European Journ al of Operational Research · ellipsoida l uncertainty sets, Kim et al. (2012) mathematicall y show how robustnes s leads to higher correlation with fundamenta l fac-

European Journal of Operational Research xxx (2013) xxx–xxx

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European Journ al of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Robust portfolios that do not tilt factor exposure

0377-2217/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.ejor.2013.03.029

⇑ Corresponding author. Tel.: +82 42 350 3129; fax: +82 42 350 3110.E-mail addresses: [email protected] (W.C. Kim), [email protected] (M.J.

Kim), [email protected] (J.H. Kim), [email protected] (F.J. Fabozzi).

1 For further contributions on robust port folio optimization, refer to Fabo(2007a,b), Fa bozzi et al. (2010), and Kim et al. (accepted for publicat ion-b).

Please cite this article in press as: Kim, W.C., et al. Robus t portfolios that do not tilt factor exposu re. Europe an Journal of Operation al Rese arch http ://dx.doi.org /10.1016/ j.ejor.2013.03.029

Woo Chang Kim a,⇑, Min Jeong Kim a, Jang Ho Kim a, Frank J. Fabozzi b

a Department of Industrial and Systems Engineering, Korea Advanced Institute of Science and Technology (KAIST), Yuseong-gu, Daejeon 305-701, Republic of Korea b EDHEC Business School, Nice, France

a r t i c l e i n f o

Article history:Available online xxxx

Keywords:Investment analysis Robust portfolio model Robustness analysis Fundamental factors

a b s t r a c t

Robust portfolios reduce the uncertainty in portfolio performance. In particular, the worst-case optimization approach is based on the Markowitz model and form portfolios that are more robust compared to mean–variance portfolios. However, since the robust formulation finds a different portfolio from the optimal mean–variance portfolio, the two portfolios may have dissimilar levels of factor exposure. In most cases, investors need a portfolio that is not only robust but also has a desired level of depende ncy on factor movement for managing the total portfolio risk. Therefore, we introduce new robust formulations that allow investors to control the factor exposure of portfolios . Empirical analysis shows that the robust portfolios from the proposed formulations are more robust than the classical mean–variance approach with comparable levels of exposure on fundamental factors.

� 2013 Elsevier B.V. All rights reserved.

1. Introductio n

Robust optimizati on was developed to solve problems where there is uncertainty in the decision environment, and therefore is sometimes referred to as uncertain optimization (Ben-Tal and Nemirovski, 1998 ). Robust models were adapted in portfolio optimization to resolve the sensitivity issue of the mean–variancemodel (Markowitz, 1952 ) to its inputs. Even though the Markowi tz model is still the basis for portfolio optimization and one of the most significant contributions to portfolio selection, its drawbacks are well documented. For example, Best and Grauer (1991) showthat mean–variance portfolio weights are extremely sensitive to changes in asset means when no constraints are imposed. Broadie(1993) points out errors in computing efficient frontiers and stres- ses that estimates of mean returns using historical data should be used with caution. Chopra and Ziemba (1993) go one step further to analyze the relative impact of estimation errors in means, variances, and covariances.

These studies naturally led to the advancement of robust portfolio optimization. Costa and Paiva (2002) optimize robust track- ing-error optimization when the expected returns and covariance matrix are not exactly known. Similarly under uncertain situations,El Ghaoui et al. (2003) solve the worst-case value-at-risk problem when only bounds on the inputs are known. In addition, Goldfarband Iyengar (2003) introduce a way to formulat e robust optimization problems as second-o rder cone programs using ellipsoidal uncertainty sets, and similar approach es are investigated by

Tütüncü and Koenig (2004). As summari zed, much effort has been put into defining uncertainty in input parameters and formulating the problems using worst-case optimization methods.1

On the same topic but on a different track, there have been recent developmen ts looking into the behavior of robust portfolios formed from worst-case optimization (Kim et al., accepted for publication-a; Kim et al., 2012; Kim et al., 2013 ). They look for unexpected properties of robust formulation s in order to not only expand our understa nding on robust portfolios but also to increase its practical use. Focusing on the worst-ca se formulation with ellipsoida l uncertainty sets, Kim et al. (2012) mathematicall y show how robustnes s leads to higher correlation with fundamenta l factors. Kim et al. (accepted for publication- a) further investigate this behavior using worst-ca se formulation s with ellipsoidal and box uncertainty sets under various settings to confirm the relationship between robustness and increased correlation with factors. Even though some worst-ca se formulat ions may result in portfolios with unintended factor loadings, robust portfolios are still required to protect portfolio returns from unexpected market movements .Kim et al. (2013) find that under the existence of market regimes,a portfolio that focuses on asset returns during the worst regime of the stock market results in a portfolio that is not only robust but has an improved performanc e based on measure s such as Sharpe ratio, drawdow n, and value-at-ris k.

Based on these research findings, we know that investors can achieve robustness through robust optimizati on techniqu es but at the same time lose control of the factor dependency of the optimal portfolio. Factors explain the underlying movement of the

zzi et al.

