Cash Flow Valuation Mode Lin Discrete Time

Transcription

1 IOSR Journal of Mahemaics (IOSR-JM) e-issn: ,p-ISSN: X, 6, Issue 6 (May. - Jun. 2013), PP Cash Flow Valuaion Mode Lin Discree Time Olayiwola. M. A. and Oni, N. O. Deparmen of Mahemaics and Saisics, The Polyechnic, Ibadan, Saki Campus. Absrac: This research consider he modelling of each cash flow valuaion in discree ime. I is shown ha he value of cash flow can be modeled in hree equivalen ways under same general assumpions. Also, consideraion is given o value process a a sopping ime and/ or he cash flow process sopped a some sopping imes. I. Inroducion I is observed ha money has a ime value ha is o value an amoun of money we ge a some fuure dae we should discoun he amoun from he fuure dae back o oday. In finance and economics, discouned cash flow analysis is a mehod of valuing a projec, company, or asse using he concep of he ime value of money. All fuure cash flows are esimaed and discouned o give heir presen values. The sum of all fuure cash flows, boh incoming and ougoing, is he ne presen value, which is aken as he value or price of he cash flows in quesion. Mos fuure cash flows models in finance and economics are assumed o be sochasic. Thus, o value hese sochasic cash flow we need o ake expecaions. 1.1 Probabiliy Space Probabiliy heory is concerned wih he sudy of experimens whose oucomes are random; ha is he oucomes canno be prediced wih cerainy. The collecion Ω of all possible oucomes of a random experimen is called a sample space. Therefore, an elemen of ω of Ω is called a sample poin. Definiion 1: A collecion ofωis called a -algebra if (i) Ω ε (ii) Ω ε, hena C ε (iii) For any counable collecion A n, n 1 we have U n 1, A n ε Definiion 2: Le Ω be a sample space and is a algebra of subses of Ω. A funcion P defined on and aking values in he uni inerval 0,1 is called a probabiliy measure if (i) P Ω = 1 (ii) P A 0 A ε (iii) for a mos counable family A n, n 1, of muually disjoin evens we have P U n 1, A n = n 1 P A n The riple (Ω,, P) is called probabiliy space. The pair (Ω, )is called he probabilizable space. Definiion 3: A probabiliy space (Ω,, P)is called a complee probabiliy space if every subse of a P- null even is also an even Filered Probabiliy Space Le (Ω,, P)be a probabiliy space and F( ) = = ε o,, be a family of sub algebraof such ha (i) conained all he P null member of (ii) s whenever > s 0 where s is sub algebra of (iii) F ( ) is righ coninuous in he sense ha + =, 0 where s> = s>0 hen F ( ) = : ε o, is called filraion of. The quadruple(ω,, P,F ( )) is called a filer probabiliy space. If F( ) = : ε o, is a filraion of, hen is he informaion available abou a ime while filraion F ( ) describes he flow of informaion wih ime. 35 Page

2 Cash Flow Valuaion Mode Lin Discree Time Definiion 4: Le (Ω,, P,F( ))be a filered probabiliy space wih he filraion express as F ( ) = : ε o, and le X π = X, ε π be a sochasic process wih values in R, hen X() is called an adaped o he filraion F( )if X() is measurable for ε π. Where X() is a random variable. 1.2 Mahemaical Expecaion Le X = e 0 (Ω, R d ), hen he mean or mahemaical expecaion of X is defined by E X = X ω μ dω ω ε Ω Ω If X has probabiliy densiy funcion f x which is absoluely coninuous wih respec o lebesgue measure hen E X = xf x x dx R d For a random variable X = (X 1, X 2,... X n ) E X = E(X 1, X 2,... X d ) = E X 1 E X 2 E(X d ) Condiional Expecaion Le X 1 and X 2 be wo random variables, hen he condiional expecaion of X 1 given ha X 2 = x 2 is defined as E(X 1 X 2 = x 2 ) = E(X 1 x 2 ) = x 1 f(x 1 x 2 ) dx 1 f(x 1, x 2 ) = x 1 dx f(x 2 ) 1 f(x 2 ) > 0 Properies of Condiional Expecaion (i) Le a, b ε R, X, Y are IR valued random variables and F is a sub algebra of hen E[(aX + by) F ] = E(aX F) + E( by F) = a E(X F) + b E( Y F) (ii) If X 0 a. s, en E(X F ) 0 a. s E( F ) is posiively preserving. (iii) E( C F ) = C where C is a consan value. (iv) If X is F - measurable, hen E( X F ) = X (v) If X is F - measurable, hen E(X F ) = E(X) (vi) If F 1 and F 2 are boh algebra of such ha F 1 F 2 hen E(E(X F 2 F 1 ) = E(X F 1 ) a.s (vii) If X ε L I (Ω,, P) and X n X 1 hene(x n F ) E(X F) in L I (Ω,, P) (viii) If φ: R R is convex and φ(x)ε L I (Ω,, P)such ha E φ(x) < hen E (φ(x) F) φe( X F ) a.s (xi) If X is independen of F 1 hen E(X F ) = E(X) 1.3 Sochasic Processes Definiion 5: Le X ω, ω ε Ω, ε o, be a family of R d valued random variable define on he probabiliy space (Ω,F, P) hen he family X, ω ε o, of X is called a sochasic process ω ε Ω A sochasic process X, ω, ω ε Ω, ε π is hus a funcion of wo variables X: T X Ω R d, X for 0 is said o be F adaped if for every 0, X is F measurable. 1.4 Maringale Process Definiion 6: A sochasic process X n n 1 is said o be a Maringale process if (i) E[X X n ] < (ii) E[X n+1 X 1, X 2,... X n ] = X n Definiion 7: Le X. π = X, ε π be an adaped R valued sochasic process on a filered probabiliy space (Ω,F, (F ) επ,p) hen X is called maringale if for each ε π R E[X() F] = X(s) a.s s <. II. Model Formulaion The discouned cash flow formular is derived from he fuure value formular for compuing he ime value of money and compounding reurns. C = V(1 + r) n 36 Page

