Förändring i Aktievärdering med Avseende på Nyckeltal

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IN

DEGREE PROJECT TEKNIK, FIRST CYCLE, 15 CREDITS

,

STOCKHOLM SWEDEN 2018

Förändring i Aktievärdering

med Avseende på Nyckeltal

En Multipel Linjär Regressionsanalys på

Finanskrisen 2008

THEODOR OHLSSON

LOVE WIKLAND

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IN

DEGREE PROJECT TECHNOLOGY, FIRST CYCLE, 15 CREDITS

,

STOCKHOLM SWEDEN 2018

Shift in Stock Valuation with

Regards to Key Ratios

A Multiple Linear Regression Analysis of the

2008 Financial Crisis

THEODOR OHLSSON

LOVE WIKLAND

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Shift in Stock Valuation

with Regards to Key Ratios

A Multiple Linear Regression Analysis of the 2008 Financial

Crisis

May 23, 2018

Abstract

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F¨

or¨

andring i Aktiev¨

ardering

med Avseende p˚

a Nyckeltal

En Multiple Linj¨

ar Regressionsanalys av Finanskrisen 2008

May 23, 2018

Sammanfattning

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1 Introduction

1.1 Background

Different ways of pricing stocks are as diverse as they are creative. Ranging from the Dividend Discount Model to Comparable Company Analysis, all seek to determine the fair price of a company. This is clearly important for investors seeking to place their money in return for a future reward; how else will they be able to judge whether a company will yield the sought after return? The methods used are reliant on public information about the com-pany. In order to ensure that relevant and accurate information on American companies is readily available to the public, a number of customs and rules have been set up and monitored by the Securities and Exchange Commission (SEC).[2] These rules dictate that listed companies ought to supply the pub-lic with, among other things, a series of relevant financial indicators which in many cases form the basis for future valuation.

This might give the wrong impression that the financial market is imperme-able of speculation. As the surging Dutch tulip prices of 1634 - 1637, or, more recently, the feverish ascents and descents of the price of Bitcoin indicate, speculation is ever-present.[10, 18] However, the relatively levelled volatility of the S&P 500 index (an index of the 500 biggest listed companies in the United States) hints at more or less consensus regarding their fair value. The different nature of the major financial upheavals has dictated various types of responses from investors and legislators alike. The Dot-com bubble resulted in a soberer valuation of companies in which the business model centred around the internet. The financial crisis of 2008, which was under-pinned by an over reliance on mortgage-backed securities in tandem with an aggressive housing bubble, was legislatively resulted by the The Dodd–Frank Wall Street Reform and Consumer Protection Act of 2010. Although the stock market was not the triggering factor of the crisis, the S&P 500 fell by 56% from its high on October 8 2007 to its low on 2 March 2009[15, p. 4, 9].

1.2 Thesis Aim

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divided by its book value of equity, using a series of performance, solvency, liquidity and efficiency ratios as regressors, for example profit margin and re-turn on assets. The data used to build this model will be from the 2009 S&P 500 Industrials. Secondly, the form of this model, that is response variable transformations and regressor variable selection from the model will be used on data from a year before to obtain a model for 2008. Finally, a comparison of the coefficient estimates will be made for the two years with the aim of in-vestigating any potential shift in investor valuation preferences, with regard to the regressor ratios, before and after the crash of 2008.

Ultimately, the research question is formulated as follows; was there a sta-tistically significant shift in investor preferences, with regard to valuation by comparable ratios for S&P 500 Industrials, before and after the 2008 financial crisis?

1.3 Purpose

The purpose of this thesis is to investigate whether the market valued cer-tain key ratios differently before and after the financial crisis of 2008. For instance, did investors punish, or perhaps reward, debt more heavily after the crisis? A well constructed model concerning the impact of key ratios, with a proper analysis of for instance leverage points, may be revealing of overall trends among investors. Through comparing the two models, (one before the financial crisis and one after) one may reveal a shift in preference for certain key ratios among investors in the wake of financial instability. Such an investigation, although very particular in its scope of just treating the 08’-09’ crisis, might yield some insight into investor behaviour in the wake of a financial crisis.

1.4 Related Works

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[5].

In 2014 Calomiris and Nissim investigated Crisis-related shifts in the market valuation of banking activities through price to book ratio analysis for banks. They conclude that the substantial decline in the ratio during the crisis can-not be explained by a delayed acknowledgement of losses on existing financial instruments. Instead, they argue that a shift in valuation of intangible as-sets, for instance customer relationships, account for most of the decline in price to book ratios. Following that result they claim that the crisis have led investors to associate less value to intangible assets as a result of changes in the business environment.[8]

Beyond the scope of investigating the pre-and post crisis valuation, research has also treated the prospect of using multiple linear regression analysis in the context of stock valuation. Qing and Ai-min (2013) attempt to find the relationship between the stock price and financial indicators. However, the results are not satisfactory, attributed to the legal framework of the Chinese stock market operates.[13]

2 Methodology

2.1 Literature Study

In order to present a somewhat comprehensive report, literature of both economics and regression analysis has been studied and drawn upon. In terms of finance, it has been done so in order to infuse validity to the starting point of this report: that one would expect some relationship between key ratios and the stock price. However, financial theory has not been used to actually answer the question of investor preferences before and after the financial crisis. Instead, a mathematical framework has been constructed for there to be a rigorous mathematical analysis of data to answer this question, and thus it constitutes the core of the method.

2.2 Data Collection

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of the world’s market cap data, 87 countries and more than 22 000 compa-nies. In addition they have more than 1300 institutional clients and around 80 contributing providers of information. As the data being used for this report is almost ten years old, some of the entries were not available and had to be manually found by the authors. To do so, quarterly and annually filings to the Securities and Exchange commission were used. In addition, Eikon data as well as SEC filings reports present rounded figures, often to the nearest million or thousand. However, since the same figures where used by the market ten years ago, this should not affect the model substantially, if at all. In fact, whether the data provided by Eikon is correct is not relevant as long as it is the same data used by investors.

There is a discrepancy in the data available to us in that the data for 2009 is more complete than the data set of 2008. As of such, using 2008 as the reference model will expose the reference model to a higher risk of being over fitted. When this model then is compared to the model of 2009, deviation in the estimates of β might be the result of change in investor behaviour but also as a consequence of the poor predictive power of the reference model. As the later hypothesis cannot be excluded for sure, the analysis would have been fruitless in answering the research question. Therefor, changing the reference model to 2009 will be an easy fix. The interpretation of this is that instead of asking “did investor behaviour change from before to after the financial crisis”, one asks “was investor behaviour different before the financial crisis”.

2.2.1 Regressor Variables

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2.2.2 Companies

The S&P 500 Industrials companies are listed companies which are among the 500 largest ones (by market capitalization) on the American stock market and that are defined as industrials by the Global Industry Classification Stan-dard (GICs). According to the GICS these are defined as manufactures and distributors of capital goods, and transportation services.[1] One might ask, why use the S&P 500 Industrials for the raw data? Firstly, the S&P 500 are large corporations and the market in which they are trade can be considered very efficient, see section 3.1.4. Secondly, it is acknowledged that key ratios are best suited when comparing similar companies and thus it seems unrea-sonable to build a model on the entire S&P 500 index. However, limiting the companies used in the model to a few, say ten, very similar companies does not supply a sufficient amount of data points to do a linear regression model. In other words, finding enough very similar companies for there to be a stable regression analysis would be hard. It is therefore reasonable to use companies from a common sector, for example the industrial sector or financial sector, providing at least to some extent similar companies but still sufficiently many to allow for regression modeling. Lastly, companies holding a high amount of ”soft” value attached to their valuation are to be avoided, as this would dilute the value of key ratios. Threading this needle, the com-panies chosen are those listed as ”Industrials” on the S&P 500, both before and after the financial crisis.

