,
which
is
To
invert
the
Markowitz
model,
with
non-binding
constraints,
we
define
the
following min-max
problem:
following
min-max
problem:
easily solvable – even
without
predefined
derivatives.
following min-max problem:
Implied returns
with
′
− 𝑝𝑝𝑝𝑝𝑤𝑤𝑖𝑖 �
𝜇𝜇 = 𝑚𝑚𝑚𝑚𝑚𝑚
�𝑚𝑚𝑚𝑚𝑚𝑚
� 𝑤𝑤 𝜇𝜇
The semi closed form solution presented
above
is𝑤𝑤in general
not �
applicable
in
𝜇𝜇
𝑤𝑤 ′ 𝜇𝜇 𝑖𝑖2 � − 𝑝𝑝𝑝𝑝𝑤𝑤𝑖𝑖 �
𝜇𝜇 = 𝑚𝑚𝑚𝑚𝑚𝑚
�𝑚𝑚𝑚𝑚𝑚𝑚
�𝑤𝑤 ′ Ω𝑤𝑤=𝜎𝜎
practise, as it is only valid with binding constraints.
The
is that a
′problem
𝜇𝜇
𝑤𝑤
𝜏𝜏=1 2
𝑤𝑤𝑤𝑤′ Ω𝑤𝑤=𝜎𝜎
𝑖𝑖
𝑢𝑢1 =1𝑤𝑤is
′ 𝜏𝜏=1
natural constraint such as positive portfolio weights
a non-binding
𝑢𝑢1 =1
3(9)
Implied Returns.docm
1. 1.
SoSo
in in
words:
Where 𝜎𝜎𝑖𝑖2 = 𝑝𝑝𝑝𝑝𝑤𝑤𝑖𝑖′ Ω𝑝𝑝𝑝𝑝𝑤𝑤𝑖𝑖 , and 𝜇𝜇1 denotes
Where
denotesthe
thereturn
returnofofasset
asset
words:
2
′
=
𝑝𝑝𝑝𝑝𝑤𝑤
,returns
and 𝜇𝜇1which
denotes
the return
of asset
1. So inweights
words:
Where
we
seek𝜎𝜎to
to
find
the𝑖𝑖 Ω𝑝𝑝𝑝𝑝𝑤𝑤
set of
of𝑖𝑖returns
which
makes
initial
portfolio
𝑖𝑖 find
we
seek
the
set
makes
the initial
portfolio
weights
we seek
to find
the setdefined
of returns
which
makes
the(the
initial
portfolio
mean
variance
optimal;
byby
a penalty
function
norm).
mean
variance
optimal;
defined
a penalty
function
(the
norm). weights
mean variance optimal; defined by a penalty function (the norm).
In
problemisisnot
noteasily
easilysolvable
solvableasasititrequires
requiresthe
theutilization
utilization
In practise
practise this
this problem
of of
somewhat
complicated
optimization
algorithms.
As
a
general
note,
we
get
In practisecomplicated
this problemoptimization
is not easily algorithms.
solvable as As
it requires
thenote,
utilization
somewhat
a general
we getof
proper
results
using
the
optimization
toolbox
of
Matlab
2014a
although
somewhat
complicated
optimization
algorithms.
As
a
general
note,
wethe
get
proper results using the optimization toolbox of Matlab 2014a although
proper
results
using
the optimization
of Matlab
2014a although the
the
convergence
without
fixed
derivatives
is not
impressive.
convergence
without
fixed
derivatives
istoolbox
not impressive.
A synthetic
A synthetic
example
example
convergence without fixed derivatives is not impressive.
To illustrate the proposed model in “practise”, we restate the properties of our
To illustrate the
proposed model in “practise”, we restate the properties of our
3-dimensional
system:
3-dimensional system:
Page 6
𝑥𝑥1
1.55
5
𝑥𝑥
� 𝑥𝑥21� ~𝑁𝑁 ��45� , � 1.55
0
0
0
2.86
+0.48
+0.48��
−0.11
3(9)
3(9)
3(9)
In
practise this
problem optimization
is not easily solvable
as itAsrequires
the note,
utilization
of
somewhat
complicated
algorithms.
a general
we get
somewhat
complicated
optimization
algorithms.
