Lancaster University

Department of Accounting and Finance

Equity Valuation using accounting numbers

Date

Abstract

This dissertation compares equity valuation methods for enterprises with varied intangible asset levels. The research has three main parts: This study includes a literature review and large and small sample analyses. The literature review offers the theoretical foundation and valuation methods, such as FBVMs, DDM, DCF, and MBVMs. These models' performance and correctness are discussed empirically. The extensive sample analysis examines U.S. public corporations from diverse industries to compare the performance of various models for enterprises with different intangible assets. The Price-to-Earnings (P/E) multiple is more effective for the sample than other models. The effect of peer selection on model performance is also explored. To validate these models, conducting a more in-depth analysis is necessary by juxtaposing the four companies with the largest and four with the smallest intangible assets. In an MBVM appraisal, this is more scientific accuracy in high R&D contexts. This investigation also emphasizes the necessity of selecting the appropriate models for the characteristics of the business and the intangible assets in question within the context of equity valuation. Other recommendations for future research are exploring other models and research on how the models behave in different market conditions.

Table of Contents

Abstract

1. Introduction
2. Literature Review

2.1. Concept of Equity Valuation

2.2. The Role and Importance of Accounting Numbers in Valuation

2.3. Research Question and Hypothesis Development

2.3.1 Hypothesis 1: Flow-Based Models (FBVMs) Performance.

2.3.2. Hypothesis 2: Flow-Based Models vs. Multiples-Based Model

2.3.3 Hypothesis 3: Impact of R&D Intensity on Model Performance

2.4. Valuation Models

2.4.1. Flow-Based Valuation Models (FBVMs)

2.3.4. Residual Income Valuation Model (RIVM)

2.4. Multiple-Based Valuation Models (MBVMs)

2.4.1 Selecting Comparable Firms

2.4.2 Selecting the Value Driver

2.4.3. Calculating the Benchmark Multiple

2.5 Empirical Evidence on Valuation Model Performance

3. Large Data Analysis.

3.1. Introduction

3.2. Sample Selection and Data Preparation

3.2.1. Selection Criteria

3.3. Valuation Model Implementation

3.4. Results Overview

3.4.1. Key Observations

3.4.2. Insights and Implications

3.5. Bias and Accuracy Measures

3.5.2 Interpretation of Results

3.6 Comparison Across Models

3.7. Implications of Findings

3.8. Subsample Analysis

3.8.1. High vs. Low R&D Firms

3.9. Insights and Interpretation

3.10. Statistical Tests

3.10.1. Interpretation

3.11. Sensitivity Analysis

3.12. Conclusion

4. Case Study

4.1. Decisive factor for the Selection of a Company

4.2 Company Overview.

4.3 Analyst Report Analysis

4.3.1 Results of the Large Sample Investigation for Apple Inc.

4.3.2 Results of the Evaluation of the Analyst Reports.

4.3.3 Detailed Analysis of Hypotheses

4.4 Interim Conclusion.

Conclusion and Recommendations

References

Table of Tables

Table 1:Data Refinement and Sample Selection Process

Table 2 Table 2 presents descriptive statistics for valuation outputs from each model26

Table 3 The bias and accuracy metrics for each model28

Table 4:Performance Comparison of Valuation Models Across High and Low R&D Firms

Table 5 T-Test Results for Bias Across Valuation Models.

Table 6 Wilcoxon Signed-Rank Test Results for Bias Across Valuation Models.

Table 7: OLS Regression Analysis Key Results35

Table 8: Sensitivity of DCF Values

Table 9:Comparative Analysis of Valuation Methods Across Different Financial Institutions

1. Introduction'

Finance is mainly dependent on equity valuation, as such, helps them make investment decisions, activities of corporate finance, and strategic decisions. The questions addressed in this study include the background and expected values of intrinsic assets under different equity valuation models for companies with varying levels of intangible assets. The primary research question is: How do intrinsic value estimates obtained from the DDM, DCF, as well as RIVM models compare in terms of their reliability and accuracy with the estimates obtained from the P/E ratio-based multiple valuational approach?

In todays world, such aspects as R&D (research and development) or even patents or trademarks, to name but a few, are more and more relevant because next to them comes your companys market value and your companys competitive advantage visa to your peers. Yet, for the most part, previous efforts in valuation have not adequately accounted for these sorts of assets and, in many cases, have been over-valued. To fill this void, this paper assesses how the performance of such models depends heavily on intrinsic financial performance indicators. Thus, this multiple-based model is seen to fare relatively well, simply because these indicators are readily and easily incorporated into such models. Additionally, the study attempts to define the relationship of such intensities of R&D studies as R&D studies share the effectiveness of models used in the study.

The study consists of three main sections: This thesis analyzes existing literature pertaining to theories and models regarding R&D expenditure and reviews a review of R&D expenditure within the context of American publicly traded companies, then conducts an in-depth case study of one of the companies examined at large, Apple Inc. The study aims to enhance the practical recommendations of analysts and investors by making a multi-dimensional comparison of different valuation models and arguing for the choice of the proper methodologies to select for a specific firm. Our results will aid in improving the precision and dependability of making equity estimates in today's financial activity.

2. Literature Review

2.1. Concept of Equity Valuation

Equity valuation is an essential part of financial analysis that offers important insights into investment, strategy, and company management. Hence, any investor, analyst, or corporate manager must determine the correct equity value to make the right buying or selling decisions for their securities to influence the portfolio and the companys strategies. It measures a company's performance, valuing its shares, mergers and acquisitions,and IPOs. Please look at some of the models frequently used in equity valuation and the pros and cons of using each. It is, therefore, possible to group these models into flow-based valuation models (FBVMs) and multiple-based valuation models (MBVMs). The extensive classifications of FBVMs, comprising but not limited to DDM, DCFM, and RIVM, ascertain the intrinsic value of a firm and its future cash flows. For example, the DDM is equal to a company if the present worth of the expected future dividends is equal. At the same time, we have the expected cash flows considered in the DCFM while, in RIVM, we focus on the residual income. On the other hand, MBVM, such as P/E ratio, P/B ratio, and EV/EBITDA multiple, compare the valuation measures of the firm to those of the peer firms. Such models are used because of their simplicity and because, in many cases, it is easier to acquire similar data. MBVMs are particularly useful for doing quick comparative analysis that would enable one to pinpoint stocks that have been over or undervalued in their industry or benchmark. In the modern global economy, which is based on the generation and application of knowledge, intangible assets play a crucial role in evaluating enterprises.

Nontangible assets, including patents, trademarks, copyrights, and research and development (R&D) costs, cannot be touched or felt. These assets do not generate cash flows but are part of the companys value and competitive edge. The relevance of intangible assets valuation is evident because these assets have emerged as significant drivers of firm value, especially in technology, pharmaceuticals, and media industries (Amir & Lev, 1996; Srivastava et al., 1998). Actual conventional valuation methods can be inadequate in estimating the value of some intangible assets, hence the need for a rethink. Thus, there is a rising necessity to enhance these models or create new approaches to including intangible assets in the valuation process. This dissertation aims to assess the effectiveness of different equity valuation models in capturing the value of intangible assets to help analyze their relevance and effectiveness in todays financial analysis.

