Multiple factor models for equities: An empirical study of the performance of factor mimicking portfolios

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Multiple factor models for equities

An empirical study of the performance of factor mimicking portfolios

Students

Christoffer Forssén Gustav Åhs

Spring 2017

Master thesis, 30 ECTS

Master of Science in Engineering and Management, 300 ECTS

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i Abstract

The trade-off between risk and return for equities has long been a challenge for portfolio and risk managers in order to create financial success and stability. This issue has led to several researchers trying to explain equity returns through various factor models. The capital asset-pricing model (CAPM) formulated by Sharpe (1964), Lintner (1965), and Black (1972) was the first model explaining the relation between cross-sectional returns relative the broad market index. Since then, factor models have evolved and fundamental multiple factor models have been found to successfully explain the risk structure of equities, through linear combinations of firm specific data and market data.

In this paper, we implement and analyze a fundamental factor model. The objective is to build a dynamic and robust model that provide portfolio and risk managers with insight of what drives returns and risks of equities and portfolios. A key to understand the advantages of factor models lies in the characteristics of factors and the concept of factor mimicking portfolios, whose return perfectly replicates those of factors. These portfolios are derived through cross-sectional regressions of security returns and standardized exposure towards factors, which results in portfolios with a desired exposure. The model implementation is applied and evaluated for both a European and Swedish estimation universe, and the result indicate that some factor mimicking portfolios yield an excess return relative the market during 2015-01-01 to 2017-01-01.

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ii Sammanfattning

Avvägningen mellan risk och avkastning för aktier har länge varit en utmaning för portföljförvaltare och riskanalytiker, vars syfte är att skapa ekonomisk framgång och stabilitet. Denna fråga har lett till att forskare försökt förklara aktiers avkastning genom olika faktormodeller. Capital-asset-pricing model (CAPM) som formulerades av Sharpe (1964), Lintner (1965) och Black (1972) var den första modellen som förklarade förhållandet mellan tvärsnittsavkastning och marknadsindexet. Modellen har sedan generaliserats och genom att beskriva aktieavkastningen som ett linjärt samband mellan flera faktorer så erhålls bättre förståelse kring den faktiska aktie- och portföljstrukturen.

I denna rapport implementerar och utvärderas en fundamental faktormodell. Målet är att skapa en dynamisk och robust modell som utgör ett verktyg för portfölj- och riskanalytiker, med syftet att ge en bättre inblick i vad som driver avkastning samt förnyar metoder för att analysera riskstrukturen för aktier och portföljer. Genom att förklara riskstrukturen med hjälp av grundläggande faktorer så ökar förståelsen för uppbyggnaden av en portfölj. För att lyckas med detta måste faktorernas egenskaper studeras. Detta görs genom att skapa portföljer som perfekt replikerar företagsspecifika faktorer, vilka bidrar till förklaringen av aktiers avkastning. Dessa portföljer är härledda genom tvärsnittsregressioner av aktieavkastningar och aktiers standardiserade exponering mot faktorer, vilket resulterar i portföljer med önskad exponering. Modellen som implementerats har utvärderats för både ett europeiskt och svenskt aktieuniversum och resultaten visar på att vissa faktorportföljer tenderar att ge en överavkastning i förhållande till marknaden under perioden 2015-01-01 till 2017-01-01.

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1. Introduction

1.1 Background

The trade-off between securities expected return and their associated risk is an important objective for portfolio and risk managers, whose purpose is to provide financial success and sustainability. Because of financial crises and stricter regulatory framework, managers are in need of robust instruments to interpret risks and enhance portfolio performance. The capital asset-pricing model (CAPM) formulated by Sharpe (1964), Lintner (1965), and Black (1972), has long been the dominant instrument to interpret the risk-return relationship and the cross-sectional return relative to the broad market index. Since the asset-pricing model was formed, researchers have studied the trustworthiness of the model and numerous empirical contradictions have been found.

One researcher who developed an alternative to the CAPM was Ross, in the Arbitrage Pricing Theory (APT). Ross (1976) proved that single factor models should be generalized to include multiple factors, and thereby created the foundation for multiple factor models. With multiple factor models, it is more reasonable that factors, which explain the cross-sectional return, are uncorrelated with the non-diversifiable risk and thus making the model more accurate. Fama and French (1992) further explored the topic and concluded that the relation between beta, 𝛽, which is a measure of systematic risk of a security compared to the market, and the average return in the CAPM is only positive related in certain periods. Therefore, 𝛽 alone fails to explain the security return.

Instead, Fama and French found empirical evidence that by including the three factors, book-to-market, market capitalization and 𝛽, the cross-sectional variation in average stock returns are more accurate explained. Since then, a vast amount of research has been done in the field to further distinguish factors whom explain securities return.

Analyzing the situation today, multiple factor models have grown in popularity and is frequently applied in several areas of financial theory. The factors do not only have the ability to explain the structure of security returns, they also provide portfolio and risk managers with a framework to categorize securities after economic, statistic and firm specific attributes (fundamentals). Madhavan (2016) describes how different factors in a multiple factor model serves as a basis for investments, which is a strategy known as factor investing or smart-beta. This investing method aims to explain the return structure of a security through different factors and thereafter find the ones whose return tend to systematic outperform the market or severely contributes to the risk. To gain these insights, factor mimicking portfolios are created, which is portfolios whose returns perfectly replicates those of factors. These portfolios are constructed so that it takes long positions in securities with high exposure towards a factor, and short positions in securities with low exposure towards a factor. Thus, by implementing a multiple factor model and analyzing the factors, portfolio and risk managers will be benefited with interpretable and robust guidelines to increase portfolio performance and a greater understanding of risk.

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1.2 Objective

In this paper, we will implement and analyze a fundamental multiple factor model for equities. The objective is to create a dynamic and efficient program that serves as a tool to explain the return and risk structures in securities. To achieve this, we will create portfolios with a weight structure that isolate securities exposure towards firm specific factors, which thereby replicates the returns of these factors. The analysis will mainly focus on time series of cumulative returns for these replicating portfolios, in order to distinguish which factors that tend to perform well and thereby are important for portfolio performance. To better understand the risk structure of securities, the relationship between portfolios will also be studied. The models’ credibility and accuracy will be evaluated through measurements of linear dependence between input parameters and how this in turn affects the coefficient of determination. Moreover, the model will be evaluated when changing both security estimation universe and rebalancing frequency of portfolios. Consequently, we aim to create a robust model that provides asset and portfolio managers with additional insights of the risk structure of securities, which thereby hopefully benefits their investors with more attractive products.