As
a
general
note,
we
get
proper results using the optimization toolbox of Matlab 2014a although the
proper
resultswithout
using the
optimization
toolbox
of Matlab 2014a although the
convergence
fixed
derivatives is
not impressive.
convergence without fixed derivatives is not impressive.
thetic
hetic
ple
ple
To illustrate the proposed model in “practise”, we restate the properties of our
To
illustrate the proposed model in “practise”, we restate the properties of our
To3-dimensional
illustrate thesystem:
proposed model in “practise”, we restate the properties of
A Synthetic Example
3-dimensional system:
our 3-dimensional system:
per is that it provides a
to the optimization.
𝑥𝑥1
1.55
0
+0.48
5
y be derived by simple
0
+0.48
�𝑥𝑥𝑥𝑥12 � ~𝑁𝑁 ��54� , � 1.55
0
2.86
−0.11��
onstraints invalidates
�𝑥𝑥𝑥𝑥23� ~𝑁𝑁 ��46� , � +0.48
0
2.86 −0.11
−0.11
3.16 ��
𝑥𝑥3
rced into the realm of
+0.48 −0.11 3.16
6
is the possibility of
We then define a set of initial portfolio weights:
nts that is the
This weights:
invalidates
Wemain
then define
definein
aa set
initial
We
then
setofofconstraint.
initialportfolio
portfolio
weights: the closed form solution and makes the inverted
inequalities
rather complex.
the constraints Markowitz problem34%
34%
𝑝𝑝𝑝𝑝𝑤𝑤𝑖𝑖 = �33%�
= �33%
𝑝𝑝𝑝𝑝𝑤𝑤
an be estimated using
To invert
the𝑖𝑖 Markowitz
33%�model, with non-binding constraints, we define the
33%
m:
following min-max problem:
the
Theproblem
problem then
then isis how
how to
to find
find the
the returns
returns
which
–– given
The
which
given
the
covariance
′ covariance
𝜇𝜇 = 𝑚𝑚𝑚𝑚𝑚𝑚
�𝑚𝑚𝑚𝑚𝑚𝑚
� − 𝑝𝑝𝑝𝑝𝑤𝑤𝑖𝑖 �
� 𝑤𝑤
the𝜇𝜇covariance
The
problem
is in
how
find thebeing
returns
which
– given
𝜇𝜇
𝑤𝑤
′
matrix
wouldthen
result
“optimal”.
matrix
––would
result
inthe
thetoportfolio
portfolio
being
“optimal”.
𝑤𝑤 Ω𝑤𝑤=𝜎𝜎𝑖𝑖2
matrix – would result in the portfolio being “optimal”.
𝑤𝑤 ′ 𝜏𝜏=1
𝑢𝑢1 =1
The
todefine
definethe
thelevel
levelofofrisk.
risk.ToTododososo
Thefirst
firstactual
actualstep
step in
in the
the optimization
optimization isisto
The
first
actual
step
in
the
optimization
is
to
define
the
level
of
risk.
To do so
we
thevariance
varianceofofour
ourinitial
initial
portfolio:
wesimply
simplyestimate
estimate the
portfolio:
we simply estimate the variance2of our initial
portfolio:
Where 𝜎𝜎𝑖𝑖 = 𝑝𝑝𝑝𝑝𝑤𝑤𝑖𝑖′ Ω𝑝𝑝𝑝𝑝𝑤𝑤𝑖𝑖 , and 𝜇𝜇1 denotes the return of asset 1. So in words:
′
𝑝𝑝𝑝𝑝𝑤𝑤the
𝜎𝜎𝑖𝑖2to=find
𝑖𝑖 Ω𝑝𝑝𝑝𝑝𝑤𝑤
we seek
set of𝑖𝑖 returns which makes the initial portfolio weights
𝜎𝜎𝑖𝑖2 = 𝑝𝑝𝑝𝑝𝑤𝑤𝑖𝑖′ Ω𝑝𝑝𝑝𝑝𝑤𝑤𝑖𝑖
Here Ω again denotes
the variance
covariance
matrix
of the
then find
mean
optimal;
defined
bysystem.