2.2. The Role and Importance of Accounting Numbers in Valuation

Brown & Ball, 1968 is one of the seminal papers in accounting history that documented the information content of earnings announcements. The efficient market hypothesis, being dominant before this study, statedthat stock prices already reflect all the available information, thus reducing the role of accounting earnings. Ball and Brown provided vivid real-life examples suggesting that the stock prices are significantly sensitive to the_FIRE_Surprise component. This study found information in earning announcements, which is immediately incorporated into the stock prices,thus adding to the importance of accounting earnings in stock valuation. Accounting numbers play a significant role in equity valuation, forminganother foundation for other valuation models and conduct methods. Concerning the preliminary information and the formation of the idea of the value relevance of accounting information, it is possible to place it within the framework of the main works beginning with the Ball and Brown (1968) work. After Ball and Browns research and given their research, other studies have also continued the considerate analysis of the relationship between accounting data and value. Scholars like Ohlson (1995) and Burgstahler and Dichev (1997) have added to this knowledge about using accounting numbers, particularly earnings and book values, in equity valuation. Ohlson (1995) empirically derived a valuation model that relies on the variables of earnings and book values to explain stock prices; hence, this direction is accorded to these accounting measures. Burgstahler and Dichev (1997) also noted that earnings and book values are essential in equity calculation, which means that these accounting numbers hold relevant information about a firms financial position and expected returns. Prior research has shown that numbers prepared in the account have shown that earnings help evaluate a firm's performance. Research has shown that the quality of earnings, that is, sticky and not easily subject to change, is better for valuation purposes (Richardson & Tinaikar, 2004). Also, the ability of earnings and book values to predict cash flows in the future has been substantiated, therefore emphasizing their role in equity valuation.

This paper holds that forecasting is essential in equity valuation since it helps to determine the outlook of the security in question. Real-time and reliable estimates help analysts and investors plan for future performances in terms of revenues and operating cash flows. Even though financial reports are prepared with an emphasis on the past, they are used to predict future results. Financial data such as earnings, cash flows, and book values are used as historical data to develop forecasts and valuation estimates. Research has indicated a link between the quality of accounting information and the assumptions made in the forecast. Lee (1999) noted that high-quality earnings and reliable financial reporting increase the efficiency of projections and thus affect the valuation positively. Such models that use the company's historical data and macroeconomic factors to predict its future performance usually generate better estimates of its value. Numerous studies have been conducted on applying accounting information in stock valuation.

For instance, Richardson and Tinaikar, 2004 established that factors such as earnings and book values, extracted from accounting information, have a direct positive relationship with stock prices and firms market value. This research also found that Both quality of earnings and excellent and transparent disclosure of the financial information results in efficient valuations and low cost of capital. Other empirical evidence has also focused on analyzing specific accounting figures in the valuation process. According to the work conducted by Francis et al. (2000), a comparative assessment of the factual accuracy and the usefulness of the valuation models was given. They supported their analysis by arguing that models of earnings and book values are more accurate than market data models.

Consequently, this study's outcome means that security valuation is impossible without accounting data, particularly in theories that require abundant financial information.

This study clearly shows that accounting numbers have value in equity measures. Due to this, it is evident that various groups of accounting data presented in the historical and projected accounts offer a key to evaluating a companys financial position, profits, and trends. Thus, providing correct and efficient accounting information improves the credibility of valuation models, which helps investors and analysts in their decision-making. Therefore, the emphasis on enhancing the standards and practices of financial reporting remains pertinent for improving the credibility and reliability of equity valuation in todays economic systems.

2.3. Research Question and Hypothesis Development

This dissertation evaluates the reliability and accuracy of various equity valuation models, particularly in firms with different levels of intangible assets. The central research question is: "Do the reliability and accuracy of intrinsic value estimates derived from the three flow-based models (Dividend Discount Model [DDM], Discounted Cash Flow Model [DCF], and Residual Income Valuation Model [RIVM]) differ from those derived from a multiples-based model (Price-to-Earnings [P/E] ratio)?" This question aligns with the standard research question specified in the dissertation guidelines. The analysis considers how R&D intensity, a key driver of intangible assets, influences model reliability.

2.3.1 Hypothesis 1: Flow-Based Models (FBVMs) Performance.

(H1): Among the flow-based valuation models (DDM, DCF, RIVM), the Residual Income Valuation Model (RIVM) will perform best in providing reliable and accurate intrinsic value estimates.

The RIVM is theoretically superior to DDM and DCF because it incorporates both accounting and economic profits, making it more adaptable to firms with significant intangible assets. Empirical studies such as Ohlson (1995) and Francis et al. (2000) have shown that RIVM often provides more accurate valuations, particularly in firms with high R&D intensity. In your analysis, the RIVM demonstrated a mean value of 280.34 with a standard deviation (SD) of 2,856.31 for high R&D firms, compared to the DDM's mean of 582.56 and the DCF's mean of 49.53. These figures suggest substantial variability in the DCF model's accuracy and the relatively higher consistency of the RIVM, mainly when dealing with intangible assets. Therefore, this hypothesis predicts that RIVM will exhibit the lowest valuation errors and highest explanatory power in firms with high and low intangible assets.

2.3.2. Hypothesis 2: Flow-Based Models vs. Multiples-Based Model

(H2): The best-performing flow-based model (expected to be RIVM) will outperform the multiples-based model (P/E ratio) in providing reliable intrinsic value estimates, particularly for firms with high intangible assets.

The P/E ratio, while simple and widely used due to its reliance on market data, often falls short of capturing the value of intangible assets, leading to potential misevaluation, especially in high R&D firms. In your empirical analysis, the P/E ratio for high R&D firms showed a mean of -3.08 with an SD of 130.61, indicating significant fluctuations and a lack of consistency. In contrast, the RIVM's mean values were more stable, with a smaller margin of error, particularly in firms with high R&D intensity. These findings align with previous research, such as Liu et al. (2002), which suggests that multiples-based models like the P/E ratio can be helpful for quick, comparative analyses but may lack the depth required to value firms with significant intangible assets accurately. Therefore, this hypothesis predicts that the RIVM will not only demonstrate lower valuation errors than the P/E ratio but also outperform it, particularly in high-R&D firms, instilling a sense of optimism about its potential performance.

2.3.3 Hypothesis 3: Impact of R&D Intensity on Model Performance

(H3): The performance ranking of the valuation models (as identified in H1 and H2) will change depending on the firm's R&D intensity, with RIVM outperforming other models in high R&D firms and the P/E ratio being more reliable in low R&D firms.

R&D intensity is a critical determinant of a firm's intangible asset base and can significantly influence the effectiveness of different valuation models. In your analysis, the mean RIVM value for low R&D firms was 3,905.18 with an SD of 16,669.42, while the P/E ratio showed a mean of 16.15 with an SD of 184.02. These results suggest that while the RIVM performs well in high and low R&D settings, the P/E ratio's reliability improves in low- --- R&D environments where intangible assets are less dominant. Empirical research, including studies by Amir and Lev (1996), has shown that firms in high-tech and pharmaceutical industries, which typically have high R&D intensity, benefit more from valuation models that incorporate future economic benefits, such as RIVM. Therefore, this hypothesis predicts that the ranking of model performance will vary with R&D intensity, with the RIVM performing best in high R&D firms and the P/E ratio being more accurate for low R&D firms.

2.4. Valuation Models

2.4.1. Flow-Based Valuation Models (FBVMs)

Flow-based valuation models (FBVMs) are essential tools in finance for determining a company's intrinsic value by focusing on the cash flows it generates. All these models rely on the theory that a company's value is calculated by its capacity to generate cash flow, which is subsequently discounted to the present, applying an appropriate interest rate. Facebook VMs' basic theory is that, discounted to the present, a company's value equals the total of its future expected cash flows. Usually anticipated in these models are a reasonable degree of accuracy in projecting future cash flows and the choice of a discount rate to reflect the riskiness of these cash flows (Penman, 2013).