1.3 Delimitations

The included factors are selected from existing literature and in agreement with the supervisors of this project. Furthermore, the common approach of building factors based on several measurements are excluded in this paper. Instead are single fundamentals used as factors in the model. Moreover, due to limited access to data are only two equity universes considered, EURO STOXX 600 and OMXS Stockholm PI. This limitation also restricts us from including the currency effects, since all securities are listed in the same currency. Lastly, categorizing securities after corresponding countries are excluded in this implementation.

1.4 Approach and Outline

In this paper, the characteristics of fundamental factor models will be studied and implemented. Furthermore, portfolios that perfectly mimic the return of specified equity fundamentals (factors) will be derived and analyzed throughout the paper. In the second chapter, the reader will be provided with an overall picture of multiple factor models to establish understanding of the application area. The continuing section explains the basic theory behind factor mimicking portfolios, thus how the returns of equity fundamentals are derived and the construction of factor mimicking portfolios. In the third chapter, focus is on the implementation of fundamental multiple factor models, along with a review of necessary steps to replicate the procedure. Thereafter, results from the model implementation will be presented in the forth chapter, which continues with a short discussion and conclusion in the final chapter.

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2. Theory

2.1 Factor models

There are several approaches to multiple factor models in financial theory. Although only the fundamental factor model will be implemented and analyzed, it is important to understand the difference between multiple factor models. Therefore, this section will be initiated with a short review of the three most commonly used model and their advantages and disadvantages respectively. The model approaches are; macro-economic, statistical and fundamental.

2.1.1 Macro-economic factor model

Connor (1995) describes that a macro-economic multiple factor models aims to explain the return of a security through a linear combination of macro-economic factors. The first step of implementing such model is to specify which factors to include. This decision depends on the investor’s empirical research and available data. For example, Benakovic et al (2010) analyze how well a macro-economic model can explain the security return by including the macro-economic factors; price of crude oil on world market, industrial production index volume index, interest rate, consumer price index, and stock exchange index. Initially, the change of the selected macro-economic factors for a given time-period are calculated. This includes calculating the changes in crude oil prices, consumer price index, industrial production, etc. for a given time-period. Once these changes are known, one can use them to estimate how exposed securities are towards changes in macro-economic factors. For example, assume a company’s operations mainly consists of transportation with vehicles using fossil fuel. If the price of crude oil on the world market raises, the company’s profitability might decrease and thereby reducing their stock’s return. Hence, the company’s security return has a significant exposure to changes in price of crude oil on the world market. To estimate each securities exposure towards changes in macro-economic factors, a multiple regression is conducted. The input parameters for the regression is security returns, 𝑟_𝑛,𝑡, as dependent variables and changes of macro-economic factors for a given time-period, 𝐶_𝑘,𝑡, as independent variables. Assume that N number of stocks and K macro-economic factors are included in the model. Then the model is expressed as,

𝑟_𝑛,𝑡 = 𝛼_𝑛+ ∑ 𝑋_𝑛,𝑘,𝐶_𝑘,𝑡

𝐾

𝑘=1

+ 𝜖_𝑛,𝑡 (2.1)

where

𝑟_𝑛,𝑡 = return for security 𝑛 at time t 𝛼_𝑛 = constant term

𝑋_𝑛,𝑘 = exposure for security 𝑛 to macro-economic factor 𝑘 𝐶_𝑘,𝑡 = change of macro-economic factor 𝑘 at time t

𝜖_𝑛,𝑡 = disturbance term for security n at time t.

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The output from the regression of equation (2.1) is the estimated exposure, 𝑋_𝑛,𝑘 for each security n, towards each macro-economic factor k, the constant term, 𝛼_𝑛, for each security and the disturbance term, 𝜖_𝑛,𝑡, for each security. For example, if the multiple regression includes 4 stocks and 5 factors, a total of 20 exposures are estimated. The disturbance term, 𝜖_𝑛,𝑡, has an expected value of zero and constant variance. The constant term, 𝛼_𝑛, is seldom statistical significant and is therefore usually zero. Once the estimated exposures and the constant terms are known, a second regression is performed to estimate the macro-economic factor returns and once again the disturbance term. The estimated factor returns are also known as risk premiums and can be viewed as the systematic return of factors. For example, a positive value of a factor return indicates that investors are being compensated for bearing a positive exposure towards that factor. The regression is preformed cross-sectional, with the security returns, 𝑟_𝑛,𝑡, as dependent variable and the estimated exposures, 𝑋_𝑛,𝑘, calculated in the first regression, as independent variable. A cross-sectional regression, in contrast to time series regression, means that all data used in the regression represents the same point in time.

Hence, it is cross-sectional over all included securities at time t. The second regression takes the form

𝑟_𝑛,𝑡 = 𝛼_𝑛 + ∑ 𝑋_𝑛,𝑘,𝑓_𝑘,𝑡

𝐾

𝑘=1

+ 𝜖_𝑛,𝑡 (2.2)

where

𝑟_𝑛,𝑡 = return for security 𝑛 at time t 𝛼_𝑛 = constant term

𝑋_𝑛,𝑘 = exposure for security 𝑛 to macro-economic factor 𝑘 𝑓_𝑘,𝑡 = factor return of macro-economic factor 𝑘 at time t 𝜖_𝑛,𝑡 = disturbance term for security n at time t

Once the factor returns are known, several of statistical tests can be applied to evaluate the model and if the output parameters from equation (2.1) and (2.2) are statistically significant. Connor (1995) emphasize the importance of identifying and measure all of the macro-economic factors affecting security returns. If an investor is uncertain about the factors affecting the exposures, or the magnitude of these factors, there is a lack of information to explain security returns. Another disadvantage of macro-economic factor models is that it requires an extensive set of historical security returns to accurately estimate exposures and factor returns. In practice, it is occasionally difficult to find long and stable time series of data for a security’s return. This is particularly the case for securities on emerging markets or securities that recently been offered to the public.