2.2.3 Time Frame

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2.3 Modeling with Financial Ratios

There is an apparent problem with modelling the stock price as a function of key ratios, which is that the stock price is dependent on the number of share outstanding and its market capitalization. Since the key ratios do not take this into account (as well as the magnitude of assets, liabilities, sales etc), two identical companies with the only difference that one company has twice the amount of shares outstanding than the other company, and thus the stock price would be half of the other. The model exclusively using key ratios would not take this into consideration, and therefore be highly unstable. To solve this problem, there has to be some normalizing factor for all the stocks. The solution for this report is to normalize the market capitalization of company with its book value of equity. In this way, the dependent variable of all companies can be compared.

3 Financial Framework

3.1 Fundamental Concepts

3.1.1 Ownership of a Corporation

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3.1.2 Market Capitalization versus Book Value of Equity

From the balance sheet of a company, a snapshot of its financial position listing all assets and liabilities it currently holds, one can compute the dif-ference of total assets and total liabilities to obtain the book value of equity, an accounting measure of the net worth of the company. It is an intuitively simple measure; sell all of the assets, pay all the liabilities and the worth of the firm is what is left. However the book value of equity does not represent a fair value of a firm’s equity because, for instance, many of the firm’s assets are not listed on the balance sheet. Consider for example the expertise of the employees, the potentially recognized brand and reputation of the firm and its relations and contracts with customers and suppliers respectively. Such entities are not defined on the balance sheet and therefore the real value of a firm’s equity in many cases significantly differ from the book value of equity. The amount investors are willing to pay for a firm’s equity is represented by the market capitalization of a firm, i.e. the price its share is traded at times the number of outstanding shares. An important feature of the ”market cap” is that it does not reflect the historical value of a firm. Rather, it reflects what investors think the current assets will produce in the future.[3, p. 27]

3.1.3 Returns and Risk

There are two fundamental ways of earning returns, or actual money, by trading or owning stocks. Firstly there is changes in a stock price. If one buys a share today and sells it to a higher price tomorrow, one has made a profit. Secondly, there are dividends. As a shareholder of a company stock you are entitled to dividend payments as previously stated.[3, p. 271] An important feature about returns is the correlation with risk; investments associated with higher risk generally generate greater returns compared to investments with lower risk. The higher return is meant to compensate investors for making a risky investment.[3, p. 86]

3.1.4 Efficient Market Hypothesis

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all encompassing term which includes a variety of information. It is in part underpinned by a company’s financial statement but also by news about for example executives and legislation and other macroeconomic factors such as interest and employment. The hypothesis states that as information arises, the news spreads quickly and is instantly incorporated into share prices. As a result, neither technical analysis, which is the study of predicting future prices based on historical ones, nor fundamental analysis, which is the anal-ysis of financial information, such as earnings, cash flows, assets, etc., would enable an investor to achieve returns greater than those that could be ob-tained by holding a portfolio consisting of randomly picked stocks.[14, p. 2-3] Indeed the EMH has historically been associated with the idea of a random walk, or the notion that an asset’s future price is a random deviation from its current price. Since information is immediately reflected in the stock price, tomorrow’s price changes will be independent of the price changes today, as price changes tomorrow only responds to tomorrow’s news. And news are by definition random thus making tomorrows price movement random.[14, p. 4] In fact, many investment methods are based on the geometric Brownian mo-tion, a famous one being Black -Scholes equation for pricing European Call options.[6] If total randomness of a stock price really was the case a total novice on the market would earn the same return as the expert, as long as their portfolio holds equivalent risk.[14, p. 4]

Intuitively it might seem strange that by throwing darts on a board with lists of all traded stocks, on can construct an equally well performing port-folio as that of an expert. Indeed, economists have debated the legitimacy of EMH for decades. Economists and statisticians alike have argued that stock prices, at least to some extent, are predictable. Critics of EMH have in particular stressed psychological and behavioral elements of stock-price prediction. Some believe that future prices are partially predictable on the basis of past stock price series or specific fundamental valuation criterion.[14, p. 5]

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market where actors are well and rapidly informed.[3, p. 300] However the concept of rapid price adjustment and efficient markets imply there are at least some consensus regarding stock valuation.

3.2 Valuation Methods

There are numerous methods of valuing companies and its shares. These may differ in sophistication and complexity. Below, three common valuation methods will be described in order to give some insight into the different ways of reasoning.

3.2.1 Dividend Discount Model

The Dividend Discount Model (DDM) is a common model for stock valuation. It considers the total return, from both share dividends and future expected share price changes, that an investor will receive in the future from owning the stock. However, as money today is worth more than money tomorrow, because money today can earn you returns for tomorrow, the future returns have to be discounted with the equity cost of capital, i.e. the expected return from an investment with equivalent risk.[3, p. 300]

3.2.2 Discounted Free Cash Flow Model

The Discounted Free Cash Flow Model prices a stock by firstly determining the value of a firm to all investors, both equity and debt holders, then ad-justing for debt and dividing by shares outstanding. The value of a firm is determined by considering its future free cash flows, the amount of actual cash a firm is able to generate, and discounting it to a net present value.[3, p. 300]

3.2.3 Trading Comparables Model

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one compares certain ratios, or ”multiples”, of measure that take into ac-count the firm’s scale. An analogy can be made when pricing an apartment; a natural measure of the total value would be to take the average per square meter price of similar apartments recently sold in the same area, and multi-ply this number by the relevant apartment area in question[3, p. 288]. There are two major advantages in comps valuation. Firstly there is simplic-ity; while some valuation methods require sophisticated financial modeling which can be quiet cumbersome to build, comps can offer an easy and quick way to value a firm based entirely on the firm’s financial statements and the share price. Secondly, comps valuation has the advantage of valuing a firm based on numbers the market already has priced since the valuation is based on real publicly traded firms and their actual prices. There is no need for sweeping assumptions about future performance and cash flows, thus elim-inating the potential mistake of making unrealistic forecasts. Rather, the valuation is now indirectly based on the assumptions the market has made for firms similar to the one up for valuation.[3, p. 291]

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regard it as a shortcut and supplement to more fundamental valuation models based on future cash flows.

3.3 Financial Ratios and Multiples

The final ratios which are going to be included in the full model can be broken up into four parts; profitability, liquidity, turnover and solvency ratios.

3.3.1 Market Value of Equity to Book Value of Equity

Also known as price-to-equity-ratio, the price-to-book-ratio is an indicator of how the market prices the equity compared to the book value of said equity. A low price-to-book ratio can be an indicator of the company being poorly managed, given that the price is right. Notice that this is a Valuation Multiple as it contains the share price of the company.

P riceT oBook = M arketCapitalization BookV alueOf Equity [3, p. 27]

3.3.2 Profit Margin

As profitability ratio, the profit margin is a ratio describing the relationship of sales and net income, and is a measure of profitability. Specifically, it is defined as:

P rof itM argin = N etIncome Revenue

The investor can use this ratio in order get an idea of how of much safety margin a company has in its activities; a low profit margin would imply low wiggle room for staying profitable. The profit margin might vary greatly among companies due to variation in business activities within and across sectors. [3, p. 36]

3.3.3 Operating Margin

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of goods sold, raw material etc, and operating expenses, salaries and similar.

OperatingM argin = OperatingIncome Revenue

The operating margin reveals how much a company earns before interest and taxes from each dollar of sales.[3, p. 25]

3.3.4 Gross Profit Margin

As a profitability ratio, it is similar to profit margin and operating margin but instead of Net Income or Operating Income in the numerator one finds Gross Profit - a firm’s revenue net of cost of goods sold.

