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# Ch5slides - Future Value and Present Value Standard Annuity...

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Future Value and Present Value Standard Annuity Modified annuities Conclusion Ch5. The time value of money JHT Kim ActSc 371 January 27, 2009 1/26

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Future Value and Present Value Standard Annuity Modified annuities Conclusion Notations Different texts use different terminology in interest theory. In this course (including this chapter): Discount rate is the same as interest rate Nominal rate of interest is the same as the Annual Percentage Rate (APR) 2/26
Future Value and Present Value Standard Annuity Modified annuities Conclusion Future Value If your invest A at an annual rate of r (100r%), the FV in your acct in one year is A (1 + r ) This can be split into the principal A and the interest amount earned A (1 + r ) - A = Ar . The FV in n yrs is A (1 + r ) n and can again be split into principal A and the interest amount A (1 + r ) n - A 3/26

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Future Value and Present Value Standard Annuity Modified annuities Conclusion Present Value Suppose your bank offers you an annual interest rate of r . If you want your FV to be B in one yr the amount you need to invest now is B / (1 + r ) We call 1 / (1 + r ) the discount factor or the PV factor. Since we use this quite often we denote 1 / (1 + r ) by v . If you want the FV to be B in n yrs, your initial investment amount now (or PV) should be PV 0 = B / (1 + r ) n = Bv n Note: the text uses term NPV when there are initial costs, but we will use them interchangeably. 4/26
Future Value and Present Value Standard Annuity Modified annuities Conclusion Present Value Example You can invest in a project with \$ B 0 now to get \$ B 1 in one year from now and \$ B 2 in 2 yrs. Compute the NPV of this project at an annual rate of r. 5/26

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Future Value and Present Value Standard Annuity Modified annuities Conclusion NPV principle For a project (say, A) If NPV(A) is positive should undertake the project (value created) If NPV(A) is negative should forgo the project (value cutback) For two projects (A and B), assuming only one of the two can be chosen, If NPV(A) > NPV(B) should choose A If NPV(A) < NPV(B) should choose B 6/26
Future Value and Present Value Standard Annuity Modified annuities Conclusion Definition of annuity An annuity is a series of payments that lasts for a fixed number of periods. Consider an annuity that has the following payment pattern: We use an actuarial symbol for the PV of this annuity. That is, a n | r = 1 (1 + r ) + 1 (1 + r ) 2 + ... + 1 (1 + r ) n = v + v 2 + ... + v n We can simplify this (how?) to arrive at an imprtant formula a n | r = 1 - v n r 7/26

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Future Value and Present Value Standard Annuity Modified annuities Conclusion Notation a n | r is commonly called the annuity factor (text uses A n r instead) When the payment amount is \$ C, the the PV of the annuity becomes C a n | r
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