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5 2.1.2 Statistical factor model

A statistical factor model aims to explain the return of a security through statistical methods, most commonly by the principal component analysis (PCA). As in macro- economic factor models, the statistical approach aims to assign a value to a factor that explains the security return. However, a key difference is that the model is strictly statistical and do not include neither economic nor financial interpretation. Hence, exposures and factor returns are created in the model only through the security returns, which is explain more in detail by Bro and Smilde (2014). Alexander (2008) states that by conducting PCA on a large set of security returns, a smaller number of principal components can be identified. The principal components are uncorrelated with the security returns and each principal component explains a proportion of their variation.

The major advantage of using a statistical factor model is that it only requires time series of security returns. Statistic factor models also have a high explanatory power since the statistical factors, i.e. principal components, are decided by maximizing the fit of the model. Although this is desirable properties, it is difficult to interpret what actions that should be carried out when analyzing the output. As Connor (1995) points out, a high or low measure in one of the principal components does not contribute to understanding what factors that explains the return.

2.1.3 Fundamental factor model

The fundamental approach of multiple factor models aims to explain security returns as linear combinations of fundamentals, industry and country factors. Additionally, a market factor is included in the model with the purpose to capture the general fluctuations of markets. The market factor is a source of risk that all securities are subject to and therefore all securities have a unit exposure towards it. These categories of factors are discussed in Section 2.3. Alexander (2008) states that a fundamental factor, also known as style factor, is usually a firm specific attribute. Amongst other, factors such as firm’s size, sales, volatility, book-to-market value and dividend yield can be included to explain security returns, since all firms have an exposure towards these attributes. The reasoning behind factor exposure is similar to the macro-economic model, although one key difference is that the exposure to fundamental factors do not have to be estimated through a multiple regression. Hence, a security’s exposure towards a fundamental factor is known since it is observable and calculated through fundamental and market data. For example, assume a fundamental factor model includes the three factors size, book-to-market value and momentum. Consider a company in the estimation universe that is much larger than the average company. This company will have a large and positive exposure to the size factor. If the same company recently have underperformed the average company in the universe, it will also have a low exposure towards the momentum factor. The industry exposure is usually expressed with an indicator variable and can either assume the value of 0 or 1, depending on whether the security can be linked to a certain industry or not. The same procedures apply for the country exposure.

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After all firm specific data have been specified, thus securities exposures towards factors have been calculated, the data have to be standardized. The reason behind this and how the standardization is performed is clarified in later sections. Thereafter, a cross-sectional regression is conducted to estimate the factor returns and the disturbance term. Assume that N is the number of stocks, Kc is the number of countries, Ki is the number of industries and Ks is the number of fundamental factors. The model can then be expressed as,

𝑟_𝑛,𝑡− 𝑟_𝑓,𝑡 = 𝑓_{𝑚𝑘𝑡,𝑡}+ ∑ 𝑋_{𝑛,𝑘,𝑡}𝑓_𝑘,𝑡

𝐾𝑐

𝑘=1

+ ∑ 𝑋_{𝑛,𝑘,𝑡}𝑓_𝑘,𝑡

𝐾𝑐+𝐾𝑖

𝑘=𝐾𝑐+1

+ ∑ 𝑋_{𝑛,𝑘,𝑡}𝑓_𝑘,𝑡

𝐾𝑐+𝐾𝑖+𝐾𝑠

𝑘=𝐾𝑐+𝐾𝑖+1

+ 𝜖_𝑛,𝑡, where

𝑟_𝑛,𝑡 = return for security 𝑛 at time t 𝑟_𝑓,𝑡 = periodic return of risk-free rate 𝑓_{𝑚𝑘𝑡,𝑡} = return of market factor at time t

𝑋_{𝑛,𝑘,𝑡} = exposure for security 𝑛 to a factor 𝑘 at time 𝑡 𝑓_𝑘,𝑡 = factor return to a factor 𝑘 at time 𝑡

𝜖_𝑛,𝑡 = disturbance term for security n at time t

(2.3)

In the cross-sectional regression, the security returns over the risk-free rate are used as dependent variables and the factor exposures are used as the independent variable. The risk-free rate is subtracted from the security returns because an investor should only be compensated if the return of a risky security outperforms a risk-free asset. To further clarify, the known input values before the cross-sectional regression is security’s exposure towards all factors, security returns and the return of the risk-free rate. The final output from the cross-sectional regression is the factor returns for each factor included in the model and the disturbance term, 𝜖_𝑛,𝑡. The disturbance term, 𝜖_𝑛,𝑡, can be interpret as the return not captured by factors and therefore is the firm specific return.

For example, assume one wants to explain the security return through the factors size, book-to-market value and momentum in one time-period. Assume that the estimation universe consists of 3 securities, with firm specific data in time t specified as,

Security 𝑟_𝑛,𝑡− 𝑟_𝑓,𝑡 Country Industry Size

Book-To-

Market Momentum

Company 1 0.01 Sweden Financial 450 1.1 0.3

Company 2 -0.05 England Industrial 800 0.8 0.1

Company 3 0.02 Sweden Energy 300 2.1 0.2

As mentioned above, the fundamental data, i.e. securities exposure towards size, book- to-market value and momentum, is observable and have to be standardize before the cross-sectional regression. For the reader to easier interpret the model, we will express the explicit form of equation (2.3) before the standardization,

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7 [

0.01

−0.05 0.02

] = [ 1 1 1

] 𝑓_𝑚𝑘𝑡+ [ 1 0 1

0 1 0

] [𝑓_𝑆𝑤𝑒 𝑓_𝐸𝑛𝑔] + [

1 0 0

0 1 0

0 0 1

] [ 𝑓_𝐹𝑖𝑛 𝑓_𝐼𝑛𝑑 𝑓_𝐸𝑛𝑒

] + [ 450 800 300

1.1 0.8 2.1

0.3 0.1 0.2

] [ 𝑓_{𝑆𝑖𝑧𝑒} 𝑓_{𝐵𝑜𝑜𝑘} 𝑓_𝑀𝑜𝑚

] + [ 𝜖₁ 𝜖₂ 𝜖3

].

