What are the main factors impacting the discount to net asset value? - A case study of Investor AB

(1)

IN

DEGREE PROJECT TECHNOLOGY, FIRST CYCLE, 15 CREDITS

STOCKHOLM SWEDEN 2017 ,

What are the main factors

impacting the discount to net asset value?

A case study of Investor AB ALVA ENGSTRÖM

FILIPPA FRITHZ

(2)

(3)

What are the main factors

impacting the discount to net asset value?

A case study of Investor AB ALVA ENGSTRÖM

FILIPPA FRITHZ

Degree Projects in Applied Mathematics and Industrial Economics Degree Programme in Industrial Engineering and Management KTH Royal Institute of Technology year 2017

Supervisor at Investor AB: Jan Lernfelt, Anders Eckerwall.

Supervisors at KTH: Pierre Nyquist, Hans Lööf Examiner at KTH: Henrik Hult

(4)

TRITA-MAT-K 2017:06 ISRN-KTH/MAT/K--17/06--SE

Royal Institute of Technology School of Engineering Sciences KTH SCI

SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

(5)

Abstract

The discount to net asset value for investment companies has puzzled both the world of academics as well as the investment companies themselves. This thesis aims to examine how the Swedish investment company Investor AB can combat this market inefficiency by analysing the main factors impacting the discount to net asset value over the time period 2006:M12-2016:M12. A multiple linear regression approach is used in order to find significant factors affecting the discount to net asset value.

The final result in this thesis finds a statistically significant negative relationship between the discount to net asset value and Investor’s cash level and debt equity ratio.

Positive statistically significant relationships are found for the share of unlisted holdings of the total net asset value of Investor, the company’s costs, and analysts’ ratings of the company.

(6)

(7)

Sammanfattning

Substansrabatten hos investmentbolag har konfunderat både den akademiska världen och investmentbolagen själva. Den här uppsatsen ämnar undersöka hur det svenska investmentbolaget Investor AB kan gå tillväga för att angripa denna marknadsineffek- tivitet genom att analysera de huvudsakliga faktorerna som påverkar substansrabatten över tidsperioden 2006:M12-2016:M12. Uppsatsens analysmetod är multipel linjär regression.

Det slutgiltiga resultatet visar statistiskt signifikanta negativa förhållanden mellan substansrabatten och Investors kassa, samt skuldsättningsgraden. Statistiska signifikanta positiva förhållanden finns mellan substansrabatten och andelen onoterade innehav, Investors kostnader, samt andelen köprekommendationer från analytiker som följer In- vestor.

(8)

(9)

Acknowledgements

We would like to express our greatest gratitude to our supervisors from the Royal In- stitute of Technology. Pierre Nyquist, at the Department of Mathematics, who has provided us with deep insights in regression analysis, financial modelling and moral support and Hans Lööf, at the Department of Industrial Economics and Management, who has provided us with guidance during the course of the work.

We would also like to thank Investor, and especially Jan Lernfelt and Anders Eck- erwall who provided valuable feedback during the process.

(10)

(11)

1 Introduction

1.1 Background

The business idea of an investment company is to own and invest in other listed and unlisted companies. For people lacking time and expertise in portfolio management it is popular to acquire shares in investment companies [1]. An investment company invests its capital in a number of different assets in order to create value for its shareholders. The value of an investment company increases when its assets grow or pay dividends. Some investment companies focus on capital growth while others focus on taking more active roles in its assets to develop them over time, as long as they see potential for further value creation.

Historically, one of the major reasons to invest in an investment company has been to easily get a diversified portfolio [2]. By acquiring shares in an investment company, one avoids the transaction costs that would have followed if investing in the assets separately. Some investment companies, whose holdings also consist of unlisted companies, present small private investors with the opportunity to invest in companies which they otherwise would not have access to. Another reason for choosing to acquire shares in an investment company over investing in its assets is that the investment company shares tend to have greater liquidity than the relatively illiquid assets they hold [3].

Since the business idea of an investment company differs from the business idea of for example a producing company to such a large extent, it is reasonable that the method for valuing the companies also differs. Investment companies do not have the type of sales and profit margins that producing companies have, thus they can not be valued on the basis of their sales or growth in profit [2]. Instead the value of investment companies is determined by the market value of its assets, which is called net asset value (NAV) [4].

The task of valuing an investment company may at first glance seem easy since the value should be the sum of it assets in an efficient market. However, investment companies are often traded at a discount to their NAV. This means that the value of its assets is higher than the market value of the investment company itself. If the investment company trades at a stock price higher than the NAV of its components, it is traded at premium. It is more common to be traded at a discount for an investment company [2].

1.2 Problem

To protect itself from hostile takeovers and maximise shareholder value, an investment company wants to decrease the discount to NAV. If there exists a great discount it is theoretically possible for a takeover to take place where the party that takes over the investment company will make a profit by splitting up the investment company and selling the pieces separately.

Thus, decreasing the discount to NAV protects the investment company’s raison d’être. An increasing discount to NAV poses a problem for the shareholders of the investment company, since the yield of their investment is lower than if they would have invested in the assets held by the investment company directly.

In Sweden, the number of stock listed investment companies have been declining since the 1980’s, from 30 listed investment companies to today’s 13. The reason for the decrease is according to Arne Karlsson, former CEO of the investment company Ratos, that the best way to realise an investment company’s value has many times been to either get bought or to be liquidized [5]. The discount to NAV thus poses a threat the possibility of running a listed investment company successfully over a longer period in Sweden [5].

(14)

The great and varying discount to NAV looks both like a market inefficiency and a threat to the raison d’être of investment companies. Thus, it also poses a threat to the popularity of investment companies as a way to easily access a diversified portfolio without the time and transaction costs associated with investing in the investment companies’ assets directly.

1.3 Purpose

The objective of this thesis is to investigate and find the factors that drive the discount to net asset value, in order to be able to explain the level of the discount. By extension, this will assist investment companies in potentially lowering their discount to NAV, where a decreasing discount would create a higher yield for their current shareholders, while also protecting the company from takeovers.

Academically it is of high interest to explain the causes of the discount to NAV in order to support the notion of rational and efficient markets. In the absence of efficiency it would be possible for investors to make arbitrage profits, which is a violation of a common assumption of financial economics [6].

1.4 Research question

This thesis will investigate what affects the discount to NAV. The thesis is limited to only examine the discount to NAV of the Swedish listed investment company Investor.