GrossP rof itM argin = GrossP rof it Revenue

The gross profit margin reveals a relationship between revenues and the cost of goods sold. [3, p. 35]

3.3.5 Return on Equity

Return on equity measure of profitability in relation to a company’s book value (in other words, net assets of the company).

ROE = N etIncome BookV alueOf Equity

It can be interpreted as a measure of how well the company utilizes its investments to generate earnings. [16, p. 181]

3.3.6 Return on Assets

Similar to ROE but instead measures a firms profitability in relation to its assets.

ROA = N etIncome + InterestExpense BookV alueOf Assets

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3.3.7 Debt to Equity

A solvency ratio. An important multiple is the Debt to Equity ratio, or the leverage of a firm.

Leverage = DebtT oEquity = T otalDebt T otalEquity =

T otalDebt BookV alueOf Equity Total debt is the sum of short term debt, including maturing long term debt,long term debt and other interest bearing liabilities. Total book value of equity is a firm’s assets net of all of its liabilities. An investor should beware of very high leverage multiples, especially if the industry is facing tough times. [20, p. 238] Lehman Brothers had a debt/equity ratio of over 30:1 in 2007 before its bankruptcy.[4]

3.3.8 Debt to Assets

A solvency ratio. Similar to the debt to equity ratio but instead uses total assets in the denominator.

DebtT oAssets = T otalDebt BookV alueOf Assets

The multiple is a measure how much of the firms assets are funded by debt. [3, p 40]

3.3.9 Current Ration

The current ratio compares a firm’s current assets, such as cash, account receivable and inventory to its current liabilities, such as accounts payable and short term debt. It is used to assess whether the firm has sufficient working capital to meet its short-term expenses.

CurrentRatio = CurrentAssets CurrentLiabilities [16, p. 179]

3.3.10 Quick Ratio

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more stringent.

QuickRatio = CurrentAssets − Inventory CurrentLiabilities

After all, inventory needs to be sold off in order for the company to be able to pay its obligations with its worth and should therefore not be regarded as liquid assets. [16, p. 180]

3.3.11 Asset Turnover Ratio

The asset turnover ratio highlighting how efficiently a company utilizes its assets in order to generate sales.[3, p. 44] It is defined as:

AssetT urnover = Sales T otalAssets

3.4 The Financial Crisis

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4 Mathematical Framework

4.1 Multiple Regression Analysis

The multiple linear regression seeks to model an outcome, y, as a linear combination of a set of p regressor variables, xi, for i = 1, 2, ..., p, from

n observations. Using matrix notation, the model can then be expressed as Y = Xβ + where can be thought of as the statistical error for each observation with a expected value of 0 and a variance of σ. For this model, one seeks to find an estimate of the regression coefficients, denoted as ˆβ. The difference between the observed value yi and its corresponding fitted value,

ˆ

yi is defined as ei. That is, ei = yi− ˆyi. Relevant metrics which will be used

later in this report are the regression sum of squares defined as:

SSR = ˆβ0X0y −

(Pn

i=1yi)2

n As well as the residual sum of squares

SSRes = y0y − ˆβ0X0y

and the total sum of squares

SST = SSR+ SSRes

Lastly, the mean square residual :

M SRes =

y0y − ˆβ0X0y n − p [17, p. 86]

4.2 Method of Least squares

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n

X

i=1

2_i = 0 = (Y − Xβ)0(Y − Xβ)

is minimized. The method of least squares yields the following formula for estimating the regression coefficients:

ˆ

β = (X0X)−1X0y [17, p. 71-73]

Furthermore, the assumption that the errors are normally distributed is needed for hypothesis testing and interval estimation. When normality is fulfilled, the confidence interval is for ˆβj is:

ˆ βj − tα₂,n−p q se( ˆβj) ≤ βj ≤ ˆβj+ tα₂,n−p q se( ˆβj) where tα

2,n−p denotes the t-distribution with n − p degrees of freedom and

a significance level of α[17, p. 98]. The null hypothesis H0 : β = 0 can be

shown to follow a F-distribution. More precisely

F0 =

SSR/k

SSRes/(n − k − 1)

follows an Fk,n−k−1-distribution. A test of significance for the model can thus

be done by comparing the F0 and Fα,k,n−k−1, for some level of significance α.

[17, p. 85]

4.3 Overfitting

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4.4 Residual Analysis

When a model has been fitted, it has been so on the assumptions that the relationship between the response and regressors is (approximately) linear, that the error terms are uncorrelated with mean equal to 0 and constant vari-ance. The latter would imply homoscedasticity, while non-constant variance is called heteroscedasticity. Furthermore, for hypothesis testing and inter-val estimation the assumption that the error term is normally distributed is also required.[17, p. 129] The inspection of the residuals may gain insight to whether these assumptions are correct.[9]

Various methods of plotting residuals exist with the aim of exposing some information of the model. Plotting the residuals against its corresponding fitted value will give an idea of whether the errors have the same variance regardless of the value of the fitted value. If so, this would indicate ho-moscedasticity and implies some relevance of the model. If this is not the case however, the model suffers from heteroscedasticity. However, it says little about whether those points with a relatively high residual are in fact to be considered as outliers. One can then introduce the standardized residuals. The standardized residuals are plotted against the fitted values. These resid-uals will have a standard deviation equal approximately to 1; a point with a standardized residual of above 3 is an indication of that point requiring closer inspection. The standardized residuals are defined as

di =

ei

M SRes

[17, p. 130-131]

4.4.1 QQ-plot

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4.4.2 Shapiro-Wilks

The Shapiro-Wilks statistic is a test of whether the sample is from a normally distributed population or not, and serves as a more formal test of normality than inspection of the QQ-plot. This implemented on the residuals in order to test that they satisfy the normality assumptions, which is essential when for example constructing confidence intervals for the estimates. More techni-cally it tests the null hypothesis H0 that the sample is taken from a normal

distribution. [19]

4.4.3 Breush-Pagan Test

The Breusch-Pagan test for homoscedasticity, i.e that the variance is con-stant. The null hypothesis is χ2_{-distributed, and so a corresponding p-value}

can reveal whether the null hypothesis should be rejected or not with some confidence, similair to the Shapiro-Wilks test. [7]

4.4.4 Outliers and Influential Points

In any case, a model should be investigated thoroughly in order to detect outliers and inf luential points. Outliers are points which are located far from the main cluster in x-space and are themselves not necessarily harmful towards the validity of the fitted model if they lie approximate to the line fitted to the cluster of values. A deletion of such a point would have little effect on the estimates of the model. However, if the removal of a point produces a significant change in the model, it is said to be a influential point. Such points should be considered to be omitted from the data set. This decision should be based on the amount of leverage the point exerts but also whether there is any underlying explanation for its strange behaviour in relation to the rest of the data set. Leverage is here defined as a measure of how much the model changes if the point is omitted and is thus a point which ’draws’ the regression line close to it.

A plot of the fitted values in relation to Cook’s distance can be used in order to identify possibly influential points. Cook’s distance of the i:th point is defined as:

Di(X0X, pM SRes) =

( ˆβi− ˆβ)0X0X( ˆβi− ˆβ)

pM SRes

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where ˆβi is the estimate without using the i:th point in the data set.