(2013),

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mailto:[email protected]





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2 W.C. Kim et al. / European Journal of Operational Research xxx (2013) xxx–xxx

market and widely studied factors include the market index, size (market capitalization), and book-to-ma rket ratio factors (Famaand French, 1993 ). In addition to these three fundamental factors,portfolio managers look for macroecono mic factors and also iden- tify statistical factors through principal component analysis. Inves- tors and portfolio managers need to control the factor exposure for managing the portion of total risk generated from each factor because it allows them to better understand the overall risk of their portfolios.2 Therefore, in this paper, we introduc e formulation s that form robust portfolios which are not tilted towards factors. Investors may consider robust counterpa rts of mean–variance portfolios for achieving robust performan ce. However, if the factor exposure of the robust count erparts is affected by the robust formula tions, investors need to compare not only the performan ce between the classical mean–variance and robust portfoli os but also their factor exposures.By forming portfolio s from our proposed optimiza tion problems,investors will be able to control the risk associated with each factor of robust portfolios. In other words, investors can hold portfolios that are robust and at the same time have a better understan ding of how the portfolio will be affected by movements in market factors.

Before we introduce and analyze the revised robust formulations, we need a measure to compare the robustness of portfolios.This robustness measure allows us to compare the optimal portfolio from the new formulations with the existing mean–varianceand robust portfolios. Therefore, we begin our analysis by defininga robustness measure and exploring its propertie s. Next, we derive new robust formulation s to form optimal portfolios that match atarget factor exposure by adding constraints to the original robust formulation . Since the additional constraints are developed based on factor exposure measure s of portfolios, the new formulations will not only add robustnes s but also keep the factor dependency to a specified level. Furthermore, we perform empirical tests to confirm the factor exposures of the new robust portfolios and findthat the new approach forms portfolios that are more robust than Markowitz portfolios without affecting the factor exposure.

The organization of the paper is as follows. Section 2 defines the robustness measure and its properties, and the new robust formulations that do not affect factor exposure are introduced in Sec- tion 3. Section 4 includes empirical tests using industry-l evel returns to confirm our developmen t, and Section 5 concludes.

2. Robustness measure

In this section, we define a robustness measure prior to devel- oping new robust formulations. By defining a measure for portfolio robustness, investors can compare the robustnes s among portfolios and also use this measure to set the desired robustnes s level when choosing an optimal portfolio. Once we define a robustness measure, we analyze its properties for the case when the geometry of the uncertainty set for the expected asset returns is an ellipsoid.

2.1. Defining a robustness measure

We begin defining a robustnes s measure by identifyin g the meaning of robustnes s. In general, X(e), which is a function of an uncertain parameter e, being robust means that the value of X(e)is relatively stable even when the uncertain parameter e fluctuates.Therefore, we can measure the robustness of X(e) by the movement of X(e) with respect to e.

The robustness in portfolio management can be defined in similar fashion. We first determine a suitable uncertain function X(e)

2 The importance of managing systematic risk through factors is illustrated, for example, by Fung and Hsieh (1997), Clark et al. (2002), Alexan der and Dimitriu (2004), and Zhu et al. (2011).

Please cite this ar ticle in press as: Kim, W.C., et al. Robus t portfolios that do nhttp://dx.doi.org/1 0.1016/ j.ejor.2013.03.029

for measuring portfolio robustness. Since the optimal portfolio weights are a function of uncertain parameters l and R, expected return and covariance of returns, respectively , the weights of the optimal portfolio are one possible choice for the function X(e).However , this measure does not reflect the portfolio robustness desired by investors because investors want robust portfolio performance but not necessarily robust portfolio weights. In other words, the stability of the portfolio performanc e is more important to investors than the stability of the optimal weights. Thus, it is more reasonable to use the portfolio performanc e for X(e) when we define portfolio robustness. Consequently, we can say that the optimal portfolio x⁄ is robust if the performance of the portfolio x⁄ is stable with respect to l and R.

Based on the above definition of robustness in portfolio optimization, we derive the robustness measure for portfolios. The stability level of portfolio performanc e determines the degree of robustness.Therefore, the robustness measure can be defined by the performance fluctuation from the change in uncertain parameters. Portfo- lios that are robust will have small performanc e fluctuations and therefore low levels for the robustness measure. For simplicit y, we assume that the only uncertain parameter is the mean of asset returns l and the uncertainty is revealed at time T. Since the portfolio performanc e depends on the mean of asset returns l and the portfolio weight x⁄, it can be represented as f(x⁄, l). The expected value of l at time 0 is l0 and the realized value at time T is lT. Hence, investors make investme nt decisions based on l0 at time 0 but they evaluate the portfolio performance using lT at time T. Thus, performanc efluctuation is the difference between the expectation and the reali- zation of portfolio performance,

dðx�Þ ¼ jf ðx�;l0Þ � f ðx�;lTÞj:

However, we cannot evaluate d(x⁄) at time 0 because lT is un- known until time T. So, instead of d(x⁄), the maximum of d(x⁄) on the uncertainty set U of lT should be the robustness measure,

Dðx�Þ ¼maxlT2U

dðx�Þ ¼ maxlT2Ujf ðx�;l0Þ � f ðx�;lTÞj:

When the realiza tion lT of the uncertain paramete r l varies in a given set U, the fluctuation of the portfolio performanc e cannot ex- ceed the robustne ss measure D(x⁄). It follows that portfolios with smaller values of D(x⁄) will have less fluctuation in portfoli o performance and can be considered to be more robust.