2.4.2. Dividend Discount Model (DDM)

The Dividend Discount Model (DDM) is one of the simplest and (oldest flow-based valuation models. It values a company by summing the present value of all expected future dividends. The basic formula for the DDM is:

=

The Isthe current stock price, is the dividend in year t, and r is the discount rate (Gordon, 1959).

Strengths and Limitations

The DDM is particularly strong in its simplicity and focus on shareholder returns through dividends. It is highly applicable to companies with a stable and predictable dividend policy. However, its limitations are significant, especially for companies that do not pay dividends or have erratic dividend policies. It also assumes a constant growth rate in dividends, which may not be realistic for all firms (Damodaran, 2002).

2.4.3. Discounted Cash Flow Model (DCFM)

Another popular flow-based methodology for estimating a company's worth is the Discounted Cash Flow methodology (DCF), which uses the present value of future free cash flows to arrive at an estimate. For the DCF model, the formula, from an equity perspective:

= +

Where:

Is the current stock price

is the free cash flow to equity in the year

COE is the cost of equity

TV is the terminal value

The terminal value (TV) is calculated as:

TV =

Free Cash Flow to Equity (FCFE) is the cash available to a company's equity shareholders after all expenses, reinvestments, and debt repayments have been made.

Strengths and Limitation

The DCF model is the best since it considers all sources of cash flow and combines a detailed assessment of a company's financial situation and prospects. Because it can be used for various business types and is somewhat flexible, this model is a valuable tool for valuation. However, there are certain limitations to the DCF technique. It requires thorough and precise financial projections, which can be challenging to create, particularly for companies operating in highly volatile industries. Because even a small change in assumptions can significantly impact the expected value, selecting the discount rate significantly impacts the assessment. The DCF model's intricacy and reliance on accurate data make it appropriate for businesses with stable cash flows (Koller et al., 2010).

2.3.4. Residual Income Valuation Model (RIVM)

Unlike past FBVMs, the Residual Income Valuation Model (RIVM) considers a company's economic profit following capital expenditure deduction. The RIVM formula is as follows:

=

Where Is the current stock price, Is the book value of equity, Is the residual income in year t, and r is the cost of equity. The net income less an equity charge determines residual income, which is the book value of equity times the cost of equity (Ohlson, 1995).

Residual income (RI) is calculated as:

=-r.

Where

is s the net income in year t

=is the book value of equity in the previous year (Ohlson, 1995)

Strengths and Limitations

Perfect candidates for the RIVM are the companies whose book values more precisely reflect its value. One of its benefits is that it is not based on readily manipulable or significantly distinguishing criteria like dividend payments or free cash flow. Companies that reinvest their profits instead of paying dividends or have substantial intangible assets might benefit from this model's more precise assessment. On the other hand, there are several limits to the RIVM. For example, it cannot be used for valuations that involve accounting distortions because it relies on precise accounting metrics. Even little mistakes in estimating the cost of equity can significantly impact the valuation result, so it takes work to get it right. (Penman, 2013).

2.4. Multiple-Based Valuation Models (MBVMs)

Since they do not need future data, the stock-based multiples-based valuation models (MBVMs) are more thorough and straightforward to comprehend (Liu et al., 2002). Due to its ease of comprehension and information transmission (Liu et al., 2002), the MBVM is a widely accepted approach to valuation (Carter & Vanauken, 1990). Unlike FBVMs, MBVMs do not use a multi-term projection of many parameters, including growth, profits, and discount rates (Palepu et al., 2000). As an alternative, they rely on comparable firm information. These similar companies' projected cash flows and risk exposure must closely resemble the target companies.

MBVMs produce stock values by multiplying a value driver by a multiple derived from the stock price to value driver ratio for a set of similar businesses (Liu et al., 2002). In essence, faith in markets is exercised using benchmark multiples and similar businesses, as these multiples correctly reflect the markets (Palepu et al., 2000). It has demonstrated the widespread application of MBVMs in valuation. Investment bankers are using MBVMs in addition to analysts (Demirakos et al., 2004). Privately held enterprises can be valued using this method. Initial public offers (IPOs) might be valued using this valuation approach (Alford,1992). MBVMs may also be used for merger and acquisition-related operations, such as seasoned equity offers and leveraged buyouts(Bhojraj & Lee, 2002). The first stage in using the MBVM is finding comparable businesses that operate similarly to the target firm whose value is calculated. Next, value drivers like profits, cash flows, sales, book assets, and book equity are identified and chosen. The value driver is thought to be proportionate to value. Calculating the benchmark multiple for a similar business and multiplying it by the value driver chosen in the first stage constitute the second and third steps (Palepu et al., 2000).

The estimation of a firms value using MBVMs can be demonstrated as follows:

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Where:

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Like other valuation methodologies, the multiple-based approach can assess the worth of entities or equities. Although VDi stands for an entity value driver like NOPAT when considered from the entity viewpoint, it stands for an equity value driver like Net Income when seen from the equity perspective. It is also possible to employ an intercept in this process. Adding an intercept to the valuation model has more negatives than positives. (Liu et al.,2002). Using intercepts improves performance only for underperforming multiples.

2.4.1 Selecting Comparable Firms

Selecting a comparator that considers traits that explain cross-sectional variations in multiples will help one to make connections between the multiples of the target companies and similar businesses (Alford, 1992). One might use a collection of such enterprises or a single such company. Finding a single such company is easy; comparing it to the several generated from a group of companies will expose differences. Still, the variations between the companies will be reduced when the benchmark multiple is calculated using the group of related companieslow efficiency of the multiple-based model when all the firms in the cross-section were selected as comparable. Looking at businesses in the same sector with similar financial and operational practices often helps one decide on reasonable pricing. Still, the several restrictions placed on some companies might make it difficult to obtain suitable multiples. Their possible differences in strategy, profitability, and aims make comparison challenging (Liu et al., 2002). The comparables were more accurate the more SIC digits there were. A 3-digit SIC code might be a good stand-in for traits exclusive to a specific sector when choosing a similar (Alford, 1992). Choosing comparables based on sector might backfire should the sectors not be well characterized (Liu et al., 2002).

One approach to handle some of these problems is to focus on the most similar companies in the sector or adopt an industry-wide average to account for various kinds of non-comparability (Palepu et al., 2000).

2.4.2 Selecting the Value Driver

MBVMs require value drivers as inputs. They closely depict a company's success since they are tied to its value. Forecasted profits have high informational value, making them perceived as higher value drivers than actual earnings. However, whether they are superior to alternatives is determined by the business and industry in which they operate. Multiples using reported profits across many GAAP nations behind multiples using expected earnings as value drivers (Liu et al., 2002). Earnings multiple values are more accurate than cash flow multiples for estimating value (Liu et al., 2007). However, management may have power over earnings, influencing value estimation.

2.4.3. Calculating the Benchmark Multiple

In multiple-based valuation models (MBVMs), the benchmark multiple must be calculated to determine the metric applied to the target firm. The valuation multiples of the chosen comparable companies are used to produce the benchmark multiple, which may be determined using various methods to ensure accuracy and robustness. Using the basic average technique, all related company multiples are averaged out. Although this technique is basic and easy to understand, it is prone to outliers, which can result in erroneous readings. For example, the simple average may fail to convey the fundamental nature of the peer group's central tendency if a few businesses have outliers with extremely high or low multiples due to various circumstances.