a penaltyWe
function
(the norm).
s the weight vector,
returns, and 𝜇𝜇the
the
returns by solving the optimization problem. Solving on the ba𝑝𝑝 is implied
Here
Ω again
denotes
covariance
matrix
ofisthe
We
thenas
find
the
Inthe
practise
thisinproblem
notsystem.
easily solvable
it requires
the utilization of
sis
of
the
specified
weights
results
the following
implied
returns:
Here
Ω
again
denotes
the
covariance
matrix
of
the
system.
We
then
find
the
implied returns by solving
the
optimization
problem.
Solving
on
the
basis
of
somewhat complicated optimization algorithms. As a general note, we get
implied
returns
by solving
the optimization
problem.
Solving on the basis of
1.0the
the specified
weights
results
the following
implied
returns:
properinresults
using
optimization
toolbox of Matlab 2014a although the
the specified weights results in the 𝜇𝜇following
implied
returns:
= �2.4�
convergence without fixed derivatives is not impressive.
4.4
P
)
4(9)
Tocan
illustrate
the proposed
modeloptimization
in “practise”,
we
restate
of our
AAs
synthetic
4(9)
run a standard
Markowitz
using
these the properties
a sanity check, one
Implied
Returns.docm
As a sanity check, one 3-dimensional
can run a standard
Markowitz optimization
using
thesystem:
Implied
Returns.docm
returns. If the optimizer has found a stable solution then this will naturally
example
se returns. If the optimizer has found a stable solution then this will naturally
result in the “optimal” portfolio being the same as our initial portfolio.
result in the “optimal” portfolio being the same as our initial portfolio.
𝑥𝑥1please note
1.55
0
+0.48
5 that
they
completely
With
regard
level
of returns,
regard
to to
thethe
level
of returns,
please
note that
they
are are
completely
aro ways. EitherWith
directly
𝑥𝑥
�
~𝑁𝑁
��
�
,
�
��
�
4
0
2.86
−0.11
2
arbitrary.
It
is
only
the
relative
returns
that
are
of
importance
to
the
to the optimizacky job – orbitrary.
merelyIt is only the relative returns that
𝑥𝑥3 are of importance
−0.11
optimization.
So all conclusions
should
like: if1 6asset
1+0.48
delivers
a return
of3.16
1,
tion.
So all conclusions
should be
like: ifbeasset
delivers
a return
of 1, does
does it makes
senseasset
that asset
2 delivers
a return
2.4?
Sameififasset
asset 11 had
had aa
it makes
sense that
2 delivers
a return
of of
2.4?
Same
We
then
define
a setIfofyou
initial
portfolio
weights:
return
of
2
and
asset
2
a
return
of
4.8.
wish
you
can
find
a
numeraire
return of 2 and asset 2 a return of 4.8. If you wish you can find a numeraire
�
using
CAPMororthe
thelike
likefor
forthe
the return
return you
about.
− 𝑝𝑝𝑝𝑝𝑤𝑤𝑖𝑖 � using
CAPM
youfeel
feelthe
themost
mostcertain
certain
about.
34%
𝑝𝑝𝑝𝑝𝑤𝑤𝑖𝑖 = �33%�
What
youdodoififthe
thereturns
returns do no
expectation?
In that
casecase
the
What
dodoyou
notmatch
matchyour
your
expectation?
In that
33%
s for which we seek
to need to be changed: you simply increase the allocation
weights
to
those
the weights need to be changed: you simply increase the allocation to those
assets
whichyou
youfeel
feelthe
the returns
returns have
It should
be
assets
forforwhich
havebeen
been“underestimated”.