The generalized form of the robustness measure is

Robustness measure ¼maxh2N0jf ðx�; �hÞ � f ðx�; hÞj;

– f(x, h): portfolio performanc e,– x⁄: an optimal portfolio,– h: uncertain paramete rs,– �h: the expected value of uncertain paramete rs,– N0: an unit uncertainty set for h.

2.2. Properties of the robustness measure

For the robustnes s measure of portfolio performanc e, we assume the uncertain parameter l to be in an uncertainty set U. Since ellipsoidal uncertainty sets are often used in robust optimization, we also assume that the uncertainty set of l to be in an ellipsoid. The robust portfolio formulation with an ellipsoidal uncertainty set is,

ðRdÞ minx2X

maxl2Udð�lÞ

12x0Rx� kl0x ¼min

x2X

12x0Rx� k�l0xþ kd

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix0Rlx

q;

where X ¼ fxjx0i ¼ 1g; Udð�lÞ ¼ ljðl� �lÞ0R�1l ðl� �lÞ 6 d2

n o,

where R is the covariance of expected returns, i is a vector of ones,

ot tilt factor exposu re. Europe an Journal of Operation al Rese arch (2013),


W.C. Kim et al. / European Journal of Operational Research xxx (2013) xxx–xxx 3

k is the risk coefficient, and Rl is the covarian ce matrix of estimation errors for the expected returns. Among many possibilit ies for represen ting portfolio performan ce, we focus on the objective function of the robust formulation . Even though the objective function of the robust formulation is not always a convention al measure of performanc e, measurin g the stability of the objective function is applicable when investo rs use portfolio optimization models for constructing portfoli os because the objective function reflects what investor s seek to optimize . In other words, investo rs will consid er the portfolio robust if the objective value is stable with respect to changes in input values. Accordin gly, from the objective function of the above formula tion, we define f ðx;lÞ ¼ 1

2 x0Rx� kl0x asthe measure for portfoli o performanc e. In addition , since d deter-mines the confidence level of the uncertaint y set, higher values of d form portfolio s that are more robust. Therefor e, an important property which the robustne ss measure should satisfy is the decreasing pattern with respect to d. We show this property of the robustne ss measure in the following two claims.

2.2.1. Claim 1

Claim 1. Let x�d be the optimal solution to the optimization problem (Rd). If x�d converges to x⁄ as d goes to infinity, then x⁄ is the portfolio that minimizes the robust measure. That is,

arg min x2X

maxl2Ud0

ð�lÞjf ðx; �lÞ � f ðx;lÞj

¼ limd!1

arg min x2X

maxl2Udð�lÞ

12x0Rx� kl0x

� �;

where

f ðx;lÞ ¼ 12x0Rx� kl0x;X ¼ fxjx0i ¼ 1g;

Ud0 ð�lÞ ¼ ljðl� �lÞ0R�1l ðl� �lÞ 6 d2

0

n o;

Udð�lÞ ¼ ljðl� �lÞ0R�1l ðl� �lÞ 6 d2

n o:

Proof. From the definition of the robustness measure,3

maxl2Ud0

ð�lÞjf ðx; �lÞ� f ðx;lÞj ¼ max

l2Ud0ð�lÞ

12x0Rx�k�l0x�1

2x0Rxþkl0x

��

¼ maxl2Ud0

ð�lÞkjðl� �lÞ0xj ¼ d0k


q:

Therefore,

arg min x2X

maxl2Ud0

ð�lÞjf ðx; �lÞ � f ðx;lÞj ¼ arg min

x2Xd0k


q:

On the other hand, the robust formulation can be derived as,

arg min x2X

maxl2Udð�lÞ

12x0Rx� kl0x ¼ arg min

x2X

12x0Rx� k�l0x

þ kdffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix0Rlx

q¼ arg min

x2X

1d

12x0Rx� k�l0x

� �þ k


q:

So proving the following equation will suffice our proof for Claim 1.

arg min x2X

d0kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix0Rlx

q¼ lim

d!1arg min

x2X

1d

12x0Rx� k�l0x

� ��

þkffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix0Rlx

q �:

In order to show the above, we prove r and s.