There are several methods that can be applied to estimate the factor returns and the firm specific return in the equation above. The approach applied in this paper is weighted least square and Lagrange multiplier, which is further described in Section 2.4. Connor (1995) states that the firm specific data, i.e. factor exposures, is a key component in the fundamental model. Therefore, a large and reliable set of data regarding the company’s fundamentals is required. This causes the model to be data intensive and a numeric efficient method is preferable upon implementing. Despite these unwanted attributes, there are several of advantages. The main advantage is that it is possible to interpret the result once the factor returns have been estimated in the cross-sectional regression. For example, assume a time series of factor return for the size factor, 𝑓_{𝑠𝑖𝑧𝑒}, are being analysed. If analysis reveals that investors tend to be compensated for bearing the size risk, hence the time series of 𝑓_{𝑠𝑖𝑧𝑒} is positive, then there may be an opportunity to obtain excess return by investing in securities with a large and positive exposure towards the size factor. An additional advantage of this approach is that new securities can be incorporated to the model very quickly. This is because a cross-sectional regression is used and thus no historical data for securities are needed. The procedure of creating portfolios that replicate factor returns are described in the following section.

2.2 Construction of factor mimicking portfolios

As mentioned in the background, the implementation of a fundamental multiple factor model aims to distinguish the risk-return structure of securities and thereafter find factors that outperform the market. This leads us to the practice of factor investing, and more specifically how we can construct portfolios that perfectly replicate the returns of the factors included in the fundamental factor model. That is, portfolios with a weighting structure that isolate securities exposure towards firm specific factors, which thereby replicates the returns of these factors. As we shall see, these portfolios take long positions in securities with a positive exposure towards a factor and short positions in securities with negative exposures towards a factor. Factor investing, as this strategy is called, is also known as smart-beta strategies or risk-premia investing. As mentioned by Koedijk et al. (2003), the financial research has studied the risk premium of security returns since the eighties. Research has shown that some investment segments of the market realize better returns than those in other segments. Well known segments are stocks with high exposures towards momentum, value, low-volatility and small-size, and these have been proven by researchers to significantly outperform the market. In order to understand the concept of factor investing, we start by explaining the construction of single factor mimicking portfolios, which provides a basis for construction of pure factor mimicking portfolios and thus the understanding of this investment strategy. As we shall see, these portfolios are derived through a weighted least squared method with constraints.

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8 2.2.1 Single factor mimicking portfolios

Fundamental factor models are often used to create factor mimicking portfolios since the input data easily can be accessed and is intuitive. This leads to results that also are possible to interpret for investors, which is a wanted property. These models include factors such as countries, industries and fundamental factors (called style factors) as explanatory variables. Additionally, a market factor is often included in the model. The construction of single factor mimicking portfolios is explained by Menchero (2010) and Clarke et al. (2017) which both initially emphasize the importance of the standardization procedure of securities exposure towards factors, i.e. factor exposures, which is denoted as 𝑋_{𝑛,𝑘,𝑡}. The style factor exposures require having a weighted mean of zero,

∑ 𝑤_𝑛,𝑡𝑋_{𝑛,𝑘,𝑡}= 0

𝑁

𝑛=1

(2.4) where the security weights, 𝑤_𝑛, are calculated as in equation (2.7) or (2.8) and where N denotes the number of securities and K denotes the number of factors. Likewise, the weighted standard deviation should sum to one. Thus,

∑ 𝑤_𝑛,𝑡𝑋_{𝑛,𝑘,𝑡}² = 1

𝑁

𝑛=1

.

(2.5)

This is applied cross-sectional, i.e. for all securities in the estimation universe at time t.

By implementing this standardization methodology, it can be guaranteed that the estimated market factor returns are neutralized against all estimated style factor returns, hence the least squares estimates of style factor returns are expressed relative to the market factor return due to this weighted standardization. And as we shall see, this procedure also results in controlling the exposure of a single factor mimicking portfolio.

Menchero (2010) describes the construction of a factor mimicking portfolio for one single style factor, which is based on a univariate cross-sectional regression of the form, 𝑟_𝑛,𝑡^𝐸 = 𝑓_{𝑚𝑘𝑡,𝑡}+ 𝑋_{𝑛,𝑘,𝑡}𝑓_𝑘,𝑡+ 𝜖_𝑛,𝑡 (2.6) where 𝑟_𝑛,𝑡^𝐸 denotes the periodic excess return over the risk-free rate, 𝑟_{𝑓,𝑡−1}Δ𝑡, of the 𝑛th security. The market factor return are denoted as 𝑓_{𝑚𝑘𝑡,𝑡} which also is referred to as the intercept term, since all securities have 100% exposure towards this factor. Furthermore, 𝑓_𝑘,𝑡 denotes the return of the kth style factor, 𝑋_{𝑛,𝑘,𝑡} denotes the exposure to the kth style factor of the nth security and 𝜖_𝑛,𝑡 denotes the idiosyncratic return of the nth security. To clarify, securities exposures towards factors are known and we aim to estimate the unknown factor returns, 𝑓_𝑘,𝑡 and 𝑓_{𝑚𝑘𝑡,𝑡} at time t. Notice it is assumed that,

𝑣𝑎𝑟(𝑓_𝑘,𝑡) = 𝜎_𝑓²_𝑘 𝑐𝑜𝑣(𝑓_𝑘,𝑡, 𝜖_𝑛,𝑡) = 0, ∀𝑘, 𝑡

𝑣𝑎𝑟(𝜖_𝑛,𝑡) = 𝜎_𝑛², 𝑛 = 1, … 𝑁,

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and thus the variance of the idiosyncratic term, 𝜖_𝑛,𝑡, is heteroskedastic, i.e. not constant over time. Therefore, the weighted least squares (WLS) becomes an appropriate and efficient method for solving the factor returns. However, this suggests that securities variances are known, since these normally are used as the weighting-scheme for such regressions. Moreover, a comparable weighting-scheme that is used in this paper are the square root of market capitalization, since it is proven to be inverted proportional to a security’s variance. Hence,

𝑤_𝑛,𝑡= √𝑀𝐶𝑛,𝑡

𝛴_𝑛=1^𝑁 √𝑀𝐶𝑛,𝑡 (2.7)

where 𝑀𝐶_𝑛,𝑡 denote the market capitalization of the nth security, i.e. a security’s number of shares multiplied with its price at time t. The weights are standardized so they sum to one over all securities in the estimation universe, hence 𝛴_𝑛=1^𝑁 𝑤_𝑛,𝑡 = 1. An alternatively weighting-scheme that will be considered in this paper is the equally weighting-scheme. Unlike the market capitalization weighting-scheme, which distributes weights proportional to the size of securities, the equally weighted-scheme gives identical weighting to each stock in the universe, resulting in all stocks having the same impact on factor returns that will be estimated. Thus,