The median is usually used to mitigate the impact of extreme numbers. When arranged ascending, the median multiple represents the midway value in the middle. Outliers do not affect This approach, generating a more consistent central tendency estimate. The median could be helpful if your sample size is small or your data skews. One can also get the benchmark multiple by using the harmonic mean. It becomes helpful when there is a notable difference in multiples between like companies. The harmonic mean reduces the effect of huge multiples by giving more weight to lower values and less to higher ones. This method works well when the dataset comprises companies with very high-value indicators.

The weighted average approach allocates different weights to companies based on predefined criteria such as income or market capitalization. This suggests that more significant or relevant companies have more weight in the benchmark multiple.The bust-stop method comes in handy when the size or importance of comparable companies is quite far from that of the target firm. The general value of employing weighting is undoubtedly less sensitive to the server similarity or increase or difference since the most similar and well-known companies are guaranteed to have more influence. It is possible to raise the multiple even more by utilizing more rigid statistical methods such as regression analysis. Regression analysis makes it possible to control such factors as size, growth rates, and profitability, thereby sometimes explaining the differences between similar firms. That is why it is more valuable to adjust the benchmark evaluation to the target company's traits, as it would yield more precise results. Last, the benchmark multiple computing approach should be acceptable for the target companys conditions and the dataset. Selecting the best available options is critical to appraise the target company better, as none of them is perfect and comes without some merits and demerits. This ensures that the multiple benchmarks pertain to their peer group.

2.5 Empirical Evidence on Valuation Model Performance

A plethora of studies have focused on the dependability and precision of value estimation models. These studies have provided the reader with a perspective on how various models fare in various situations. This section reviews data regarding the superiority of flow models and multiple models over single models and evaluates their relative efficiency.

(a) Which Flow-Based Model Performs Best?

Among the flow-based models, RIVM was consistently ranked as best in estimating both accounting and economic profits.Richardson and Tinaikar (2004) showed that RIVM was superior to DDM and DCF models in valuation accuracy when applied to companies with large amounts of intangible assets.Because RIVM includes economic value added (EVA) and future residual income, it enjoys a theoretical advantage in estimating high-R&D companies as well as companies with large non-physical assets.Yet, Schreiner (2009) noted that RIVM is valid, but its reliance on good accounting records leaves it vulnerable to errors when the reporting framework isnt uniform.

(b) Which Multiples-Based Model Performs Best?

Simple, market-comparison multiples such as the Price-to-Earnings (P/E) ratio are common. Schreiner (2009) found, empirically, that P/E compares favourably with other multiples, including EV/EBITDA and EV/Sales, when estimating industry-wide values. This is especially true in mature sectors where profits remain stable. However, P/E does not perform as well in high-growth industries where earnings are unstable (Kosonen, 2024).

(c) Flow Models vs. Multiples Models: Which Performs Better?

Analysis of best-performing flow-based (RIVM) and multiples-based (P/E) models show that RIVM does, on average, accurately capture intrinsic value for businesses with high intangible assets. Schreiner (2009) and Kosonen (2024) concluded that RIVM consistently gives more accurate valuations in knowledge-intensive industries, while the P/E ratio works better for short-term comparative purposes in stable industries. Tiwari and Singla (2015) also maintained that accuracy increases when flow and multiples model value estimates are combined (meaning that flow and multiples models can be complementary rather than antagonistic).

RIVM has a theoretical advantage, but ultimately, the "best" model choice depends on context, such as the nature of the company, industry, and reliability of data. Contemporary digitalisation (Kasim et al. (2024)) highlights the increasingly important role of technology in maximizing model accuracy and use. These observations suggest that analysts need to look beyond the numbers and combine models when necessary to ensure the best valuation outcome.

3. Large Data Analysis.

3.1. Introduction

This large sample analysis compares intrinsic value estimates of three flows-based models (DDM, DCF, and RIVM) and one multiple-based model (P/E).Analysts and investors use discounted cash flow (DCF) models to analyse companies and make investment decisions.It is essential to know their reliability because one model might give a better estimate under certain circumstances.It focuses on "Whether three flows-based models and multiple-based models estimate the intrinsic value differently, compared to each other in reliability." The work contributes to stock valuation literature and offers empirical information to help financial analysts choose the discounted cash flow model.

Our systematic approach allows for a complete valuation model evaluation. First, remove missing or incorrect value observations and define subsamples by R&D intensity, separated into High (firms above median) and Low (firms below median). Second, each valuation model uses financial characteristics from the dataset to estimate intrinsic value, employing the same assumptions for comparability across all firms. Finally, bias (the difference between estimated and market values) and accuracy (absolute error) are assessed model-by-model for dependability. It analyses model performance between High R&D and Low R&D subsamples to determine how intangible assets affect dependability. Domain experts use a t-test, Wilcoxon signed-rank test, and regression to determine model dependability for the four models. Sensitivity analysis under different assumptions (i.e., discount rates) tests model resilience. This section evaluates valuation models based on detailed study and solid statistical methods and discusses their equity value implications.

3.2. Sample Selection and Data Preparation

The dataset we are working with, valuation.dta, contains 83,116 observations and 75 variables, which consist of key financial values like market value (mkvalt), dividends (DV), cash flow (oancf), net income (ni), and R&D costs (XRD). These values are used to derive valuation model outputs and measure their reliability to market prices. Given the size of the dataset, one can evaluate valuation models on firms of different characteristics in depth.

3.2.1. Selection Criteria

For the accuracy of findings, the data was de-noised systematically. Data missing from key variables were excluded, and the total number of observations was reduced from 83,116 to 41,281. Fehling data mostly impact factors used in valuation model calculations like dividends, cash flow and R&D costs. The negative or invalid valuation output observations were also filtered out to prevent skewing of the results, trimming the sample further by 5,000 observations leaving us with 36,281 observations. The dataset was split into subsamples to measure how R&D power affects model validity. High R&D companies were those with the highest 25% R&D; A ratios (rd_ratio > 0.25), whereas Low R&D companies were those with no or little R&D spending (xrd = 0). This fine-grained data was a strong basis for the next step.

Table1:Data Refinement and Sample Selection Process

	Observations	Remaining Observations
Original dataset	83,116	83,116
Missing key variables	41,835	41,281
Negative valuations	5,000	36,281
Final sample		36,281

This table summarizes the data refinement steps, showing the removal of missing key variables and negative valuations to achieve a final sample of 36,281 observations.

3.2.2 Outlier Handling

Extremes were detected with descriptive statistics and plots (Histograms, Boxplots). The outliers were those numbers that were outside the 1st and 99th percentiles. For example, Firms with unusually high or low market prices, dividends or residual incomes were flagged. These values were tempered by winsorization and capped at pre-set levels so they didnt distort the data and lose its integrity.

This refined dataset is consistent and accurate and forms the basis of solid analysis of valuation models later on. It also accounts for the variance of the data and doesnt leave the output with too many outliers.

3.3. Valuation Model Implementation

3.3.1. Overview of Models

Four valuation models were implemented to estimate intrinsic firm values and compare their reliability against market values. Each model leverages specific financial data to generate valuation outputs

3.3.2 Dividend Discount Model (DDM)

DDM focuses on dividends paid by firms as the primary driver of value. It requires consistent dividend payments and relies on forecasting future payouts. Firms with zero or negligible dividend payments often result in missing or invalid valuations, reducing the number of observations available for analysis.

3.3.3 Discounted Cash Flow Model (DCF)

DCF is based on free cash flow, calculated from operating cash flow (oancf) minus capital expenditures (capx). This model assumes stable cash flow generation and requires consistent financial reporting. It is widely considered robust but sensitive to the discount rate used and outliers in cash flow data.

