Discrete dynamic systems109(a)FV=A(1+r)n−1r=1000(1+0.065)10−10.065=£13494.40(b)PV=A1−(1+r)−nr=10001−(1+0.065)−100.065=£7188.83Discounting is readily used in investment appraisal and cost–benefit analysis.SupposeBtandCtdenote the benefits and costs, respectively, at timet. Thenthe present value of such ﬂows areBt/(1+r)tandCt/(1+r)t, respectively. Itfollows, then, that thenet present value,NPV, of a project with financial ﬂows overn-periods isNPV=nt=0Bt(1+r)t−nt=0Ct(1+r)t=nt=0Bt−Ct(1+r)tNotice that fort=0 the benefitsB0and the costsC0involve no discounting. Inmany projects no benefits accrue in early years only costs. IfNPV>0 then aproject (or investment) should be undertaken.Example 3.9Bramwell plc is considering buying a new welding machine to increase its output.The machine would cost£40,000 but would lead to increased revenue of£7,500each year for the next ten years. Half way through the machine’s lifespan, in year 5,there is a one-off maintenance expense of£5,000. Bramwell plc consider that theappropriate discount rate is 8%. Should they buy the machine?NPV= −40000+10t=17500(1+r)t−5000(1+
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