Transcription of 6.5 Present and Future Value of a Continuous Income Stream
1 Present AND Future Value OF A Continuous Income Present and Future Value of a ContinuousIncome StreamWhen an Income Stream flows into an investment, the investment grows be-cause of the Continuous flows of money and the interest compounded on themoney invested. Thus, two functions are required: a function defining theflow of money, and a function defining a function Income StreamMany business deals involve payments in the Future . For example, when acar or a home is bought on credits, payments are made over a period of Value ,F V,of a paymentPis the amount to whichPwould havegrown if deposited today in an interest bearing bank account.
2 Thepresentvalue,P V,of a Future paymentF V,is the amount that would have to bedeposited in a bank account today to produce exactlyF Vin the account atthe relevant time interest is compoundedntimes a year at an annual raterfortyears, thenthe relationship betweenF VandP Vis given by the formulaF V=P V(1 +rn) the case of Continuous compound interest, the formula is given byF V=P V need $10,000 in your account 3 years from now and the interest rate is8% per year, compounded continuously. How much should you deposit now? haveF V= $10,000, r= , t= 3 and we want to findP theformulaF V=P V ertforP Vwe findP V=F V e to obtain,P V= 10,000e $7, Income StreamIn the above discussion we introduced the Present and Future Value of a sin-gle payment.
3 Next, we want to calculate the Present and Future Value of acontinuous Stream of revenues earned by a huge corporation (such as an electricity company),for example, come in essentially all the time, and therefore they can be rep-resented by a Continuous Income the rate at which revenue is earned may vary from time to time, theincome Stream is described by a functionS(t) which represents the flow ratein dollars per year. Note that the rate depends on time, t,usually measuredin years from the find the Present Value of a Continuous Income Stream over a period ofMyears we divide the interval [0, M] intonequal subintervals each of length t=Mnand with division points 0 =t0< t1< < tn= is,over each time interval we are assuming a single payment is made.
4 Assuminginterestris compounded continuously, the Present Value of the total moneydeposited is approximated by the following Riemann sum:P V S(t1)e rt1 t+S(t2)e rt2 t+ S(tn)e rtn t=n i=1S(ti)e rti t 0, ,we obtainP V= M0S(t)e Future Value is given byF V=erM M0S(t)e investor is investing $ million a year in an account returning Assuming a Continuous Income Stream and Continuous compoundingof interest, how much will these investments be worth 10 years from now? the formula for the Future Value defined above we findF V= $ what constant, Continuous rate must money be deposited into an accountif the account contain $20,000 in 5 years?
5 The account earns 6% interestcompounded Present AND Future Value OF A Continuous Income V= $20,000, M= 5, r= assumed to be constantthen we have20,000 =S 50e forSwe findS=20,000 50e $4,630per year.
