Need question E15-24 within this pdf document answered please.

2015 Cambridge Business Publishers | For the Personal Use of Amanda Rosena. Explain company valuation using market multiples based on balance sheet measures. (p. 15-6) 3. Identify comparable companies for use in company valuation with market multiples. (p. 15-13) 2. Explain company valuation using market multiples based on income statement measures. (p. 15-9) 4. Interpret and reverse engineer market multiples to assess the reliability of market expectations. (p. 15-20) 55775 iStock LEARNING OBJECTIVES 1. 2015 Cambridge Business Publishers | For the Personal Use of Amanda Rosena. M O D U L E 15 Market-Based Valuation 55775 Family Dollar operates more than 7,000 self-service retail discount stores aimed at middle- to low-income consumers in the United States. Its annual revenues for the recent year exceeded $9 billion. During the past five years, revenues have grown by over 35% and profits by over 70%. Its stock price has tripled during that same period, exhibiting a three-year upward trend as revealed in the following graphic. FAMILY DOLLAR Family Dollar Stock Price $75 $65 $55 $45 $35 $25 $15 Jan. 2009 Jan. 2010 Jan. 2011 Jan. 2012 Jan. 2013 During this most recent 5-year period, Family Dollar's assets increased over 25%, while its equity was stable. We might ask: which financial metric, or combination, is a good measure of its shareholder value? Revenues? Profits? Assets? Equity? Other? Can we use one or more of these financial metrics to estimate Family Dollar's shareholder value? To help in this valuation, what company or companies might we use to benchmark Family Dollar's performance and/or financial condition? Should we consider a competitor such as Dollar General? How about Dollar Tree Stores or Big Lots? Analysts typically utilize a number of approaches to estimate the value of a company's stock price. In prior modules, we have covered two general methods that focus on expected free cash flows and residual operating income. In this module, we illustrate an alternative methodology that focuses on multiples of summary financial measures, such as assets, equity, and profit. This methodology seeks to determine the proper multiple of these summary measures to yield a value of the company's equity. This methodology is referred to as market-based valuation. Sources: Family Dollar Corporation 2013 Annual Report and 10-K Filing; Finance.yahoo.com, July 2013. 15-2 2015 Cambridge Business Publishers | For the Personal Use of Amanda Rosena. 15-3 Module 15 | Market-Based Valuation MODULE O R G A N I Z AT I O N Market-Based Valuation Valuation Using Balance Sheet Multiples Valuation Using Income Statement Multiples 55775 Valuation Model Valuation Model Application Using a Application Using a NOA Multiple Application Using a BV Multiple Selecting Comparables for Market Multiples NOPAT Multiple Application Using a NI Multiple Deriving PB from the ROPI Model PB Ratios and Profitability, Growth, and Risk Deriving PE from the ROPI Model PE Ratios and Interpreting and Reverse Engineering Market Multiples Interpreting and Reverse Engineering the PB Ratio Interpreting and Reverse Engineering the PE Ratio Valuation with Market Multiples Profitability, Growth, and Risk This module focuses on valuation techniques using market multiples. Unlike the valuation techniques introduced in the previous modules, valuation using market multiples does not have rigorous theoretical underpinnings. It is also problematic in that it does not require explicit forecasts of future performance for the company to be valued. Further, it requires that we assume that observed market prices for the target company are not informative about intrinsic value; and yet it requires us to assume simultaneously that observed market prices for comparable companies accurately reflect intrinsic value. Despite these inherent shortcomings, valuation techniques using market multiples are widely applied. They are commonly used as shortcut valuation methods, as screening devices, and as means to create summary forthey assessing relative Some precisely who use because these market-based valuation techniquesstatistics argue that are superior to valuations. other techniques they do not rely on subjective forecasts of future performance. Whatever the view, valuation using market multiples, also referred to as the method of comparables, is a common valuation technique. Valuation Model using Market Multiples Valuation using market multiples is popular chiefly because of its simplicity. We simply select a relevant summary measure of performance (such as earnings, book values, cash flows, or sales) from the financial statements of a target company that we wish to value. Next, we identify companies that are \"comparable\" to the target company on relevant dimensions (described later). For each comparable company, we compute the ratio of market value to the selected summary performance measure. The average of those ratios is the market multiple. (While an average of ratios from comparable companies is commonly employed, if a specific comparable company is believed to be more relevant as a benchmark, then we can place more than average weighting on its ratio.) Then, we multiply the summary measure for the target company by the market multiplethat product is the estimated value of the target company. The following model reflects this process. Value 5 Summary performance measure 3 Market multiple This model does not specify whether the value obtained is the company value or equity value. Recall that the DCF and ROPI models in Modules 13 and 14 were used to value the entire company. We then find equity value by deducting the value of net nonoperating obligations (NNO) from company value. When using market multiples to estimate value, the choice of summary performance measure determines whether the output is company value or equity value. If an equity performance measure is selected (such as earnings or book value), then the output of the model is equity value. If a company performance measure is selected (such as NOPAT or NOA), then the output of the model is company value, from which we must deduct net nonoperating obligations 2015 Cambridge Business Publishers | For the Personal Use of Amanda Rosena. Module 15 | Market-Based Valuation 15-4 to obtain equity value. The more commonly used market multiples are those that generate equity value directly because investors are most often interested in equity value. However, as we shall see, using market multiples to find equity value directly means capital structure must be considered in selecting comparable companies. Application of the Model Using Market Multiples There are five steps in estimating value using a market multiple. (The steps can be applied using performance measures, or performance measures on a per share basis.) 55775 1. Select the summary performance measure to use as the basis for valuationsuch as earnings, book value, NOPAT or NOA; the model can be applied using current performance measures or forecasted performance measures. 