Another statistic for discovering leverage is COVRATIO. Let Si be the

stan-dard error estimated with the i:th point excluded, the COVRATIO is defined as:

COV RAT IOi =

|(X_i0Xi)−1Si2|

|(X_i0Xi)M SRes|

[17, p. 219]

4.4.5 Transformation and the Box-Cox Method

In order to avoid heteroscedasticity, a variance stabilizing transformation can be employed. The transformation seeks to change the relationship between the predictor and its regressor variables in such a way that the variance be-comes constant (and the model thus fulfills homoscedasticity). For instance, one can guess that the true relationship is:

y = β0eβ1x

and in such case a logarithmic transformation can be done to achieve the linear model

ln(y) = ln(β0) + β1x + ln()

[17, p 176]

This transformation is a data transformation and is ideally chosen as a result of an understanding of the underlying dynamics of what should be modeled. However, if there is no theoretical framework to hint at what an appropriate transformation should be, one can systematically try a number of transfor-mations. This is the idea underpinning the Box-Cox method. The Box-Cox method is a procedure to find the optimal power transformation, that is op-timizing the transformation yλ where λ is some constant. More specifically, a λ is found such that one maximizes the log-likelihood function:

L(λ) = −1

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4.5 Variable Selection

4.5.1 Multicollinearity

Multicollinearity is said to exist when there is a near linear dependency among the regressor variables. Colloquially, this means that a high or low value of one of the regressor variables will imply a high or low value in one or more of the remaining regressor variables. Unwanted effects arise from multicollinearity in that the presence of it will highly inflate the variance and covariances of the estimates ˆβ. Furthermore, if perfect collinearity between the variables exist, the matrix X’X will not even be invertible since it will be of dimensions p × p but with a rank less than p. Subsequently a solution for the ordinary least squares estimator will not even exist. [17, p. 285-286, 288-289]

4.5.2 Diagnostics of Multicollinearity

The simplest and most straightforward way of detecting multicollinearity is to examine the off-diagonal elements of the matrix X’X, where X is on corre-lation form. A high value of X0X(i,j)(close to 1) indicates a high correlation

between the i:th and j:th regressor. However, simple as this diagnostic is, it only reveals correlation between two variables. When more than two vari-ables are involved in near linear dependencies there is no guarantee that this will express itself as pairwise high correlation. Such collinearity might then escape unnoticed. [17, p 292]

An alternative way of detecting multicollinearity is inspecting the variance inflation factors (VIF). The VIF for a regressor variable measures the aggre-gate effect from dependencies of some subset of the regressors on the variance of said regressor. Although there is no theoretical value for which the VIF should be lower than, it is recommended that VIF exceeding 5 to 10 is an indicator of poor estimation due to multicollinearity. [17, p. 292]

4.5.3 All Possible Regression

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com-putationally epensive to say the least. However, if the number of candidate regressor variables can be shrunk by the means of multicollinearity diagnos-tics and treatment presented above, the all possible regression method is a feasible step in the the regression analysis. [17, p. 338]

4.5.4 AIC and BIC

The Akaike Information Criteron seeks to maximize the entropy of a model, where entropy here refers to the Kullback-Leibler information measure. when L denotes the likelihood function for a given model, the AIC is defined as:

AIC = 2 ln(L) + 2p

which in the ordinary least squares regression translates to:

AIC = n ln(SSRes n ) + 2p Similarly to R2

Adj, AIC penalizes addition of variables and therefore reveals

whether such an addition is justified by the decrease in SSRes

BIC is similar to AIC but adds greater emphasize on penalizing the model for adding regressor variables when the sample size increases. It is defined as:

BIC = −2 ln(L) + p ln n

which (analogous to the AIC statistic) for the ordinary least squares estima-tion translates to:

BIC = n ln(SSRes

n ) + p ln(n) [17, p. 336]

4.5.5 Mallows’s Cp

Related to the mean square error is Mallows’s Cp which, given that ˆσ is a

good estimate of σ is defined as:

Cp =

SSRes(p)

ˆ

σ2 − n + 2p

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4.6 Model Performance

4.6.1 R2

A common measure of model performance is the R2-statistic. It is sometimes described as the proportion of variation explained by the regressors. The R2

-statistic is defined as:

SSR

SST

= 1 − SSRes SST

[17, p. 35-36]

Although this measure provides an indication of the models performance (on a scale between 0 and 1), it does not indicate whether this is due to appropriately selected regressor variables or simply because of a high number of regressor variables. Indeed, for every regressor variable that is included into the model R2 _{will increase. For comparing models with different number}

of regressor variables, R2-adjusted can be utilized. R2_Adj is defined as: R2_Adj = 1 − SSRes/(n − p)

SST/(n − 1)

This statistic will only increase when an inclusion of an additional variable reduces the residual mean square. As of such, R2

Adj can be used to compare

models with a different number of regressors.

4.6.2 Hidden Extrapolation

When using a model, constructed on one data set, on a different data set one ought to be aware of extrapolation, i.e using the model on a domain which is not included in its training set. This can yield misleading results. Furthermore, the deceiving phenomenon of hidden extrapolation should be investigated. This is done by making sure that the second data set is con-tained by the convex hull of training set. [17, p. 107,110]

4.6.3 Bootstrapping

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on which the model Y = X ˆβ has been fitted, one re-samples with replace-ment n data-points which are used as the basis for a new estimate of . This procedure of re-fitting is repeated k times, after which a lower 100(α₂). and upper 100(1 −α₂) percentiles of the confidence interval can be produced. [17, p. 518-519]

4.6.4 Welch’s t-test

The Welch’s t-test is test of whether the mean of two different populations are the same, whilst not being dependent on that the variances of the two populations are the same. Under the null hypothesis, t should approximately follow a Student tv-distribution. The t-value is defined as:

tv = ¯ µ1− ¯µ2 q s2 1 n1 + s2 2 n2

with degrees of freedom, v, is estimated and rounded to the nearest integer by: v = ( s2 1 n1 + s2 2 n2) 2 s4 1 n1v2 + s4 2 n2v2 were v1 = n1 − 1 and v2 = n2− 2. [11]

5 Results

5.1 Regression Analysis on Full Model

Using all regressor variables and fitting the model to our data set for the year of 2009 by the method of least squares yields the following model.

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Estimate Std. Error t value Pr(>|t|) (Intercept) -3.2979 1.3956 -2.36 0.0228 ROE 0.0768 0.0109 7.04 0.0000 ROA 0.4329 0.1206 3.59 0.0009 ProfitMarg -45.1569 18.5800 -2.43 0.0194 OperatingMarg 17.8612 15.6025 1.14 0.2588 GrossMarg 4.3171 3.4602 1.25 0.2191 CurrentRat -2.0003 0.8838 -2.26 0.0288 QuickRat 3.1755 1.3665 2.32 0.0250 DebtToEquity -0.3252 0.0737 -4.41 0.0001 DebtToAssets 0.2673 0.5947 0.45 0.6554 TurnoverRat -0.1758 0.1947 -0.90 0.3717 Table 1: Full model with all available data and all regressors

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5.1.1 Residual Analysis

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Figure 2: Residual-,QQ-and Leverage Plots with outliers removed.

Judging by these plots, there are no more serious need to deal with outliers, and it is thus opted to not remove anymore points for the sake of not infusing too much bias in the data sets. It now consists of 47 data points.

5.1.2 Transformation

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Figure 3: QQ-plot of model with appropriate boundaries

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Figure 4: Box-Cox Method maximizing with respect to λ

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Figure 7: Residual-,QQ-and Leverage Plots for the transformed model

5.1.3 Multicollinearity and Variable Selection

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Variables 1st Round 2nd Round 3d Round 4th Round ROE 6.418 6.387 1.557 1.550 ROA 6.833 6.822 - -ProftiMarg 16.3053 1.966 1.965 1.946 OperatingMarg 17.1444 - - -GrossMarg 1.717 1.549 1.536 1.519 CurrentRat 6.839 6.551 6.538 -QuickRat 6.8016 6.801 6.491 1.186 DebtToEquity 4.163 4.162 1.374 1.368 DebtToAssets 4.4762 3.837 3.724 3.723 TurnoverRat 3.920 3.655 3.630 3.630

Table 2: Elimination of variables using VIF

Not surprisingly, operating margin has a high multicollinearity. So does ROA and current ratio. These are all eliminated one by one, resulting in a reduced model. Now with only 7 regressor variables, all possible regression can be applied with a manageable 27 = 128 models to be computed. The all possible regression is used with 4 different metrics; Mallows’s Cp, R2 and BIC. The

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Figure 8: Model measures as functions of number of variables. Obtained using All Possible Regression.

