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July 2017. Ivo W Elch,Corp Lch,Corporate Finance 3 Stock and Bond Valuation: Annuities and Perpetuities Important Shortcut Formulas The present value formula is the main workhorse for valuing investments of all types, including stocks and bonds. But these rarely have just two or three future payments. Stocks may pay dividends forever. The most common mortgage bond has 360 monthly payments. It would be possible but tedious to work with NPV formulas containing 360 terms. Fortunately, there are some shortcut formulas that can speed up your PV computations if your projects have a particular set of cash flow patterns and the opportunity cost of capital is constant. The two most prominent are for projects called perpetuities (which have payments lasting forever) and annuities (which have payments lasting for a limited number of years). Of course, no firm lasts forever, but the perpetuity formula is often a useful “quick-and-dirty” tool for a good approximation. In any case, the formulas in this chapter are widely used and can help you understand the economics of corporate growth. 3.1 Perpetuities A simple perpetuity is a project with a stream of constant cash flows that repeats forever. If “Perpetuities” are projects the cost of capital (i.e., the appropriate discount rate) is constant and the amount of money with special kinds of cash remains the same or grows at a constant rate, perpetuities lend themselves to fast present value flows, which permit the use solutions—very useful when you need to come up with quick rule-of-thumb estimates. Though of shortcut formulas. the formulas may seem intimidating at first, using them will quickly become second nature to Ivo Welch, Corporate Finance (4th Edition), July 2017. Ivo Welch, Corporateyou. Finance (4th Edition), July 2017. Ivo Welch, Corporate Finance (4th Edition), July 2017. Ivo Welch, Corporate Finance (4th Edition), July 2017. Ivo Welch, Corporate Finance (4th Edition), July 2017. Ivo Welch, Corporate Finance (4th Edition), July 2017. Ivo Welch, Corporate Finance (4thThe Edition), Simple Perpetuity July 2017. Formula Ivo Welch, Corporate Finance (4th Edition), July 2017. Ivo Welch, Corporate Finance (4th Edition), July 2017. Ivo Welch, Corporate Finance (4th Edition), July 2017. Ivo Welch, Corporate Finance (4th Edition), July At a constant interest rate of 10%, how much money do you need to invest today to receive the Here is an example of a 2017. Ivo Welch, Corporate Finance (4th Edition), July 2017.same Ivo dollar Welch, amount Corporate of interest of $2Finance each year, (4th starting Edition), next year, forever? July Exhibit 2017. 3.1 Ivoshows Welch, Corporate Finance (4th Edition), July 2017. Ivo Welch, Corporate Finance (4th Edition), July 2017. Ivo Welch, Corporate Finance (4th Edition), July 2017. Ivo Welch, Corporate perpetuity that pays $2 Finance (4th Edition), July 2017. Ivo Welch, Corporate Financethe present (4th values Edition), of all future payments July 2017. for a perpetuity Ivo Welch, paying $2 forever,Corporate if the interest Finance rate (4thforever. Edition), July 2017. Ivo Welch, Corporate Finance (4th Edition), July 2017. Ivo Welch, Corporate Finance (4th Edition), July 2017. Ivo Welch, Corporate Finance (4th Edition), July is 10% per annum. Note how there is no payment at time 0, and that the individual payment 2017. Ivo Welch, Corporate Finance (4th Edition), Julyterms 2017. becomeIvo smaller Welch, and smaller Corporate the further out Finance we go. (4th Edition), July 2017. Ivo Welch, Corporate Finance (4th Edition), July 2017. Ivo Welch, Corporate Finance (4th Edition), July 2017. Ivo Welch, Corporate Finance (4th Edition), July 2017. Ivo Welch, Corporate To confirm the table’s last row, which gives the perpetuity’s net present value as $20, you can The shortcut perpetuity Finance (4th Edition), July 2017. Ivo Welch, Corporatespend from Finance here to eternity (4th to add Edition), up the infinite July number 2017. of terms. Ivo But Welch, if you use a Corporate spreadsheet Finance (4th Edition), July 2017. Ivo Welch, Corporate Finance (4th Edition), July 2017. Ivo Welch, Corporate Finance (4th Edition), July 2017. Ivo Welch, Corporate Finance (4th Edition), July formula. to compute and add up the first 50 terms, you will get a PV of $19.83. If you add up the first 2017. Ivo Welch, Corporate Finance (4th Edition),100 terms, July you will 2017. get a PV Ivo of $19.9986. Welch, Mathematically, Corporate the Finance sum eventually (4th converges Edition), to July 2017. Ivo Welch, Corporate Finance (4th Edition), July 2017. Ivo Welch, Corporate Finance (4th Edition), July 2017. Ivo Welch, Corporate Finance (4th Edition), July 2017. Ivo Welch, Corporate $20 sharp. This is because there is a nice shortcut to computing the net present value of the Finance (4th Edition), July 2017. 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