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2014

Implied ReturnsSEB Investment ManagementHouse View Research

Editorial

SEB Investment ManagementSveavågen 8,SE-106 Stockholm

Authors:

Portfolio Manager, TAA: Peter Lorin RasmussenPhone: +46 70 767 69 36E-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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Table of Contents

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3Sensitivity of the Markowitz Model to Input Parameters . . . . . . . . . . .4Complexity of Non-Binding Constraints. . . . . . . . . . . . . . . . . . . . . . . . . .5Implied Returns Without Inequalities in the Constraints . . . . . . . . . . .5Implied Returns with Inequalities in the Constraints. . . . . . . . . . . . . . .6A Synthetic Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7A Real Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9Uniqueness of the Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Litterature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12

Editorial

IntroductionThis 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 gi-ven 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 dif-ferent “optimal” portfolio weights. This is naturally neither intuitive nor de-sirable, 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 mo-del. 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 optimi-zation of the Kuhn-Tucker problem. The approach present can even be used for Mixed Integer Nonlinear Programming (MINLP) problems, and it is the-refore 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 pre-sent a short discussion on optimization problems with and without binding constraints. Finally, we show by example how the model can be used.

Sensitivity of the Markowitz Model to Input Parameters

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 ex-pected returns is the dominating parameter of the Markowitz model, this assumption is not as restrictive as one might initially believe. For a discus-sion on the sensitivity of the Markowitz model with regard to the different moments, we refer to Chopra and Ziemba (1993).

In order to illustrate the sensitivity of the Markowitz model with regard to the expected returns, we create a synthetic system of three assets:

Implied Returns.docm

2(9)

assumption is not as restrictive as one might initially believe. For a discussion on the sensitivity of the Markowitz model with regard to the different moments, we refer to Chopra and Ziemba (1993). In order to illustrate the sensitivity of the Markowitz model with regard to the expected returns, we create a synthetic system of three assets:

𝑥𝑥1𝑥𝑥2𝑥𝑥3~𝑁𝑁

546 ,

1.55 0 +0.480 2.86 −0.11

+0.48 −0.11 3.16

Where the “covariance matrix” is specified as the correlation matrix with the standard deviations in the diagonal. Note that the normality assumption is not relevant in anything else but a utility evaluation, but to satisfy the detail oriented reader, we have included it. That way we can talk about “optimal” portfolios more or less freely. To illustrate the sensitivity of the optimization, we estimate the efficient frontier for three different sets of expected returns. We keep the expected returns of asset 1 and 3 fixed, and only allow the expected return of asset 2 to vary: 𝑥𝑥2 ∈ 4.0; 4.5; 4.9. Intuitively, changing the expected return of only one asset by such small amounts should not result in very different “optimal” portfolios/results. However, as Figure 1 illustrates: for some parts of the efficient frontier, the allocation changes significantly as a consequence of this small change. This is not a desirable result since in practise there is a very large intuitive difference between holding a third of one’s position in each asset or focusing entirely on asset 1 and 3; see the middle part of the efficient frontiers. Figure 1: Efficient frontiers for varying expected returns of the second asset

Despite the simplicity of the example, it illustrates that being only slightly off in the expected returns can generate very different results in the Markowitz model. As stated this is neither an intuitive nor desirable effect of the model.

0

50

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X2=4

.00 Asset 1

Asset 2Asset 3

0

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100

X2=4

.50 Asset 1

Asset 2Asset 3

0

50

100

X2=4

.90 Asset 1

Asset 2Asset 3

Where the “covariance matrix” is specified as the correlation matrix with the standard deviations in the diagonal. Note that the normality assumption is not relevant in anything else but a utility evaluation, but to satisfy the detail oriented reader, we have included it. That way we can talk about “optimal” portfolios more or less freely.