“underestimated”.
It should
noted
that
this
potentially
changes
the
volatility
of
the
portfolio.
This
is
one
be noted that this potentially
changes
volatility
ofthe
thereturns
portfolio.
This–ofisgiven the covariance
The problem
thenthe
is how
to find
which
be noted thatthethe
drawbacks
of the–of
inverse
Markowitz
It is“optimal”.
not possible
one ofmajor
the major
drawbacks
the inverse
optimization.
It is not
would
result
inMarkowitz
theoptimization.
portfolio
being
LP-problem, which
is the risk level matrix
to
keep
–
in
terms
of
standard
deviations
–
constant.
possible to keep the risk level – in terms of standard deviations – constant.
The first actual step in the optimization is to define the level of risk. To do so
ToTo
illustrate
can iterate
the
to
find
aa set
of
illustratehow
howone
one
iterate
theoptimization
optimization
to our
findinitial
setportfolio:
of returns
returns
wecan
simply
estimate
the variance of
neral not applicable
in
which
matches
ones
expectations,
say
that
we
find
the
return
of
asset
to
which matches ones expectations, say that we find the return of asset 3 to3be
The problem isbethat
a
relativelytoo
toohigh.
high.We
Wetherefore
thereforereduce
reduce
exposure
towards
asset
relatively
thethe
exposure
thisthis
asset
by
𝑝𝑝𝑝𝑝𝑤𝑤𝑖𝑖′ Ω𝑝𝑝𝑝𝑝𝑤𝑤
𝜎𝜎 2 = towards
ts is a non-binding
by 10%-points,
10%-points,which
whichwe
weallocate
allocateevenly
evenlytotothe
thetwo
two𝑖𝑖 others.
others.That
Thatis,
is,𝑖𝑖the
the new
new
portfolio weights
3(9) are given as:
Implied Returns.docm
Here Ω again denotes the covariance matrix of the system. We then find the
39%
implied returns by solving the optimization problem. Solving on the basis of
𝑝𝑝𝑝𝑝𝑤𝑤𝑛𝑛𝑛𝑛𝑛𝑛 = �38%�
7
the specified weights results in the following implied Page
returns:
23%
To illustrate how one can iterate the optimization to find a set of returns
To illustrate how one can iterate the optimization to find a set of returns
which matches ones expectations, say that we find the return of asset 3 to be
which matches ones expectations, say that we find the return of asset 3 to be
relatively too high. We therefore reduce the exposure towards this asset by
relatively too high. We therefore reduce the exposure towards this asset by
10%-points, which we allocate evenly to the two others. That is, the new
10%-points, which we allocate evenly to the two others. That is, the new
portfolio weights are given as:
portfolio
as:
portfolioweights
weightsare
are given
given as:
39%
39%
𝑝𝑝𝑝𝑝𝑤𝑤𝑛𝑛𝑛𝑛𝑛𝑛 = �38%�
𝑝𝑝𝑝𝑝𝑤𝑤𝑛𝑛𝑛𝑛𝑛𝑛 = �38%�
23%
23%
Using these weights, we find that the vector of implied returns is given as:
Usingthese
theseweights,
weights, we
we find
find that
returns
is given
as: as:
Using
thatthe
thevector
vectorofofimplied
implied
returns
is given
1.0
1.0
𝜇𝜇𝑛𝑛𝑛𝑛𝑛𝑛 = �2.6�
𝜇𝜇𝑛𝑛𝑛𝑛𝑛𝑛 = �2.6�
2.4
2.4
We see that the new portfolio weights imply two things: First, the relative
We see that the new portfolio weights imply two things: First, the relative
We
see thatreturn
the new
portfolio
weights
imply two
First, the relative
expected
of asset
3 drops.
Naturally,
thisthings:
is a consequence
of ourexexpected return of asset 3 drops. Naturally, this is a consequence of our
pected
return
of
asset
3
drops.
