hAs decribed in the chapter, the abnormal earnings approach for estimating commob share value is P$0=$BV$0+\backslash$sum$\_\{$t$=1\}^\{\backslash$infty $\}\{$E$\_\{0\}($x$\_\{1\}-$r$\backslash$times BV$\_\{$t$-1\}\})/\{(1+$r$)^\{$t$\}\}$where P$0$is the total of all outstanding shares. BV$0$ is the current bookfolders' equity, BVt$-1$ is the book value of shareholders equity in the beginning of the period t$,$ r is the cost of equity capitol, E$0$ is the exceptions operator, and X$1$ 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 the 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 small present value and are essesntially 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 $10.$ Comparitive income statements and retained earning statements for Illinois Tool Works $($ITW$)$ for Year $1-$Year $3$ follow.

REQUIRED:

$1)$ Assume a $10$ year forcasting horizon. ALso assume ITW's Year $3$ return on beginning stockholders' equity $($net income without extraordinary items divided by beginning Year $3$ stockholders' equity$)$ of $9\mathrm{.}5\%$ is expected to persist throughout the forecasting horizon $($that is$,$ that expected net income is always equal to $0\mathrm{.}095$ multiplied by beginning of the year stockholders equity$).$ Also assume that no additional stock issuances or repurchases are made and dividends equal $25\%$ of net income in each year. $($This is ITW's approximate historical dividend payment 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 $(1-0\mathrm{.}25)$ times Year $4$ net income. Finallyassume 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 ITW's common shares as of the end of Year $3.$ Ignore termianl 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 the ITW will maintain a $20\%$ return on beginning stockholders' equity over the $10$ year forecast horizon. What would the company's shares then be worth?