𝑤_𝑛,𝑡= 1 𝑁

(2.8) where N denotes the total number of securities in the estimation universe. Furthermore, consider any of the two weighting-schemes (2.7) and (2.8), then single factor returns can be derived and expressed as,

𝑓_𝑘,𝑡 = ∑(𝑤_𝑛,𝑡𝑋_{𝑛,𝑘,𝑡})𝑟_𝑛,𝑡^𝐸

𝑁

𝑛=1

(2.9)

whereas style factor mimicking portfolios are given by 𝑤_𝑛,𝑡𝑋_{𝑛,𝑘,𝑡}. This can be interpreted as taking long positions in all stocks with positive exposure towards a given style factor and short positions in all stocks with negative exposures towards the same factor. With an accurate standardization procedure, the weights of single style factor mimicking portfolios sum to zero and have 100% exposure to the given factor as of equation (2.4) and (2.5) respectively. Moreover, the return of the market factor can be derived from the regression model (2.6) and is given by,

𝑓_{𝑚𝑘𝑡,𝑡}= ∑ 𝑤_𝑛,𝑡𝑟_𝑛,𝑡^𝐸

𝑁

𝑛=1

(2.10)

which simply becomes the return of the weighted market portfolio, 𝛴_𝑛=1^𝑁 𝑤_𝑛,𝑡, since the properties of idiosyncratic returns tend to diversify away, 𝐸[𝜖_𝑛,𝑡] = 0. Furthermore, derivation of factor mimicking portfolios including indicator variables such as countries and industries, differ from how the style factor mimicking portfolios are derived.

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Consider the regression model defined as in equation (2.6), where 𝑋_𝑛,𝐼 ∈ {0, 1} and now denotes the exposure of the nth security to either industry or country 𝐼. This leads to a perfect linear dependency between the exposure of the market factor and all industry factors. This can be interpreted as the market being divided into these different industries or countries. Consequently, an issue occurs with multicollinearity and the regression cannot result in a unique solution. In order to manage this issue, Heston and Rouwenhorst (1994) introduces linear constraints to obtain a unique solution to the regression model. The most commonly used constraint is that the sum of all weighted factor returns sum to zero, hence 𝛴_𝐼𝑊_𝐼,𝑡𝑓_𝐼,𝑡= 0. In this case, 𝑊_𝐼,𝑡 denote the market capitalization weights of all stocks included within a specific industry or country. The market factor return can now be expressed as,

𝑓_{𝑚𝑘𝑡,𝑡}= ∑ 𝑊_𝐼,𝑡

𝐾𝑖

𝐼

∑ (𝑤_𝑛,𝑡𝑟_𝑛,𝑡^𝐸 𝑉_𝐼,𝑡 )

𝑁

𝑛∈𝐼

(2.11)

where 𝑉_𝐼,𝑡 denote the regression weight of industry or country I. Notice the difference between equation (2.9) and (2.11) where each indicator variable is market capitalization weighted, but the stocks within the groups are regression weighted. As a result, the single industry or country factor returns are given by,

𝑓_𝐼,𝑡= ( 1

𝑉_𝐼,𝑡∑^𝑁 𝑤_𝑛,𝑡𝑟_𝑛,𝑡^𝐸

𝑛∈𝐼 ) − 𝑓_{𝑚𝑘𝑡,𝑡} (2.12)

which can be interpreted as these factor mimicking portfolios going long the indicator factor mimicking portfolio and goes short the market factor mimicking portfolio defined in equation (2.11). Thus, industry factor returns are estimated relative to the market, just as for style factor returns.

2.2.2 Pure factor mimicking portfolios

Unlike the construction of a single factor mimicking portfolios, which is derived through a univariate cross-sectional regression model, the derivation of pure factor mimicking portfolios is formed throughout a multivariate cross-sectional regression model defined as the fundamental model approach. Hence,

𝑟_𝑛,𝑡^𝐸 = 𝑓_{𝑚𝑘𝑡,𝑡}+∑𝑋_{𝑛,𝑘,𝑡}𝑓_𝑘,𝑡

𝐾𝑐

𝑘=1

+ ∑ 𝑋_{𝑛,𝑘,𝑡}𝑓_𝑘,𝑡

𝐾𝑐+𝐾𝑖

𝑘=𝐾𝑐+1

+ ∑ 𝑋_{𝑛,𝑘,𝑡}𝑓_𝑘,𝑡

𝐾𝑐+𝐾𝑖+𝐾𝑠

𝑘=𝐾𝑐+𝐾𝑖+1

+ 𝜖_𝑛,𝑡. (2.13)

That is, by simultaneously processing every country, industry and style factor along with the market factor. With this approach, we have two exact multicollinearities, that is, both for the industry and country factors exposures relative to the market factor exposure. As mentioned before, Heston and Rouwenhorst (1994) manages this issue with the constraints that the weighted country and industry factor returns sum to zero, 𝛴_𝐼𝑊_𝐼,𝑡𝑓_𝐼,𝑡 = 0. This methodology ensures that these factors together do not contribute to

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the market factor return. The same procedure for standardization of style factor exposures applies when deriving pure factor mimicking portfolios, which makes them orthogonal to the market factor and thus calibrates the model so that the return of the market factor mimicking portfolio is style factor neutral, ∑^𝑁_𝑛=1𝑤_𝑛,𝑡𝑋_{𝑛,𝑘,𝑡} = 0 . The factor returns are then estimated using a weighted least squared method with constraints and the general solution can be expressed as,

𝑓_𝑘,𝑡 = ∑ 𝛺_{𝑛,𝑘,𝑡}𝑟_𝑛,𝑡^𝐸

𝑁

𝑛=1

.