The research questions this thesis will answer are:

• What factors impact the discount to NAV?

• How can Investor decrease the discount to NAV?

1.5 Scope

The limitations of the thesis consist of only focusing on quantitatively explaining the discount to NAV of Investor. The reason for limiting the thesis to focusing on one investment company is that the discount to NAV is firm specific, thus making a cross-section study more difficult to perform. In addition, the composition of Investor’s portfolio, with both listed and unlisted assets, is unique on the Swedish stock market hence making a cross-section study more problematic.

The time period studied for the thesis is limited to the ten year period 2006:M12-2016:M12.

A ten year time period enables us to study how the discount to NAV changes over a business cycle, since a business cycle has duration of less than ten years [7].

1.6 Investor AB

Investor AB, founded in 1916 by the Wallenberg family, is the largest listed Swedish investment company. Since the start it has been controlled by the Wallenberg family through the Wallenberg Foundations, which are the main owners of Investor AB. The business concept of Investor is to generate long-term value for their shareholders by owning and developing companies with high value potential. The work for continuous improvement of the perfor- mance of its assets is done through board representation.

(15)

The investments consist of both listed companies, Listed Core Investments, unlisted companies, Patricia Industries, financial investments and investments in EQT, a venture capitalist firm. Investor is a major shareholder in a number of Swedish companies such as ABB, Astra Zeneca, Atlas Copco, Ericsson, SAAB och SEB. Investor also has holdings in companies that primarily runs business in both North America and Asia.

Investor’s long term objective is to steadily grow its NAV, operate efficiently and maximise its operating cash flow. Furthermore, their objective is to pay a steadily increasing dividend to its shareholders. The dividend policy is not only that the dividend continu- ously shall grow, but also to distribute the majority of the dividends received from its core investments [8].

1.7 Methodology

1.7.1 Literature

To be able to understand what affects the discount to NAV, previous research about the discount is studied. The purpose of the literature study was not only to select variables, but also to able to analyse the outcome of the regression models and their usefulness.

Scientific studies for this thesis was found by using KTHB, Econlit and Scopus. The articles were chosen based on relevance of abstract and number of peer reviews.

1.7.2 Interviews

In order to get a deeper understanding of what impacts the discount to NAV, interviews were conducted with two employees of the finance department at Investor. These interviews were conducted as open conversations with discount to NAV as topic. These meetings yielded complementary understanding about the company Investor and of previous research within the field, since theory and reality could differ. In addition, the interviews provided knowledge of what Investor suspects impact the discount to NAV.

1.7.3 Statistical Analysis

To estimate the relationship between the selected variables and the discount to NAV, a multiple linear regression analysis was performed. Different techniques for dealing with incorrectness and other problems, together with variable selection methods were used in order to create simple yet accurate models that explain the relationship.

(16)

2 Economic theory

2.1 Efficient market hypothesis

If in a market, the prices always fully reflect the information that is available, the market is said to be efficient. This theory is called the efficient market hypothesis (EMH), and it implies that it is not possible to outperform the market through stock picking.

The hypothesis exists in three forms, the weak, semi-strong, and strong form. In the weak form, the EMH, concludes that the prices reflect historical prices and information. In the semi-strong form, the prices reflect historical prices and information, and also prices in new information that is publicly available instantly. Publicly available information is usually earnings announcements and similar forms of information the publicly traded companies release. The strong form of the efficient market hypothesis states that the prices fully reflect all past and current information, even insider information that is not publicly available [6].

Since publicly available information such as earnings reports are published online, the access when published is immediate and available to all. Thus, the semi-strong form of market efficiency is assumed in this thesis.

2.2 Agency costs

The principal-agent problem arises when when one person’s (the agent) actions affect another person (the principal). A common corporate situation where this problem occurs is when the agent makes decisions on behalf of the principal, where the agent is an employee (e.g. the CEO) and the principal is the owners of the company. The problem is that the agent has the opportunity to act in a way that benefits herself rather than acting to maximise the payoff of the principal. One of the major causes of the problem is the information asymmetry between the agent and the principal, where the agent holds an advantage [9].

One way that these agency costs can exhibit themselves in the corporate application is by empire building. The most common occurrence of agency costs is that the CEO of the company tries to expand the company as much as possible, even when it is not optimal or benefits the company’s mission. The objective for the CEO in this application is to maximise her own status or wage [9].

2.3 Previous research on the subject

The discount to NAV for investment companies has puzzled academics for decades and previous research about discount to NAV agrees that the phenomenon exists, but no full explanation of its existence has been presented. Scientists present several factors that impact the discount to NAV and there is no consensus between them. A majority of the previous research within the field has been done on the American and British market, thus closed-end funds (CEF) have mainly been investigated. A CEF is a mutual fund with a fixed number of shares and it has many similarities with an investment company. It can manage both listed and unlisted securities and is often traded at a stock market with a discount to its NAV.

The main differences between a CEF and an investment company are that a CEF charges a management fee and has a set expiration date, while investment companies do not [10].

Lee et al. examine if the discount to NAV reflects the expectations of the investors. They present a theory that so called noise traders, which are investors who buy and sell securities without using fundamental analysis, increase the resale price risk. They increase the risk

(17)

since they overestimate the expected return during certain time periods and underestimate it during others. In contrast, rational investors have rational expectations about the their holding’s return. Lee et al. show that there is a correlation between the discount to NAV and the investor sentiment, i.e. noisy traders increase the risk and hence the discount to NAV [11]. A study by Morri et al. finds on the topic of investor sentiment, that funds with a large share of institutional investors have a lower discount [12].

The conflict of interest between shareholders and management in a company is an interesting and widely research phenomenon. The article by Cronqvist et al. examines the agency costs of minority shareholders, who have control over the voting rights but not the capital, and finds that family minority shareholders are associated with the largest discount to NAV [13].

Benedetto et al. investigate the relationship between leverage and discount, and they even- tually find a negative relationship between the two, even without including the interest rate tax shield, which is a benefit of leverage [12].

There are multiple theories of how unlisted companies in the portfolio affect the discount to NAV. One theory is about the illiquidity of a portfolio, and another theory is about replicability. The logic behind a illiquidity discount is that illiquid assets pose a greater risk than liquid assets, since they can be difficult to sell. According to Chan et al., the illiquidity of an asset held by a CEF has a positive relationship with the discount to NAV in open markets.