2. Select the comparable companies to use in determining the market multiple. 3. Compute the market multiple from the comparable companies' market values and performance measures. 4. Compute the target company's value using its performance measure and the market multiple. 5. If its performance measure is an equity performance measure (such as earnings and book value), divide by shares outstanding to get equity value per share. If the performance measure is a company performance measure (such as NOPAT and NOA), subtract net nonoperating obligations, and then divide by shares outstanding to get equity value per share. (If a per share performance measure is used, we do not divide by shares outstanding.) Although the steps are simple, key decisions are made in steps 1, 2 and 3. These decisions reveal the inherent weakness of this valuation method. First, in step 1, which performance measure is the right one? This question has no answer as there is no right measure. Company value depends on future company performance. Previous modules of this book show that no single measure entirely summarizes current period performance, much less provides a sufficient basis for forecasting future company performance. Still, we could argue that particular are more suited to particular companies industries. For example, book value might such bemeasures more suitable companies assets whoseor reported values approximate market values as with manyfor financial firms.with Earnings might be more suitable for companies that are growing rapidly with relatively little volatility. Free cash flows might be more suitable for companies with negative earnings. Second, in step 2, what companies should we use as comparable companies? If comparable companies are under- or overvalued, the computed market multiple will under- or overvalue the target company. We might appeal to market efficiency and claim that the comparable companies are likely fairly priced, but this begs the question of why we are trying to find a value for our target company if it is also fairly priced? Even if comparable companies are accurately priced, when they differ in ways that matter for valuation (such as on profitability, expected growth, and risk), the computed multiple will under- or overvalue our company. RESEARCH INSIGHT Efficient Markets and Valuation Using Comparables There is an inherent inconsistency in using market multiples to value publicly traded companies. By choosing to value a specific company using market multiples, the investor presumes that the target company is not fairly valued. However, by selecting comparable companies to determine the multiple, the investor relies on the markets to accurately value those companies. How does the investor know which companies are fairly valued and which are valued with error? Some assert that a group of companies is valued correctly on average. However, if a company deviates from the average, is that a pricing error or a valid variance from the distribution of intrinsic values? Third, in step 3, how do we combine the comparable company data to produce a multiple? We could compute the multiple for each company and then use an equal-weighted average, or combine the data for all companies and compute a value-weighted average. Or, we could use a median of all comparable company ratios, or use an average after eliminating the highest and lowest observations. 2015 Cambridge Business Publishers | For the Personal Use of Amanda Rosena. 15-5 Module 15 | Market-Based Valuation Again, there is no correct answer to this question and the others as this valuation technique is ad hoc and the decisions commonly made to apply it are ad hoc. Despite deficiencies in valuing companies using market multiples, it is commonly applied in practice. The following portion of a ValuEngine Inc. analyst report uses market multiples to explain and justify its current assessment of a company's stock price. We see references to peers, or \"comparables\" (comps), and the market multiple in analyzing current performance and predicting future performance, including stock price performance. Analyst Report 55775 MARKET RATIO BASED VALUATION Portfolio managers and professionals traditionally rely on market ratios to gauge whether a stock is fair valued or overvalued. On this page, we present such a valuation based on one of three market ratios: PEG (price to trailing 4 quarter earnings ratio, divided by the consensus analyst forecasted next year EPS growth), P/E (price to forward 4 quarter earnings ratio), and P/S ratio (price to trailing 4 quarter sales). Among the three, PEG is the most informative as it reflects both the price/earnings ratio and expected future EPS growth, while P/E is better than P/S. For each given stock, we apply the PEG to give a fair value assessment if both its trailing 4 quarter EPS and forecasted EPS growth rate are positive. If its forecasted EPS growth is negative but its forward 4 quarter EPS is positive, we apply the P/E to give a fair value for the stock as of today. Otherwise, we resort to the P/S to assess its fair value. To establish a valuation standard, we use both (i) the average historical market ratio of the stock over the past 10 years (or however long there is data available for the stock), and (ii) the average market ratio today of five comparable stocks in the same sector and from companies of similar size. These two alternative perspectives should give you a good idea about where this stock's valuation stands today. 1. Valuation Based on FDO's Past PEG Over the past 10 years, FDO's average PEG is 2.96. FDO earned $3.81 per share in its recent 4 quarters. The analyst consensus estimate is $4.02 for its 4 quarter forward EPS. FDO's current price sales ratio is 0.78. Source: ValuEngine Inc. The following assessment is based on multiplying the historical PEG with recent 4 quarter EPS and the forecasted EPS growth rate over the next 4 quarters for FDO. Fair Value Historical Average PEG . . . PEG-Based Fair Value . . . . 2.96 $61.27 2.Valuation Based on Comparables' PEG FDO's comparables are BIG, DLTR, MMRTY, PSMT and SHLD. The current PEG average of these comparables is 1.77. FDO earned $3.81 per share in its recent 4 quarters. The analyst consensus estimate is $4.02 for its 4-quarterforward EPS. FDO's current price sales ratio is 0.78. The following assessment is based on multiplying comparable stocks' average PEG today with FDO's recent 4 quarter EPS and the forecasted EPS growth rate over the next 4 quarters. Comparables' PEG Comparables BIG . . . . . . . . . . . . . . . . . DLTR. . . . . . . . . . . . . . . . MMRTY . . . . . . . . . . . . . PSMT . . . . . . . . . . . . . . . SHLD . . . . . . . . . . . . . . . Current PEG 1.5 1.49 n/a 2.32 n/a Fair Value Comparable Stocks' Avg PEG . . . . . . . . . . . Comparable PEG-Based Fair Value . . . . . . . . . . 