Scarica Financial math theory (bocconi, math2, second partial) e più Sintesi del corso in PDF di Matematica Applicata solo su Docsity! Elementary ENANCA OPERATIONS exchange between z amountof money aveilable on differentdate I =lifet/ME usually MIC ① ACCUMULATION:C, M. 10- exchange between ( today and MatI C =PRINCIPAL M =FINAL VALUE 3*8(e) -ACCUMULATIONFACTOR M=C8(E) VE10 1 =INERESt-1 =M-C ② viscouns S =NOMINAL VALLE A =PRESEN VALLE *est) =orscouwrfacror A =Se(t) Vt=0 0 =DISCOUN- 0 =S-A *financial factors-i gle)elt) -fllo e coniugarefin FAcrors: ·NERES RAE:interestgeneratedby 1Ein year Slave (-apply 8-applye-hane (ogain i =M - C =(f(t)- 2 =f(1) - 1 Eme Suppose 8 (,z)iscont ->CHOUCH3:f(x +3) =f(x)+f(3) Xx,y==f(x)=ax willae Xxe d atleastat(03ER ↓ CONSEP:f(x+3,z) =f(x,z) +f(3,z) Xx,y,ze =f(x,z)=a(z)xwilla(t) Sandian afz UxE↓ (HOUCH32andf(x)<0 =f(x+3) =f(x)8(3) fx,yeREf(x)=emwith me fxER <20 at =0 ↑accumulation primitivemorions - 10 PROCESS M IatI 20 I A1:M=M(,t) AxoMSA2: Xc..(2101M(x +,t) =M(x,t) +M(x,t) A3: Vt,,t2 =0[,[z- M(,()=M(3,2) As L =>f(t) Si. INCR.I A3" k(=0 - M(t) 20 =8(t) =0 Vt10 AGift=0 =M(c,0) =[ TH: (A3 +A2):VC20, kt20 -M(t):ME, TH:(A2+A3+ Aa):i(( - M(x,t) >M(x,t) TH:M chatratific AXOMS -M=(f(t) s T-50 f(0) =1 ② 8INCREASING FINANCIAL OPERATIONS: countable sequence of poi:{Its,as3== Ello.ad....,(in,am] unlerets are dateand as ass ach flau ba,INFCOW Elte, a3= {ltoo, ..., ((m,an)....3 as0 Ourflow ① ANNWITIES: Sinite sequence where as,..., am ha same sign ② PERPETUITIES: infinitesequence where as,..., an Lane same sign ③ INVESTMEN: Sinite sequence willatleast 2 terms, asan,...,an0 *LOANS: Sinite sequence willatleast 2 terms, asan,...,amo FORMULAE ON ANNUITIES:ifthemare many paymentstheparedure is o LONo-some formulas com help 1) ordinary annuities, constant payments analese incompound INResi Spongmentitale place atdatesr.2,..., n - even is no paymentato time gas from o tom-paymentirefer tohepreveding timeperiod 2) OVE annurties, constant payments analese incompound INERES Spoymentitale place atdote0,1,2,..., n---ropaymentathot goes from otoc-apay for following perico TH SERIES Consider geometric sequence: Am=q, m =0,1,2... Consider Sr =cotant... +am =1 +9+... +=ifqfe Smith THS2-PARTICULAR Cave of THSERLES n - 7 1) ifan= q =G+i)" with ic:Sme =Sor =(1 +i)-- i 1 -1 2) 8am:q =Film with ico:Sm-1 =Ear =(+1) ( +i) i TH 53 Given an ordinary amilywithconstantpaymenti, educatedincompound interete (wich isd;t'inyear, paymentiall equal we fare: evo.E TH 57 givenadueautoincolantparmenti, evaluatedincompound interest(milie Es i S=Vn=(1+i)- 1 when R=1 we we thefollowing symbole for ordinary annuitia:A= Vo= ansi S =Vn =G+i)*- 1 =Smy: ad thefollowing for OVE ANNUNES A =Vo=Geti)a .. i =dei i 20 S =Vn=(+i)(1 + i)*1 =Sn! so general formulafor ordinary amuitiescan be writtenas:A: Vo=ansi S =Vm=RSmei - - due ~ ...... A =Vo=Ränei S:Vn=RSne i ex:see slides formulaeon perpetuitIEs Set's see twocaves: 1) Ordinary pere, constant PAMEN, analizedinCompound INERES although thereisno paymentato, thecontractstadifrom o;every paymentrefore tohefollowing time period 2) OVE PERP., const.PA., analizedinCompound INTERES We want ofind formular for:voiardinaryReicheedei We obtain: TH after 58 rends a ordinary s corner:for ordinary perpetutia:A =Vo= Raseitana oporary so angle: ↳where a:I Tr after def 60 rende a due so come i Son due perpetuties:A=Vo=Re =R(+1) =Rassia a dire a anglei Swhea asei* = 1+ OCF, NPV, internal rates Considera Sinancial operation: 9o .... qu 7 Es:O In In DiscountCASHFLOW(OCE) GA) =otget... in dotatente. unlarex=VARAELECompound annual int. Rate We'll acceptall values of 1 six-1- corresponde toacceptwe can love a partof themoney me inventeroall of it htnotmore than tat. Su axiomatic terme, itmereissubefilutedby As" In terms of thepropertiesof theaccumulation factor 8(t)=Gtil"-> theprety8(t) insig V10 become 8(t)=0 X (=0 IS me Sixan annual ratex=i, thenumber: 6) -tit... tm =dotit...te incalled thenet present Valle(Nor NERNAL RArE;any x =1*sr:G(x4)=0 -> 6may have No, or more interval rates ocf of an investment domare isfr,tad Saz every intestment ex. -1000 600 quo (+6(x) =+0(and) =x= -redical asemplate 0 1 2 6() =-100+ 8006sacoaro) 8():asso fantantial ampiaemillyaxiex =0- 6(r) = - 1000 +600 +700=300it's aloe SUno CASHFowb 6x) =j -xfr, +0) - 6x) strcly decoringan fr,+d 6(x)........ forte INV. I I 7 ! BOLZAN TH:continuous and strictlydeweasig 45!x =x*s5f(xt) =0 di ate... Sfaam investmenthisunique intermal rate icalled :internal rateof returnofthe ↓representsthe rate INC.. Itesur of theflame isNEGATI- X * is o -x*0-swe're losing money atwhich we're insertig 1 I ⑨ 1 !x =x*s6(x4) =0coriero robitahambegawatte(5!internal rotel loar. REMARK contineit's difficulttofind theexactinternalratebutwe can always say ift's than a given i inanestreNT- Dece G(x) is trict DECREAbind * * 6(x)-6(i)=6(i)c0 ina LOAN- fece 6(x) is trin * 6(x)((6(i)eG(i)p