3 The proo f is based on the study by Kanniappan and Sastry (1983).

Please cite this article in press as: Kim, W.C., et al. Robus t portfolios that do nhttp ://dx.doi.org /10.1016/ j.ejor.2013.03.029

rlimd!1

minx2X

1d

12x0Rx� k�l0x

� �þ k


q� �¼min

x2Xkffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix0Rlx

q:

Proof of r. Let,

fdðxÞ ¼1d

12x0Rx� k�l0x

� �þ k


q; f1ðxÞ ¼ k


q

x�d ¼ arg min x2X

fdðxÞ; x�1 ¼ arg min x2X

f1ðxÞ:

Then r can be written as,

limd!1

fd x�d� ��

¼ f1 x�1� �

:

From our assumption that x�d converg es to x⁄ as d ?1, it follows that x�d

is a bounded sequence. Furthermor e, x�d

� X; x� 2

X; x�1 2 X, so there is a closed bounded subset D of X which con- tains x�d

; x�, and x�1. Since fd(x) unifo rmly converges to f1(x)

on the set D as d ?1, there exists n 2 N satisfying the following for all e > 0.

d P n ) jfdðxÞ � f1ðxÞj 6 e; 8x 2 D:

By the uniform convergence of fd(x), the following inequal ities are satisfied for all d P n,

fd x�d� �

� f1 x�d� �� 6 e; fd x�1

� �� f1 x�1

� �� 6 e:

Thus,

�e 6 fd x�d� �

� f1 x�d� �

6 fd x�d� �

� f1 x�1� �

6 fd x�1� �

� f1 x�1� �

6 e

and for all d P n,

fd x�d� �

� f1 x�1� �� 6 e:

Therefore, it follows that r holds

limd!1

fd x�d� ��

¼ f1 x�1� �

:

s If x�d converges to x⁄ as d ?1, then x⁄ is equal to x�1.Proof of s. Let x�d

be the sequenc e of the optimal solutions of

minx2X fd(x). Since f1(x) is continuous in x and x�d converg es to x⁄ as d!1; f1 x�d

� �conve rges to f1 (x⁄). Therefor e, for all e > 0,

there exists n1 2 N such that

d P n1 ) f1 x�d� �

� f1ðx�Þ�� 6 1

3e: ðiÞ

Because lim d!1

fd x�d� ��

¼ f1 x�1� �

by r, for all e > 0 there exists n2 2 Nsuch that

d P n2 ) fd x�d� �

� f1 x�1� �� 6 1

3e: ðiiÞ

In addition, fd(x) uniform ly conve rges to f1 (x) on the set D asd ?1, and therefore there exists n3 2 N for all e > 0 such that

d P n3 ) fdðxÞ � f1ðxÞj j 6 13e; 8x 2 D: ðiiiÞ

By defining n0 = max (n1, n2, n3), the inequal ities (i), (ii), and (iii) can be expressed by x�n0

as,

f1 x�n0

� �� f1ðx�Þ

�� 6 13e; ðivÞ

fn0 x�n0

� �� f1 x�1

� �� 6 13e; ðvÞ

fn0 x�n0

� �� f1 x�n0

� �� 6 13e: ðviÞ

Now, adding the three inequalities (iv), (v), and (vi) gives us,

f1 x�1� �

� f1ðx�Þ�� 6 e:



Fig. 1. Robustness measure for 10 portfolios for values of d from 0 to 3.

Fig. 2. Portfolio x is chosen so that that (xmax �xmin) and (x �x0) are perpendicular.


Thus, f1 x�1� �

¼ f1ðx�Þ and because f1(x) is a convex function, x⁄

is equal to x�1. In conclusion, if x�d converges to x⁄, x⁄ is the optimal solution of min x2X f1(x). By verifying s, we show that

arg min x2X

maxl2Ud0

ð�lÞjf ðx; �lÞ � f ðx;lÞj

¼ limd!1

arg min x2X

maxl2Udð�lÞ

12x0Rx� kl0x

� �:

Hence, we prove Claim 1. h

2.2.2. Claim 2

Claim 2. The robustnes s measure is a decreasing function of d. That is,the optimal portfolio becomes more robust as d increases and the robustness measure maxh2N0 f x�d; �h

� �� f x�d; h� �� decreases.

Proof. We provide a proof under the stylized assumption used in Kim et al. (2012) due to analytical limitatio ns for analyzing the general case. They show that (Rd) and the below optimization problem (QPa) have the same optimal portfolio under the proper choice of a.

ðQPaÞ minx

12 x0ðRþ aRlÞx� k�l0x

s:t: x0i ¼ 1:

The analytic solution for (QPa) can be found under the below stylized assumptions .

� Let R have the same value for all diagonal terms (r2) and also the same value for all off-diago nal terms (qr2)

Pleasehttp:/

R ¼r2 qr2

. ..

qr2 r2

2664

3775:

Then, since the inverse of R will have identica l diagonal terms and identica l off-diagonal terms, it can be expressed as

Fig. 3. Portfolio x is chosen so that xmax �x0 and x �x0 are perpendicular.