1.77 $36.61 2015 Cambridge Business Publishers | For the Personal Use of Amanda Rosena. Module 15 | Market-Based Valuation 15-6 VALUATION USING BALANCE SHEET MULTIPLES The balance sheet is one source of performance measures to use in estimating company value. This section illustrates the mechanics of using balance sheet summary measures to estimate value. First, we estimate company value for Family Dollar using a multiple of net operating assets. Next, we estimate its equity value using a multiple of book value. The comparable companies we use are Dollar General and Big Lots, both of which compete with Family Dollar. Summary information for all three companies, using reported fiscal year-end data for Family Dollar (August 25, 2012), Dollar General (February 1, 2013) and Big Lots (February 2, 2013), is in Exhibit 15.1.1 Market values for each company are collected on the fiscal year-end date from Yahoo!, Google, and/or the 10-K filing. Our task is to estimate the intrinsic value of Family Dollar's equity. L O 1 Explain company valuation using market multiples based on balance sheet measures. 55775 EXHIBIT 15.1 Data for Valuation Using Balance Sheet Multiples* (in millions) Company assumed value . . . . . . . . . . . . . . . . . . . . . . . . Equity assumed value . . . . . . . . . . . . . . . . . . . . . . . . . . Net operating assets . . . . . . . . . . . . . . . . . . . . . . . . . . . Book value of equity . . . . . . . . . . . . . . . . . . . . . . . . . . . . Net nonoperating obligations (assets) . . . . . . . . . . . . . . Common shares outstanding . . . . . . . . . . . . . . . . . . . . . Family Dollar Dollar General Big Lots $1,582 $1,298 $ 284 115.4 shares $17,769 $15,138 $ 7,616 $ 4,985 $ 2,631 327.1 shares $1,964 $1,853 $ 869 $ 758 $ 111 57.3 shares *Financial statements for these companies report amounts in thousands; we round to millions for ease in computation. Valuation Using a Net Operating Asset (NOA) Multiple We use the data from Exhibit 15.1 to compute a market multiple based on net operating assets to estimate the value of Family Dollar. We begin by determining the NOA market multiple for both Dollar General and Big Lots, which is computed as the company assumed value divided by net operating assets. Exhibit 15.2 shows the results of this computation. For example, Dollar General's NOA market market multiples multiple is(from 2.33, Dollar computed as $17,769/$7,616. then2.30, average the comparables' NOA General and Big Lots) We to get computed as (2.33 1 2.26)/2. This market multiple is used to estimate the company intrinsic value of Family Dollar as follows: Company intrinsic value 5 Net operating assets 3 NOA market multiple $3,639 5 $1,582 3 2.30 To obtain Family Dollar's equity intrinsic value we subtract net nonoperating obligations from its company value, which is then divided by shares outstanding to yield its per share intrinsic value of common equity as follows: Equity intrinsic value 5 Company intrinsic value 2 Net nonoperating obligations $3,355 5 $3,639 2 $284 Then, Equity intrinsic value Common shares outstanding $3,355 5 115.4 shares Equity intrinsic value per share 5 $29.07 1 Recall the following key definitions (readily available from services such as Yahoo! and Google): Company assumed value 5 Firm value (market value of a company's net operating assets), which is also equal to the market value of its common stock and net nonoperating obligations. Equity assumed value 5 Market capitalization (market value of a company's common stock) Book value of equity 5 Book value of a company's common stockholders' equity From market From nancial statements 2015 Cambridge Business Publishers | For the Personal Use of Amanda Rosena. 15-7 Module 15 | Market-Based Valuation Family Dollar stock closed at $62.35 on Friday, August 24, 2012, the company's fiscal year-end, and at $65.10 on Tuesday, October 19, 2012, the date on which its 10-K was filed with the SEC (the filing date). Accordingly, this estimate suggests that Family Dollar was markedly overvalued on those dates. Of course, the estimated value would have been markedly different had we used Dollar General's market multiple alone, and yet again different if we had used Big Lots' multiple alone. This shows the key role that the market multiple plays in determining intrinsic value under this method. EXHIBIT 15.2 Estimating Intrinsic Value Using a Net Operating Asset Multiple 55775 (in millions, except per share amounts) Family Dollar Dollar General Big Lots Company assumed value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Net operating assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Common shares outstanding . . . . . . . . . . . . . . . . . . . . . . . . . . $1,582 115.4 shares $17,769 $ 7,616 327.1 shares $1,964 $ 869 57.3 shares NOA market multiple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Company intrinsic value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equity intrinsic value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equity intrinsic value per share. . . . . . . . . . . . . . . . . . . . . . . . . 2.30 $3,639 $3,355 $29.07 2.33 2.26 Valuation Using a Book Value (BV) Multiple We can repeat the analysis above using the book value of equity as the multiple for valuing the company. This approach yields the intrinsic value for equity, not for the entire company. This method relies on different data and, as such, we will not get the same intrinsic values as computed in the previous section. EXHIBIT 15.3 Estimating Intrinsic Value Using a Book Value Multiple (in millions, except per share amounts) Family Dollar Dollar General Big Lots Equity assumed value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Book value of equity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $1,298 $15,138 $ 4,985 $1,853 $ 758 Common shares outstanding . . . . . . . . . . . . . . . . . . . . . . . . . . 115.4 shares 327.1 shares 57.3 shares BV market multiple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.74 3.04 2.44 Equity intrinsic value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $3,557 Equity intrinsic value per share. . . . . . . . . . . . . . . . . . . . . . . . . $30.82 We begin by computing the book value market multiple for both Dollar General and Big Lots, which is computed as equity assumed value divided by book value of equity. Exhibit 15.3 shows the results of this computation. For example, Dollar General's book value market multiple is 3.04, computed as $15,138/$4,985. We then average the book value market multiples (from Dollar General and Big Lots) to get 2.74, computed as (3.04 1 2.44)/2. This BV market multiple is used to estimate the equity intrinsic value of Family Dollar as follows: Equity intrinsic value 5 Book value of equity 3 BV market multiple $3,557 $1,298 2.74 5 3 To obtain Family Dollar's equity intrinsic value per share we divide by shares outstanding as follows: Equity intrinsic value Common shares outstanding $3,557 5 115.4 shares Equity intrinsic value per share 5 $30.82 2015 Cambridge Business Publishers | For the Personal Use of Amanda Rosena. Module 15 | Market-Based Valuation 15-8 The $30.82 stock price estimate suggests that Family Dollar stock was markedly overvalued given its $62.35 closing price at its fiscal year-end (and vis--vis the $65.10 price on its filing date). The estimated value would have been different had we used Dollar General's market multiple alone, and yet again different if we had used Big Lots' multiple alone. Again, this shows the key role that the market multiple plays in determining intrinsic value under this method. It is useful for us to compare estimates of Family Dollar's intrinsic value of equity using the net operating assets multiple vis--vis the book value multiple. Using the NOA multiple, we estimated the intrinsic value of a share of Family Dollar to be $29.07, while the BV market multiple gave an estimate of $30.82. For Family Dollar, these estimates differ by less than 10%, which is a much smaller difference than is commonly seen between various multiplesbased estimates. How do we assess the quality of the different estimates? To answer this, let's consider the comparables. When choosing comparables, we should select companies that are similar in terms of profitability, growth, and risk. In this case, we did not control for profitability, growth, or financial riskwe did control for operating risk by choosing companies in the same industry. Still, we might have done better by choosing other retail firms with profitability, growth, and financial risk characteristics similar to those of Family Dollar. (Later in this module we provide guidance on choosing comparables for valuation purposes.) Consequently, the estimate based on the NOA market multiple is likely \"better.\" The reason is that company financial risk does not materially affect the value of net operating assets, but it does affect the value of equity. This means the different capital structures of Family Dollar, Dollar General, and Big Lots do not matter when applying NOA multiples in valuation. But, when using book value multiples to estimate intrinsic value, it is important to select comparables with similar capital structures. 55775 ANALYSIS DECISION You Are an Entrepreneur You are the sole owner of a software firm that has developed a system to handle back-office processing for municipalities. During its first three years, your firm focused on developing the software and reported sizeable research and development expenses and annual losses. An interested buyer has approached you, and the investment bankers representing the buyer want to value your firm using market multiples. Because your firm has reported losses, they suggest basing the valuation on book value per share and using Oracle and SAP as comparables, which provide a book value multiple of 8. You believe their methodology is flawed, and their offering price is too low. What arguments can you make in support of your position? [Answer, p. 15-24] MID -MODU LE R E VIE W 1 The table below provides summary data for Johnson & Johnson (JNJ), along with Procter & Gambleand Merck, two large cap firms that compete in many segments of JNJ's medical and consumer products businesses. (in millions) Company assumed value . . . . . . . . . . . . . . . . . Equity assumed value . . . . . . . . . . . . . . . . . . . Net operating assets . . . . . . . . . . . . . . . . . . . . Book value of equity . . . . . . . . . . . . . . . . . . . . . Net nonoperating obligations (assets) . . . . . . . Common shares outstanding . . . . . . . . . . . . . . Johnson & Johnson Procter & Gamble Merck $45,894 $39,318 $ 6,576 2,893 shares $238,532 $203,325 $101,972 $ 66,760 $ 35,212 3,132 shares $ 99,560 $103,315 $ 13,809 $ 17,560 $ (3,750) 2,168 shares 2015 Cambridge Business Publishers | For the Personal Use of Amanda Rosena. 15-9 Module 15 | Market-Based Valuation Required a. Estimate the net operating assets market multiple for Procter & Gamble and Merck. Then, compute the average of these as the NOA market multiple for valuation and assess its reliability. b. Use the results from part a to estimate the company intrinsic value, the equity intrinsic value, and the equity intrinsic value per share for Johnson & Johnson. c. Compute the book value multiple (also called price-to-book or PB) for Procter & Gamble and Merck. Then, compute the average of these as the BV market multiple for valuation. d. Use the results from part c to estimate equity intrinsic value and the equity intrinsic value per share for Johnson & Johnson. 55775 The solution is on page 15-37. VALUATION USING INCOME STATEMENT MULTIPLES L O 2 Explain company valuation using market multiples based on income statement measures. The most commonly used performance measure for estimating company value with market multiples is earnings. Price-to-earnings (PE) ratios are cited in news articles and analysts' reports, used in stocks screens and trading strategies, and even appear as the subject of academic research. The intuition in using earnings as the basis for valuing a company is straightforwarddividends are paid out of earnings, and potential dividend payouts are the basis for company value. Accordingly, we should pay more for companies that generate more earnings. This section illustrates the mechanics of using net operating profit after tax (NOPAT) and net income (NI) as the basis for valuation by market multiples. (For simplicity, NOPAT computations assume a tax rate of 35% for companies in this module.) First, we estimate company value forFamily Dollar using a multiple of NOPAT. Next, we estimate its equity value using a multiple of NI. The comparable companies we again use are Dollar General and Big Lots, both of which compete with Family Dollar. Summary information for all three companies, using reported fiscal year-end data for each company (see Exhibit 15.1 for further is in Exhibit estimate the equity and company values for Family explanation), Dollar using Dollar General15.4. and Our Big task Lots is astocomparables. EXHIBIT 15.4 From market From nancial statements Data for Valuation Using Income Statement Multiples (in millions) Family Dollar Dollar General Company assumed value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equity assumed value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Net operating profit after tax . . . . . . . . . . . . . . . . . . . . . . . . . . Net income . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Common shares outstanding . . . . . . . . . . . . . . . . . . . . . . . . . . $438 $422 115.4 shares $17,769 $15,138 $ 1,036 $ 953 327.1 shares Big Lots $1,964 $1,853 $ 180 $ 177 57.3 shares Valuation Using a Net Operating Profit After Tax (NOPAT) Multiple We use the data from Exhibit 15.4 to compute a market multiple based on net operating profit after tax (NOPAT) to estimate the value of Family Dollar. We begin by determining the NOPAT market multiple for both Dollar General and Big Lots, which is computed as company assumed value divided by NOPAT. Exhibit 15.5 reports the results of this computation. For example, Dollar General's NOPAT market multiple is 17.15, computed as $17,769/$1,036. We then average the comparables' NOPAT market multiples (from Dollar General and Big Lots) to get 14.03, computed as (17.15 1 10.91)/2. This market multiple is used to estimate the company intrinsic value of Family Dollar as follows: 2015 Cambridge Business Publishers | For the Personal Use of Amanda Rosena. Module 15 | Market-Based Valuation 15-10 Company intrinsic value 5 Net operating profit after tax 3 NOPAT market multiple $6,145 $438 14.03 5 3 EXHIBIT 15.5 Estimating Intrinsic Value Using a Net Operating Profit after Tax Multiple (in millions, except per share amounts) Family Dollar Dollar General Company assumed value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Net operating profit after tax . . . . . . . . . . . . . . . . . . . . . . . . . . Common shares outstanding . . . . . . . . . . . . . . . . . . . . . . . . . . $ 438 115.4 shares $17,769 $ 1,036 327.1 shares $1,964 $ 180 57.3 shares NOPAT market multiple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Company intrinsic value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equity intrinsic value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equity intrinsic value per share. . . . . . . . . . . . . . . . . . . . . . . . . 14.03 $6,145 $5,861 $50.79 17.15 10.91 55775 Big Lots To obtain Family Dollar's equity intrinsic value we subtract net nonoperating obligations (in Exhibit 15.1) from company value, which is then divided by shares outstanding to yield the per share intrinsic value of its common equity as follows: Equity intrinsic value 5 Company intrinsic value 2 Net nonoperating obligations 5 2 $5,861 $6,145 $284 Then, Equity intrinsic value Common shares outstanding $5,861 5 115.4 shares Equity intrinsic value per share 5 $50.79 This estimate suggests that Family Dollar was overvalued with respect to its $62.35 closing price at its fiscal year-end had (andwe vis--vis its $65.10 price on the filing date).alone The estimated value would have been different used Dollar General's market multiple (Family Dollar would have been estimated to be undervalued at its fiscal year-end), and yet again different if we had used Big Lots' multiple alone, and yet again different if we had computed a weighted average. This shows the key role of the market multiple. Valuation Using a Net Income (NI) Multiple We can repeat the analysis above using a net income multiple as the basis for valuing the company. This approach produces the intrinsic value of the equity, not of the entire company. This method relies on different data and, as such, we will not get the same intrinsic values computed in the prior section. EXHIBIT 15.6 Estimating Intrinsic Value Using a Net Income Multiple (in millions, except per share amounts) Family Dollar Dollar General Equity assumed value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Net income . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Common shares outstanding . . . . . . . . . . . . . . . . . . . . . . . . . . $ 422 115.4 shares $15,138 $ 953 327.1 shares NI market multiple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equity intrinsic value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equity intrinsic value per share. . . . . . . . . . . . . . . . . . . . . . . . . 13.18 $5,562 $48.20 15.88 Big Lots $1,853 $ 177 57.3 shares 10.47 2015 Cambridge Business Publishers | For the Personal Use of Amanda Rosena. 15-11 Module 15 | Market-Based Valuation We begin by computing the net income market multiple for both Dollar General and Big Lots, which is computed as equity assumed value divided by net income. Exhibit 15.6 shows the results of this computation. For example, Dollar General's net income market multiple is 15.88, computed as $15,138/$953. We then average the net income market multiples (from Dollar General and Big Lots) to get 13.18, computed as (15.88 1 10.47)/2. This NI market multiple is used to estimate the equity intrinsic value of Family Dollar as follows: Equity intrinsic value 5 Net income 3 NI market multiple $5,562 5 $422 3 13.18 55775 To obtain Family Dollar's equity intrinsic value per share we divide by shares outstanding as follows: Equity intrinsic value Common shares outstanding $5,562 5 115.4 shares Equity intrinsic value per share 5 $48.20 The $48.20 stock price estimate suggests that Family Dollar stock was overvalued based on a $62.35 closing price at its fiscal year-end (and vis--vis its $65.10 price on the filing date). The estimated value would have been different had we used Dollar General's market multiple alone, and yet again different if we had used Big Lots' multiple alone, and again different if we had computed a weighted average. It is again useful for us to compare estimates of Family Dollar's intrinsic value of equity using the NOPAT multiple vis--vis the NI multiple. Using the NOPAT multiple, we estimated the intrinsic value of a share of Family Dollar to be $50.79, while the NI market multiple gave an estimate of $48.20. How do we assess the quality of the different estimates? The estimate using the NOPAT market multiple is likely better (for the same reasons that the NOA multiple was superior to the BV multiple for balance sheet methods). That is, we did not select comparables with similar capital structures. When selecting comparables, we should selectfor firms that are profitability, growth, do control operating similar risk, byonchoosing companies in and the risk. sameAlthough industry, the we comparables did not control for financial risk. Consequently, because financial risk affects the equity value, but not the company value, we prefer the NOPAT multiple. This means the different capital structures of Family Dollar, Dollar General, and Big Lots do not matter when applying NOPAT multiples in valuation. But, when using net income multiples to estimate intrinsic value, it is important to select comparables with similar capital structures. The estimates of equity value we have computed for Family Dollar thus far using multiples highlight the first issue we raised about the use of this method. The two methods based on net operating assets and book value market multiples yielded similar values of $29.07 and $30.82 per share respectively. The two methods based on net operating profit after tax and net income market multiples yielded similar values of $50.79 and $48.20 per share respectively. Given that the balance sheet and the income statement based estimates differ by approximately 65% we are faced with the question: which performance measure is the right one? Again, this question has no answer (at least not until price is determined at some future point). Often in practice upon finding different estimates, a process of justifying the desired answer ensues. Beware as this is a dangerous path to take. Valuation Using Industry-Based Multiples Some industries have specific measures related to characteristics of the industry that are closely watched by investors and analysts. For example, in the retail industry sales per square foot of selling space is a common measure used to understand the sales each location is able to generate 2015 Cambridge Business Publishers | For the Personal Use of Amanda