3 Cash Flow Valuaion Mode Lin Discree Time Discouned presen value (DPV) for one cash flow (FV) in one fuure period is expressed as C V = = FV(1 d)n (1 + r) n Where, V is he discouned presen value; C is he norminal value of a cash flow amoun in a fuure periods; r is he ineres rae, which reflecs he cos of ying up capial and may also allow for he risk ha he paymen may no be received in full; r d is he discoun rae, which is (1+r) n is he ime in years before he fuure cash flow occurs. For muliple cash flows in muliple ime periods o be discouned i is necessary o sum hem and ake he expecaion. C V 0 = E (1 + r) k (1) k=1 We ake he expecaion because we assume ha mos cash flows models are sochasic. Inroducion of he value a ime 0 we have a dynamic model V = E k=+1 C k (1 + r) k Here E ( ) is he expecaion given informaion up o and including ime. 1 leing m k = 1 + r k V 0 = E k=1 2.1 General definiions Definiion 2.1 : A cash flow (C ) ε N is a process adaped o he filraion F and ha for each ε N, C < a. s A cash flow process ha is non-negaive a.s will be referred o as a dividend process. Definiion 2.2 A discoun process is a process m: N N Ω R saisfying (i) 0 < m s, < a. s s, ε N (ii) m s,, ω is F max(s,) measurable s, ε N and (iii) m s, = m s, u m u, a. s s, u, ε N A discoun process fulfilling i I 0 < m s, < 1 a. s s, ε N wi s will be referred o as normal discoun process m s, is inerpreed as he sochasic value a ime s of geing one uni of currency a ime, while m, s will be referred o as he growh of one uni of currency invesed a ime, a ime s. We wrie m = m 0,, N as a shor-hand noaion. Definiion 2.3 The discoun rae or insananeous rae a ime implied by discoun process denoed by r for = 1, 2,... is defined as 1 Λ 1 r = 1 = 1 m( 1, ) Λ Where is he deflaor associaed wih m. The relaionship beween he discoun process and rae process is deduced from he following Lemma. Lemma 1: Le m be a discoun process and le r be he discoun rae implied by m. Then he following holds: (i) 1 < r <, ε N (ii) r 0 iff m is normal discoun process (iii) For any given λ 0 we have for ε N 0 < λ < r iff In 1+λ 0 < m e (iv) The insananeous rae process and discoun process uniquely deermine each oher. C k m k III. Valuaion of Cash Flow Definiion 3: Given a cash flow process C for ε N and a discoun process m s, for s, ε N we define for ε N he value process as V = E[ k=+1 C k m(, k) F ] 37 Page