R�1 ¼

a b

. ..

b a

2664

3775:

cite this ar ticle in press as: Kim, W.C., et al. Robus t portfolios that do n/dx.doi.org/1 0.1016/ j.ejor.2013.03.029

� Let Rl be a diagonal matrix with the same diagonal terms r2l

� �,

ot tilt

Rl ¼

r2l 0

. ..

0 r2l

2664

3775:

� Let Ra = R + aRl, therefore, Ra for a > 0 represents a covariance matrix with extra penalization on the diagonal terms imposed on the Markowitz problem � Let the inverse of Ra = R + aRl be

R�1a ¼

aa ba

. ..

ba aa

2664

3775:

Under the stylized assumptions , Kim et al. (2012) find the optimal portfolio x�a for (QPa),

x�a ¼ ðaa � baÞu0 þ v0; where u0 ¼ kðl� �liÞ and v0 ¼1ni:

Now, we use the analytic optimal portfolio x�a for the calculatio n of the robustness measure, and show that the robustne ss measure decreases as a increases. The robustne ss measure becom es

factor exposu re. Europe an Journal of Operation al Rese arch (2013),


Fig. 4. Relax (R4) by using xa to expand the feasible region.

Table 1Summary of the new robust formulations.

Classical robust formulation ðRÞminxmaxl2Udð�lÞ12 x0Rx� kl0x

s.t. x0i = 1

Additional constraints Based on mean ðR1Þ x00B ¼ x0B

ðR10Þ x00B� e 6 x0B 6 x00Bþ e

Based on variance ðR2Þ x0BRf B0xx0Rx ¼ x00 BRf B0x0

x00Rx0

(R3) (xmax �xmin)0 (x �x0) = 0(R4) (xmax �x0)0 (x �x0) = 0(R40) (xmax �xa)0 (x �xa) 6 0

where xa = axmax + (1 � a)x0


maxh2N0

f x�a; �h� �

� f x�a; h� �� ¼ d0k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix�0a Rlx�a

q:

Since we already know the optimal solution of the robust optimization under the stylized assumption, the robustne ss measure can be calculate d analyticall y. Thus, we will focus on the change of the value x�0a Rlx�a with respect to the chang e of a. Thus, we have

x�0a Rlx�a ¼ ððaa � baÞu0 þ v0Þ0Rlððaa � baÞu0 þ v0Þ

¼ ðaa � baÞ2u00Rlu0 þ v 00Rlv0 þ 2ðaa � baÞu00Rlv0:

Because the third term 2ðaa � baÞu00Rlv0 is zero,

x�0a Rlx�a ¼ ðaa � baÞ2u00Rlu0 þ v 00Rlv0:

Since (aa � ba) is strictly positive and a decreasin g function of a; x�0a Rlx�a is also a decreasing function of a. Furthermore , because the value of a is proportio nal to the value of d, the robustness measure is a decreasing function of d under the stylized assumption . h

4 We use this simplified assumption because the scaled covariance matrix of returns provides the estimation error covariance in a stationary return process and the diagonals of the covariance matrix of estimation errors are known to contain enough information for forming robust portfolios. For detailed approaches employed in practice, see Stubbs and Vance (2005).

2.2.3. Simulation for Claim 2We already showed that the robustness measure is a decreasing

function of d under the stylized assumption. The remaining proof for Claim 2 is to show its validity for the general case and we do so using a simulation. We focus on the robust optimizati on problem (Rd) with an ellipsoidal uncertainty set and perform the following steps for the simulation.


Step 1: Generate �l and a positive definite matrix RStep 2: Let Rl be a diagonal matrix whose diagonal is the same as

R4

Step 3: Solve Rd and calculate the robustness measure for the optimal portfolio

We repeat the above three steps for various values of d to ob- serve any pattern. Fig. 1 presents results when n = 10, k = 0.5, and d from 0 to 3 with 0.1 increments. Results for 10 simulations are shown and all cases clearly show that the robustness measure decreases as d increases from 0 to 3. Therefore, we can conclude that Claim 2 is also satisfied even without the stylized assumptions .

3. New robust portfolio formulation

In the previous section, we introduced the robustnes s measure and its properties. The main purpose for presenting the robustness measure is to compare the robustness of portfolios from the original robust formulation with the robust portfolios from the new formulation s. Therefore, we return to our focus of this paper and derive robust formulation s that do not influence the factor exposure of portfolios .

Kim et al. (accepted for publication-a) and Kim et al. (2012) ob-serve that robust formulat ions may indirectly affect the factor loading of portfolios . Therefore, although mean–variance investors can protect the uncertainty risk by using the robust formulation sbased on the mean–variance model, they will not be able to sustain the expected factor exposure of the mean–variance portfolios. Con- sequently , we introduce new robust formulation s that control the depende ncy on fundamental factors by having the same factor exposure relative to a target portfolio. We again base our model on the robust formulation with an ellipsoidal uncertainty set since it is one of the first and most researched formulations.