Rosena. Module 15 | Market-Based Valuation 15-12 relative to its size. Another example is the airline industry, which focuses on revenues, expenses or profits per available seat mile (one aircraft seat flown one mile, whether occupied or not). These measures can be used to create additional \"industry-based\" multiples that can be examined in valuing a company. To illustrate the use of an industry-based multiple to estimate value, and using data from the MD&A section of the 10K, we find in fiscal year 2012 that Family Dollar sales per square foot of selling space was approximately $181. In contrast, Dollar General and Big Lots had sales per square foot of selling space of $216 and $163, respectively. Exhibit 15.7 shows the results of this analysis for these three companies. The $63.88 stock price estimate suggests that Family Dollar stock was fairly priced based on a $62.35 closing price at its fiscal year-end (and vis--vis its $65.10 price on the filing date). EXHIBIT 15.7 55775 Estimating Intrinsic Value Using an Industry-Based Multiple (in millions, except per share amounts) Family Dollar Dollar General Equity assumed value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sales per square foot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Common shares outstanding . . . . . . . . . . . . . . . . . . . . . . . . . . $ 181 115.4 shares $15,138 $ 216 327.1 shares Sales per square foot market multiple . . . . . . . . . . . . . . . . . . . Equity intrinsic value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equity intrinsic value per share. . . . . . . . . . . . . . . . . . . . . . . . . 40.73 $7,372 $63.88 70.08 Big Lots $1,853 $ 163 57.3 shares 11.37 The limitations of the use of an industry-based multiple must be recognized. In this case, we are using a measure that focuses on sales rather than focusing on profitability. Further, if the company we are trying to value has a different cost structure than its comparables, this will add error to our estimation. Combining Estimates from Differing Multiples In prior sections we the arrived at aofvariety of estimates of the value of Family Dollar. Rather them than attempting to weigh merits the different estimates, some investors prefer to combine in the hope that estimation error for a specific multiple is mitigated by other multiples. This combination is typically done using a simple average of the estimates. Consider the following example in Exhibit 15.8, which combines the various estimates that we previously computed for Family Dollar. In the first column we exclude the estimate based on sales-per-square-foot, whereas the second column excludes the estimate based on NOA. We do this to show the sensitivity to this combination method. EXHIBIT 15.8 Combining Estimates of Equity Intrinsic Value per Share Multiple Used Net Operating Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Book Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Net Operating Profit after Tax. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Net Income. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sales per Square Foot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Average of Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimate #1 Estimate #2 $29.07 $30.82 $50.79 $48.20 - $39.72 $30.82 $50.79 $48.20 $63.88 $48.42 By choosing to compute an estimate that includes sales-per-square-foot and not NOA, we get an average estimate of $48.42, which is more than 20% higher than the $39.72 estimate of the intrinsic value of Family Dollar that excludes sales-per-square-foot. Before deciding what multiples to use and how we might appropriately weight them, we must understand more about the purpose and role of selecting comparable companies. 2015 Cambridge Business Publishers | For the Personal Use of Amanda Rosena. 15-13 Module 15 | Market-Based Valuation MID -MO DULE R EVI EW 2 The table below reports summary data for Johnson & Johnson (JNJ), along with that for Procter & Gambleand Merck, two large cap firms that compete in many segments of JNJ's medical and consumer products businesses. (in millions) Company assumed value . . . . . . . . . . . . . . Equity assumed value . . . . . . . . . . . . . . . . Net operating profit after tax . . . . . . . . . . . Net income . . . . . . . . . . . . . . . . . . . . . . . . . Net nonoperating obligations (assets) . . . . Common shares outstanding . . . . . . . . . . . Johnson & Johnson Procter & Gamble Merck $10,563 $11,053 $ 6,576 2,893 shares $238,532 $203,325 $ 11,188 $ 10,340 $ 35,212 3,132 shares $ 99,560 $103,315 $ 4,181 $ 4,434 $ (3,750) 2,168 shares 55775 Required a. Estimate the NOPAT market multiple for Procter & Gamble and for Merck. Then, compute the average of these as the NOPAT market multiple for valuation. If we believe that one company's multiple was more reliable than another, how could we adjust our estimation? b. Use the results from part a to estimate the company intrinsic value, the equity intrinsic value, and the equity intrinsic value per share for JNJ. c. Compute the NI multiple (also called equity-to-income) for Procter & Gamble and for Merck. Then, compute the average of these as the NI market multiple for valuation. d. Use the results from part c to estimate equity intrinsic value and the equity intrinsic value per share for JNJ. The solution is on page 15-38. SELECTING COMPARABLES FOR MARKET MULTIPLES L O 3 Identify comparable companies for use in company valuation with market multiples. When valuing companies using multiples, it is by important to selecthow comparables similar on growth, risk.market Wefrom begin this section demonstrating theoretically correctprofitability, market multiples canand be derived the residual income operating (ROPI) model. We then examine how such multiples change with variations in key characteristics of the comparables. It is important for us to remember that all market multiples, except industry-based multiples, illustrated in the previous sections can be derived from the residual income operating model. However, in this section we limit ourselves to the Price-to-Book (the BV multiple used earlier) and Price-to-Earnings (the NI multiple used earlier) multiples. The reason is that they are the multiples most commonly used in practice. This section describes how key factorssuch as profitability, growth, and operating riskshould guide our selection of comparables. However, as we explained in previous sections, choosing comparables for NOA and NOPAT multiples does not depend on similar capital structures. Deriving Price-to-Book from Residual Operating Income Model Recall from Module 14 the following residual operating income (ROPI) model. Firm value 5 NOA 1 Present value of expected ROPI, discounted using rw Specifically, residual operating income is operating income in excess of a fair rate of return on beginning net operating assets: ROPI 5 NOPAT 2 1 NOABeg 3 rw 2 where NOA r Beg 5 w Net operating assets at beginning of period 5 Weighted average cost of capital 2015 Cambridge Business Publishers | For the Personal Use of