(2.14)

Where the 𝑛×𝑘 matrix 𝛺_{𝑛,𝑘,𝑡}, denotes the weight of the nth stock in the kth pure factor portfolio. Hence, we derive all pure factor mimicking portfolios simultaneously. Since all factors now are neutralized towards the market factor mimicking portfolio, due to the standardization procedure and including two linear constraints, the return of the market factor can be expressed as,

𝑅_𝑀,𝑡= 𝑓_{𝑚𝑘𝑡,𝑡}+ ∑ 𝑤_𝑛,𝑡𝜖_𝑛,𝑡

𝑁

𝑛=1

.

(2.15)

Where 𝑤_𝑛,𝑡 corresponds to the weight of the nth security in the market portfolio. The contribution of idiosyncratic returns to the market factor returns are minimal, since the properties states that, 𝐸[𝜖_𝑛,𝑡] = 0, 𝐶𝑜𝑣(𝜖_𝑛,𝑡, 𝜖_𝑛+1,𝑡) = 0 and 𝐶𝑜𝑣(𝜖_𝑛,𝑡, 𝑓_𝑛,𝑡) = 0. Notice that this implies that the pure market factor mimicking portfolio is represented by the weighted market portfolio, 𝛴_𝑛=1^𝑁 𝑤_𝑛,𝑡.

So, the main difference between construction single factor mimicking portfolios and pure factor mimicking portfolios is that we process one factor at the time, which results in unknown secondary exposure to other factors. In contrast, pure factor mimicking portfolios have a 100% exposure to one single factor and zero exposure to all other factors included in the model, since we process all factors simultaneously. The objectives for portfolio and risk managers using this approach are thus to get an overview over the risk structure of portfolios, hedge secondary exposure and also make investment based on certain factors that tend to outperform the market.

2.3 Factor composition

As mentioned as one limitation of this paper, we exclude the step of combining several risk measurements explaining one single factor exposure due to the scope of this project.

Instead, basic firm specific data are used as factor exposures. For example, the natural logarithm of market capitalization characterizes the size of a security and are thus used as a factor exposure towards size.

To reduce the sensitivity of a security’s exposure towards a factor, most factor exposures are calculated on a 12-month basis of historical data. So, if a security would

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temporarily report bad figures one quarter, the effect on the factor exposure would not be that crucial. Hence, a security’s exposure towards a factor are slowly changing with this methodology. However, this counteracts the advantage of the fundamental factor model and the cross-sectional regression, because we now need historical data for each security in the estimation universe, which also makes the implementation even more data intensive. All style factors used in this implementation have a significant role in fundamental equity research and will be presented in Section 2.3.2.

Alexander (2008) remarks an alternative approach of creating robust factor exposures by merging several measurements into one single factor exposure. Consequently, each factor and its exposure consists of n measurements, where the number of measurements describing the factor, depends on how much information that is needed to make it robust.

For example, consider securities size as a factor. For example, consider securities size as a factor. Instead of only using market capitalization to describe the exposure towards size, one can build a more robust exposure by combining several of measurements related to size. For example, the natural logarithm of a security’s market capitalization, total assets and total sales, can be used to better capture the size factor.

To provide a basic summary of this methodology of creating robust factors, a weighting algorithm is needed in order to merge different measurements into factor exposures.

Before this procedure is applied, each measurement must be standardized to be comparable with each other. Thereafter, to achieve reasonable weightings between risk measurements, principal component analysis is often used. Since the first principal component explains most of the variance in a data set, it is reasonable to use its coefficients as weights for creating the factor exposure. A complete explanation of this process for creating security’s exposures towards factors is beyond the scope of this project.

2.3.1 Market factor

All equities in the estimation universe have a unit exposure to an overall factor, called the market. This factor corresponds to the well-known phenomena in finance called undiversifiable risk. In many contexts are changes in interest rates, inflation or events that affect the broad market return associated with this factor. Intuitively, this factor is both unpredictable and impossible to completely avoid, thus a source of risk which all securities in the estimation universe are subject to. Moreover, when considering factor models where several risk factors are considered, then the market factor is identified as the net of portfolio exposures to each other factor.

2.3.2 Style factors

Alexander (2008) remarks that a key to succeed with multiple factor models is to accurately select factors. Faboozzi et al (2010) further states that factors should be selected throughout economic reasoning in order to be intuitive for investors. The

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factors must also be measurable and a reliable data set must be available. A common way to select factors is therefore by analyzing firm’s annual statements and sources of security market data. However, before adding a factor to the model, it is important to measure the linear dependency between the existing factor exposures and the new one, since this could interfere with the least squares estimates. A credible way to measure the dependency is through Spearman’s rank correlation since the difference between factors exposures can be significant. The following section will highlight the style factors that we have chosen to consider in this implementation and also give a short review of how securities exposure towards these factors are calculated, i.e. the exposure of security n to a factor k at time t, as well as the economical reasoning behind them.

2.3.2.1 Book-value

The exposure towards book-to-market value at time t for a security is based on 12- months of the security’s historical data and is given by,

𝐵𝑜𝑜𝑘 − 𝑇𝑜 − 𝑀𝑎𝑟𝑘𝑒𝑡_𝑡 = 𝐵𝑉12𝑀

𝑀𝑉_𝑡

(2.16)

where 𝐵𝑉_12𝑀 denotes the average book-value the last 12 months and 𝑀𝑉_𝑡 denotes the market value at time t. Fama and French (1992) found empirical evidence that there exists a cross-sectional relation between the book-to-market value and security returns.

The study, which was conducted by analyzing portfolios with different book-to-market values, showed that securities with high book-to-market value tend to have higher return than those with a low value. The economic reasoning behind the value factor is that securities with high book-to-market ratio are more likely to generate excess return than securities with a low ratio.

2.3.2.2 Momentum

The exposure towards momentum at time t for a security is based on 12-months of the security’s historical data and is given by,

𝑀𝑜𝑚𝑒𝑛𝑡𝑢𝑚_𝑡 = 𝑃𝑡

𝑃_𝑇

(2.17) where 𝑃_𝑡 denotes the price today and 𝑃_𝑇 denotes the price 12-months ago for a security.

The momentum factor is a measure of a security’s performance and has been recognized by several researchers. Amongst others, Jegadeesh and Titman (1993) found empirical evidence of predictability of portfolio returns through momentum. By analyzing past security prices, they could measure the serial correlation of security market returns, which is a measure of how well security returns are related to recent performance. A positive serial correlation means that it is likely for that security to continue to exhibit positive returns. A negative serial correlation on the other hand, indicates that positive returns are likely to be followed by negative returns. Jegadeesh and Titman (1993) further found that significant abnormal returns could be achieved over certain periods through creating trading strategies which buys securities that have performed well in the past and sells securities that performed badly. Faboozzi et al (2010) states that the

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economic reasoning behind the momentum factor is that investors are attracted to stocks that have performed well in past time periods.

