The present value of a single payment of B dollars in year T when the interest rate is R is given by the formula:

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1 Notes on Present Value Page 1 Web Notes > Microeconomics HE MAXWELL SCHOOL SYACUSE UNIVESIY Here are the key equations for doing present value calculations. A single payment he present value of a single payment of dollars in year when the interest rate is is given by the formula: = /(1+) he cash flow diagram that corresponds to this case is: A series of payments he present value of a series of payments is just the sum of the present values of the individual payments. For example, the of a sequence of payments of 1 dollars in year 1, 2 dollars in year 2, zero dollars in year 3 and 4 dollars in year 4 would be: = 1/(1+) + 2/(1+) 2 + 4/(1+) 4 he cash flow diagram that corresponds to this case is: An infinite series of payments he present value of receiving dollars per year forever starting one year from now is: = / he cash flow diagram that corresponds to this case is: An easy way to remember the formula is to turn it around this way: = * his says that you would need to put dollars in the bank at interest rate in order to earn dollars in interest each year forever.

2 Notes on Present Value Page 2 Net present value If the stream of payments involves costs as well as benefits then the present value is the sum of the present values of the individual payments treating the costs as negative payments. his is called "net present value". It can be calculated in two equivalent ways. One approach is to compute the present value of the benefits minus the present value of the costs: N = () - (C) he other approach is to subtract each year's costs from that year's benefits and take the present value of the stream of net payments: N = (-C) hese give identical results and you can use whichever one is most convenient for the particular problem at hand. Additional esources Exercises Problems involving present value calculations, with solutions. Present Value 2: Combined Forms Some extensions built from the fundamentals above. UL: Peter J Wilcoxen, he Maxwell School, Syracuse University evised 11/21/2011

3 Notes on Present Value Page 3 Peter J Wilcoxen Economics and Public Administration he Maxwell School Syracuse University Present Value 2: Combined Forms he fundamental equations for present value (see Present Value 1: Fundamentals ) can be combined to analyze more complex cash flows. Here are a couple of important special cases. In all examples, the interest rate is given by. Infinite stream with a delayed start Suppose an infinite stream of equal payments of dollars begins in year +1. he cash flow diagram would be: he present value can be computed in two steps. First, the infinite stream is converted to an equivalent lump sum in the year before the first payment arrives in this case, year. hat s easy because from year s perspective, the stream of payments is very simple: the payment at +1 occurs 1 year in the future, the payment at +2 is 2 years in the future, and so on. hus, from period s perspective it s an infinite stream with payments beginning in one year in the future: From period s perspective, therefore, the value of that stream is /. hus, the original infinite stream from +1 on can be replaced by a single lump-sum payment of / in period : he second step is to compute the present value (at year 0) of the lump sum payment in period. hat s a straightforward application of the usual present value formula but with / inserted for the payment at : 1

4 Notes on Present Value Page 4 As an example, suppose a stream pays $1M per year beginning in year 11, and the interest rate is 10%. In step 1, the lump sum payment in year 10 would be $1M/0.1 or $10M. In step 2, the final would be $10M/(1.1)^10 = $3.86M. Long but finite stream Now suppose a stream has equal payments from year 1 to but no payments after that: he present value could be computed by summing the present values of all payments. However, there s a faster way that s more versatile as well. he first step is observe that the original series (call it stream I ) could be combined with one beginning at +1 (call that stream II ) to give an infinite series beginning at 1 (stream III ): Stream I I? Stream II II 1 Stream III III Since stream III has exactly the same payments as streams I and II combined, its present value will be the sum of the present values of the other two: III I II hus, the present value of stream I can be computed from the other two: I III II 2

5 Notes on Present Value Page 5 Substituting in the equations for III and II : I 1 Factoring out / gives the following: I Essentially, this says that the present value of a long stream is equal to the present value of an infinite stream less a correction for the fact that the stream doesn t actually last forever. For example, suppose a stream pays $1M per year from years 1-10 and the interest rate is 10%. he value of an infinite stream would be $1M/0.1 or $10 million. he value of the finite stream is somewhat lower: 1 I 10M 1 10M M 11. Notice that the sum of this value and the previous example total exactly $10M, the value of receiving $1M per year forever starting in year 1. hat s because together they produce exactly that set of cash flows. 3