4 Cash Flow Valuaion Mode Lin Discree Time The value process is defined as ex dividend if cash flows from ime +1 and onwards is included in he value a ime. I is oherwise considered as cum divided if he cash flow is included a ime. The following lemma gives a sufficien condiion as o when is he value process will be finie a.s Lemma 2: If C is a cash flow process and E k=1 C k m k < a. s Then V a.s for all ε N Proof: Since E k=1 C k m k < for ε N V = E[ k=+1 C k m(, k) F ] = E[ k=+1 C k m(, 0)m(0, k) F ] m(, 0)E[ k=+1 C k m(0, k) F ] 1 E[ C m (0,) k=+1 km k F ] 1 E[ C m k=1 km k + k=+1 C k m k k=1 C k m k F ] 1 E[ C m k=1 km k F + k=1 C k m k ]< a. s Thus, if C is a discoun process such ha V 0, hen V a.s for every ε N Noe ha if X is a random variable wih E X hen E[X F ] for = 1, 2,..., is a uniformly inegrable (U.I) maringale. Hence, if [ C k m k ] a.s hen E [ k =0 C k m k F ] is a U.I maringale. Below proposiion gives he characerisics of value process Proposiion 2: Le C and m be a cash flow and discoun process respecively. If [ C k m k ] hen he discouned value process (V m ) can be wrien as V m = M A for ε N Where M is a U.I maringale and A an adoped process. Furhermore, lim V m = 0 a. s Proof: Noe ha [ k =0 C k m k ] a. s sincee [ k =0 C k m k ] Le M = E [ C k m k F ] for ε N k =0 A = C k m k k =0, ε N I is hen immediae ha V m = M A. Since [ C k m k ], M is a U.I maringale and A ia adaped. As, he UI maringale M = E [ k =0 C k m k F ] = k =0 C k m k = A a. s This implies ha as lim V m = lim M = lim A = 0 Since M = A = k =0 C k m k is finie a.s. Now le C be dividend process. Then A is an increasing process and E A = E [ k =0 C k m k ] < by assumpion. Thus: V m = E [ k = +1 C k m k F ] = E A F ] - A Characerisics relaion beween C (cash flow process), m(discoun process) and V(value process) is given in he following heorem. Theorem 3: Le C be a cash flow process and m a discoun process such ha E[ C k m k ] <. Then he following hree saemens are equivalen. (i) For every ε N V = E [ k = +1 C k m(, k ) F ] (ii) For every ε N M = V m + C k m k is a UI maringale (b) V m 0 a. s wen 0 (iii) For every ε N (a) V = E [m(, + 1)(C +1 + V +1 ) F ] (b) lim T E [m, T V T F ] = 0 Proof: Noe ha E C k m k E [ C k m k ] < which implies ha C k m k show ha < a. s Now o 38 Page

5 Cash Flow Valuaion Mode Lin Discree Time (i) (ii) and (i) (iii) (i) (ii) The if par follows from he proposiion. For he only if par he expression in (ii) a is wrien as m k +1 C k +1 = m k +1 V k +1 m k V k M k +1 M k And ake he sum from o T 1 T k = +1 m k C k = m T V T m V M T + M If we le T he erm m T V T 0 a. s by he assumpion and M T M a.s from he convergence resul of UI maringales. Thus, we have V m = C k m k M + M (1) k = +1 I is observed ha E M F ] = M from he convergence resul of UI maringale. Taking condiional expecaion wih respec o F of (1) E V m F ] = E [ k = +1 C k m k F ] - E M F ] + E M F ] V m = E [ k = +1 C k m k F ] -M + M V m = E [ k = +1 C k m k F ] 1 m V = E [ C m k = +1 k m k F ] = E [ C k k = +1 k F ] m = E [ k = +1 C k m(, k ) F ] Noe: (i) M = A = C k m k E M F ] = E [ C k m k F ] = M (ii) E M F ] = E [ C k m k F F ] = E [ C k m k F ] Regarding (i) (iii) We begin wih he if par. Fix ε N V = E [m(, + 1)C +1 + k = +2 C k m(, k ) F ] V = E [m(, + 1)C +1 + m(, + 1) k = +2 C k m( + 1, k ) F ] V = E [m(, + 1)C +1 + k = +2 C k m( + 1, k ) F ] V = E [m(, + 1)C +1 + V +1 F ] Now le T. From V T = k =T +1 C k m(t, K) F We have V T = E [ k =T +1 C k m(t, k )m(, k ) F T] m(,k ) = E [ k =T +1 C FT] k m(,t ) V T m(, T ) = E [ k =T +1 C k m(, k ) F T] (2) Taking condiional expecaion wih respec o F T of (2) E [V T m(, T )/F T ] = E [ k =T +1 C k m(, k ) F T F T] = E [ k =T +1 C k m(, k ) F T] 1 = E [ C m k = +1 k m k F ] Since C k m k C k m k And he las random variable is inegrable by assumpion we ge for every ε N and A ε F lim E m, T V T 1 A = lim m, T V T 1 A = 0 T T To prove he only if par we ierae (iii) a o ge T V = E [ k =T +1 C k m, k + m(, T )V T F ] 1 V = E [ C m k = +1 k m k F ] + E [m(, T )V T F ] As T lim E [m(, T )V T F ] 0 a.s from (iii)b since T k = +1 C k m k C k m k And he las random variable is inegrable by assumpion. Thus, we ge V T = E [ C k m(, k ) F ] using he heorem of dominaed convergence. k = +1 IV. Sopping he cash flow and value process Theorem3: Show ha he hree equivalen represenaions of he value process a deerminisic imes. The heorem also assumes ha cash flow sream is defined for all 0. In he sream, we shall consider he value process sopping ime and / or he cash flow process sopped a some sopping ime. Using he fac ha he maringale 39 Page