The theory of replicability however, suggests that unlisted assets are preferred over listed ones, since any investor can easily replicate a portfolio. The inclusion of unlisted companies could also increase the diversification, which would lower the risk and thus should decrease the discount, according to Chan et al. [14].

Kumar et al. examine the effects of the expenses associated with the operations of CEF:s, and find that the costs have a significant positive impact on the discount. This relationship is intuitive, as for example the value created for an investment company’s shareholders is the value created by its assets subtracted by the costs associated with the operation [15].

Multiple studies about the payout policies of CEF:s, and their relationship to the discount to NAV have been conducted, and the main finding is that a CEF that applies a minimum dividend policy (MDP) has a significantly lower discount [16]. A MDP also has the effect of dividend signaling, as the company states a lower bound for its dividends. This effect was investigated by Johnson, et al. and was found to have a negative impact on the discount [17].

The former CEO of the Swedish investment company Ratos, Arne Karlsson, famously constructed a theory called "the black box", where he describes the problem of the discount to NAV for investment companies. Karlsson argues that an investment company’s dividend payout ratio should be high to avoid the discount-on-discount effect that arises if the required yield is not achieved and a small share of the profit is paid out as dividend [5].

(18)

3 Mathematical theory

In this section, a brief description of the mathematical procedures that are used in the thesis is given.

3.1 Multiple linear regression

A multiple linear regression model is defined as [18]:

y_i = β₀+

k

X

j=1

β_jx_ij+ ε_i, i = 1, ..., n

It models the relationship between the response variable y_i and the k regressors x_ij. The regression coefficient β_j determines the change in the response variable y_i per unit change in x_ij when all other regressors are constant. β₀ is the intercept of the regression space.

The error term εi is the difference between the observed value, yi, and the fitted value for observation i, ˆyi, and can be interpreted as a random statistic error [18]. n refers to the number of observations and k is the number of regressor variables that have been measured in order to explain or predict the response variable.

The multiple linear regression model can alternatively be expressed in matrix notation y = Xβ + ε

where

y =





 y1

y2

... y_n





 X =







1 x11 x12 ... x1k

1 x21 x22 ... x2k

... ... ... ... 1 x_n1 x_n2 ... x_nk





 and

β =





 β₀ β₁ ... β_k





 ε =





 ε₁ ε₂ ... ε_n







3.2 Ordinary least squares

Ordinary least squares (OLS) is a method that estimates the regression coefficients for a multiple linear model. The OLS estimate of β is [18]:

β = (Xˆ ⁰X)⁻¹X⁰y

The idea of OLS is to minimise the difference between the fitted values, ˆyi, and the observed value, yi. Alternatively formulated, OLS minimises the sum of squares of the residuals ε⁰ε = (y − Xβ)⁰(y − Xβ). Moreover, ˆβ is an unbiased estimator, E[ ˆβ] = β, and OLS is also the best linear unbiased estimator. The covariance matrix of the OLS estimate of the regressions coefficients is Cov( ˆβ) = (X⁰X)⁻¹σ².

(19)

3.3 Key assumptions

In order to draw correct conclusions from the linear regression analysis, certain assumptions need to be fulfilled. If the assumptions are violated it is likely that the model is incorrectly specified. The key assumptions are [18]:

1. There is approximately a linear relationship between the response variable and the regressors.

2. The expected value of the error is zero, E[ε] = 0.

3. The variance of the error term is constant, V ar[ε] = σ². 4. The errors are uncorrelated.

5. The errors follow the normal distribution.

3.4 Potential problems

3.4.1 Multicollinearity

A possible complication of the model is the existence of multicollinearity. Multicollinearity implies that there exists near linear dependencies among the regressors variables. If multicollinearity is present then for some set of constants a1, a₂, ..., a_p that are not all zero, the following approximately holds for some subset of regressors:

p

X

j=1

a_jX_j = 0

Where Xj is the column of X representing the j:th regressor variable.

All data sets will have some multicollinearity, unless the regressors are orthogonal, which would generally only be the case in designed experiments. The primary sources of multicollinearity are the data collection method that is used, constraints of the model, the way the model is specified, and an overestimated model [18].

Multicollinearity is a problem for the regression model because the existence of multicollinearity can both inflate the variance of the OLS coefficient estimates, and also inflate the absolute value of the OLS coefficient estimates.

3.4.2 Heteroscedasticity

If the error terms of a regression model do not have constant variance, they are said to be heteroscedastic. This violates the OLS assumption that the errors have constant variance and thus it is necessary to control that there is no heteroscedasticity.

Plots of the residuals against the fitted value of the dependent variable can be used to identify heteroscedasticity [19]. If the residuals can be contained within a horizontal band there are no indications of violation of the normality assumption. However, an opening funnel pattern or a double bow pattern indicate heteroscedasticity.

Another approach for detecting heteroscedasticity is by studying the Q-Q-plot where the sample quantiles of the residuals are plotted against the theoretical quantiles. If the sample quantiles of the residuals are plotted approximately on the normal probability line, there is no heteroscedasticity present. Other patterns are indicators of violation of the normality assumption [18].

(20)

3.4.3 Endogeneity

Endogeneity exists when a regressor is correlated with the error term of the regression model.

Endogeneity thus violates the assumption that the expected value of all error terms e_i is equal to zero, i.e. endogeneity exists when:

E[ei] 6= 0

This problem can occur because of multiple reasons. Possible causes of endogeneity is omitted variables, simultaneous causality problems in the model, bias in the sample selection, and measurement errors.

Omitted variable bias exists when variables that should be included in the model are over- looked. This causes the omitted variable to be included in the error term, and thus, the assumption about the expected value of the error term is violated.

Simultaneous causality occurs when the dependent variable influences some subset of the regressor variables, i.e. the dependent variable is influenced by the regressor variables, of which one or more are in turn influenced by the dependent variable.

Bias in the sample selection is the result of a selection process that in some way influences the available data [20].

3.5 Transformation

The assumption of normally distributed error terms is a requirement for performing regression analysis. A common reason of violation of this assumption is that the data set comes from another distribution. These violations need to be corrected before performing the regression analysis which can be done analytically with the Box-Cox method.

3.5.1 Box-Cox method

In order to correct non-constant variance, power transformation presented by Box and Cox in 1964 can be performed. It is an analytical procedure where the response variable is being transformed. By transforming the response y to y^λ, constant variance can hopefully be retrieved. For example, if y is Poisson distributed, the transformation y¹² can be used. The objective of the Box-Cox method is to determine the value of λ, and this is done by using the method of maximum likelihood.