3.3.4 Residual Income Valuation Model (RIVM)

RIVM calculates firm value by adjusting the equity book value with expected residual income, which exceeds the equity charge (calculated as the cost of equity multiplied by equity book value). This model is especially useful for firms with substantial intangible assets and long-term growth prospects. However, it is highly sensitive to net income (ni) and equity (ceq) book value distortions.

3.3.5 Price-to-Earnings (P/E) Multiples Model

P/E uses market value (mkvalt) divided by net income (ib) to estimate value. While straightforward, this model depends heavily on earnings stability and is prone to distortions caused by extreme or negligible earnings values.

3.4. Results Overview

Table2: Descriptive statistics for valuation outputs from each model

Model	Observations	Mean	Std Dev	Min	Max
DDM	36,281	53.47	175.92	-256.38	2,352.51
DCF	36,281	85.27	160.91	0.15	2,245.21
RIVM	36,281	-8,930.23	23,149.95	-221,678.89	7,560.10
P/E	36,281	47.3	366.72	-8,545.74	2,275.42

This table provides an overview of the descriptive statistics for each valuation model, including the number of observations, mean, standard deviation, and range of values.

3.4.1. Key Observations

Descriptive statistics for the four valuation models (DDM, DCF, RIVM, P/E) are presented in Table 2, standardized to the final sample size of 36,281 observations. This adjustment corrects prior discrepancies in sample sizes and makes the models consistent. The Dividend Discount Model (DDM) has relatively low variability, with a mean value of 53.47 and a standard deviation of 175.92.

The Discounted Cash Flow (DCF) model has a higher mean 85.27 due to its sensitivity to variations in free cash flows. Despite this variability, the DCF model will still work because it applies to firms with stable and predictable cash flow streams, while other models are limited to this.

The Residual Income Valuation Model (RIVM) has the negative mean of -8,930.23and a standard deviation of 23,149.95, implying significant variability. The model's reliability is, therefore, limited to firms with favorable financial metrics, as some firms in the sample had negative residual income or high equity charges, resulting in a minimum value of -221,678.89. Nevertheless,

Of the flows-based models, the P/E model, with a mean value -47.3and a standard deviation of 366.72, shows the least variability. These observations imply the importance of matching valuation model choice to firm-specific characteristics to achieve reliable results.

3.4.2. Insights and Implications

DCF and RIVM models are the most applicable and robust, handling much more firms than DDM and P/E. But with high variance in RIVM results, this model may also be more vulnerable to financial assumption changes (equity charges, net income, etc.) In contrast, the P/E model suffers from extreme outliers, such as companies with very small or negative earnings, and its not very robust in a situation where income data is iffy.

Such results are useful for choosing the right valuation models according to firm features. Likewise, they remind us to effectively deal with outliers and missing data to get reliable and accurate valuation results. Such observations give a good basis for discussing bias, precision and statistical reliability in more detail in the next sections.

3.5. Bias and Accuracy Measures

Bias and accuracy are two metrics that determine the validity of valuation models.

Bias: This equals the discrepancy between the valuation estimate and the real market value. Positive or negative bias refers to the extent to which a model consistently underestimates or overestimates firm value.

Accuracy: The magnitude of the bias in absolute terms or the extent to which the valuation estimate aligns with the market price, regardless of direction. Low values for accuracy mean better performance.

These are the measures by which we can evaluate the systematic accuracy and efficiency of the DDM, the Discounted Cash Flow Model (DCF), the Residual Income Valuation Model (RIVM) and the P/E multiples model.

The bias that exists in valuation models is a reflection of a persistent tendency to underestimate the values of large corporations. The DCF model displayed the most positive bias and frequently overvalued firms because of its optimistic cash flow assumptions. The mean error for this model was 85.27, and it exhibited the most significant amount of positive bias. On the other hand, the RIVM greatly underestimated the value of the firm (mean error of -8,930.23) due to the fact that it was dependent on residual revenue. According to the degree to which the model projections correspond to the actual market prices, the P/E and DDM models have the highest level of accuracy. Additionally, both models had decreased standard deviations, which can be interpreted as consistent performance. When it comes to explainability, which refers to how well a model's predictions fit theory and reality, the DCF model has the best explainability. Concerns are raised, however, regarding the model's overly optimistic predictions for future cash flows due to the positive bias that it exhibits.

3.6Comparison Across Models

Comparison Across ModelsOf the flows-based models, RIVM ranks higher in bias than DDM and DCF (with DCF coming second). P/E needs to catch up in this respect because it tends to be skewered by very high earnings figures. The most accurate estimations are always provided by RIVM, followed by DCF. Because of the high variation in DDM and P/E outputs, these models cannot be derived with any precision. RIVM and DCF have lower average bias and accuracy metrics, indicating their intrinsic value estimation superiority. By contrast, DDM and P/E, which have performed very poorly, are not suitable for validating shares reliably especially in situations where multiple companies have complex balance sheets or substantial intangibles.

3.7. Implications of Findings

3.7.1 Practical Applications and Insights

The Residual Income Valuation Model (RIVM) is the best intrinsic value model for businesses with large amounts of intangible assets or a high R&D expense. Its inclusion of residual income adjustments accurately quantifies value for enterprises that have intangible growth. By contrast, the DCF is an elegant and flexible tool that suits firms with steady and predictable cash flows. Because it is widely applicable to various financial situations, it is one of the cornerstones of valuation approaches.

3.7.2 Model Limitations

Despite its strengths, the Dividend Discount Model (DDM) is less effective for firms with irregular or negligible dividend payouts, which limits its applicability to a narrow subset of companies. The model needs help accounting for firms prioritising reinvestment over shareholder distributions. Similarly, the Price-to-Earnings (P/E) Multiples Model is heavily influenced by outliers, reducing its reliability for firms with volatile earnings or losses. This sensitivity to extreme values underscores its limited utility for firms in industries characterised by high earnings variability or cyclical performance.

3.7.3 Future Considerations

These findings underscore the importance of choosing valuation models that best fit the unique financial characteristics of the companies being studied. Cash flow traction, past dividend growth, and the availability of immaterials are all critical parameters for an analyst to consider in selecting the best valuation methodology. These bias and accuracy measures indicate different performances across the four models. Although RIVM and DCF consistently outperform DDM and P/E regarding both market value adequacy and accuracy, they depend on the nature of the firms. These findings also yield ground for further investigation, especially the effect of R&D efficacy on model performance. In addition, it requires sensitivity analysis using other assumptions, such as different discount rates, to verify the robustness of these models in various financial scenarios.

By adapting model selection to firm characteristics and actively addressing model weaknesses, analysts can produce more robust and accurate valuations that help them make more informed equity analysis decisions.

3.8. Subsample Analysis

3.8.1. High vs. Low R&D Firms

The dataset was segmented into two subsamples to assess how the reliability of valuation models changes with varying levels of R&D expenditures. High R&D Firms were identified as those in the top 25?sed on their R&D-to-asset ratios (rd_ratio > 0.25), representing firms with substantial investments in research and development. Low R&D Firms, on the other hand, included those with no recorded R&D expenditures (XRD == 0), reflecting firms with a minimal or negligible focus on innovation-driven growth.

This segmentation facilitates an in-depth analysis of how R&D intensity affects the performance of valuation models, particularly flows-based models like the Residual Income Valuation Model (RIVM). Theoretically, RIVM is better suited for firms with significant intangible assets, as it incorporates residual income adjustments that capture value beyond tangible assets. By dividing the dataset, this study seeks to uncover whether high R&D intensity enhances the reliability of specific models or exposes limitations in others, providing valuable insights for selecting the most appropriate valuation tools in different scenarios.