This indicates that two different models are to be preferred: using 2 or 3 regressor variables respectively. More specifically, the variables that are to be included are RE, DebtT oEquity for the 2-variable model, with DebtT oAssets added for the 3-variable model. These two models are now compared to one another using AIC and mean-squared error. It can here be seen that the difference in MSE extremely small, preferring the 3-variable model. AIC and Cp both prefer the 2-variable model. The latter is thus chosen as the

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ROE Prof.Mrg Gros.Mrg QuickR. DtE DtA TurnoverR.

BIC x - - - x -

-Cp x - - - x -

-R2_adj x - - - x x

-Table 3: All possible regression variable selection according to different cri-teria.

AIC MSE 2 Regressors -50.01 0.0174 3 Regressors -49.85 0.0167

Table 4: Further measures comparing the fitted models using two and three regressors respectively.

5.1.4 Final Model The final model is:

P riceT oBook1/4 = 0.972886 + 0.013229 × ROE − 0.043669 × DebtT oEquity Estimate Std. Error t value Pr(>|t|)

(Intercept) 0.9729 0.0426 22.84 0.0000 ROE 0.0132 0.0019 7.12 0.0000 DebtToEquity -0.0437 0.0199 -2.19 0.0337

5.2 2008 Comparison

For the comparison between before and after the financial crisis, a null hy-pothesis is postulated: there is no difference in investor behaviour. This translates to the fact that the regression coefficients do not differ signifi-cantly. Formally this is expressed as:

H0 : β2009 = β2008

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in such a way that it only includes companies used for the fitting of the final 2009 model. Furthermore, one company (United Continental Holdings), is outside of the convex hull created by the 2009-data and is thus removed.

Figure 9: Convex Hull. Note that the black hull is containing all magenta points except one.

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Figure 10: Bootstrapping estimates for 2009 ˆβ

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2.5% E[ ˆβ] 97.5% Intercept 2008 0.9000 1.0666 1.2305 Intercept 2009 0.8900 0.9729 1.0577 ROE 2008 0.0111 0.0188 0.0264 ROE 2009 0.096 0.0132 0.0169 DebtToEquity 2008 -0.1601 -0.0834 -0.0063 DebtToEquity 2009 -0.0852 -0.0437 -0.0068

Table 5: Estimates of relevant regressors for both models including confi-dence intervals. Note that the intervals overlap.

To further test whether the coefficients have changed, and not solely rely on the confidence intervals for the estimates, the Welch’s t-test is now applied to all coefficients. 32 is used as an approximation of the the corresponding degrees of freedom. This yields the following t-statistic and p-values.

Coefficient t-value. Pr (Intercept) 0.9352311 0.3567

ROE 1.259151 0.2171 DebtToEquity -0.8307545 0.4123

Table 6: Welch’s t-test on the regression coefficients. Low absolute t-values supports the null hypothesis.

6 Discussion

The low t-values of the Welch t-test suggest that one cannot reject the null hypothesis. This is furthermore supported by the more informal inspection of the confidence intervals (see table 5). It is therefore, on the basis of this report, concluded that there is no statistically significant change in the use of key ratios in valuing S&P500 Industrials companies before and after the financial crisis.

6.1 Implication of Results

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in the same domain. This underscores the fact that valuation indeed might have changed. One has to keep in mind the relatively poor predictive value of the model in the first place (with R2_Adj of 0.5142) combined with the large confidence intervals for the estimates. The fact that R2

Adj is fairly low means

that the model can only account for roughly half of the variability in the results. Thus, further investigation in the matter should be conducted, see section 6.2 for critique, suggested improvements and further research.

6.1.1 Management

Based on the final model, which was distilled from 10 regression variables into 2, one can hastily draw the conclusion that the 8 other variables are irrelevant when valuing a company and thus implying that managers need not care about the other ratios if they seek to increase the value of the stock. This conclusion, however, is a false one. It assumes the model to be a perfect predictive one, neglecting its relatively poor explanatory performance (i.e low R2). In fact, key ratios may play a very important role in the valuation of individual companies, but has an ambiguous common interpretation when comparing multiple companies from different industries.

6.1.2 Extrapolation

The results, that there is no change in the relationship between market-to-book and other key ratios from before and after the financial crisis for industrials companies, cannot be extrapolated to other sectors. As the rele-vance of key ratios varies from industry to industry, the composition of key ratios in the valuation would differ. Thus, such conclusions should require several separate analyses of different sectors. Moreover, the results should not be extrapolated in time either, by concluding that the final model holds today as well.

6.1.3 Chosen Variables

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be noted though that there is more to it than that; it has been proposed that highly indebted companies can be preferable for investors in order to avoid highly entrenched managers, i.e managers that act in there own interest rather than the shareholder’s.[12]

6.2 Critique

6.2.1 Companies Chosen

The ratios, in this report used as regressor variables, are ideally used to compare similar companies. As mentioned in the financial framework of this report, companies from different industries will vary greatly in their com-position of key ratios. It is then not very surprising that the final model had a poor predictive value with very large confidence intervals for the es-timates. One could then speculate that better estimates and performance would have been obtained by constricting the data set to similar companies. However, a large enough data set of similar companies for there to be any reliable estimates would be hard to find. A potential solution to this prob-lem, and that of the large confidence intervals, would be to gather data from several quarters both before and after the financial crisis. Indeed more data points are associated with narrower confidence intervals and the estimates would perhaps prove to become more accurate, resulting in a more precise conclusion.

6.2.2 Linear Regression Analysis

One might question whether a linear regression model is a suitable choice in the first place. The matter of the fact is that the intrinsic relationship in the valuation of stocks is not one derived from any laws of physics; rather it only expresses the aggregate opinion of more or less independent investors. As of such, a clear-cut mathematical expression (especially a linear one), that would be stable over some period of time, would be a rare find.

6.3 Further Research

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References

[1] Global Industry Classification Standard (GICS ).R https://www. msci.com/documents/1296102/1339060/GICSSectorDefinitions.pdf/ fd3a7bc2-c733-4308-8b27-9880dd0a766f. Accessed: 2018-05-05. [2] The Laws That Govern the Securities Industry. https://www.sec.gov/

answers/about-lawsshtml.html. Accessed: 2018-04-27.

[3] Berk, J. and DeMarzo, P. (2014). Corporate Finance. Pearson.

[4] Berman, K. and Knight, J. (September 16 2009). Lehman’s three big mistakes. Harvard Business Review.

[5] Bertsatosa, G., Sakellarisa, P., and Tsionas, M. (2017). Did the financial crisis affect the market valuation of large systemic u.s. banks? Journal of Financial Stability, 32:115–123.

[6] Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. The Journal of Political Economy, 81(3):637–654.

[7] Breusch, T. S. and Pagan, A. R. (1979). A simple test for heteroscedas-ticity and random coefficient variation. Econometrica, 47(5):1287–1294. [8] Calomiris, C. W. and Nissim, D. (2014). Crisis-related shifts in the market

valuation of banking activities. Journal of Financial Mediation, (23):400– 435.

[9] Cook, R. D. and Weisberg, S. (1983). Diagnostics for heteroscedasticity in regression. Biometrika, 70(1):1–10.