Often when using the Box-Cox method an approximate confidence level for the transfor- mative parameter λ is calculated. The reason for this is that rounded numbers can be easier to explain. For example, if λ = 1 is within the 100(1 − α) confidence level, no transformation is needed [18].

3.6 Validation of the model

3.6.1 ANOVA

ANOVA, or analysis of variance is computed to derive the F-statistics of the regression model which is used for hypothesis testing. The F-statistics is used for hypothesis testing, described in the following section [18].

(21)

Table 1: ANOVA table

Source Sum of Squares Degrees of Freedom Mean Square F₀

Regression SSR=Pn

i=1( ˆyi− ¯y)² k M SR=^SS_k^R M SR/M SRes

Residual SSRes=Pn

i=1(yi− ˆyi)² n − k − 1 M SRes= _n−k−1^SS^Res Total SST =Pn

i=1(yi− ¯y)² n − 1 3.6.2 Hypothesis testing

Hypothesis testing is used to test the significance of either a single regressor or the entire model. The hypothesis test is conducted by computing a test statistics and comparing it with a critical value of its distribution with the null hypothesis H0.

F-statistics

The F-statistics can be used for testing the significance of an entire model. Then a the null hypothesis is set where all OLS estimates of the coefficients are equal to zero, i.e.:

H₀: β = ¯0 versus the alternative hypothesis H₁: β 6= ¯0

The F-statistics, F₀, of the model is obtained from the ANOVA table and is defined as:

F0= M SR

M SRes

∈ Fk,n−k−1under H0

The null hypothesis, H0, can be rejected at an α% confidence level if : F₀> F_{α,k,n−k−1}

A rejection of the null hypothesis, H0, means that there is a linear relationship between the response variable and the regressors [18].

t-statistics

To test the significance of a single parameter, a t-test is appropriate. The t-statistics is defined as:

t₀= βˆ_j se( ˆβj)

The hypotheses for testing the significance of a single parameter, ˆβ_j, are:

H₀: ˆβ_j= 0 versus the alternative hypothesis H₁: ˆβ_j6= 0

The null hypothesis, H₀, can be rejected at an α% confidence level if the following holds:

|t0| > t^α

2,n−k−1

If H0 cannot be rejected, this means that the regressors j should not be deleted from the model. In addition, the t-statistic is a partial test which means it tests the contribution of βˆ_j given the other regressors included in the model [18].

3.6.3 Coefficient of determination

The coefficient of determination, R², is defined as:

R²=SS_R SST

= 1 −SS_Res SST

(22)

This is interpreted as the proportion of variance in the dependent variable that is explained by the regressors. R²∈ [0, 1], and as high as possible R² is desired, since the regressors in that case explain the variance of the dependent variable to a large extent.

R² will by design increase when more regressor variables are added, because of this it is not appropriate to only use R² as a measure of the model’s adequacy. Adjusted R²on the other hand does not necessarily increase when more regressor variables are included and is defined as:

R²_Adj = 1 −SSRes/(n − p) SS_T/(n − 1)

Since R²_Adj only increases if an added regressor variable reduces the model’s M SRes, R²_Adj is preferred over R² as a variable selection criteria [18].

3.6.4 Residual analysis

Standardised residuals are used to detect outliers. An outlier is an observation in the data sample that is in some way extreme compared to the rest of the data sample. Standardised residuals is a way of scaling residuals, it is defined as:

di= ei

√M SRes

A common cutoff for when an observation is viewed as an outlier is d_i> 3 [18].

Since the residuals of observations that are distant in x-space are often very small, even though those observations often violate the model assumptions, it is desirable to include the measure of location hii of the observation. This is done when calculating the studentised residuals, defined as:

ri= ei

pM SRes(1 − hii)

The studentised residuals are constructed to detect outliers and possible influential points, as an observation with a large residual that also has a large h_iiis likely to be influential [18].

This thesis will apply a cutoff of 2, i.e. if r_i> 2 it will be further investigated as an outlier.

3.6.5 Variance inflation factor, VIF

To detect multicollinearity, the variance inflation factor, VIF, can be computed. The VIF is defined as:

V IFj = (X⁰X)⁻¹_jj = 1 1 − R²_j

Where R²_j is the coefficient of multiple determination, which is obtained by regressing the variable xjon all other regressor variables. If the VIF is larger than 10, it implies that there exists a serious problem with multicollinearity in the model [18].

3.6.6 Condition number

One additional approach for detecting multicollinearity is to calculate the condition number of the estimated model. The condition number is defined as:

κ = λmax

λ_min

(23)

Where λmin and λmax are the smallest and largest eigenvalue of the matrix X⁰X respec- tively. An eigenvalue that is below 100 indicates that there is no serious problem with multicollinearity [18].

3.6.7 Diagnostics of influential points

An outlier that greatly influences the OLS estimates is considered to be an influential point.

These observations are of interest to detect to be able to examine further whether they are representative and should be included in the model or not.

One possible approach to detect influential observations is by calculating Cook’s distance.

Cooks’s distance measures how much the OLS estimate of a model differs when excluding a potentially influential observation. Cook’s distance is defined as:

D_i= ( ˆβββˆˆ_(i)_(i)_(i)− ˆβˆβˆβ)⁰XXX⁰XXX( ˆβββˆˆ_(i)_(i)_(i)− ˆβˆβˆβ) pM SRes

= r_i² p

V ar(ˆy_i) V ar(ˆei) = r_i²

p h_ii 1 − hii

, i = 1, 2, · · · , n

Where ˆβ(i) is the coefficient estimated with OLS when observation i is excluded from the data. Observations with Di> 1 are usually considered to be influential and should be more closely examined [18].

Another method to detect outliers is DFFITS. Just like Cook’s distance, DFFITS is also a deletion diagnostics that measures the influence of an observation by comparing the results of the regression when the observation is excluded. DFFITS is defined as:

DF F ITi= yˆi− ˆy_(i) qS²_(i)h_ii ,

Where S²_(i)is an estimate of the variance of the residual [18]. Observations where DF F ITi>

|2pp

n| are considered to be influential and should be further examined [18].

3.7 Variable selection and model building

Often when analysing historical data not all the candidate regression variables are important for the model composition [18]. Simplicity of the model is also desirable since it is easier for researchers to interpret. By using all possible regressions method, stepwise selection and comparing the models with selection criteria as Bayesian Information Criterion (BIC), adjusted R² or Mallows’ Cp one can find an appropriate subset of variables that optimises the trade-off between the model’s accuracy and its simplicity.