Amanda Rosena. Module 15 | Market-Based Valuation 15-14 If we divide both sides of the ROPI model by book value of equity (EQ), we have a formula for the price-to-book, or PB, ratio (we exclude the algebraic derivation): Present value of expected ROPI, discounted using rw EQ This formula reveals the parameters determining PB ratios: residual operating income, the expected growth rate in residual operating income, the weighted average cost of capital, and the company's leverageEQ is less for companies with debt in their capital structure. From this formula we can show that companies which are expected to generate no future residual operating income should sell at book value and carry a PB of 1. Companies with lower profitability, lower expected growth rates, and higher discount rates should sell at lower PB ratios. Companies with higher profitability, higher expected growth rates, and lower discount rates should sell at higher PB ratios. Further, levered companies will sell at higher PB ratios than unlevered companies. We explain these relations in the next section. PB 5 1 1 55775 PB Ratios in Relation to Profitability, Growth, and Risk The parameters determining PB are readily identified if we assume a simple model for future residual operating income. One simple model assumes that residual operating income grows at a constant rate, g, in perpetuity. Using the formula for the present value of a perpetuity, we express the present value of residual income as follows: Present value of expected ROPI 5 where Expected ROPI rw 2 g Expected ROPI is for period t 1 1 g is the growth rate in ROPI each period r w is the weighted average cost of capital We use this model in the following sections to value the present value of expected residual operating purposes of illustrating the relation between PB ratios and the factors of profitability,income growthfor rates, operating risk, and leverage. PB Increases with Company Expected Profitability Exhibit 15.9 reports data for two companies with identical net operating assets (of $100) and identical capital structures (all equity). Company A has RNOA (and ROE) of 22%, while Company B has RNOA (and ROE) of 14%. We assume these companies have the same operating risk (with a weighted average cost of capital of 10%), and we expect perpetual growth in residual operating income equal to 2% for both. Using the model for the present value of a perpetuity with growth of g (which is Value 5 Amount/[r 2 g]), we determine the present value of expected future ROPI for Company A as $150, computed as $12/(0.10 2 0.02). The present value for the low profitability Company B is $50. EXHIBIT 15.9 PB in Relation to Profitability Company A: High profitability . . . Company B: Low profitability. . . . Net operating assets Stockholders' equity RNOA $100 $100 $100 $100 22% 14% ROE Weighted average cost of capital Residual operating income Expected growth rate in ROPI Present value of expected ROPI using rw Value of equity 22% 14% 10% 10% $12 $ 4 2% 2% $150 $ 50 $250 $150 Next, we compute the value of the equity for the high profitability Company A as $250, computed as $100 in NOA plus the $150 present value of future ROPI. Similarly, the low profitability Company B's value of equity is $150, computed as $100 in NOA plus the $50 present value. From these results we obtain the theoretically correct equity multiple for the high profitability Company A of 2.5, computed as 1 plus the quantity of the $150 present value divided by the $100 2015 Cambridge Business Publishers | For the Personal Use of Amanda Rosena. 15-15 Module 15 | Market-Based Valuation stockholders' equity. Similarly, the low profitability Company B's multiple is 1.5, computed as 1 plus the quantity of the $50 present value divided by the $100 stockholders' equity. These results yield the following inference: PB increases with company expected profitability. PB Increases with Company Expected Growth Exhibit 15.10 reports data for two companies with identical net operating assets ($100), capital structures, and profitability. Both companies have RNOA (and ROE) of 22%, and have similar operating risk (with a weighted average cost of capital of 10%). However, Company C is a high growth firm, with a perpetual growth rate of 4%, compared to a growth rate of 0% for Company D. (Recall that a zero growth rate in ROPI does not imply a zero growth rate in net income.) Using the formula for the present value of a perpetuity with growth of g, we find the present value of expected future residual income for the high growth Company C is $200, computed as $12/(0.10 2 0.04). For the no growth Company D, the present value is $120, computed as $12/ (0.10 2 0.00). 55775 EXHIBIT 15.10 PB in Relation to Growth Net operating assets Stockholders' equity RNOA $100 $100 $100 $100 22% 22% Company C: High growth. . . . . . Company D: Low growth . . . . . . ROE Weighted average cost of capital Residual operating income Expected growth rate in ROPI Present value of expected ROPI using rw Value of equity 22% 22% 10% 10% $12 $12 4% 0% $200 $120 $300 $220 Next, we compute the value of the equity for the high growth Company C as $300, computed as $100 in NOA plus the $200 present value. Similarly, the low growth Company D's value of equity is $220, computed as $100 in NOA plus the $120 present value. From these results we obtain the theoretically correct equity multiple for the high growth Company C of 3.0, computed as 1 plus the quantity of the $200 present value divided by the $100 stockholders' equity. Similarly, the low growth Company D's multiple is 2.2, computed as 1 plus the of the inference: $120 present divided the $100expected stockholders' equity. These results yieldquantity the following PB value increases withby company growth. PB Decreases with Increasing Company Operating Risk Exhibit 15.11 reports data for two companies with identical net operating assets ($100), identical capital structures (no debt), and identical RNOA of 22%. However, Company E is a high operating risk firm, operating in an industry with high technological uncertainty, while Company F has low operating risk. Accordingly, high risk Company E has a higher cost of capital (15%) compared with the low risk Company F (10%). Using the formula for the present value of a perpetuity with growth of g, we find the present value of expected future residual income for the high risk Company E is $46.7, computed as $7/(0.15 2 0.00). For the low risk Company F, the present value is $120, computed as $12/ (0.10 2 0.00). EXHIBIT 15.11 PB in Relation to Operating Risk Net operating Stockholders' assets equity RNOA Company E: High operating risk . . . Company F: Low operating risk . . . . $100 $100 $100 $100 22% 22% ROE Weighted average cost of capital 22% 22% 15% 10% Residual Expected operating growth rate income in ROPI $7 $12 0% 0% Present value of expected ROPI using rw Value of equity $46.7 $120 $146.7 $220 Next, we compute the value of equity for the high risk Company E as $146.7, computed as $100 in NOA plus the $46.7 present value. Similarly, the low risk Company F's value of equity is $220, computed as $100 in NOA plus the $120 present value. 