[10] Day, C. C. (2004). Is there a tulip in your future?: Ruminations on tulip mania. Journal des Economistes et des Etudes Humaines, 14(2).

[11] Fagerland, W. and Sandvik, L. (2009). Performance of five two-sample location tests for skewed distributions with unequal variances. Contempo-rary Clinical Trials, 30(5):490–496.

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[13] Li, Q. and min Li, A. (2013). Empirical study on stock valuation model based on multiple linear regression analysis. The 19th International Con-ference on Industrial Engineering and Engineering Management, pages 479–486.

[14] Malkiel, B. G. (2003). The efficient market hypothesis and its critics. Journal of Economic Perspectives, 17(1):59–82.

[15] Malliaris, A., Shaw, L., and Shefrin, H. (2006). The Global Financial Crisis and Its Aftermath: Hidden Factors in the Meltdown. Oxford Schol-arship.

[16] McMillan, E. J. (2010). Not-for-Profit Budgeting and Financial Man-agement. John Wiley Sons, Inc., 4th edition.

[17] Montgomery, D., Peck, E., and Vining, G. (2012). Introduction to Linear Regression Analysis. John Wiley Sons, Inc.

[18] Popper, N. and Bowles, N. (January 17, 2018). Bitcoin falls below $10,000 as virtual currency bubble deflates. New York Times.

[19] Shapiro, S. and Wilk, M. B. (1965). An analysis of variance test for normality (complete samples). Biometrika, 52(3/4):591 – 611.

[20] Silvia, J. E. (2011). Dynamic Economic Decision Making: Strategies for Financial Risk, Capital Markets, and Monetary Policy. John E. Silvia. [21] Steyerberg, A. (2014). The number of subjects per variable required in

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2009

RIC Company Name Price Close Total RevenueGross ProfitEBITDA Operating IncomeNet Income After TaxesDiluted EPS Excluding Extraordinary Items CHRW.OQ CH ROBINSON WW ORD52,98 8,58E+09 1,37E+09 5,98E+08 5,72E+08 3,59E+08 2,07938

WM.N WASTE MANAGEMENT ORD28,88 1,34E+10 6E+09 3,45E+09 2,23E+09 1,13E+09 2,19419

BA.N BOEING U ORD47,7 6,09E+10 1,06E+10 4,88E+09 3,95E+09 2,65E+09 3,64609

MAS.N MASCO ORD9,711735 9,48E+09 2,36E+09 8,03E+08 32000000 -3,3E+08 -1,05666

EXPD.OQ EXPEDITORS INTERNATIONAL OF WASN ORD34,21 5,63E+09 1,6E+09 5,15E+08 4,73E+08 3,03E+08 1,37343

FLR.N FLUOR ORD50,5 2,23E+10 1,24E+09 1,2E+09 1,14E+09 7,49E+08 3,88592

RTN.N RAYTHEON ORD45,88 2,32E+10 4,69E+09 2,91E+09 2,62E+09 1,7E+09 3,92497

HON.N HONEYWELL INTL ORD35,14879 3,66E+10 5,44E+09 2,22E+09 6E+08 8,26E+08 1,08392

SRCL.OQ STERICYCLE ORD51,04 1,08E+09 4,85E+08 3,16E+08 2,74E+08 1,49E+08 1,68344

PCAR.OQ PACCAR ORD30,39885 1,5E+10 2,84E+09 2,03E+09 1,38E+09 1,02E+09 2,78191

PWR.N QUANTA SERVICES ORD23,6 3,78E+09 6,35E+08 4,03E+08 2,89E+08 1,58E+08 0,86903

SNA.N SNAP ON ORD32,7 2,85E+09 1,28E+09 4,76E+08 3,89E+08 2,4E+08 4,07401

JBHT.OQ JB HUNT TRANSPORT SERVICES ORD31,97 3,73E+09 7,14E+08 5,63E+08 3,58E+08 2,01E+08 1,56063

CMI.N CUMMINS ORD35,74 1,43E+10 2,94E+09 1,37E+09 1,27E+09 8,18E+08 3,84164

PNR.N PENTAIR PLC26,22 3,35E+09 1,01E+09 4,75E+08 3,2E+08 2,59E+08 2,58775

AAL.OQ AMERICAN AIRLINES GROUP INC.59,31 2,93E+10 5,37E+09 4,1E+09 3,65E+09 2,48E+09 6,21448

GD.N GENERAL DYNAMICS ORD12 1,4E+10 3,43E+09 2E+09 6,29E+08 3,24E+08 1,29425

TXT.N TEXTRON ORD5,176669 1,45E+09 3,74E+08 1,28E+08 75100000 47500000 0,21441

AOS.N A O SMITH ORD46,67 1,21E+10 2,81E+09 1,76E+09 1,33E+09 9,58E+08 5,53161

PH.N PARKER HANNIFIN ORD22,10724 2,69E+10 4,73E+09 3,34E+09 1,09E+09 4,5E+08 0,81551 NSC.N NORFOLK SOUTHERN ORD44,06164 2,63E+10 5,22E+09 3,14E+09 2,08E+09 1,02E+09 2,98009

DE.N DEERE ORD 14,77779 2,53E+09 8,01E+08 5,36E+08 4,33E+08 2,47E+08 1,02153

GE.N GENERAL ELECTRIC ORD33,84 2,38E+10 8,94E+09 4,57E+09 3,83E+09 2,52E+09 3,09856

AME.N AMETEK ORD55,27 5,91E+10 1,55E+10 8,37E+09 7,51E+09 5,05E+09 4,90276

EMR.N EMERSON ELECTRIC ORD26,19331 4,47E+09 1,58E+09 6,61E+08 6,16E+08 4,45E+08 2,56952

RHI.N ROBERT HALF ORD27,81246 7,57E+09 2,73E+09 1,32E+09 1,03E+09 6,95E+08 3,67074

EFX.N EQUIFAX ORD57,88 3,8E+10 2,89E+10 4,82E+09 2,08E+09 1,13E+09 3,60577

UTX.N UTD TECHNOLOG ORD23,20056 3,94E+09 1,68E+09 7,69E+08 5,77E+08 3,35E+08 2,14898

FLS.N FLOWSERVE ORD23,61 3,69E+09 1,27E+09 8,13E+08 2,83E+08 73900000 0,37198

DOV.N DOVER ORD5,15 3,27E+09 1,12E+09 1,06E+09 -6,3E+08 -7E+08 -12,6181

FDX.N FEDEX ORD 29,30153 7,14E+08 3,86E+08 3,25E+08 2,99E+08 1,33E+08 2,65181

CTAS.OQ CINTAS ORD82,93 6,85E+09 2,81E+09 9,22E+08 7,77E+08 4,75E+08 5,97002

RSG.N REPUBLIC SVCS ORD44,98 4,77E+09 1,44E+09 1,09E+09 9,74E+08 6,78E+08 4,16206

URI.N UNITED RENTALS ORD44,33 2,31E+09 1,19E+09 5,84E+08 4,83E+08 2,82E+08 3,00829

TDG.N TRANSDIGM GROUP ORD18,085 2,34E+09 1,24E+09 4,9E+08 4,5E+08 2,8E+08 0,93967

GWW.N WW GRAINGER ORD11,23666 1,13E+10 6,13E+09 3,65E+09 2,75E+09 1,49E+09 1,21139

COL.N ROCKWELL COLLINS ORD31,16 4,6E+09 1,45E+09 8,37E+08 7,04E+08 4,54E+08 3,30617 ROP.N ROPER TECHNOLOGIES INC13,21884 1,97E+10 6,95E+09 3,28E+09 1,9E+09 1,06E+09 2,38615