To illustrate the sensitivity of the optimization, we estimate the efficient frontier for three different sets of expected returns. We keep the expected returns of asset 1 and 3 fixed, and only allow the expected return of asset 2 to vary:


2(9)

assumption is not as restrictive as one might initially believe. For a discussion on the sensitivity of the Markowitz model with regard to the different moments, we refer to Chopra and Ziemba (1993). In order to illustrate the sensitivity of the Markowitz model with regard to the expected returns, we create a synthetic system of three assets:

𝑥𝑥1𝑥𝑥2𝑥𝑥3~𝑁𝑁

546 ,

1.55 0 +0.480 2.86 −0.11

+0.48 −0.11 3.16

Where the “covariance matrix” is specified as the correlation matrix with the standard deviations in the diagonal. Note that the normality assumption is not relevant in anything else but a utility evaluation, but to satisfy the detail oriented reader, we have included it. That way we can talk about “optimal” portfolios more or less freely. To illustrate the sensitivity of the optimization, we estimate the efficient frontier for three different sets of expected returns. We keep the expected returns of asset 1 and 3 fixed, and only allow the expected return of asset 2 to vary: 𝑥𝑥2 ∈ 4.0; 4.5; 4.9. Intuitively, changing the expected return of only one asset by such small amounts should not result in very different “optimal” portfolios/results. However, as Figure 1 illustrates: for some parts of the efficient frontier, the allocation changes significantly as a consequence of this small change. This is not a desirable result since in practise there is a very large intuitive difference between holding a third of one’s position in each asset or focusing entirely on asset 1 and 3; see the middle part of the efficient frontiers. Figure 1: Efficient frontiers for varying expected returns of the second asset


0

50

100

X2=4

.00 Asset 1

Asset 2Asset 3

0

50

100

X2=4

.50 Asset 1

Asset 2Asset 3

0

50

100

X2=4

.90 Asset 1

Asset 2Asset 3

. Intuitively, changing the expected return of only one asset by such small amounts should not result in very different “opti-mal” portfolios/results. However, as Figure 1 illustrates: for some parts of the efficient frontier, the allocation changes significantly as a consequence of this small change. This is not a desirable result since in practise there is a very large intuitive difference between holding a third of one’s position in each asset or focusing entirely on asset 1 and 3; see the middle part of the efficient frontiers.

As mentioned in the introduction, the novelty of this paper is that it provi-des a recipe for incorporating non-binding restrictions into the optimiza-tion. Without such constraints, the efficient frontier can easily be derived by simple linear algebra. However, using non-linear or integer constraints invalidates this approach. To solve the resulting problem one is forced into the realm of Kuhn-Tucker optimization and numerical methods. It is the pos-sibility of inverting the Markowitz model under such constraints that is the main contribution of this paper.

Complexity of Non-Binding Constraints

Implied Returns Without Inequalities

in the Constraints

Without non-binding constraints, the efficient frontier can be estimated using simple linear algebra. Say we have the following problem:


3(9)

Complexity of non-binding constraints

As mentioned in the introduction, the novelty of this paper is that it provides a recipe for incorporating non-binding restrictions into the optimization. Without such constraints, the efficient frontier can easily be derived by simple linear algebra. However, using non-linear or integer constraints invalidates this approach. To solve the resulting problem one is forced into the realm of Kuhn-Tucker optimization and numerical methods. It is the possibility of inverting the Markowitz model under such constraints that is the main contribution of this paper.

Implied returns without inequalities in the constraints


𝑤𝑤 = min𝑤𝑤

𝑤𝑤′Ω𝑤𝑤

subject to:

𝑤𝑤′𝜇𝜇 = 𝜇𝜇𝑝𝑝 𝑤𝑤′𝜄𝜄 = 1

Where ι is a column vector of ones, same dimension as the weight vector, Ω is the covariance matrix, 𝜇𝜇 is the vector of expected returns, and 𝜇𝜇𝑝𝑝is the required portfolio return. The solution to this problem is:

( ) ( )2

1

BACBABCw PP

−−+−

Ω= − µιµµ

Where µµ 1' −Ω=A , ιµ 1' −Ω=B and µµ 1' −Ω=C Now the implied returns can be deducted in one of two ways. Either directly isolate µ in the equations – a very tedious and tricky job – or merely optimize:

𝜇𝜇 = min𝜇𝜇Ω−1

𝜇𝜇𝐶𝐶𝜇𝜇𝑝𝑝 − 𝐵𝐵 + 𝜄𝜄𝐴𝐴 − 𝐵𝐵𝜇𝜇𝑝𝑝𝐴𝐴𝐶𝐶 − 𝐵𝐵2

− 𝑝𝑝𝑝𝑝𝑤𝑤𝑖𝑖

Where 𝑝𝑝𝑝𝑝𝑤𝑤𝑖𝑖are the initial portfolio weights; the weights for which we seek to find the implied returns. Either way results in the same solution. It should be noted that the optimization problem presented above is a simple NLP-problem, which is easily solvable – even without predefined derivatives.