2015 Cambridge Business Publishers | For the Personal Use of Amanda Rosena. Module 15 | Market-Based Valuation 15-16 From these results we obtain the theoretically correct equity multiple for the high risk Company E of 1.47, computed as 1 plus the quantity of the $46.7 present value divided by the $100 stockholders' equity. Similarly, the low risk Company F's multiple is 2.2, computed as 1 plus the quantity of the $120 present value divided by the $100 stockholders' equity. These results yield the following inference: PB decreases as company operating risk increases. BUSINESS INSIGHT Fair Value Accounting and PB Ratios 55775 The FASB's recent writings suggest a preference for fair value accounting, which it regards as more relevant than historical cost accounting. Beginning in fiscal 2008, companies were permitted, but not necessarily required, to value any financial asset or liability at fair value and to record changes in fair value in income. We should consider this rule when choosing comparables for valuation by market multiples because companies using fair value accounting will tend to have lower PB ratios and more volatile earnings than those using historical cost accounting. PB Increases with Company Leverage PB ratios are higher than Price-to-NOA ratios when companies carry debt. The Price-to-NOA ratio is often referred to as an unlevered PB ratio. Exhibit 15.12 reports data for two companies with identical net operating assets ($100), identical weighted average cost of capital (10%), identical RNOA of 22%, and identical growth of 0%. However, Company G finances its operations with equity, while Company H finances its operations with $60 of debt with an after-tax rate of 6%, and 2 $40 of equity. The cost of equity capital for Company G is 10%, while it is 11.5% for Company H. Using the formula for the present value of a perpetuity with growth of g, we find the present value of expected future residual income for both the unleveraged and leveraged companies is $120, computed as $12/(0.10 2 0.00). Consequently, the theoretically correct Price-to-NOA multiple for both companies is 2.2, computed as 1 plus the quantity of the $120 present value divided by the $100 NOA. The unleveraged Company G's PB ratio is the same as its 2.2 Price-to-NOA multiple. However, the PB multiple for the leveraged Company H is 4.0, computed as 1 plus the quantity of the $120 present value divided by the $40 stockholders' equity. EXHIBIT 15.12 PB in Relation to Leverage Net Debt operating (6% Stockholders' assets rate) equity RNOA ROE Weighted Cost of Residual Growth Present value average cost equity operating rate in of expected of capital capital income ROPI ROPI using rw Company G: Unleveraged . . . . . $100 $ 0 $100 22% 22% 10% 10% $12 0% $120 $220 Company H: Leveraged . . . . . . . $100 $60 $ 40 22% 46% 10% 11.5% $12 0% $120 $160 Why would we pay more for a dollar of equity than for a dollar of net operating assets? The reason is that for every dollar of net operating assets, the company generates ROPI with a present value of $1.20 ($120/$100). However, for every dollar of equity, the company generates ROPI with a present value of $3 ($120/$40). Accordingly, this scenario shows that levered PB ratios are higher than unlevered PB ratios and yields the following inference: PB increases with company leverage. In sum, when selecting comparable firms to use in valuation based on a multiple of book value, we should select firms with similar operating profitability, similar expected growth, and similar riskboth operating and financial. IVEquity IV Debt b 1 a re 3 b . The IVFirm IVFirm high leverage Company H has debt of $60, with an after-tax interest rate of 6%. That company's intrinsic value is $220, its intrinsic value of debt is $60, and its intrinsic value of equity is $160. 2 Value of equity Recall from Module 12 that the weighted average cost of capital is rw 5 a rd 3 2015 Cambridge Business Publishers | For the Personal Use of Amanda Rosena. 15-17 Module 15 | Market-Based Valuation BUSINESS INSIGHT Quality of Comparables and Simple Controls Selecting firms with similar operating risk, leverage, and profitability can improve our valuations. Yet, it is unlikely to compensate for the substantive deficiencies in the technique. The table below shows RNOA, leverage, and PB ratios for a set of consumer stocks (pretax ROA is between 8% and 9% for all). We see from this small sample that simple controlsfor industry, current profitability, and leverageexplain little of the variation in PB ratios. 55775 Company RNOA MDC Holdings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dow Jones & Company . . . . . . . . . . . . . . . . . . . . . . . . . . . Iconix Brand Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oxford Industries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brown Shoe Company . . . . . . . . . . . . . . . . . . . . . . . . . . . . Burger King Holdings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cabelas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ConAgra Food . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ruddick Corporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . News Corporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lifetime Brands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . McCormick & Schmick's Seafood Restaurant . . . . . . . . . . 8.6% 8.6 8.7 8.8 8.8 8.8 8.8 8.8 8.9 8.9 8.9 8.9 Debt-to-Equity 50.7% 0.0 127.1 44.2 26.8 131.0 54.3 74.6 36.6 36.9 61.7 5.2 PB Ratio 0.9 8.6 2.7 1.5 1.6 4.9 2.1 2.8 2.2 1.4 1.7 2.1 MID -MO DULE R EVI EW 3 The following table reports data for Companies A through E with identical net operating assets ($100), but with differences in debt levels, returns, cost of capital, and growth. Net Debt Weighted Cost of operating (6% Stockholders' average cost equity Company assets rate) equity RNOA ROE of capital capital A. . . . . . B. . . . . . C. . . . . . D. . . . . . E...... $100 $100 $100 $100 $100 $ 0 $ 0 $ 0 $ 0 $60 $100 $100 $100 $100 $ 40 16% 12% 16% 16% 16% 16% 12% 16% 16% 31% 10% 10% 10% 12% 10% 10% 10% 10% 12% 12.1% Residual operating income Growth Present value Value of rate in of expected ROPI ROPI using r w company Value of equity 2% 2% 4% 2% 2% Required a. Assume that the present value of expected ROPI follows a perpetuity with growth g (Value 5 Amount/[r 2 g]). Determine the theoretically correct PB ratio for each company A through E. b. State the relation between the PB ratio and that of profitability, expected growth in ROPI, expected operating risk, and financial leverage. Identify two companies for each of the four relations that illustrate that relation (companies can be listed more than once). The solution is on page 15-38. Deriving Price-to-Earnings from Residual Operating Income Model We can derive a model for the theoretical value of the price-to-earnings (PE) ratio using the residual operating income model in a way similar to what we did for the PB model. PE