2.3.2.3 Volatility

The exposure towards volatility at time t is derived from time series of historical daily returns on a 12-months basis for a security. Hence,

𝜎_𝑡^{𝑑𝑎𝑖𝑙𝑦} = √Σ_𝑖=1^𝑇 (𝑟_𝑖− 𝑟̅)² 𝑇 − 1

(2.18)

where 𝑟_𝑖 denotes the daily return and 𝑟̅ denotes the average of daily returns over the past 12-months for a security. Moreover, since the volatility usually is expressed on an annual basis, we apply the rule of scaling the daily volatility as, √𝑇𝜎_𝑡^{𝑑𝑎𝑖𝑙𝑦}, where T is assigned a value of 252, corresponding to number of business days annually. Moreover, applying this rule also assumes that variables are independent and identical distributed and that logarithmic returns are small and approximately the same as actual returns, hence ln (_𝑃^𝑃^𝑡

𝑡−1) ≈_𝑃^𝑃^𝑡

𝑡−1− 1 where 𝑃_𝑡 denotes the price. Volatility has for long been an important topic in the security valuation literature and is a measurement of a security’s price fluctuation for a given time period. One of the pioneers in the topic was Markowitz (1952) who found that the risk of an individual security can be explained through the volatility of its returns. An investor should view securities with larger volatility of returns as riskier and therefore should expect a greater return.

2.3.2.4 Beta

The exposure towards beta at time t for a security is given by, 𝛽_𝑡 = 𝐶𝑜𝑣(𝑅_𝑡, 𝑅_{𝑟𝑒𝑓,𝑡})

𝜎_{𝑟𝑒𝑓,𝑡}²

(2.19)

where 𝐶𝑜𝑣(𝑅_𝑡, 𝑅_{𝑟𝑒𝑓,𝑡}) denotes the covariance between a security and a reference portfolio at time 𝑡. The 𝜎_{𝑟𝑒𝑓,𝑡}² denotes the variance of the reference portfolio at time 𝑡. In this paper, two reference portfolios are used, namely the OMXS 30 Index and EURO STOXX 50 Index. The beta, 𝛽, is one of the most commonly used risk-factors in finance.

It is a measure of systematic risk and usually measures the volatility of a stock relative to a market as a whole. The economic reasoning behind the beta factor is the same as for volatility - an investor should view securities with lager beta as riskier and therefore should expect a greater return.

2.3.2.5 Dividend yield

The exposure towards dividend yield at time t is calculated as the average dividends over the past 12-months for a security and is divided by the current stock price. Hence,

Dividend yield_t=𝐷𝑖𝑣_12𝑀 𝑃t

(2.20)

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where 𝑃_𝑡 denotes the current stock price at time t and 𝐷𝑖𝑣_12𝑀 denote the average annual dividends per share for a security. Ball (1978) discuss the inefficiencies in markets and how new information regarding earnings and dividends attracts investors to gain excess returns. The excess returns tend to exceed the transaction costs and processing costs for investors acting on this new information, thus making dividend yield a factor that could capture excess return.

2.3.2.6 Size

The exposure towards size at time t for a security is based on 12-months of the security’s historical data and are given by,

𝑆𝑖𝑧𝑒_𝑡 = ln (𝑆𝑂_𝑡∗ 𝑃_𝑡) (2.21)

where 𝑃_𝑡 denotes the current stock price at time t and 𝑆𝑂_𝑡 denotes the total number of outstanding shares at time t, for a security. The size factor is found to be strongly related with returns of a security. Banz (1980) found that securities related to companies with small market capitalization on average, have higher return than companies with larger market capitalization. Further analysis also revealed that this is a non-linear relationship since the difference between adjusted returns for very small companies and large companies are sufficient, whilst the difference between medium and large companies is not as significant. This creates an issue with skewness in the distribution between return and size, which commonly is reduced by the natural logarithm. Faboozzi et al (2010) states that the economic reasoning behind the size factor is that companies with smaller market capitalization are likely to outperform larger companies.

2.3.2.7 Share turnover

The exposure towards share turnover at time t is calculated as the natural logarithm of the sum of daily share turnover during the previous 252 trading days for a security. The factor is given by the following formula,

Share turnover_t= ln (∑V_d S_d

252

d=1

) (2.22)

where 𝑆_𝑑 denotes the amount of shares outstanding and 𝑉_𝑑 denotes the trading volume of day 𝑑 over the past 252 days for a security. Several researchers have noted that the level of liquidity in a security influence its return. For example, Amihud and Mendelsen (1986) conducted an empirical study where they measured the relationship between security returns and the security’s bid-ask spread. The results proved that securities with large difference between the bid and ask price, generally have higher average return.

Chordia, Subrahmanyam, and Anshuman (2001) further explored this subject and studied the relation between expected security return and fluctuation in liquidity. By using share turnover as a proxy for liquidity, they found that securities with high fluctuation generally have lower expected return. In line with previous research, we will use share turnover as a measure of liquidity and the economic reasoning behind it, is that investors should expect lower return from securities with high liquidity.

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The exposure towards sales at time t for a security is based on 12-months of the security’s historical data and are given by,

𝑆𝑎𝑙𝑒𝑠_𝑡=𝑆𝑃𝑆_3𝑀

𝑃_𝑡 (2.23)

where 𝑆𝑃𝑆_3𝑀 denotes the current quarterly figures of sales-per-share of a security and is divided with 𝑃_𝑡 which denotes the current stock price at time t for a security. Barebee et al (1996) argued sales-to-price ratios as being a significant factor describing security returns. They further argue that this factor may absorb the role of book-to-market as Fama and French (1992) proved as the best factor to explain security returns.