3.7.1 All possible regressions

The all possible regressions algorithm considers all possible combination of regressors involv- ing one regressors, two regressors, ..., k regressors. Linear regression is performed on every combination of regressors and then evaluated according to some criteria. If the intercept β0

is included the algorithm will consider 2^p possibilities which makes the method impractical to use if the number of candidate regressors is large [19].

3.7.2 Stepwise selection

As calculating all possible regressions can be computationally burdensome for large models there are methods that investigate a smaller set of models by adding or removing one variable at a time. These stepwise procedures can be divided into three categories [19]:

(24)

• Forward selection

• Backward selection

• Hybrid approaches Forward selection

The forward selection algorithm starts with a model that does not include any regressors and then for every step adds one variable until all variables are included. At each step the model will select the variable which gives the best improved fitted model. The forward selection approach is not as computer intensive as all possible regressions method since the number of models considered is 1 + p(p + 1)/2 but the algorithm could fail to find the best subset selection when only looking at what gives the best improvement in each step.

Backward selection

The backward selection approaches the problem of finding the best subset of regressors by beginning with all regressors included in the model and then step-by-step removing one. For every iteration the algorithm deletes the regressor whose loss gives the least deterioration of the model fit. Just as forward selection, backward selection searches through 1 + p(p + 1)/2 models and is thus preferable to use when there are many candidate regressors.

Hybrid approaches

In the hybrid approach regressors are added to the model in analogy with forward selection.

When a variable has been added, all other variables are tested to see if they provide improvement of the fit and if not, they are removed. Thus, the hybrid approach is imitating the concept of all possible regressions while remaining computationally feasible for a large subset [19].

3.7.3 Bayesian Information Criterion

The Bayesian Information Criterion (BIC) is a measure of the quality of a model and can thus provide means for variable selection. For ordinary least squares regression BIC is defined as [18]:

BIC = n ln(SSRes

n ) + p ln(n)

The BIC value measures the quality of the model by rewarding a good fit and penalising overfitting. When choosing between models, the model with lowest BIC value is preferred.

3.7.4 Mallows’ Cp

Mallows’ Cp is a model selection tool just like BIC and assesses the fit of a model. It is defined as [18]:

C_p=SS_Res(p) ˆ

σ² − n + 2p

Mallows’ Cpaddresses the issue of overfitting by penalising a large number of regressors, because introducing more regressors will always improve MSE and other measures of goodness of fit. Hence, a low Mallows’ Cp is preferable.

3.7.5 Strategy for model building

Montgomery et al. present a strategy for model building and variable selection when the basic form of the model is unknown and when it is unclear which variables that should be used [18]. The strategy is as follows:

(25)

1. Fit the full model where all regressors are included.

2. Perform a thorough analysis of the residuals from the full model and see if there is any multicollinearity between the regressors of the full model.

3. Determine if a transformation of any of the variables is needed.

4. Perform all possible regressions if it is possible and select the subset models according to criteria such as Mallows’ C_p, Adjusted R² and BIC. If all possible regressions is not possible to perform, use a stepwise technique like forward selection to generate the largest possible model that can undergo all possible regressions. Then perform all possible regressions and select the subset models according to criteria such as Mallows’

Cp, Adjusted R² and BIC.

5. Compare the models recommended by the criteria in in all possible regressions.

6. Perform a thorough analysis, i.e. residual analysis and check for multicollinearity, of the models recommended by each criterion in all possible regressions to determine their adequacy.

7. Determine if there is need for any further transformation.

8. To be able to choose a final model, discuss the final set of models’ advantages and disadvantages with experts and other subject matter competent people.

(26)

4 Method

4.1 Data collection

The sample used to conduct the statistical analysis consists of 121 monthly observations.

The discount to NAV is investigated from 2006:M01-2016:M12. Since this thesis assumes the semi-strong form of market efficiency, variables are included to match when they are publicly available rather than to include them for the period they are actually calculated for. The data is mainly collected from Investor’s quarterly and annual reports, exceptions of this are commented in the explanation of variables below [21] [22]. For all observations that are only available with a quarterly or yearly frequency, it is assumed that the market perceives them as constant during intermediate months. All monetary variables are included in SEK.

4.2 Data processing

The data is initially processed in Microsoft Excel in order to create a single data set. Some values are calculated manually, such as Investor’s NAV for the months between the interim reports. The statistical calculations are made in RStudio using the programming language R.

4.3 Data

In this section, the variables used in this thesis and the justification for why they are examined are presented.

4.3.1 Discount to NAV

The discount to NAV is the response variable in this thesis, i.e. it is the variable that is explained by the final models. The discount to NAV is calculated as:

Discount to NAV = − Market Cap

NAV − 1

Since the NAV includes both the NAV of listed companies, that are included with a monthly frequency, and unlisted companies, that has a constant NAV for every quarter, the frequency of the discount to NAV is calculated and included monthly. The NAV for the unlisted companies are their book values, whilst the NAV for the listed companies are their market value and have been calculated from their stock price collected from Nasdaq OMX [23].

4.3.2 Change variables

To observe whether any asset’s change in net asset value has a disproportionate impact on the discount, which would be interpreted as a reflection of the markets belief or disbelief in the asset, the change of net asset value is included for all listed companies, EQT, Investor Growth Capital (IGC), and financial investments. The stock listed holdings of Investor will henceforth be called listed core investments (LCI). These variables are calculated with monthly frequencies for the LCI and quarterly for EQT, IGC and financial investments. The share prices for the LCI are collected from Nasdaq OMX [23].

(27)

4.3.3 Share variables

For a portfolio to be well diversified, it should consist of a number of different assets, and no single asset should dominate the portfolio. The share of the net asset value of a holding compared to the total NAV of Investor is included for LCI, EQT, IGC, and Patricia Industries (PI). These variables are included in order to observe the market’s opinion about the diversification of Investor, and also to include when Investor increases or decreases its holding in an asset. These variable are included with monthly frequencies and the LCI’s stock prices have been collected from Nasdaq OMX [23].

4.3.4 Unlisted companies’ profitability

Since the unlisted companies do not have market values reported in Investor’s interim reports, it is not possible to include market valuations of the unlisted companies. However, according to the interviews conducted with Investor, their values are often calculated as a multiple of its profitability. The unlisted assets’ profitabilities are included as:

EBITDA Net Sales

Where EBITDA is the earnings before interest payments, taxes, depreciation and amortisa- tion. This variable has quarterly observations.