FAST.OQ FASTENAL ORD25,985 1,8E+10 1,21E+10 5,44E+09 4,07E+09 2,34E+09 2,26699

CSX.OQ CSX ORD 59,29 2,53E+10 1,19E+10 6,64E+09 5,22E+09 3,52E+09 4,89253

JCI.N JOHNSON CONTROLS INTERNATIONAL ORD4,96 2,02E+10 1,12E+09 -8,8E+08 -4,4E+09 -5,4E+09 -42,5118 KSU.N KANSAS CITY SOUTHERN ORD72,43055 1,49E+10 1,56E+09 1,76E+09 1,69E+09 9,29E+08 7,42647 UPS.N UNITED PARCEL SERVICE-CL B ORD37,55 5,13E+10 1,29E+10 7,64E+09 4,41E+09 3,55E+09 5,66491

UNP.N UNION PACIFIC U ORD35,85 1,71E+10 5,91E+09 3,05E+09 2,5E+09 1,69E+09 3,24453

MMM.N 3M ORD 3,9625 3,66E+09 2,61E+08 45300000 -1,7E+08 -1,4E+08 -0,93484

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LLL.N L-3 COMMUNICATIONS HOLDINGS ORD36,55 4,43E+09 1,67E+09 6,35E+08 3,76E+08 2,21E+08 2,74182

CAT.N CATERPILLAR ORD6,43 2,27E+10 6,55E+09 1,44E+09 -8,3E+09 -8,9E+09 -19,0641

ITW.N ILLINOIS TOOL ORD32,71 5,7E+09 2,34E+09 1,03E+09 8,09E+08 5,78E+08 3,89001

ALK.N ALASKA AIR GROUP ORD23,58 1,54E+10 4,19E+09 1,94E+09 1,3E+09 1,07E+09 3,25015

IR.N INGERSOLL RAND ORD27,96 2,03E+09 8,16E+08 3,09E+08 2,61E+08 1,49E+08 3,57211

LUV.N SOUTHWEST AIRLS ORD84,16 4,14E+10 4,57E+09 5,3E+09 5,05E+09 3,17E+09 7,73571

2008

RIC Company Name Price CloseTotal RevenueGross ProfitEBITDA Operating IncomeNet Income After TaxesDiluted EPS Excluding Extraordinary Items CHRW.OQ CH ROBINSON WW ORD64,5 7,32E+09 1,24E+09 5,37E+08 5,1E+08 3,24E+08 1,86314

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DPS - Common Stock Primary IssueFree Cash Flow Per Share - ActualTotal Debt Total EquityTotal AssetsInterest ExpenseReturn On Assets - ActualReturn On Equity - ActualEnterprise Value (Daily Time Series)

0,9 2,454 83591000 1,11E+09 1,82E+09 -6801000 19,782 32,44 8,53E+09

1,08 2,73 8,33E+09 5,9E+09 2,02E+10 4,55E+08 5,43 17,77 2,23E+10

1,62 -2,94 7,51E+09 -1,3E+09 5,38E+10 2,02E+08 4,99 -212,34 3,96E+10

0,93 1,69 3,99E+09 2,82E+09 9,48E+09 2,28E+08 1,18 3,93 7,29E+09

0,32 1,593 0 1,37E+09 2,1E+09 183000 14,328 21,751 6,39E+09

0,5 3,67 1,51E+08 2,67E+09 6,42E+09 18439000 10,47 25,17 7,24E+09

1,12 4,04 2,31E+09 9,09E+09 2,31E+10 1,29E+08 7,19 17,904 1,83E+10

1,1 3,907 8,38E+09 7,19E+09 3,55E+10 4,56E+08 7,867 38,848 3,25E+10

0 1,85 7,93E+08 6,7E+08 1,76E+09 33104000 8,64 22,67 5,08E+09

0,82 2,3 7,48E+09 4,85E+09 1,62E+10 7,37E+08 6,26 21 1,7E+10

0 0,28 1,23E+08 2,68E+09 3,56E+09 32002000 4,69 6,27 4,3E+09

1,2 2,43 5,15E+08 1,19E+09 2,71E+09 33800000 8,73 19,95 2,32E+09

0,4 1,57 6,34E+08 5,29E+08 1,79E+09 35337000 11,3 38,29 4,66E+09

0,6 2,26 6,68E+08 3,23E+09 8,52E+09 42000000 9,43 22,71 7,73E+09

0,68 1,61 9,54E+08 1,9E+09 4,05E+09 61464000 5,37 10,77 3,66E+09

1,4 6,61 4,02E+09 1,01E+10 2,52E+10 7,7E+08 8,73 24,65 2,56E+10

0,92 0,89 9,96E+09 2,37E+09 2,84E+10 1,33E+08 3,946 33,39 1,23E+10

0,12334 0,22333 3,35E+08 2,39E+08 2E+10 4,48E+08 4,7 13,8 1,68E+09

0,84 6,04 2,07E+09 5,26E+09 1,93E+09 11000000 9,28 18,32 1E+10

2,04002 -8,49 1,06E+10 1,17E+10 1,04E+10 98996000 2,58 6,81 1,7E+10

1,57 7,4 3,94E+09 1,19E+10 2,63E+10 4,44E+08 5,89 14,92 1,88E+10

0,10667 0,84444 1,11E+09 1,29E+09 3,87E+10 1,14E+09 9,15 21,53 4,53E+09

1,2 3,267 4,52E+09 9,11E+09 7,98E+11 1,19E+09 11,663 26,929 2,95E+10

1,35 5,173 1,15E+10 1,58E+10 3,06E+09 63652000 8,304 27,703 6,06E+10

0,33333 1,62667 5,73E+08 1,37E+09 2,1E+10 2,44E+08 11 32,19 4,78E+09

0,9 3,58 2,09E+09 3,79E+09 1,41E+09 13127000 8,83 18,32 7,48E+09

0,3 1,72 2,01E+09 1,45E+10 3,26E+09 71300000 7,13 12,54 1,84E+10

0,46 2,27 9,44E+08 2,25E+09 5,68E+10 5,73E+08 8,81 14,88 4,22E+09

0,72 0,63 7,7E+09 7,28E+09 4,02E+09 51293000 1,83 5,4 1,63E+10

0 0,703 3,35E+09 -2,9E+07 7,88E+09 1,3E+08 5,37 -775,86 3,45E+09

- 3,561 1,36E+09 6,54E+08 2,56E+10 98000000 6,209 21,42 3,13E+09

1,55 4,59 5,29E+08 2,03E+09 3,81E+09 52823000 13,63 23,54 6,34E+09

0,8 2,756 5,15E+08 1,41E+09 1,99E+10 1,32E+08 16,361 48,153 7,55E+09

0,3 4,32 1,27E+09 2E+09 4,19E+09 9000000 7,21 14,3 5,07E+09

0,395 0,555 0 1,14E+09 2,26E+09 92677000 21,84 24,93 5,26E+09

0,25666 0,95667 7,83E+09 8,05E+09 3,52E+09 14485000 4,52 14,75 2,04E+10

0,6 2,96 8,29E+08 2,27E+09 4,14E+09 21000000 10,14 20,33 4,69E+09

0,7127 -3,44607 4,26E+09 1,55E+10 3,97E+09 60819000 5,183 9,627 1,65E+10

0,49 1,03 8,93E+09 1,54E+10 1,3E+09 -930000 5,89 15,14 3,44E+10

2 4,33 6,78E+09 9,88E+09 2,63E+10 5,19E+08 14,25 36,99 4,57E+10

0 -13,1 8E+09 -2,3E+09 2,88E+10 3,96E+08 -7,68 60,65 6,21E+09

1,2 9,63 4,49E+09 5,86E+09 5,44E+09 1,39E+08 6,49 16,03 1,28E+10

1,62 1,24 3,55E+10 6,09E+09 3,19E+10 4,42E+08 5,25 53,8 5,45E+10

1,18 3,58 3,68E+09 7,66E+09 3,97E+10 5,11E+08 10,41 20,66 2,07E+10

0 -1,7095 1,84E+09 6,62E+08 2,58E+10 2,15E+08 0,091 0,665 1,32E+09

(55)