3.1. Parameter assumptions

For simplicity, we assume the following case,



Fig. 5. Yearly value of factor loading (top: k = 0.01, bottom: k = 0.05).


n risky securities r: (random) return vector of risky securities E[r] = lVar (r) = R is strictly positive-definite r = Bf + e where f 2 Rm is the returns of m(<n) factors, B 2 Rn�m

is stock betas and e 2 Rn is the idiosyncratic risk x0: a portfolio that satisfies the target factor exposure (we set it to the Markowi tz portfolio).

3.2. Measuring the proportion invested in factors

The portfolio return of x can be expresse d as r = Bf + e by the factor model, where B is the matrix of stock betas and f is the factor returns. Therefore, the expected portfolio return and the variance of returns become,

E½x0r� ¼ E½x0Bf þx0e� ¼ x0Blf

Varðx0rÞ ¼ Varðx0Bf þx0eÞ ¼ x0BRf B0xþx0Rex

where lf = E[ f] and Rf = Var (f).The exposure to factors can be measured by how much the

mean and variance of portfolio returns are affected by factors,respectively . In other words, we can use the amount of return invested on factors, x0B, or the proportion of variance depended on factors, Varðx0Bf Þ

Varðx0rÞ ¼x0BRf B0x

x0Rx . Therefore, by controlling the factor exposure with constrain ts that incorporate these values, we can con-


struct portfolios that possess the same level of factor depende ncy as the target portfolio x0.

3.3. New robust formulatio ns

Consider the worst-case robust optimization of the form,

ðRÞ minx

maxl2Udð�lÞ

12 x0Rx� kl0x

s:t: x0i ¼ 1;

where Udð�lÞ ¼ ljðl� �lÞ0R�1l ðl� �lÞ 6 d2

n o. Based on (R), we intro-

duce new robust portfoli o formulation s that include additional constraint s on factor exposure.

3.3.1. Formulat ion based on the expected return of a portfolio We consider a constraint that matches the factor loading with

the target portfolio x0. The expected return of the target portfolio x0 is x00Blf , where x00B is the part of the return that depends on the factors. Thus, an arbitrary portfolio x will have the same factor exposure as portfolio x0 if x satisfies the following constraint,

x0B ¼ x00B:

Notice that the above constrain t has a convex feasibl e set because of linearity. By including this constra int into the original robust optimiza tion problem (R), the new robust formulation (R1) can be derived,



Fig. 6. Yearly value of proportion of variance from factors (top: k = 0.01, bottom: k = 0.05).


ðR1Þ minx

maxl2Udð�lÞ

12 x0Rx� kl0x

s:t: x0i ¼ 1x00B ¼ x0B:

However, since the equality constrain t of (R1) could restrict the feasible region too much at times, a variat ion using an inequal ity constraint can be used,

ðR10Þ minx

maxl2Udð�lÞ

12 x0Rx� kl0x

s:t: x0i ¼ 1x00B� e 6 x0B 6 x00Bþ e;

where e is a vector that sets the bound, which may be represented as a desired percentage of x00B. The new robust formula tion main- tains not only the factor exposure but also the robustne ss of the original formulation . Furthermor e, even though (R1) and (R10) have additional constrai nts, the optimal solution can be found efficiently since the extra constra ints are linear in both cases.

3.3.2. Formulation based on the variance of portfolio returns Next, we look at the variance of portfolio returns, which can be

represented using the factor model as x0BRf B0x + x0Rex for a


portfolio x. If the portfolio return is heavily affected by the factor returns, it means that the value of x0BRf B0x is relatively greater than x0Rex. Likewise, a greater value of x0Rex relative to x0BRf-

B0x shows that the portfolio return depends more on the idiosyncratic risk of individual asset movements. Therefore, the proportio n of variance explained by the variance of factors,x0BRf B0x

x0Rx , can also be used to match the factor exposure of portfolios.Thus, portfolio x will have the same factor exposure as portfolio x0 if the following constraint is satisfied,

x0BRf B0xx0Rx

¼ x00BRf B0x0

x00Rx0� � � ð�Þ

The new robust formulation with this constra int becomes,

ðR2Þ minx

maxl2Udð�lÞ

12 x0Rx� kl0x

s:t: x0i ¼ 1x0BRf B0x

x0Rx ¼ x00BRf B0x0

x00Rx0

Although the above robust formulation fulfills our goal, the non- convexit y from the new constra int makes it difficult to find the optimal solution even when using optimiza tion solvers. Therefor e,



Fig. 7. Robustness measure (95% confidence, top: k = 0.01, bottom: k = 0.05).


we look for a relaxed modification that can still achieve similar outcomes .