4.3.5 Costs

The costs of an investment company are likely to affect the discount to NAV [24]. The yield of Investor’s shareholders will, somewhat simplified, be the gains from Investor’s assets subtracted by the costs of Investor. For an investment company to create value for its shareholders, the costs have to be sufficiently low to meet the required yield from the market.

The costs are reported on a quarterly basis and included as MSEK.

4.3.6 Cash

The amount held in cash, bank and short term financial investments is included as a variable, hereby called cash. This variable can affect the discount because a high amount of cash can cause negative carry, when the interest rate received on the cash is lower than the interest rate on loans the company holds. To hold cash includes an opportunity cost, as it could yield a higher return by being invested, or by using it to amortise on loans. The level of cash can also be viewed as a proxy for when the investment company makes exits and entries in investments, which causes the level of cash to rise or fall quickly. The amount of cash is reported quarterly, and included as MSEK.

4.3.7 Change in NAV

The change in Investor’s NAV is included since it is one of Investor’s strategies to create shareholder value. The objective of including it in the analysis is to observe if the strategy is successful and appreciated by the market. Since the NAV changes every month, the change is NAV is included with a monthly frequency.

4.3.8 Debt equity ratio

In order to observe if the leverage of Investor affects the discount to NAV, the debt to equity ratio is included. Increasing leverage increases the risk of the firm, however, since it is a cheaper way to finance investments than with equity, it is usually beneficial for firms to have

(28)

some leverage [9]. Another advantage of financing through loans is the interest tax shield [9]. The debt equity ratio is reported quarterly.

4.3.9 Dividend payout ratio

The dividend payout ratio, which is the ratio of the dividends received by Investor to the dividends paid by Investor is included, in accordance with the theories by Arne Karlsson and the theories of a minimum dividend policy [16]. There is also a positive tax effect from having a high dividend payout ratio in Sweden, as it helps the company avoid double taxation. This variable is included with a yearly frequency.

4.3.10 Share of foreign capital

The share of foreign capital is included after discussions with Investor, who suspects that the share of Investor owned by foreign investors could affect the discount. It is likely that foreign investors to a larger extent are institutional actors, since private investors are generally less informed and according to the interviews with Investor less likely to invest abroad [25].

Observing foreign investors as institutional actors opens the analysis for the notion that institutional investors are assumed to act rationally, and invites discussion about the noise trader phenomenon. This variable is reported yearly.

4.3.11 Business tendency indicators

In order to control for the effect of the business cycle on the discount to net asset value, the business cycle indicator collected at Konjunkturinstitutet is included. The economic tendency indicator is created by interviewing 6000 firms about production, the present situation, and future expectations, to get a measure of the market sentiment and production [26]. Two different business cycle indicators are included, one for the entire Swedish economy, and one for the health sector. The indicator for the health sector is included since many of Investor’s holdings in Patricia Industries belong to the health sector.

4.3.12 Analyst ratings

As an attempt to include market sentiment in the analysis, the share of positive analyst ratings of Investor is used. As analysts put a buy, hold or sell recommendation on the company, the share of positive analyses is calculated as the ratio of buy recommendations of the total number of analyses. These analyst ratings are the result of the 15 analyst from different banks who follow Investor [27]. The data is collected at Factset, and has a monthly frequency.

4.3.13 Abbreviations of data variables

In table 2, the abbreviations of the variables with corresponding description are presented in order to to facilitate the reading comprehension of this report.

(29)

Table 2: Data Data variable Description of variable ABBchange Change in stock price of ABB ATCOchange Change in stock price of Atlas Copco AZNchange Change in stock price of Astra Zeneca ELUXchange Change in stock price of Electrolux ERICchange Change in stock price of Ericsson HUSQchange Change in stock price of Husqvarna OMXchange Change in stock price of OMX NDAQchange Change in stock price of Nasdaq SAABchange Change in stock price of SAAB SCVchange Change in stock price of Scania SEBchange Change in stock price of SEB SOBIchange Change in stock price of Sobi WRTchange Change in stock price of Wärtsilä EQTchange Change in value of holdings in EQT IGCchange Change in value of holdings in IGC

FINVESTchange Change in value of holdings in financial investments MolnlyckeEBITDA Profitability of Mölnlycke

GrandEBITDA Profitability of Grand Group TreEBIDTA Profitability of Tre

AlerisEBITDA Profitability of Aleris PermobilEBITDA Profitability of Permobil BraunEBITDA Profitability of Braun Ability VecturaEBITDA Profitability of Vectura LindorffEBITDA Profitability of Lindorff GambroEBITDA Profitability of Gambro LaborieEBITDA Profitability of Laborie

ABBshare ABB’s share of the total NAV of Investor ATCOshare Atlas Copco’s share of the total NAV of Investor AZNshare Astra Zeneca’s share of the total NAV of Investor ELUXshare Electrolux’s share of the total NAV of Investor ERICshare Ericsson’s share of the total NAV of Investor HUSQshare Husqvarna’s share of the total NAV of Investor OMXshare OMX’s share of the total NAV of Investor NDAQshare NDAQ’s share of the total NAV of Investor SAABshare SAAB’s share of the total NAV of Investor SCVshare Scania’s share of the total NAV of Investor SEBshare SEB’s share of the total NAV of Investor SOBIshare Sobi’s share of the total NAV of Investor WRTshare Wärtsilä’s share of the total NAV of Investor EQTshare EQT’s share of the total NAV of Investor IGCshare IGC’s share of the total NAV of Investor

PIshare Patricia Industries’ share of the total NAV of Investor Cash Amount of cash held by Investor

ChangeNAV Change in Investor’s NAV

Costs Investor’s costs

DebtEquity Debt to equity ratio DivPayoutRatio Dividend payout ratio ForeignCapital Share of foreign capital

ETImedical Business tendency indicator for the health sector ETItotal Business tendency indicator for the entire economy Rating Analyst ratings of Investor

(30)

5 Results

Initially, a linear regression model including all regression variables is analysed. This model is called full model. By applying selection criteria, an improved model from the initial full model is obtained, referred to as statistically chosen model. Further on, by only including the theoretically supported variables from previous research within the field, a theoretically supported model is obtained.