1,26 5,25 1,61E+09 1,71E+09 1,45E+10 2,9E+08 5,624 16,256 4,38E+09

0 -6,9 1,78E+10 8,74E+08 6,78E+10 1,43E+09 -1,117 -57,55 1,77E+10

1,16 24,064 1E+09 1,69E+09 1,52E+10 1,54E+08 13,236 36,002 5,1E+09

1 2,61 4,27E+09 6,32E+09 4,84E+09 81600000 6,66 17,56 1,16E+10

0,54 4,678 3,64E+08 5,76E+08 2,09E+10 2,43E+08 11,197 27,406 1,28E+09

1,83 8,537 3,81E+09 2,87E+09 1,41E+10 1,05E+08 9,622 112,286 3,41E+10

DPS - Common Stock Primary IssueFree Cash Flow Per Share - ActualTotal Debt Total EquityTotal AssetsInterest ExpenseReturn On Assets - ActualReturn On Equity - ActualEnterprise Value (Daily Time Series) 0,76 1,52 87443000 1,04E+09 1,81E+09 -1,4E+07 17,9 31,11 1,06E+10

(56)

Number of EmployeesTotal Current AssetsTotal Current LiabilitiesTotal InventoryTotal Common Shares OutstandingGrowth

15074 1,35E+09 6,98E+08 0 1,7E+08 13,86

42300 2,34E+09 3,04E+09 1,1E+08 4,91E+08 12

140800 2,6E+10 3,08E+10 1,56E+10 7,27E+08 8,4

26000 3,3E+09 1,55E+09 9,41E+08 3,51E+08 10,33333

16500 1,57E+09 6,7E+08 0 2,12E+08 14,05

56706 4,67E+09 3,16E+09 9,81E+08 1,82E+08 18,5

64000 7,42E+09 5,15E+09 3,25E+08 4E+08 12

131000 1,33E+10 1,23E+10 3,85E+09 7,35E+08 9,75

25000 2,24E+08 1,8E+08 0 85252880 20

25000 1,17E+10 5,65E+09 6,58E+08 3,63E+08 12,5

32800 1,39E+09 4,52E+08 80192000 1,98E+08 8

12600 1,14E+09 5,48E+08 3,59E+08 57441940 11,33333

24681 3,96E+08 4,08E+08 18214000 1,26E+08 7,76667

58600 4,71E+09 2,64E+09 1,78E+09 2,01E+08 10,33333

18400 1,03E+09 4,99E+08 4,17E+08 98276920 12

98600 1,2E+10 1,04E+10 2,03E+09 3,87E+08 17,5

37000 1,38E+10 4,78E+09 3,09E+09 2,42E+08 8,4

16100 8,24E+08 5,15E+08 2,82E+08 1,64E+08 8,75

56690 4,1E+09 2,18E+09 1,49E+09 1,68E+08 10

41500 8,15E+09 7,28E+09 3,24E+09 2,67E+08 13,33333

70000 8,24E+09 7,45E+09 1E+09 3,27E+08 12,33333

16900 9,55E+08 4,48E+08 3,5E+08 2,4E+08 7,16667

76500 9,33E+09 6,57E+09 2,35E+09 7,71E+08 8,16667

205000 2,45E+10 1,98E+10 8,34E+09 9,44E+08 14,25

17000 2,33E+09 1,61E+09 8,35E+08 1,66E+08 9,4

29000 2,63E+09 1,25E+09 6,36E+08 1,86E+08 5

117000 7,24E+09 5,37E+09 4,35E+08 3,11E+08 12

42000 1,28E+09 3,67E+08 2,39E+08 1,54E+08 9

35000 1,33E+09 2,57E+09 37100000 3,79E+08 11

14800 7,03E+08 2,71E+08 59000000 59890230 14,33333

9200 4,23E+08 88502000 1,44E+08 48600850 7,8

24400 2,14E+09 7,62E+08 1,01E+09 74781030 10,66667

29000 2,34E+09 1,74E+09 9,7E+08 1,59E+08 12,5

14236 8,58E+08 6,19E+08 1,86E+08 89721000 8

20565 9,75E+08 1,48E+08 5,64E+08 2,97E+08 10,2

24000 2,39E+09 2,4E+09 2,17E+08 1,17E+09 11,07143

17000 2,05E+09 9,91E+08 4,76E+08 1,34E+08 17,66667

121000 8,43E+09 5,69E+09 1,88E+09 4,31E+08 11,25

41992 2,81E+09 2,88E+09 4,5E+08 1,01E+09 17,25

91536 9,6E+09 5,84E+09 3,01E+09 6,94E+08 9,95

89800 4,87E+09 7,28E+09 4,56E+08 1,4E+08 16,75

31000 4,96E+09 2,71E+09 2,59E+08 1,19E+08 16,25

98400 3,19E+10 2,56E+10 8,78E+09 6,02E+08 8,71429

50000 5,75E+09 4,83E+09 1,77E+09 4,99E+08 12,3

20183 1,51E+09 1,36E+09 51900000 1,45E+08 10,1

(57)

57765 1,5E+09 1,19E+09 5,15E+08 78876030 9,475

86564 8,97E+09 1,11E+10 3,88E+08 6,95E+08 8,33333

22000 2,44E+09 1,3E+09 5,76E+08 1,43E+08 4,3

96000 4,8E+09 3,75E+09 1,55E+09 3,3E+08 14

12500 7,56E+08 5,19E+08 1,46E+08 40201710 9,33333

100000 1,07E+10 1,05E+10 1,9E+09 3,95E+08 12,66667

Number of EmployeesTotal Current AssetsTotal Current LiabilitiesTotal InventoryTotal Common Shares OutstandingGrowth 15074 1,39E+09 7,58E+08 0 1,71E+08 20,675

Förändring i Aktievärdering med Avseende på Nyckeltal

Förändring i Aktievärdering

med Avseende på Nyckeltal

En Multipel Linjär Regressionsanalys på

Finanskrisen 2008

THEODOR OHLSSON

LOVE WIKLAND

Shift in Stock Valuation with

Regards to Key Ratios

A Multiple Linear Regression Analysis of the

2008 Financial Crisis

THEODOR OHLSSON

LOVE WIKLAND

Shift in Stock Valuation

with Regards to Key Ratios

A Multiple Linear Regression Analysis of the 2008 Financial

Crisis

May 23, 2018

Abstract

F¨

or¨

andring i Aktiev¨

ardering

med Avseende p˚

a Nyckeltal

En Multiple Linj¨

ar Regressionsanalys av Finanskrisen 2008

May 23, 2018

Sammanfattning

Contents

1

Introduction

1.1

Background

1.2

Thesis Aim

1.3

Purpose

1.4

Related Works

2

Methodology

2.1

Literature Study

2.2

Data Collection

2.3

Modeling with Financial Ratios

3

Financial Framework

3.1

Fundamental Concepts

3.2

Valuation Methods

3.3

Financial Ratios and Multiples

3.4

The Financial Crisis

4

Mathematical Framework

4.1

Multiple Regression Analysis

4.2

Method of Least squares

4.3

Overfitting

4.4

Residual Analysis

4.5

Variable Selection

4.6

Model Performance

5

Results

5.1

Regression Analysis on Full Model

5.2

2008 Comparison

6

Discussion