We first define g(x) to be the proportio n of the portfolio variance that depends on factor movement,

gðxÞ ¼ x0BRf B0x

x0Rx

Then, the portfoli o whose variance depends the most and the least on factors can be expressed respectivel y as,

xmax ¼ arg max gðxÞ such that x0i ¼ 1xmin ¼ arg min gðxÞ such that x0i ¼ 1:

Using the Rayleigh’s principle (Strang, 1988 ), the two values, xmax

and xmin, can be easily found by calculating the maximum and min- imum eigenvalues . Then the following inequali ty is satisfied for the given portfoli o x0,

gðxminÞ 6 gðx0Þ 6 gðxmaxÞ

and this can be used to find a constraint that relaxes constraint (⁄).Even if the problem (R2) cannot be solved directly, the alternativ eapproac h should at least find an optimal portfoli o that is not too close to either xmax or xmin. In other words, the factor exposure of portfolio x should not be maximized or minimize d unless that is the case for portfoli o x0. As shown in Fig. 2, portfoli o x willnot be close to either xmax or xmin if (xmax �xmin) and (x �x0)are orthogonal . This approach gives the third robust formula tion that does not have extreme factor exposures,


ðR3Þ minx

maxl2Udð�lÞ

12 x0Rx� kl0x

s:t: x0i ¼ 1ðxmax �xminÞ0ðx�x0Þ ¼ 0:

Since (R3) has only linear constra ints, it can be solved efficiently.Although it will not control factor exposure as well as (R2), it will be more effective than the original robust formulation .

We can modify (R3) based on the findings by Kim et al. (ac-cepted for publication-a) and Kim et al. (2012) that robust formulations tend to increase the dependency on factors. Hence, it is more important for the optimal portfolio to not move towards xmax than xmin. In this context, the orthogonality of (xmax �x0)and (x �x0) restricts the convergence of the optimal portfolio to xmax as shown in Fig. 3.

Therefore, the final robust formulation that only focuses on xmax is written as,

ðR4Þ minx

maxl2Udð�lÞ

12 x0Rx� kl0x

s:t: x0i ¼ 1ðxmax �x0Þ0ðx�x0Þ ¼ 0:

Furthermore, as we have shown in (R1), the additional constraint of (R4) to control factor exposure may be overly restric- tive for finding optimal portfolios. Therefore, the feasible region can be expanded using the modification of the constraint as shown in Fig. 4 and the resulting robust formulation becomes,



Fig. 8. Average standard deviation of returns of 10 portfolios using past 10 years data (top: k = 0.01, bottom: k = 0.05).

5 Note that the measure RM is derived from the robustness measure presented in Section 2. This is a special case when the ellipso idal uncertainty set is considered and Section 2.2.1 shows the derivation.


ðR40Þ minx

maxl2Udð�lÞ

12 x0Rx� kl0x

s:t: x0i ¼ 1ðxmax �xaÞ0ðx�xaÞ 6 0;

where xa = axmax + (1 � a)x0 for a 2 [0, 1] and larger values of aresult in optimal portfolios closer to xmax.

3.3.3. Summary Table 1 summari zes the results on the new robust formulat ions

presented in this section.

4. Empirical analysis with stock market returns

In the previous section, we propose new robust formulations having the same factor exposure relative to a given target portfolio.We now empirica lly confirm the effect on factor exposure of our proposed formulation s using historical market returns. We analyze robust portfolios formed from (R1) and (R40) using several test measures to compare the performance and factor exposure with the portfolios constructed from the original robust model (R) as well as the classical mean–variance portfolios from (MV).

4.1. Test measures

We define the following measures to evaluate the new robust optimal portfolios.


(1) Robustness measure To compare portfolio robustness, we use the robustnes smeasure defined in Section 2. Portfolios that are more robust have smaller values of this measure,5

ot tilt

RM ¼ kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix�0Rlx�

q:

(2) Factor exposure based on portfolio mean The factor loading of the optimal portfolio x⁄ is x⁄

0B, so the

2-norm of x⁄0B can be used to measure the magnitude of the

factor exposure,

F1 ¼ kx�0Bk2:

(3) Factor exposure based on portfolio variance The proportion of portfolio movement that depends on factor movement can be represented as x�0BRf B0x�

x�0Rx� as shown in the previous section. Therefore, we use this ratio as the second measure of the factor exposure of portfolios,

F2 ¼x�0BRf B0x�

x�0Rx�:

factor exposu re. Europe an Journal of Operation al Rese arch (2013),


Fig. 9. Average Sharpe ratio of 10 portfolios using past 10 years data (top: k = 0.01, bottom: k = 0.05).


4.2. Data description

In addition to using simulated returns, we use historical returns in the US stock market to check how the new robust portfolios perform against the classical mean–variance portfolio and the original robust portfolio with an ellipsoida l uncertainty set. The three-factor model proposed by Fama and French (1993, 1995) is used for the factor returns and the 49 industry portfolios proposed by Famaand French (1997) to validate their factor model are considered as the asset universe. We collect daily returns from January 1970 to December 2011 for both factor and industry-level returns.6

4.3. Empirical results

Using actual market data, we compare the portfolios from the four formulations: (MV), (R), (R1), and (R40) with a = 0.5. We run the empirical tests using a 1-year and 5-year rebalancing periods,values of k between 0.01 and 0.1, and confidence levels between 10% and 95%. We use the test measures defined in Section 4.1 forcomparing the portfolios and we present the results for a few cases but the overall observation is the same in all settings. All figures in-

6 These are the same data used in the empirical analysis by Kim et al. (accepted for publication-a). Both sets of data are available online from French’s data library (http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_lib rary. html).


cluded use a 1-year rebalancing period since it gives most data points, one portfolio every year.