5.1 Full model

When including all 51 regression variables the following full model is obtained. The λi:s are binary interaction terms, and are used as indicators when a variable should be included in the model since Investor has made exits and entries in companies over the investigated time period. The coefficients of the regressors are presented in table 8 in appendix.

discount = β₀

+β₁ ABBchange + β₂ATCOchange

+β3 AZNchange + β4ELUXchange

+β5 ERICchange + β6HUSQchange +β7λ7OMXchange + β8λ8 NDAQchange +β9 SAABchange + β10λ10 SCVchange +β11 SEBchange + β12λ12 SOBIchange +β13λ13WRTchange + β14EQTchange +β15λ15IGCchange + β16FINVESTchange +β17 MolnlyckeEBITDA + β18GrandEBITDA +β19 TreEBITDA + β20λ20 AlerisEBITDA +β21λ₂₁PermobilEBITDA + β22λ₂₂ BraunEBITDA +β₂₃λ₂₃VecturaEBITDA + β₂₄λ₂₄ LindorffEBITDA +β₂₅λ₂₅GambroEBITDA + β₂₆λ₂₆ LabEBITDA

+β₂₇ ABBshare + β₂₈AZNshare

+β₂₉ ATCOshare + β₃₀ELUXshare +β₃₁ ERICshare + β₃₂HUSQshare +β33λ33NDAQshare + β34 SAABshare +β35 SEBshare + β36λ36 SOBIshare +β37λ37WRTshare + β38λ38 OMXshare +β39λ39SCVshare + β40λ40 IGCshare +β41 EQTshare + β42PIshare

+β43 Cash + β44ChangeNAV

+β45 Costs + β46DebtEquity

+β47 DivPayoutRatio + β48ForeignCapital +β49 ETImedical + β50ETItotal +β51 Rating

As presented in table 3, the model explains the observations well. R² and adjusted R² is high and there is significance of regression as the p-value of the F-statistic is infinitesimal.

These values that indicates a good fit likely occurred due to overfitting, since there are 51 variables and only 121 observations, thus variable selection is necessary [28].

(31)

Table 3: Summary of full model

Observations R² Adjusted R² Residual Std. Error F-Statistic Prob > F

121 0.9581 0.9272 0.02054 30.67 < 2.2e-16

Furthermore, multicollinearity is present in the full model. The condition number is 2481.92 which implies serious problems with multicollinearity. Also, 29 of the variables have a VIF over 10 which confirms the presence of multicollinearity. The VIF numbers are shown in table 12 in the appendix. To battle the multicollinearity and the linear dependence between the variables, variable selection is performed.

5.2 Statistically chosen model

Since the full model has too few observations in relation to the number of variables and shows near linear dependencies, variable selection is performed. It is not possible to perform all possible regressions on the complete dataset as it too computationally expensive, hence forward selection is used to generate a subset of 30 the variables that can undergo all possible regressions. The models obtained by all possible regressions are selected by using the criteria; BIC, Mallows’ Cp and adjusted R².

In figure 1, the separate best model of each size with the selection criteria, Mallow’s Cp, BIC and adjusted R-squared and SSR are presented. The red spot represents the size of the models recommended by each criteria. Mallows’ Cp recommends a subset model including 24 variables, BIC recommends a subset model of 12 variables and Adjusted R², a model including 27 variables.

(32)

Figure 1: All possible regressions and the recommended sizes of the model

When further investigating the three recommended models, only the subset model recommended by BIC has variables that all are significant on a 95 % confidence level. Additionally, as recommended by Harrell there should be at least 10 observations for every variable included in the model and this criterion is only fulfilled by the subset model recommended by BIC [28]. By studying the plots in Figure 1, after 12 variables all plots rates of change decrease, which is interpreted as that a model including 12 variables is adequate. Thus, the BIC model is further analysed as the final chosen statistical model.

The final statistically chosen model is presented below. The λi:s are again binary interaction terms, and are used to indicate when a variable should be included in the model since Investor has made exits and entries in companies over the investigated time period. The coefficients of the model are presented in table 9 in the appendix.

(33)

discount = β₀+

+β₁EQTchange +β₂λ₂ IGCchange +β3FINVESTchange +β4λ4 VecturaEBITDA +β5λ5 LindorffEBITDA +β6 AZNshare

+β7SEBshare +β8λ8 WRTshare +β9EQTshare +β10λ10IGCshare +β11ForeignCapital +β12 ETItotal

In table 4, a summary of the statistically chosen model is presented. Notable is that R² is scarcely lowered and adjusted R² has increased compared with the full model.

Table 4: Summary of statistically chosen model

121 0.9357 0.9285 0.02035 130.9 < 2.2e-16

Further analysis of the model implies that there are no violations of the normality assumption as the QQ-plot does not indicate any heteroskedasticity. The QQ-plot is presented in figure 2. This is also supported by the Box-Cox method as λ = 1 is within the 95 % confidence interval.

Figure 2: Q-Q plot of the final statistical model

The residuals and the coefficients of the estimated model are analysed in order to detect potential outliers and influential points. To justify omitting an outlier, it is necessary to have strong statistical evidence that it is a defective value [18]. Two observations are indicated to be potential outliers according to the standardised residuals, whilst the studentised residuals indicate that there is one such observation. When searching for potential influential points, no such point is detected by Cook’s distance, while DFFITS detects nine influential points. Some of these extreme observations correspond to when Investor has made entries or exits in companies. It is evident that observations corresponding to exits or entries can appear as outliers or influential points. No measurement errors or similar are detected when

(34)

investigating the observations further, and thus it is not justified to remove any of these observations from the data set.

One of the goals of the variable selection was to reduce the presence of multicollinearity.

The condition number of the statistically chosen model is 56.86 and none of the VIF numbers are over 10. The VIF numbers are presented in table 5. Thus, there are no indications that the regression coefficients are poorly estimated because of multicollinearity.

Table 5: VIF EQTchange 1.832

IGCchange 1.641 FINVESTchange 1.676 VecturaEBITDA 6.786 LindorffEBITDA 2.760

AZNshare 1.562

SEBshare 4.895

WRTshare 7.121

EQTshare 4.975

IGCshare 6.417

ForeignCapital 2.660 ETItotal 3.936

5.3 Theoretically supported model

The model below is obtained by including the independent variables suggested by previous research and interviews with Investor (e.g. [11], [12], [14], [15], [16], [17], [24],). The model includes ten independent variables that are all believed to affect the discount to NAV through their economic impact. The coefficients of the regressors are presented in table 10 in the appendix.

discount = β0

+β1PIshare +β2 Cash +β3ChangeNAV +β4 Costs

+β5DebtEquity +β6 DivPayoutRatio +β7ForeignCapital +β8 ETImedcial +β9ETItotal +β10 Rating

However, since these variables were selected manually, it is of interest to perform some variable selection in order to assure that only significant variables is included in the final analysis of the model. By applying the variable selection method all possible regressions, this model is reduced to contain five variables. The reason why five variables are included is that both Mallow’s Cpand BIC recommended it, as represented by the red dots in figure 3.

