Chapter 4 The Time Value of Money and Discounted Cash Flow Analysis

Transcription

1 Chaper 4 The Time Value of Money and Disouned Cash Flow nalysis 4.5 Muliple Cash Flows Suppose ha an invesmen plan has muliple ash inflows as follows. Cash inflows are $,000 a year from now and $2,000 wo years from now. Cash ouflow is $2,500 now. We desribe muliple ash flows using a diagram alled "ime line." Suppose ha your bank pays 0% per year if you keep money in he bank. Wha is NPV for his invesmen? The opporuniy os of apial is 0%. NPV is obained by alulaing he following. In[77]:= Clear[NPV] NPV= (+0.) ; Prin["NPV =", NPV] NPV = NPV is posiive. You should underake his invesmen. 4.6 nnuiies PV and FV of nnuiies Cash flows of he same amoun is alled "annuiy". Paying 0,000 yen per monh for gif erifiae of he deparmen sore onsiues an annuiy. If ash flow sars immediaely, i is alled an "immediae annuiy." If he ash flow sars a he end of he urren period, raher han immediaely, i is all an "ordinary annuiy." Fuure Value Le r be ineres rae per monh. Suppose you keep paying amoun $ eah monh for a year. Saring oday, you pay welve imes. pply monhly ompounding. This is an immediae annuiy. FV is given as a sum of geomeri sequene; FV = (), where () = +() () In ase of he gif erifiae program, ash flows of 0,000 yen onsiue immediae annuiy. Their fuure value is 3,000 yen. Presen Value Nex suppose ha you will reeive amoun $ per monh imes for a year. The firs paymen is a monh from now. Wha is PV of his ordinary annuiy? pply monhly ompounding. PV of he annuiy is expressed as a sum of geomeri sequene. is he firs erm and is he ommon raio Ch04.nb

2 PV of he firs ash flow = PV of he seond ash flow = PV of he las ash flow = () () 2 Then PV of his annuiy is given by ExampleofPVofannuiy. () Suppose ha you will reeive amoun $00 per monh imes for a year, saring he nex monh. Ineres rae per monh is peren. Then PV of his annuiy is $,5.5 as shown below. In erms of PV, having $,5.5 oday is equivalen o reeiving $00 eah monh for a year. In[80]:= Clear[, r]; =00; r=0.0; Prin " () = ", () () = 5.5 Example Finding IRR of Muliple Cash Flows In he example in 4.5 Muliple Cash Flows, NPV was posiive. So by NPV mehod, we onlude ha we should underake ha invesmen was good enough o underake. Le s see wheher we would have he same resul if we use IRR mehod. IRR is ineres rae suh ha NPV beomes zero. So o find IRR we solve he following equaion; 000 +x = 0 (+x) 2 In[82]:= Clear[x, ans] ans=nsolve 000 +x , {x} 2 (+x) Ou[83]= {{x }, {x 0.655}} In[84]:= ans ans[[]] ans[[2]] Ou[84]= {{x }, {x 0.655}} Ou[85]= {x } Ou[86]= {x 0.655} Ch04.nb 2

3 In[87]:= Clear[IRR] IRR=x/.ans[[2]] Ou[88]= In[89]:= Prin["IRR = ", IRR] IRR = Example 2 : Finding PV and IRR of nnuiy Gif erifiae program of he deparmen sore is an example of annuiy. In his ase, you pay onsan amoun regularly; monhly paymens. Paymen sars oday. So his is immediae annuiy. Fuure value is paid all a one. We wan o evaluae his immediae annuiy. We have wo mehods; NPV and IRR. NPV mehod We assume bank pays % per monh. This ineres rae is opporuniy os for you. So i beomes os of apial o find NPV. In[90]:= Clear[r, NPV] r=0.0; NPV= 3 () - =0 Prin["NPV= ", NPV] () ; NPV= NPV is posiive. The gif erifiae program is beer han keeping money in he bank aoun. IRR mehod IRR is ineres rae suh ha NPV beomes zero. So we solve In[94]:= Clear[ans, x] 3 ans=nsolve - (+x) =0 0, {x} (+x) 3 - (+x) =0 = 0. (+x) Ou[95]= {{x }, {x }, {x }, {x }, {x }, {x }, {x }, {x }, {x }, {x }, {x }, {x }} In[96]:= Clear[IRR] IRR=x/.ans[[]] Ou[97]= Ch04.nb 3