Implied returns with

The semi closed form solution presented above is in general not applicable in practise, as it is only valid with binding constraints. The problem is that a natural constraint such as positive portfolio weights is a non-binding

subject to:


3(9)







subject to:



( ) ( )2

1

BACBABCw PP

−−+−

Ω= − µιµµ








Where


3(9)







subject to:



( ) ( )2

1

BACBABCw PP

−−+−

Ω= − µιµµ








is a column vector of ones, same dimension as the weight vector,


3(9)







subject to:



( ) ( )2

1

BACBABCw PP

−−+−

Ω= − µιµµ








is the covariance matrix,


3(9)







subject to:



( ) ( )2

1

BACBABCw PP

−−+−

Ω= − µιµµ








is the vector of expected returns, and


3(9)







subject to:



( ) ( )2

1

BACBABCw PP

−−+−

Ω= − µιµµ








is

Figure 1: Efficient frontiers for varying expected returns of the second asset

0

50

100

X2=4

.00

Asset 1Asset 2Asset 3

0

50

100

X2=4

.50


0

50

100

X2=4

.90



the required portfolio return.

The solution to this problem is:


3(9)







subject to:



( ) ( )2

1

BACBABCw PP

−−+−

Ω= − µιµµ








Where


3(9)







subject to:



( ) ( )2

1

BACBABCw PP

−−+−

Ω= − µιµµ








Now the implied returns can be deducted in one of two ways. Either directly isolate


3(9)







subject to:



( ) ( )2

1

BACBABCw PP

−−+−

Ω= − µιµµ








in the equations – a very tedious and tricky job – or merely op-timize:


3(9)







subject to:



( ) ( )2

1

BACBABCw PP

−−+−

Ω= − µιµµ








Where


3(9)







subject to:



( ) ( )2

1

BACBABCw PP

−−+−

Ω= − µιµµ








are the initial portfolio weights; the weights for which we seek to find the implied returns.

Either way results in the same solution. It should be noted that the optimi-zation problem presented above is a simple NLP-problem, which is easily solvable – even without predefined derivatives.

Implied returns with inequalities in the constraints

The semi closed form solution presented above is in general not applicable in practise, as it is only valid with binding constraints. The problem is that a natural constraint such as positive portfolio weights is a non-binding con-straint. This invalidates the closed form solution and makes the inverted Markowitz problem rather complex.

To invert the Markowitz model, with non-binding constraints, we define the following min-max problem:


4(9)

inequalities in the constraints

constraint. This invalidates the closed form solution and makes the inverted Markowitz problem rather complex. To invert the Markowitz model, with non-binding constraints, we define the following min-max problem:

𝜇𝜇 = 𝑚𝑚𝑚𝑚𝑚𝑚𝜇𝜇

𝑚𝑚𝑚𝑚𝑚𝑚𝑤𝑤

𝑤𝑤′𝜇𝜇𝑤𝑤′Ω𝑤𝑤=𝜎𝜎𝑖𝑖

2

𝑤𝑤′𝜏𝜏=1


𝑢𝑢1=1

Where 𝜎𝜎𝑖𝑖2 = 𝑝𝑝𝑝𝑝𝑤𝑤𝑖𝑖′Ω𝑝𝑝𝑝𝑝𝑤𝑤𝑖𝑖, and 𝜇𝜇1denotes the return of asset 1. So in words: we seek to find the set of returns which makes the initial portfolio weights mean variance optimal; defined by a penalty function (the norm). In practise this problem is not easily solvable as it requires the utilization of somewhat complicated optimization algorithms. As a general note, we get proper results using the optimization toolbox of Matlab 2014a although the convergence without fixed derivatives is not impressive.