(35)

Figure 3: All possible regressions and the recommended sizes of the model

The reduced model this results in, is called the final theoretically supported model. The variables included in the final theoretically supported model are presented below and the coefficients of the regressors are presented in table 10 in the appendix.

discount = β₀ +β₁PIshare +β₂Cash +β₃Costs +β₄DebtEquity +β₅Rating

Where all variables are significant on a 100% confidence level. As presented in table 6 below, the R²is 0.50, the adjusted R²is 0.48, and the probability of making a mistake when rejecting the null hypothesis of the model’s significance is very low.

(36)

Table 6: Summary of final theoretical supported model

121 0.5004 0.4787 0.05496 23.04 5.768e-16

The condition number of the final theoretically supported model amounts to 34.55, and as it is below 100 it implies that there is no serious problem with multicollinearity present in the model. This is supported by the VIF-values presented in table 7, which are all below 10, and thus also implies that multicollinearity is not present.

Table 7: VIF of the final theoretical supported model PIshare 1.995

Cash 5.837

Costs 1.278

DebtEquity 8.442 Rating 1.083

When the model is estimated, its residuals are analysed. By plotting the theoretical quantiles against the sample quantiles of the residuals, it is possible to observe that the Q-Q plot does not indicate any presence of heteroskedasticity. Thus, the normality assumption of the analysis is not violated. This is also supported by the Box-Cox method as λ = 1 is within the 95 % confidence interval.

Figure 4: Q-Q plot of the final theoretically supported model

The residuals and the estimated coeffiecients of the final theoretically supported model are investigated in order to find potential outliers and influential points. By Cook’s distance

(37)

and DFFITS, there are no influential points in the data set of the five included independent variables. According to the standardised residuals, there are five potential outliers. When computing the studentised residuals only one observation appears as a potential outlier.

However, further investigation of these observations do not reveal any nonstatistical evidence that they would be bad values, and thus they are not to be excluded from the analysis [18].

(38)

6 Discussion

6.1 Model discussion

In the following sections, the results of the final models are discussed, their plausibility is analysed, and after a discussion of potential sources of error, a final recommendation is made.

6.1.1 Statistically chosen model

The full model has several deficiencies - it is complex and shows near linear dependencies among the regressors. The near linear dependencies are discovered by calculating the variance inflation factors for the full model, which are presented in table 12 in the appendix. To combat this multicollinearity problem, variable selection is performed and a reduced model including twelve variables is obtained. The variable selection leads to a reduced improved model and there are no problems of multicollinearity. Nor does this statistically chosen model violate the normality assumption. What concludes the improvement after performing variable selection is that R² barely decreases from 0.9581 to 0.9357 and adjusted R² increases from 0.9272 to 0.9285. Hence the simpler model, where eighteen variables are removed, explains the variance almost as well and is less complex compared with the full model.

In the statistically chosen model, a change in EQT’s share of Investor’s total net asset value has the largest impact on the discount and a change in IGC’s share of Investor’s total NAV has the second largest impact. Growth in these variables increases the discount.

Changes in the shares of the holdings in ABB, Astra Zeneca and Wärtsilä makes up to in Investor’s net asset value have the opposite effect and growth in these variables decreases the discount. A peak in the business cycle, increase in profitability of Lindorff, and a positive change in the financial investments’ value have a positive impact on the discount. While increases in foreign capital and net asset value of EQT and IGC have a negative impact on the discount, i.e. decreases the discount.

The statistically chosen model explains the data well, as the R² and adjusted R² are high.

The Achilles’ heel of this model is the signs of some of the coefficients as they are inconsistent and counterintuitive. An increase in EQT’s or IGC’s net asset value decreases the discount, while an increase in EQT’s or IGC’s share of Investor total net asset value increases the discount. This is inconsistent since a change in the EQT or IGC net asset value cannot, ceterus paribus, both decrease and increase the discount.

In addtion, it is highly counterintuitive that the profitability of Lindorff has a positive impact on the discount, as it implies that an increase in the profitability of Lindorff increases the discount. It is not probable that the stock market punishes Investor when its holdings become more profitable, and it would not be a reasonable strategic action for Investor to try to decrease Lindorff’s profitability in order to decrease the discount to NAV.

6.1.2 Theoretically supported model

The theoretically supported model was down-sized from its original ten variables to five by variable selection. The resulting model has no problems with multicollinearity nor does it violate the normality assumption. In addition, there is significance of regression, and all remaining five variables are significant on a 100 % confidence level, which is an improvement from the original theoretically chosen model. The adjusted R²slightly increases from 0.4652 to 0.4787 while R² decreases from 0.5097 to 0.5004, when comparing the reduced theoret-

What are the main factors impacting the discount to net asset value? - A case study of Investor AB

IN

DEGREE PROJECT TECHNOLOGY, FIRST CYCLE, 15 CREDITS

STOCKHOLM SWEDEN 2017 ,

What are the main factors

impacting the discount to net asset value?

A case study of Investor AB ALVA ENGSTRÖM

FILIPPA FRITHZ

What are the main factors

impacting the discount to net asset value?

A case study of Investor AB ALVA ENGSTRÖM

FILIPPA FRITHZ

Contents

1 Introduction

1.1 Background

1.2 Problem

1.3 Purpose

1.4 Research question

1.5 Scope

1.6 Investor AB

1.7 Methodology

2 Economic theory

2.1 Efficient market hypothesis

2.2 Agency costs

2.3 Previous research on the subject

3 Mathematical theory

3.1 Multiple linear regression

3.2 Ordinary least squares

3.3 Key assumptions

3.4 Potential problems

3.5 Transformation

3.6 Validation of the model

3.7 Variable selection and model building

4 Method

4.1 Data collection

4.2 Data processing

4.3 Data

5 Results

5.1 Full model

5.2 Statistically chosen model

5.3 Theoretically supported model

6 Discussion

6.1 Model discussion