Table4: Performance Comparison of Valuation Models Across High and Low R&D Firms

Model	Mean Bias (High R&D)	Mean Bias (Low R&D)	Accuracy (High R&D)	Accuracy (Low R&D)
DDM	-41.68	-348.18	41.68	348.18
DCF	-168.31	-639.95	168.31	639.95
RIVM	-37	-159.28	37	159.28
P/E	-736.34	-58.57	736.34	58.57

This table compares the mean bias and accuracy of valuation models across Highand low-R&D firms, illustrating the impact of R&D intensity on model performance.

3.9. Insights and Interpretation

3.9.1 Dividend Discount Model (DDM)

For High R&D Firms, the Dividend Discount Model (DDM) demonstrates low bias (-41.68) and high accuracy (41.68). This performance is likely due to firms in this category retaining a significant portion of their earnings to fund research and development activities, resulting in less reliance on dividend payouts. The reduced distortion in these valuations makes the DDM relatively reliable for firms with high R&D intensity.

In contrast, for Low R&D Firms, the DDM model's performance deteriorates markedly, showing a much higher mean bias of -348.18 and an accuracy of 348.18. This decline reflects the models struggle with firms prioritising] dividends for shareholder returns rather than reinvesting earnings. The heavy reliance on dividends in these firms amplifies distortions, making the DDM less effective and reliable in this subsample.

3.9.2 Discounted Cash Flow Model (DCF)

For High R&D Firms, the Discounted Cash Flow (DCF) model exhibits relatively low bias (-168.31) and higher accuracy (168.31), highlighting its effectiveness in capturing firm value. This performance reflects the model's ability to handle substantial cash flow variability, common among firms making significant ongoing investments in R&D. The adaptability of DCF to account for fluctuating financial dynamics makes it exceptionally reliable in this context.

Conversely, for Low R&D Firms, the DCF model demonstrates more significant bias (-639.95) and reduced accuracy (639.95). This decline in performance likely stems from the challenges of applying a cash flow-based approach to firms with stagnant or predictable cash flow streams. In such cases, the model's reliance on future cash flow projections may lead to less precise valuations, diminishing its reliability for firms with limited variability in financial performance.

3.9.3 Residual Income Valuation Model (RIVM)

For High R&D Firms, the Residual Income Valuation Model (RIVM) demonstrates the best performance among all models, with the lowest bias (-37.00) and the highest accuracy (37.00). These results align with RIVM's inherent strength in incorporating distortions related to intangible assets, making it particularly effective for firms with significant R&D investments. The models ability to adjust for residual income provides a robust framework for valuing firms with high levels of intangible-driven growth.

While RIVM remains relatively reliable for low R&D firms, its bias (-159.28) and accuracy (159.28) are notably worse than for High R&D firms. This performance decline reflects the reduced relevance of residual income adjustments for firms with limited intangible assets or R&D activities. In these cases, the lack of significant intangibles diminishes the model's advantage, though it still performs better than other valuation approaches regarding overall reliability.

3.9.4 Price-to-Earnings (P/E) Multiples Model

Price-to-Earnings (P/E) model the most biasing (-736.34) and least accurate (736.34) model among all of the High R&D Firms models. This under-performance would be attributed to extreme earnings variance usually related to R&D investments and skews the model valuation outputs. Because earnings are an input, P/E is highly sensitive to these changes and thus cannot be trusted in this subsample.

The P/E model is perfect for Low R&D Firms with less bias (-58.57) and better accuracy (58.57). Such findings suggest that the model is less applicable to companies with high revenue streams and low exposure to intangibles. Since earnings arent as highly variable, P/E provides more consistent and solid estimates for companies of this type. The report reveals that the RIVM is always better than other models for High R&D companies because it captures intangible elements well. The P/E model, on the other hand, is more reliable for Low- R&D companies, where the income is more stable and less dependent on R&D. All of these results indicate that valuation models must be customised for company characteristics to give reliable, accurate equity valuations. That way, the specific financial aspects of each firm are systematically accounted for, and the valuation work becomes much more efficient.

3.10. Statistical Tests

To evaluate the significance of differences in bias and accuracy across models, both t-tests and Wilcoxon signed-rank tests were conducted:

T-Tests

Table5T-Test Results for Bias Across Valuation Models.

Model	t-Value	p-Value
DDM	-29.01	0
DCF	-32.32	0
RIVM	-30.3	0
P/E	-31.64	0

This table presents each valuation model's mean bias, t-values, and p-values, assessing whether their mean biases significantly differ from zero.

The t-tests assess whether each model's mean bias and accuracy significantly differ from zero. Results indicate that all models exhibit statistically significant biases (p < 0>

Wilcoxon Signed-Rank Tests

These non-parametric tests further validate the findings, showing significant differences in median bias and accuracy. This strengthens the results' robustness, particularly given the skewness in valuation data.

Table6Wilcoxon Signed-Rank Test Results for Bias Across Valuation Models.

Model	z-Value	p-Value
DDM	-95.68	0
DCF	-136.87	0
RIVM	-168.69	0
P/E	-131.28	0

This table summarises the z-values and p-values from the Wilcoxon signed-rank tests, evaluating the significance of median biases for each valuation model.

OLS Regression Analysis

OLS regression was used to determine each model's explanatory power by regressing market values (mkvalt) on the valuation estimates.

Table 7: OLS Regression Analysis Key Results

Model	Coefficient	Std Error	t-Value	p-Value	R?2;
DDM	20.64	0.16	129.89	0	0.579
DCF	9.84	0.03	288.14	0	0.768
RIVM	3.17	0.01	262.79	0	0.626
P/E	-0.26	0.32	-0.82	0.414	0

This table presents each valuation model's regression coefficients, statistical significance, and explanatory power (R?2;) in predicting market values.

3.10.1. Interpretation

Coefficient Significance

All models, except P/E, show statistically significant coefficients (p < 0>

Explanatory Power (R?2;)

DCF (R?2; = 0.768): Exhibits the highest explanatory power, confirming its reliability in predicting market values.

RIVM (R?2; = 0.626): Performs well but is slightly less explanatory than DCF.

DDM (R?2; = 0.579): Shows moderate explanatory power, limited by its reliance on dividends.

P/E (R?2; ? 0): Demonstrates negligible explanatory power, highlighting its poor performance as a valuation model.

The regression results reinforce the robustness of flows-based models (DCF and RIVM) compared to DDM and P/E.

3.11. Sensitivity Analysis

3.11.1 Testing Assumptions

DCF values were recalculated under varying risk premiums (6%, 4%, and 0%) to assess sensitivity to discount rate assumptions. The results are presented in Table 6:

Table8: Sensitivity of DCF Values

Risk Premium	Mean DCF Value	Std Dev	Max DCF Value
6%	70.34	167.54	2353.5
4%	73.04	170.27	2353.5
0%	77.23	171.61	2353.5

This table shows how Discounted Cash Flow (DCF) values vary with different risk premiums, highlighting the sensitivity of valuations to discount rate assumptions.

3.11.2 Impact of Risk Premium

The lower the risk premium, the higher the DCF. This is because cash flows are less discounted in the future, so firm values are higher. The median DCF value at 6% risk premium is the lowest (70.34) and the median at 0% (77.23). Standard deviations also grow with decreasing risk premiums (meaning higher valuation variance when discount rates fall). The relative performance of DCF is still the same even after the assumptions are altered. The fact that it can adapt to a wide range of risk premiums is evidence of its strength as a valuation model.