6 Cash Flow Valuaion Mode Lin Discree Time M = V m + C k m k from heorem 3, is uniformly inegrable we can ge he following resuls. Proposiion 4: Le C and m be cash flow process and discoun process respecively such ha E [ C k m(, k ) < for every ε N. Furher le and be (F ) sopping imes such ha a. s hen he following wo saemens are equivalen. (i) We have V = E [ k = +1 C k m, k + V m(, )1 < F ] { } (ii) (a) for every ε N M = V m + C k m k is a UI maringale, and (b) V m 0 a. s W en Proof: We sar wih (ii) (i) considering he sopping imes n Where n ε N because he sopping ime may be unbounded we ge n M n = V n m n + C k m k (3) as n, V n m n V m 1 < a. s Since V m 1 < 0 a. s assuming we le n Thus, we have from (3) M = V m 1 < + C k m k Since M is uniformly inegrable we ake condiional expecaion of M wih respec o he algebraf o ge on { } E M F ] = E [ C k m k + V m 1 < F ] Noe ha E M F ] = M Thus, M = E [ C k m k + k = +1 C k m k + V m 1 < F ] M = C k m k + E [ k = +1 C k m k + V m 1 < F ] M = E [ C k m k + V m 1 < M = C k m k + E [ k = +1 C k m k + V m 1 < F ] I implies C k m k + V m 1 < = C k m k + E k = +1 C k m k + V m 1 < F ] Hence, V m 1 < = E [ k = +1 C k m k + V m 1 < F ] m V = E [ k = +1 C k m k + V m 1 m < F ] V = E [ k = +1 C k m(, k ) + V m(, )1 < F ] This is our desired resul. To prove (i) (ii) Le = and = for ε N in i i. e V = E [ k = +1 C k m(, k ) + V m(, )1 < F ] V = E [ k = +1 C k m(, k ) + V m(, ) F ] m V = E [ k = +1 C k m k + V m F ] m V m = E [ k = +1 C k m k + V m F ] Noe a V m 0 a. s a. s V m = E [ k = +1 C k m k F ] V m = E [ C k m k + k = +1 C k m k C k m k F ] V m = E [ C k m k C k m k F ] V m = E [ C k m k F ] C k m k Since E [ m k F ] is a UI maringale i.e M Thus, we have C k V m = M Hence, M = V m + C k m k as desired. C k m k 40 Page

7 Cash Flow Valuaion Mode Lin Discree Time V. Conclusion We have been able o show ha cash flow models can be wrien in equivalen forms using a suiable discoun facor. We proceed o consider he value process sopping a sopping ime and / or cash flow process sopped a some sopping imes wih he use of a proposiion. References: [1]. Akepe A. O (2008): Opion Pricing on Muliple Asses M.Sc. Projec Universiy of Ibadan [2]. Armerin F. (2002): Valuaion of Cash flow in Discree Time working paper Ayoola E. O: Lecure noe on Advance Analysis Lebesgue Measure Universiy of Ibadan, Unpublished [3]. Ugbebor O. O (1991): Probabiliy Disribuion and Elemenary Limi Theorems. [4]. Universiy of Ibadan Exernal Sudies Programme Universiy Of Ibadan [5]. Ellio R. J (1982): Sochasic Calculus and Applicaions, Springerverlag [6]. Pra, S. e al (2000): Valuing a Business. McGraw-Hill Professional. McGraw Hill [7]. Williams, D. (1999): Probabiliy wih Maringales, Cambridge Universiy Press ΦKsendal, B. (1998): Sochasic differenial Equaion, Fifh Ediion, Springer - verlag 41 Page