The values of F1 representi ng the magnitude of factor loading are plotted for the four formulat ions for each of the 42 years. It is clearly observed, as shown in Fig. 5, that the solid line, which represents the portfolios from (R), has values different from the other three portfolios. Even though we did not directly apply a constraint for this measure, we see that the portfolios from (R40) also have values that match the portfolios from (MV) although not as exact as the ones from (R1). The pattern is shown more visibly when looking at the bottom figure, which is when the value of k is set to 0.05. The first set of results demonstrat e that the two proposed robust formulation s in this paper can indeed control the factor exposure of portfolios .

We next check the second measure of factor exposure F2, which is the proportio n of portfolio variance that is explained by the variance of factors. Similar to the results for F1, levels of F2 that match the classical mean–variance portfolios from (MV) are desired. As shown in Fig. 6, since the additional constraint for (R40) is based on this ratio of variance, its portfolios have values that are closer to the portfolios from (MV), but not exact since a relaxed version of the constrain t is used in (R40). In addition, although (R1) does not restrict its weights based on variance, its portfolios are shown to have values much closer to the portfolios from (MV) than (R);the characteri stic of (R1) to control its factor loading is shown even when analyzed using another measure of factor exposure. This set




of results also confirms that the two formulation s (R1) and (R40)match the factor exposure with a target portfolio.

To measure robustness of the four formulation s as defined by the deviation in the objective function of the portfolio optimizati on model as introduce d in Section 2, we calculated the robustness measure RM defined in Section 4.1. As shown in Fig. 7, the portfolios from (R) are the most robust and the ones from (MV) are the least robust. Although the portfolios from (R1) and (R40) have lower robustness than the portfolios from (R), they protect the uncertainty risk better than (MV). Therefore, we can conclude that (R1)and (R40) have the target factor exposure as well as improved robustness.

We also use standard performance measures to test the robustness of portfolio returns and the risk-return trade-off of portfolio performanc e. We use out-of-sampl e data to see whether the proposed robust models satisfy the basic objectives of robust portfolios. More specifically, for year t, 10 portfolios are created from year t-10 to t-1 using each year’s returns and the returns of these 10 portfolios at year t are observed . This directly shows how a portfolio optimization technique performs on average at year t whendifferent sets of inputs are used.

As a measure of robustness of portfolio returns, we investiga te the annualized standard deviation of portfolio returns. In our set- ting, for year t, the volatility of the 10 portfolios created from returns in year t-10 to t-1 is measure d to see how much portfolio returns change with respect to the change in input. As shown in Fig. 8, the portfolios from (R) are the most robust. Portfolios from (R1) and (R40) are less robust than the ones from (R) because they each have one additional constraint, which restricts the choice of feasible solutions, and the extra constrain t is used to limit factor exposure but not included for further strengthening the robustness.

Similar to robustness, the average Sharpe ratio of the 10 portfolios for each year is calculated to compare the performance of the four formulation s. Note that this is not a pure measure of performance but rather incorporates robustness since it is looking at the average performance of 10 portfolios, each formed using sepa- rate inputs. Again, as shown in Fig. 9, the portfolios from (R) dom- inate other portfolios and they are able to capture the rally of 1985 and 1995, which can be explained by its robust nature and its ten- dency to move towards market factors. More importantly , even though the factor exposures of (R1) and (R40) are controlled, the two robust portfolios do not perform any worse than the portfolios from (MV), mainly because they are less sensitive to input changes.

In summary , the portfolios from (R1) and (R40) more closely match the target factor exposure when compared against (MV).Even though the additional constraints of (R1) and (R40) result in portfolios that are not as robust as the portfolios from (R), they still form portfolios that are relatively more robust than (MV).

5. Conclusion

In this paper, we derive new robust formulation s which main- tain a desired level of factor exposure while conserving the robust effect of robust optimizati on. It is previously shown by Kim et al.(accepted for publication-a) and Kim et al. (2012) that robust portfolios from worst-ca se optimization have higher factor dependency than the classical mean–variance model. For solving this problem,we introduce several approaches by adding constraints into the well-known robust formulation with an ellipsoidal uncertainty set. We define two measures for calculating the level of factor


exposure of portfolios that are based on the mean and variance of portfolio returns, and derive formulat ions that include restric- tions on these measure s. Our empirical results clearly show that the improved robust portfolios resolve the tilting effect of current robust formulat ions while still making the portfolios more robust than mean–variance portfolios.

Acknowledgeme nts

The authors are grateful for helpful comments of the anony- mous referees and for the support by the Public Welfare & Safety Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technolo gy (2011-0029883).

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