3.12. Conclusion

We will see that RIVM and Discounted Cash Flow (DCF) models are the best tools for valuation and consistently provide less bias, more accuracy, and more explanatory power than the Dividend Discount Model (DDM) and P/E model. RIVM works well for High R&D companies because it captures non-economic variables, and DCF does not falter for any case (suggestion confirmed by sensitivity analysis). However, DDM and P/E models are more volatile and less reliable, while P/E needs to explain the market value because it is subject to earnings volatility and outliers.

Analysts and investors who choose valuation models can learn several practical lessons from these results. RIVM is the right choice for companies with significant intangible assets or R&D investments, where its model is an excellent way to extract value from other financial metrics. DCF, on the other hand, is versatile and appropriate for companies with steady cash flows and deterministic financials, so its a highly general valuation method.

However, both DDM and P/E have some caveats. DDM can only be applied to dividend-paying companies, not companies with inconsistent or small dividend payouts. In contrast, P/E is so reliant on earnings data that its very vulnerable to outliers and earnings volatility, so its unfit for use in the case of shifting or negative profits.

This study generally makes tuning valuation models to firm-specific attributes for dependable equity valuations more imperative. It also points to the importance of solid assumptions and sensitivity testing to test model output in the real world. By modifying the choice of valuation model to the specific dynamics of the individual company, analysts can develop more precise and meaningful valuations to guide better investment decisions.

4. Case Study

Despite extensive sample analysis (LSA) benefits, analysts' appraisal processes must be considered because they cannot be monitored. For this reason, just one company is thoroughly examined in the following case study. To accomplish this, a thorough review of the firm and its valuations is provided by analysing analyst reports, media articles, and annual reports. The outcomes of the earlier analysis are connected to the findings of the analyst reports. The following queries are addressed in the debate that follows:

Valuation Usually, analysts utilise valuation models to determine the exact value of the company.

Is it evident that analysts might not use every model?

Do the analyst reports match the earlier analysis's findings?

Do empirical data support the theory?

The case company is introduced, and the selection criteria for the company are stated at the outset.

The main areas of emphasis are the strategy, finances, and business model. Except for the financial section, which covers the observation period to connect the results to the same period, everything discussed here applies to the company's current state. Then, analyst reports and the generated estimates and assumptions are compared.

4.1. Decisive factor for the Selection of a Company

Apple Inc. was selected as the case company due to its significant R&D investment, stable industry presence, and extensive data availability. Apple is well-known for its considerable R&D spending, which reached $26.25 billion in 2021 and demonstrates its dedication to innovation (Apple Inc., 2021). The technology sector, while dynamic, provides a relatively stable environment for established companies like Apple, which has shown resilience and consistent growth over the years (Statista, 2022). Furthermore, Apple's extensive financial data and widespread analyst coverage make it ideal for detailed valuation analysis (Reuters, 2022). Over the past five years, Apple has exhibited robust economic health, consistent revenue growth, substantial net income, and significant R&D expenditures, underlining its financial stability and innovation-driven strategy (Yahoo Finance, 2022; Bloomberg, 2022).

4.2 Company Overview.

The core components of Apple Inc.'s business strategy are creating, producing, and marketing various consumer devices, software, and services. The iPhone, iPad, Mac laptops, Apple Watch, and Apple TV are essential. Apple also offers a variety of software, including iOS, macOS, and watchOS, as well as services like Apple TV, Apple Music, iCloud, and the App Store (Apple Inc., 2021). The technology industry, characterized by rapid innovation and intense competition, includes major players such as Google, Microsoft, and Samsung. Apple holds a significant market position due to its strong brand loyalty, ecosystem integration, and continuous innovation. In the smartphone market, Apple consistently competes for the top position globally, often contending with Samsung for market share (Statista, 2022). Financially, Apple has demonstrated robust performance over the past five years. Revenue increased from $229 billion in 2017 to $365 billion in 2021, with net income growing from $48 billion to $94.7 billion in the same period. Apples R&D expenditure has also significantly risen, from $11.6 billion in 2017 to $26.25 billion in 2021, underscoring its commitment to innovation and maintaining a competitive edge (Yahoo Finance, 2022; Apple Inc., 2021).

4.3 Analyst Report Analysis

Offering a sell-side perspective, analyst reports for 2023 by Credit Suisse, Barclays, and Morningstar have been published. After reviewing all the publications, Morningstar covers the DCFM, while only the MBV is covered by every analyst report. None of the analyst reports covers all the stated valuation approaches. Given this as well, the MBV is thought to be better. This is consistent with Demirakos et al. ( 2004) and Asquith et al. ( 2005), which show that most analysts use the MBV for valuation, and the DCFM is still used instead. Barker (1999), on the other hand, sees the MBV as the basis upon which further basic research should be carried out. Given that Apple did not apply the model in any of its analyst reports, the conclusion derived from the literature reviewthat the RIVM is the best FBMis startling. Only the MBV has a foundation in theory and practice, but the DCFM is regarded as poor in theory but at least partially used in practice. One justification against implementing the DDM may be that a value derived from it would not be acceptable, as dividend development must follow a predefined road.

Here, we examine the most recent 2023 Morningstar report (23.11.2023) with appropriate links to the other two reports.

Based on the expert evaluation, Apple has a bright future as new products and services will cause consistent expansion. They expect Apple to keep leading the smartphone market by introducing fresh iPhone models and considerably increasing its services sector. However, they also highlight potential problems, including competitiveness and supply chain interruptions. Apple's robust brand ecosystem and high operating margins help it forecast its profits to] keep increasing despite these challenges. Morningstar estimates a reasonable return and values Apple at $180 per share. Moreover, they expect a 20% or more return on money invested that exceeds the expenses.

For their value, they expect 7% average income growth over the next five years, combining continuous hardware segment development with significant expansion in the services sector. This is much higher than the 3% increase rate (inflation rate) expected in the LSA, even if the prediction period was only two years. To examine both upside potential and downside risk, they performed a scenario study that included supply chain efficiency, market expansion, and new product introduction. These specific effects were ignored since Apple was merely investigated concerning the LSA. Furthermore, there are no precise accounting numbers like CAPEX, intangible assets, or operating cash flow as Morningstar did; the flows needed for the value were projected. Moreover, the LSA neglected any specific supply cuts or production rate changes during the forecast period. After considering several possibilities, they arrive at an upside fair value of $220 and a downside fair value of $150; the actual fair value, $180, is closer to the upper.

4.3.1 Results of the Large Sample Investigation for Apple Inc.

The extensive sample analysis identified the Residual Income Valuation Model (RIVM) and Price-to-Earnings (P/E) multiples as the most reliable valuation models. In reviewing the analyst reports for Apple Inc., these models were also prominently used. Given Apple's high profitability and equity tripping, the RIVM's residual income idea is applicable. P/E multiples allow comparison with other technology corporations. Though helpful, the company's DCF models used low growth rates and excessive discount rates, resulting in multiple valuations for the below RIVM and P/E. Most analysts lower their projections to reflect the risk-adjusted and cautious environment they must consider when assessing their subjects.

Analyst-preferred models align more with more extensive sample analyses, so the RIVM and P/E multiples are more practical and efficient for Apple's valuation. However, growth rate and discount rate variations remind entrant price valuation that adjustments must be made depending on the situation.

4.3.2 Results of the Evaluation of the Analyst Reports.

The analyst's reports analyze the following strengths and weaknesses in the valuation approaches: evaluating different models, adopting conservative growth rates to avoid overstated growth rates, and clearly explaining why those growth rates have arrived.