Naturally,
this
is
a
consequence
of
our
lowelowering the weight hereto. Second, the relative return of asset 2 rises
lowering the weight hereto. Second, the relative return of asset 2 rises
ring
the weight
hereto.
Second,
thedifficult
relativeeffect
return
asset 2The
risesreason
compared
compared
to level
1. This
is a more
to of
interpret.
for
compared to level 1. This is a more difficult effect to interpret. The reason for
tothelevel
1.
This
is
a
more
difficult
effect
to
interpret.
The
reason
for
the
rise
rise in the relative return comes from the fact that we at the same time
the rise in the relative return comes from the fact that we at the same time
inreduce
the relative
fromInthe
facttothat
we at the
the allocation
same timeofreduce
the riskreturn
of thecomes
portfolio.
order
increase
the
reduce the risk of the portfolio. In order to increase the allocation of the
the
risk ofrisky
the asset
portfolio.
order
theinallocation
the relatively
relatively
2, weInneed
to to
getincrease
an increase
its return. of
Otherwise
the
relatively risky asset 2, we need to get an increase in its return. Otherwise the
optimal
portfolio
with this
levelanofincrease
risk would
consist
more
of asset 1.the optimal
risky
asset
2,
we
need
to
get
in
its
return.
Otherwise
optimal portfolio with this level of risk would consist more of asset 1.
portfolio with this level of risk would consist more of asset 1.
As a final illustration of the model we can present the problem in terms of a
Asa afinal
finalillustration
illustration of
of the
the model we
can
present
the
in in
terms
of a
As
canthe
present
theproblem
problem
terms
standard
mean-variance
plot.model
Figure 2we
show
efficient
frontier
based
on the of
standard
mean-variance
plot.
Figure
2
show
the
efficient
frontier
based
on
the
a standard mean-variance plot. Figure 2 show the efficient frontier based
on the original return estimates and the “location” of our initial portfolio;
denoted by the red cross. It can clearly be visualised that the
Impliedportfolio
Returns.docmis
Implied Returns.docm
suboptimal, as it is below the efficient frontier.
Figure 2: The efficient frontier based on the original return estimates and the
location of the initial portfolio
1.25
Efficient Frontier
PF
1.2
Expected return
1.15
1.1
1.05
1
0.95
Page 8
1.4
1.6
1.8
2
2.2
2.4
Expected std
2.6
2.8
3
3.2
5(9)
5(9)
Figure 3 show the efficient frontier based on the set of implied returns. As
can be visualized the portfolio is now on the efficient frontier; as it should
be. The interesting thing to note is that the efficient frontier has now become steeper, and that it has shifted upwards compared to the old set of
implied returns.
Figure 3: The efficient frontier of both the implied and the original return
estimates
4.5
Old set of returns
New set of returns
PF
4
Expected return
3.5
3
2.5
2
1.5
1
0.5
1.4
1.6
1.8
2
2.2
2.4
Expected std
2.6
2.8
3
3.2
To also illustrate the method in practise we focus on the allocation of one
of our funds as of October 2014. The fund can invest solely in equities, Investment Grade, High Yield and government bonds and is only restricted
in so far that it has a maximum Value At Risk limit; implying a maximum
allocation towards equities of ~25%. As of October 2014 the allocation of
fund
waswas
given
as: as:
fund
was
given
as:
the
fund
given
𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸
20%
𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸
20%
10%
𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺
𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏
𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏
10%
𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼
=�
𝑝𝑝𝑝𝑝𝑤𝑤𝑟𝑟𝑟𝑟 =
=�
�=
�
�
�
𝑝𝑝𝑝𝑝𝑤𝑤
�
20%�
𝐻𝐻𝐻𝐻𝐻𝐻ℎ
𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌
𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏
20%
𝐻𝐻𝐻𝐻𝐻𝐻ℎ 𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌𝑌 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏
50%
𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏
𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏
50%
𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺
Nowwhat
whatdoes
doesthis
thissay
say about
about our
our view
view on
on
the
market?