A synthetic example

To illustrate the proposed model in “practise”, we restate the properties of our 3-dimensional system:

𝑚𝑚1𝑚𝑚2𝑚𝑚3~𝑁𝑁

546 ,

1.55 0 +0.480 2.86 −0.11

+0.48 −0.11 3.16

We then define a set of initial portfolio weights:

𝑝𝑝𝑝𝑝𝑤𝑤𝑖𝑖 = 34%33%33%

The problem then is how to find the returns which – given the covariance matrix – would result in the portfolio being “optimal”. The first actual step in the optimization is to define the level of risk. To do so we simply estimate the variance of our initial portfolio:

𝜎𝜎𝑖𝑖2 = 𝑝𝑝𝑝𝑝𝑤𝑤𝑖𝑖′Ω𝑝𝑝𝑝𝑝𝑤𝑤𝑖𝑖 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 the specified weights results in the following implied returns:

Where


4(9)






2



𝑢𝑢1=1


A synthetic example



546 ,

1.55 0 +0.480 2.86 −0.11

+0.48 −0.11 3.16





, and


4(9)






2



𝑢𝑢1=1


A synthetic example



546 ,

1.55 0 +0.480 2.86 −0.11

+0.48 −0.11 3.16





denotes the return of asset 1. So in words: we seek to find the set of returns which makes the initial portfolio weights mean variance optimal; defined by a penalty function (the norm).

In practise this problem is not easily solvable as it requires the utilization of somewhat complicated optimization algorithms. As a general note, we get proper results using the optimization toolbox of Matlab 2014a although the convergence without fixed derivatives is not impressive.

A Synthetic ExampleTo illustrate the proposed model in “practise”, we restate the properties of our 3-dimensional system:


4(9)






2



𝑢𝑢1=1


A synthetic example



546 ,

1.55 0 +0.480 2.86 −0.11

+0.48 −0.11 3.16







4(9)






2



𝑢𝑢1=1


A synthetic example



546 ,

1.55 0 +0.480 2.86 −0.11

+0.48 −0.11 3.16





The problem then is how to find the returns which – given the covariance matrix – would result in the portfolio being “optimal”.

The first actual step in the optimization is to define the level of risk. To do so we simply estimate the variance of our initial portfolio:


4(9)






2



𝑢𝑢1=1


A synthetic example



546 ,

1.55 0 +0.480 2.86 −0.11

+0.48 −0.11 3.16





Here


3(9)







subject to:



( ) ( )2

1

BACBABCw PP

−−+−

Ω= − µιµµ








again denotes the covariance matrix of the system. We then find the implied returns by solving the optimization problem. Solving on the ba-sis of the specified weights results in the following implied returns:


5(9)

𝜇𝜇 = 1.02.44.4

As a sanity check, one can run a standard Markowitz optimization using these 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. With regard to the level of returns, please note that they are completely arbitrary. It is only the relative returns that are of importance to the optimization. So all conclusions should be like: if asset 1 delivers a return of 1, does it makes sense that asset 2 delivers a return of 2.4? Same if asset 1 had a return of 2 and asset 2 a return of 4.8. If you wish you can find a numeraire using CAPM or the like for the return you feel the most certain about. What do you do if the returns do no match your expectation? In that case the weights need to be changed: you simply increase the allocation to those assets for which you feel the returns have been “underestimated”. It should be noted that this potentially changes the volatility of the portfolio. This is one of the major drawbacks of the inverse Markowitz optimization. It is not possible to keep the risk level – in terms of standard deviations – constant. 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 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 portfolio weights are given as:

𝑝𝑝𝑝𝑝𝑤𝑤𝑛𝑛𝑛𝑛𝑛𝑛 = 39%38%23%

Using these weights, we find that the vector of implied returns is given as:

𝜇𝜇𝑛𝑛𝑛𝑛𝑛𝑛 = 1.02.62.4

We see that the new portfolio weights imply two things: First, the relative expected return of asset 3 drops. Naturally, this is a consequence of our lowering the weight hereto. Second, the relative return of asset 2 rises compared to level 1. This is a more difficult effect to interpret. The reason for the rise in the relative return comes from the fact that we at the same time reduce the risk of the portfolio. In order to increase the allocation of the relatively risky asset 2, we need to get an increase in its return. Otherwise the optimal 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 standard mean-variance plot. Figure 2 show the efficient frontier based on the

As a sanity check, one can run a standard Markowitz optimization using the-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.

With regard to the level of returns, please note that they are completely ar-bitrary. It is only the relative returns that are of importance to the optimiza-tion. So all conclusions should be like: if asset 1 delivers a return of 1, does it makes sense that asset 2 delivers a return of 2.4? Same if asset 1 had a return of 2 and asset 2 a return of 4.8. If you wish you can find a numeraire using CAPM or the like for the return you feel the most certain about.