4 In[98]:= Prin["IRR per monh = ", IRR] IRR per monh = IRR is.2% per monh. This is higher han os of apial. In[99]:= Prin["ineres rae as annual perenage rae is equal o ", 00 IRR, "%"] ineres rae as annual perenage rae is equal o % 4.8 Loan morizaion Many loans are repaid in equal periodi insallmens. Fraion of eah paymen is ineres on he ousanding balane of he loan. nd he remaining par is repaymen of prinipal. fer eah paymen, he ousanding balane is redued by he amoun of prinipal repaid. The proess of paying off a loan s prinipal gradually over is erm is alled loan amorizaion. Example $00,000 home morgage loan. nnual perenage rae is 9%. nnual ompounding is applied. To be paid in hree equal annual insallmens. How muh do you have o pay per year? Le be onsan annual paymen. This mus saisfy he following equaion ; loan prinipal = oal of PV of monhly paymens = (+0.09) 2 + (+0.09) 3 whereisunknownvariable In[200]:= Clear[, ans]; ans = NSolve == = /. ans[[]] Prin["nnual paymen is ",, " dollars"] {} (+0.09) 2 (+0.09) 3, Ou[20]= {{ }} Ou[202]= nnual paymen is dollars We an inerpre he above quesion as follows: We have hree separae loans. We apply he same ineres rae. The sum of he prinipals of hese loans is he amoun you borrow oday.these loans have hree onseuive mauriies. eah mauriy you pay he same amoun. We denoe i. This is unknown variable. In shor, sum of PV s of paymen is equal o he amoun you borrow. Eah paymen is 39,505.5 dollars. You an alulae in he following way oo. I is, however, ompliaed.. he firs paymen, Ou of $39,505.5, $9,000 is ineres; $00, The remaining 30,505.5 is for repaymen of prinipal. The remaining ousanding balane is 00,000-30,505.5 = $69, Ineres is alulaed on his amoun during he nex year. 69, = 6,254.5 is he nex ineres paymen. The remaining 39, ,254.5 = 33,25 is for repaymen of he prinipal. 4.The similar alulaion is repeaed for he hird paymen Suppose we have one year loan of $,000 a an PR of %. The loan is o be repaid in equal monhly paymens. Wha is he monhly paymen? The following equaliy should hold; loan prinipal = Ch04.nb 4

5 monhly paymens. Wha is he monhly paymen? The following equaliy should hold; loan prinipal = oal of PV of monhly paymens. In his example, we have monhly ompounding. Be aware we apply monhly ompounding. Ineres rae per monh is given by PR/. In[204]:= Clear[ans,, r]; r = 0.; ans = NSolve 000 == Ou[205]= {{ }} {} + r, The equaion we are solving has only one unknown variable, whih is. So we use ans[[ ]]. In[206]:= = /. ans[[]]; Prin["monhly paymen = $", ] monhly paymen = $ PV of $ annuiy Ofen PV of $ annuiy is alulaed firs. Then i is used for appliaion. Our example of annuiy has regular paymens. We onsider paymen of $. So PV of annuiy of $ is given by The onsan paymen is equal o 000 value of one dollar annuiy In[208]:= Clear[v] v = + 0. ; Prin["value of $ annuiy of paymens of $ = $", v] value of $ annuiy of paymens of $ = $.255 In[2]:= Prin "monhly paymen = $", 000 v monhly paymen = $ Homework No. 2, Due May 7h Q. Consider an example given in p.6. Change 80 years o 85 and hange $0,000 o $8,000. Calulae NPV, IRR and he number of year long enough o be break even. Q2. p.44, Problem 25 Q3. p.45, Problem 28 Q4. p.42, Modify Problem 9 as follows. How muh you an spend per year, if you like o oninue o spend he fixed amoun a leas five years. Q5. ()You are hinking of buying a house whih oss yen. Bank offers you a loan program. ording o he loan program, saring from he nex monh, you pay every monh 44,000 yen nex 0 years. You pay 0 imes. Wha is ineres rae, as per monh and as PR? (2) Five years laer, righ afer you make he paymen of he monh, how muh sill do you owe? Ch04.nb 5