To
answer
this
we
Now
what
does
this
say
Now
our
view
onthe
themarket?
market?To
Toanswer
answerthis
thiswe
we
simply
type
in
the
constraints
on
the
allocation
and
run
the
inverse
Markosimplytype
typeininthe
the constraints
constraints on the
simply
the allocation
allocationand
andrun
runthe
theinverse
inverseMarkoMarkowitzmodel.
model.This
Thisresults
resultsin
inthe
the following
following set
set of
of
implied
returns:
witz
model.
This
results
in
the
witz
following
set
ofimplied
impliedreturns:
returns:
7.8%
7.8%
1.07%
=�
�1.07%�
�
𝜇𝜇𝜇𝜇𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 =
2.2%
2.2%
1%
1%
So at
at the
the time
time of
of the
the allocation,
allocation, we
we expected
expected equities
equities to
to deliver
deliver aa return
return
So
significantly
higher
than
both
credits
and
government
bonds.
Note
that
significantly higher than both credits and government bonds. Note that
the horizon
horizon isis somewhat
somewhat fluid
fluid as
as the
the implied
implied returns
returns would
would change
change the
the
the
Page 9
moment
we
changed
the
allocation.
We
also
expected
that
High
Yield
moment we changed the allocation. We also expected that High Yield
bonds should only deliver a return twice as high as that of Government
A Real Example
So at the time of the allocation, we expected equities to deliver a return
significantly higher than both credits and government bonds. Note that the
horizon is somewhat fluid as the implied returns would change the moment
we changed the allocation. We also expected that High Yield bonds should
only deliver a return twice as high as that of government bonds, illustrating
our view that much of the return potential of the asset class had evaporated
over the last couple of years. All in all, we expected a scenario where the
return would primarily come from equities.
Uniqueness of the
Solution
Finally we investigate the uniqueness of the proposed method. First off, note
that solution, i.e. the implied returns, is unique. That is no set of weights
results in two different sets of implied returns. Naturally this property also
lies behind the original Markowitz model as it would not be desirable if you
could input two different sets of returns and get the same portfolio; in anything else than a corner solution that is.
With that being said, the likelihood function is very flat and it is therefore
somewhat difficult to find the exact solution. To illustrate this we plot the
likelihood function in Figure 4 for our equally weighted portfolio and the
covariance matrix that we have used throughout the paper. As we keep the
return of asset 1 constant we plot it as a function only of the returns of asset
2 and asset 3. The large red dot shows the minimum.
Figure 4: Return combinations with resulting portfolio weights close to the
solution. Calculated on the basis of the synthetic scenario
The most noticeable feature of Figure 4 is that it is declining in the combined absolute returns of asset 2 and asset 3. So in this example is not so
Page 10
much the relative return between asset 2 and asset 3 that matters, but more
their combined difference towards asset 1. This could in some sense also
be implied by the fact that the efficient frontier of Figure 3 shifted upwards.
The other thing to note is that the likelihood function is rather flat around
our proposed solution in the direction of asset 3. That is: it doesn’t matter all
that much whether we say the expected return hereof is 3 or 5. Yet, it is not
flat in terms of asset 2. Whether we say the expected return hereof is 2.4 or
3 does have a large impact. This comes to show that the larger the relative
difference of an asset’ return is compared to the others, the less sensitive
the output of the Markowitz model becomes to the exact figure.
The paper has presented a numerical approach to find the Markowitz implied returns of a given portfolio. Both a semi closed form and a numerical
solution are presented. It is described under which conditions the two solutions are appropriate.
Conclusion
By example it is shown how one can use the approach to obtain a portfolio
which is consistent with ones views.
Chopra, Vijay K. and Ziemba, William T. (1993), The Effect of Errors in Means,
Variances, and Covariances on Optimal Portfolio Choice, The Journal of
Portfolio Management, Winter 1993, Vol. 19, No. 2, pp. 6-11
Meucci, Attilio (2007), Risk and Asset Allocation, Springer
Page 11
Litterature