What do you do if the returns do not match your expectation? In that case the weights need to be changed: you simply increase the allocation to those assets for which you feel the returns have been “underestimated”. It should be noted that this potentially changes the volatility of the portfolio. This is one of the major drawbacks of the inverse Markowitz optimization. It is not possible to keep the risk level – in terms of standard deviations – constant.

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 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

portfolio weights are given as:


5(9)

𝜇𝜇 = 1.02.44.4








5(9)

𝜇𝜇 = 1.02.44.4






We see that the new portfolio weights imply two things: First, the relative ex-pected return of asset 3 drops. Naturally, this is a consequence of our lowe-ring the weight hereto. Second, the relative return of asset 2 rises compared to level 1. This is a more difficult effect to interpret. The reason for the rise in the relative return comes from the fact that we at the same time reduce the risk of the portfolio. In order to increase the allocation of the relatively risky asset 2, we need to get an increase in its return. Otherwise the optimal 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 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 portfolio is 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

1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.20.95

1

1.05

1.1

1.15

1.2

1.25

Expe

cted

retu

rn

Expected std

Efficient FrontierPF

A Real Example

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 be-come 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

1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.20.5

1

1.5

2

2.5

3

3.5

4

4.5

Expe

cted

retu

rn

Expected std

Old set of returnsNew set of returnsPF

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, In-vestment 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 the fund was given as:


7(9)

fund was given as:

𝑝𝑝𝑝𝑝𝑤𝑤𝑟𝑟 =

𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐼𝐼𝐼𝐼𝐼𝐼𝐸𝐸𝐸𝐸𝐸𝐸𝐼𝐼𝐸𝐸𝐼𝐼𝐸𝐸 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐸𝐸 𝑏𝑏𝑏𝑏𝐼𝐼𝐺𝐺𝐸𝐸

𝐻𝐻𝐸𝐸𝐻𝐻ℎ 𝑌𝑌𝐸𝐸𝐸𝐸𝑌𝑌𝐺𝐺 𝑏𝑏𝑏𝑏𝐼𝐼𝐺𝐺𝐸𝐸𝐺𝐺𝑏𝑏𝐼𝐼𝐸𝐸𝐺𝐺𝐼𝐼𝐼𝐼𝐸𝐸𝐼𝐼𝐸𝐸 𝑏𝑏𝑏𝑏𝐼𝐼𝐺𝐺𝐸𝐸

=

20%10%20%50%

Now what does this say about our view on the market? To answer this we simply type in the constraints on the allocation and run the inverse Marko-witz model. This results in the following set of implied returns:

𝜇𝜇𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 =

7.8%1.07%2.2%1%

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 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



7(9)

fund was given as:

𝑝𝑝𝑝𝑝𝑤𝑤𝑟𝑟 =

𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐼𝐼𝐼𝐼𝐼𝐼𝐸𝐸𝐸𝐸𝐸𝐸𝐼𝐼𝐸𝐸𝐼𝐼𝐸𝐸 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐸𝐸 𝑏𝑏𝑏𝑏𝐼𝐼𝐺𝐺𝐸𝐸

𝐻𝐻𝐸𝐸𝐻𝐻ℎ 𝑌𝑌𝐸𝐸𝐸𝐸𝑌𝑌𝐺𝐺 𝑏𝑏𝑏𝑏𝐼𝐼𝐺𝐺𝐸𝐸𝐺𝐺𝑏𝑏𝐼𝐼𝐸𝐸𝐺𝐺𝐼𝐼𝐼𝐼𝐸𝐸𝐼𝐼𝐸𝐸 𝑏𝑏𝑏𝑏𝐼𝐼𝐺𝐺𝐸𝐸

=

20%10%20%50%


𝜇𝜇𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 =

7.8%1.07%2.2%1%

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 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

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 any-thing 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 com-bined absolute returns of asset 2 and asset 3. So in this example is not so

The paper has presented a numerical approach to find the Markowitz im-plied returns of a given portfolio. Both a semi closed form and a numerical solution are presented. It is described under which conditions the two solu-tions are appropriate.

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

Conclusion

Litterature

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.

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