2.3.2.9 Earnings

The exposure towards earnings at time t for a security is based on 12-months of the security’s historical data and are given by,

𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠_𝑡=𝐸𝑃𝑆_12𝑀

𝑃_𝑡 (2.24)

where 𝑃_𝑡 denotes the current price of a security and 𝐸𝑃𝑆_12𝑀 denotes earnings-per-share on a 12-month basis for a security. As Jaffe (1989) remarks, there are different perceptions between financial researchers whether or not earnings-to-price significantly explains the returns of securities. Some researchers’ states that the size factor accounts for the effects of earnings yield. A potential bias with this measure is when comparing figures of a company’s earnings, with stock prices from the same day. These figures are usually delayed to the public and would thus have a lagged effect on stock prices.

2.3.3 Industry and country factors

King (1966) states that movements of equity returns can be decomposed into market and industry factors. Lessard (1994) continued to investigate this topic and revealed a significant relation between securities categorized into countries and equity returns. The country factors thus tend to be more important in defining groups of securities that share common return elements. However, as mentioned as one limitation of this paper, we do not consider the country factors in the implementation.

Moreover, the common practice as described by Burmeister et al (1994) is to categorize securities into different industries and countries with indicator variables. That is, variables that either can assume the value of 0 or 1 depending on whether or not the security can be categorized into a specific country or industry. Another approach is to denote securities by percentage points, which would give a more ‘true’ model since large companies tend to be active in several industries and countries. The securities in this paper are categorized with indicatory variables according to a system provided by Bloomberg. The classification system consists of the industries Financial, Utilities, Industrial, Consumer non-cyclical, Basic Material, Technology, Energy, Communication, Consumer cyclical and Diversified.

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2.4 Cross-sectional weighted least squares

The weighted least squares (WLS) is a special case of the generalized least squares (GLS) method, which is applied in this paper due to the heteroskedastic of idiosyncratic returns. That is, securities variance is not constant over time. Another purpose of using this methodology is to give weights proportional to securities size, and thus give small securities less impact on the least squares estimates. This is, because small securities tend to be much more volatile than larger securities. As we shall see, using WLS in combination with the Lagrange multiplier (LM) becomes an efficient method for estimating the factor returns and thus derive the weights of factor mimicking portfolios, since the method can be adapted for matrix calculations.

2.4.1 Principles of Lagrange multiplier method

Nocedal and Wright (1999) describes that the general purpose with the method is to find the maximum or minimum of a multivariate function 𝑓(𝑥, 𝑦) which is subject to one or more multivariate equality constraints 𝑔(𝑥, 𝑦) . In order to explain how the LM technique works, we first introduce the mathematical formulation of a general minimizing problem,

{𝑥,𝑦}∈ℝmin^𝑛𝑓(𝑥, 𝑦) 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 {𝑔_𝑖(𝑥, 𝑦) = 𝑐₁, 𝑖 ∈ 𝔼, 𝑔_𝑖(𝑥, 𝑦) ≥ 𝑐₂, 𝑖 ∈ 𝕀,

(2.25)

where both f and g have continuous first partial derivatives, are real-valued functions on a subset of ℝ^𝑛, and where 𝔼 and 𝕀 are two finite sets of indices. In the formulation (2.25) of the problem, 𝑓 is denotes the objective function and 𝑔_𝑖, 𝑖 ∈ 𝔼, denotes the equality constraints and 𝑔_𝑖, 𝑖 ∈ 𝕀, denote the inequality constraints. The feasible set of possible points (𝑥, 𝑦) that satisfy the constraint and optimize the objective function are expressed as

Ω = {(𝑥, 𝑦)|𝑔_𝑖(𝑥, 𝑦) = 𝑐₁, 𝑖 ∈ 𝔼; 𝑔_𝑖(𝑥, 𝑦) ≥ 𝑐₂, 𝑖 ∈ 𝕀}, (2.26) where Ω now is defined on a subset of ℝ^𝑛. Consequently, equation (2.25) can be

rewritten in a more compact way using (2.25)

{𝑥,𝑦}∈Ωmin 𝑓(𝑥, 𝑦). (2.27)

In order to explain the basic principles behind the characterization of solutions of constrained optimization problems, we work through a simple example. The main idea using the LM optimization technique is to look for points where the contour lines (called level curves) of 𝑓 and 𝑔 are tangent to each other. The contour lines are constructed by letting 𝑓 be equal to some constant 𝑘. Consider the simple example of,

{𝑥,𝑦}∈ℝmin^𝑛𝑥𝑦 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜: 𝑥²+ 𝑦²= 1 (2.28) then the optimization problem can be illustrated as a geometrical representation shown in Figures 1-3. The feasible region set by the constraint and the different contour lines of the objective function can be observed in Figure 1 and 2 respectively.

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Figure 1 Contour line for the constraint 𝒈(𝒙, 𝒚) = 𝒙^𝟐+ 𝒚^𝟐 where 𝒄 = 𝟏.

Figure 2 Contour lines for function 𝒇(𝒙, 𝒚) = 𝒙𝒚 for different values of 𝒌.

Figure 3 Illustration of where the contour lines for 𝒇 and 𝒈 are tangent.

As can be observed in Figure 3, it is reasonable to assume that there exists a point (𝑥, 𝑦) in which the contour lines for both functions are tangent to each other. The properties of gradients suggest that the gradient always is orthogonal for points (𝑥, 𝑦) that coincide with contour lines. That is, instead of finding the points (𝑥, 𝑦) where both functions are tangent to each other, it is equivalent to find the points where the gradient vectors of 𝑓 and 𝑔 are parallel to each other. This property is illustrated in Figures 4-6.

Figure 4 - Gradients and contour line for constraint 𝒈(𝒙, 𝒚) = 𝒙^𝟐+ 𝒚^𝟐 where 𝒄 = 𝟏.

Figure 5 - Gradients and contour lines for function 𝒇(𝒙, 𝒚) = 𝒙𝒚 for different values of 𝒌.

Figure 6 - Illustration of where the contour lines for 𝒇 and 𝒈 are tangent and gradients are parallel for both functions.

As can be observed in Figure 4-6, the gradients are perpendicular to the contour lines of both functions. In Figure 3, we can find four different points where the gradients are parallel to each other and where the two functions are tangent to each other. Hence, we have four different possible solutions to consider for the optimization problem. The gradient is a representation of the direction of which the function has the greatest increase and is defined as vectors containing the partial derivatives of both functions.

Hence,

Multiple factor models for equities: An empirical study of the performance of factor mimicking portfolios