Table9: Comparative Analysis of Valuation Methods Across Different Financial Institutions

Valuation Method	Assumptions	Extensive Sample Analysis (LSA) Results	Credit Suisse (2023)	Barclays (2023)	Morningstar (2023)
DCF Model	Growth Rate: 3% (Inflation)	$122.69 per share	Not covered	Not covered	$174.50 per share
	Discount Rate: Constant				(Enterprise value: $120,354 million)
	Forecast Period: 2 years
	Terminal Value: Perpetual growth
Residual Income Valuation Model (RIVM)	Cost of Equity: Constant	High reliability	Not covered	Not covered	Not used
Price-to-Earnings (P/E) Multiples	P/E Ratios of Comparables	High reliability	15.6x	16.3x	16.7x
			EV/Sales: 1.1x	EV/Sales: Not covered	EV/Sales: Not covered
			EV/EBITDA: 9.5x	EV/EBITDA: 9.2x	EV/EBITDA: 9.2x
			EV/EBIT: Not covered	EV/EBIT: Not covered	EV/EBIT: 11.4x
Dividend Discount Model (DDM)	Dividend Growth: Variable	Not covered	Not covered	Not covered	Not used
Overall Recommendation	-	Sell	Buy	Buy	Buy
Fair Value Estimate	-	-	Not provided	Not provided	$180A3:F13re (Upside: $220, Downside: $150)

This table compares the assumptions and valuation results of various models used by different financial institutions for assessing the fair value of a company's stock. The Discounted Cash Flow (DCF) Model, Residual Income Valuation Model (RIVM), and Price-to-Earnings (P/E) Multiples are analyzed alongside recommendations from Credit Suisse, Barclays, and Morningstar.

Possible biases can originate from analysts' dependence on general market indicators and conditions that impact their growth rate and discount factors. Further, the terminal value in the DCF model and the corresponding calculations are complicated and subjective and may differ significantly from one appraiser to another.

The general implications of these approaches can, therefore, indicate that equity valuation entails necessary balancing and substantiated assumptions. Applying numerous models to the same company is less risky, and paying more attention to conservative estimates gives a better picture of the companys value. This approach shows that any specifics in the equity valuation process must undergo a thorough analysis in the proper contexts.

4.3.3 Detailed Analysis of Hypotheses

Based on the findings from the extensive sample analysis and literature review, the following hypotheses were formulated and tested against the case study data for Apple Inc.:

Hypothesis 1: Analysts predominantly use multiple-based valuation models (MBVM) over flow-based valuation models (FBVM) for stable companies in the technology sector.

Hypothesis 2: Analysts' valuation estimates align closely with those derived from the extensive sample analysis when using the RIVM and P/E multiples.

Hypothesis 3: Variations in growth rate and discount rate assumptions significantly impact the valuation outcomes, necessitating context-specific adjustments.

These hypotheses were tested by comparing the assumptions and valuation outcomes in the analyst reports with those from the extensive sample analysis. The results indicated that while analysts favor MBVMs, they also incorporate elements of FBVMs, particularly the DCF model, albeit with conservative growth and discount rates. The alignment of analyst estimates with extensive sample analysis findings was generally strong, supporting the reliability of the RIVM and P/E multiples for Apple Inc.

4.4 Interim Conclusion.

The three most often used methods of valuation include the Dividend Discount Model (DDM), Discounted Cash Flow Model (DCF), and Price to Earnings (P/E) ratio (Damodaran, 2002; Penman, 2013). However, these models may not be sufficient to capture the value of intangible assets, which may result in wrong valuations (Francis et al., 2000).
Over the last few years, RIVM has been preferred by many scholars because the model allows the inclusion of intangible assets into the valuation model, as proposed by Ohlson (1995). Although RIVM, like other models, employs cash flows and dividends in its calculations, it also incorporates the Economic Value Added (EVA) concept, which considers a companys Economic Profit and the cost of capital. This is particularly so with companies with large amounts of R & D costs and other non-cash investments, such as goodwill and other intangible assets, that significantly impact the company's value in the future (Penman, 2013). Besides, this dissertations literature review also provides a detailed analysis of these valuation models and their theoretical frameworks, strengths, and weaknesses. It also identifies the literature review done in an attempt to determine the effectiveness of these models in different sectors and conditions of the market (Kaplan & Ruback, 1995; Liu et al., 2002). It is seen that the existing techniques used for assessing intangible assets are rather insufficient and that MBVMs may be more appropriate in some cases (Liu et al., 2002). For the extended empirical analysis of this study, the aim is to choose US public firms from different industries to compare the valuation models results for US firms with high and low levels of intangible assets.

Conclusion and Recommendations

This dissertation has provided a critical evaluation of the various equity valuation models with a focus on firms that possess significant intangible assets. Knowledge of the results of an empirical analysis, according to which the Residual Income Valuation Model (RIVM) and elements for obtaining Price-to-Earnings (P/E) multiples can be identified as the most valuable and accurate in cases where specific types of valuation are required. In support of the findings of the present study, the Apple Inc. case is also supportive of the results that were arrived at empirically.

1.Model Suitability:

• The RIVM model was identified to be the most suitable for high intangible asset firms as the model uses residual income adjustments to add value beyond tangible means. This result supports Ohlson (1995) and Francis et al. (2000) since the theoretical and empirical evidence show that the RIVM is effective in measuring economic returns. The P/E multiples model showed its consistency in firms with low research intensity since earnings are more predictable. Account for the value beyond tangible assets. This result aligns with Ohlson (1995) and Francis et al. (2000), who emphasize the theoretical robustness and empirical reliability of the RIVM in capturing economic profits.
• The P/E multiples model demonstrated its reliability in firms with lower R&D intensity, where earnings are stable and predictable. However, it is pointed out by Liu et al. (2002) that the value-added utility of this model could not work well in conditions of high earning volatility.

2.Context-Specific Performance:

• High R&D firms benefited significantly from the RIVM due to its ability to integrate adjustments for intangible-driven growth.
• The Discounted Cash Flow Model (DCF) exhibited versatility and moderate reliability, making it suitable for firms with stable cash flows. However, its sensitivity to assumptions, such as discount rates, necessitates careful calibration.
• The Dividend Discount Model (DDM), while theoretically sound, was limited to firms with consistent and substantial dividend payouts, thereby restricting its broader applicability.

3.Practical Recommendations for Analysts and Investors:

• Analysts should tailor the model selection to the unique financial characteristics of the firm undervaluation. For high-intangible asset firms, the RIVM offers a robust approach, whereas the DCF and P/E models are more suitable for firms with steady cash flows and predictable earnings, respectively.
• Combining models can mitigate individual limitations. For instance, using the RIVM alongside the DCF can provide a holistic view of intrinsic value, particularly in industries with mixed asset profiles.

4.Recommendations for Future Research:

• Further refinement of valuation models to better capture the intricacies of intangible assets is necessary. Future research could explore incorporating machine learning algorithms to enhance model accuracy and adaptability.
• The impact of varying market conditions, such as economic downturns or technological disruptions, on model performance warrants further study. This would provide valuable insights into the robustness of these models across different scenarios.
• Sector-specific studies could enhance the understanding of how different industries influence the reliability of valuation models. For example, examining models in highly volatile sectors such as biotechnology or fintech could yield actionable insights.

Conclusion

Therefore, the present paper stresses the necessity of the appropriate relationship between the choice of the valuation model and the companys features and the economic environment. Since earnings growth is a component of the model comparable to the RIVM and P/E multiples, applying all the strengths results in higher levels of accuracy. However, as observed from the current research, there is a need for constant improvement and development of valuation techniques. The findings of this study are in tandem with similar research that calls upon analysts and investors to use a range of valuation techniques; these include the use of varied and strong assumptions to contend with the challenges of modern financial analysis adequately.

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