As described in the chapter, the abnormal earnings approach for estimating common share value is

where P0 is the total value of all outstanding shares, BV0 is the current book value of stock-holders€™ equity, BVt-1 is the book value of shareholders€™ equity at the beginning of period t, r is the cost of equity capital, E0 is the expectations operator, and Xt is period t net income. The model says that share value equals the book value of stockholders€™ equity plus the present value of future expected abnormal earnings (where abnormal earnings is net income minus the cost of equity capital multiplied by the beginning-of-period book value of stockholders€™ equity).
The model is silent on how one comes up with expected net income for future years (and therefore future expected abnormal earnings) and just how many future years should be used. Because of the way present value is calculated, abnormal earnings amounts expected for years in the distant future have a small present value and are essentially irrelevant to valuation, especially if abnormal earnings are close to zero in the long run, as many analysts assume. Thus, professional analysts rarely use more than 15 years, often fewer than 10.
Comparative income statements and retained earnings statements for Illinois Tool Works (ITW) for Year 1€“Year 3 follow.
Required:
1. Assume a 10-year forecasting horizon. Also assume that ITW€™s Year 3 return on beginning stockholders€™ equity (net income without extraordinary items divided by beginning Year 3 stockholders€™ equity) of 9.5% is expected to persist throughout the forecasting horizon (that is, that expected net income is always equal to 0.095 multiplied by beginning-of-the-year stockholders€™ equity). Also assume that no additional stock issuances or repurchases are made and that dividends equal 25% of net income in each year. (This is ITW€™s approximate historical dividend payout ratio.) Given these assumptions, the book value of stockholders€™ equity at the end of Year 4 equals book value at the beginning of Year 4 plus (1 - 0.25) times Year 4 net income. Finally assume that the cost of equity capital is 9%. (This is ITW€™s approximate cost of equity capital.) With these relatively simple assumptions, use the abnormal earnings model to estimate the total value of Illinois Tool Works€™ common shares as of the end of Year 3. Ignore terminal values at the end of the 10-year forecast horizon in your calculations.
2. As of the end of Year 3, 307 million common shares were outstanding. Convert your estimate in requirement 1 to a per share estimate. For purposes of comparison, the actual market value of ITW€™s common shares ranged from $56 to $64 during the first quarter of Year 4.
3. Now assume that ITW will maintain a 20% return on beginning stockholders€™ equity over the 10-year forecast horizon. What would the company€™s shares then beworth?

Illinois Tool Works Consolidated Statement of Income in millions Year 3 Year 2 Year I $9,293 5,936 5,910 6,192 3,383 1,691 $9,984 Sales Cost of goods sold Gross profit Selling and administrative expenses Depreciation and amortization Operating profit Interest expense Nonoperating income (expense) Special items Pre-tax income Total income taxes $9,468 3,532 1,720 306 1,506 68 3.792 1,814 414 1,564 72 (13) 385 1,307 68 1,479 521 1,235 429 (219) 0 $ 806 1,434 502 terms Net income S 713 S 958 Illinois Tool Works Consolidated Statement of Shareholders' Equity in millions) Year 3 Year 2 Year I Balance at beginning of year Net income loss) Stock issued (repurchased) Common stock dividends $ 9,823 713 361 (273) $10,624 $9,604 806 (337) (250) $9,823 $9,061 958 (192) (223) $9.604 Balance at end of year