Abraham Zaks. Technion I.I.T. Haifa ISRAEL. and. University of Haifa, Haifa ISRAEL. Abstract
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1 Preset Value of Autes Uder Radom Rates of Iterest By Abraham Zas Techo I.I.T. Hafa ISRAEL ad Uversty of Hafa, Hafa ISRAEL Abstract Some attempts were made to evaluate the future value (FV) of the expected value ad the varace for some classcal cash flows (CF). The motvato stemmed from some recursve formulas. Smlar recursve formulas do ot hold for the case of the preset value (PV) due to the lac of depedece of some radom varables. Oe ca get some estmates for the PV usg the results about the FV. We study the PV ad we overcome the dffculty of depedece by reversg the order of the CF. It turs out that we get smlar recursve formulas for the PV as for the FV the classcal cash flows. It turs out to be smlar to that used to evaluate the FV. Furthermore t maes t possble to study the PV of the classcal CF drectly, ad t may suggest a method to study the PV of other CF as well. Keywords : Radom Rates of Iterest; Idepedet varables; Expected value; Varace; Autes; Future value; Preset value.
2 . Itroducto We study paymets C,..., C made years t, =,...,. Varous classcal sets of paymets are metoed secto 4. The auty s due f the paymets are made the begg of each year, ad f the paymets are made at the ed of each year we term the auty as arrear. Let the terest the year t be j, ad assume that these j for =,..., are depedet radom varables wth : E(+ j ) = +j advar(+ j ) = s for all, =,...,. (.) We vestgate the expected value ad the varace of the preset value (PV). Dufrese (989) ad Bedard ad Dufrese (00) study smlar PV uder a dfferet set of assumptos. The future value (FV) s dscussed McCutcheo ad Scott (986), by Zas (00) ad by Burec, Marcu ad Wero (003). For a seres of yearly paymets PV() deotes the preset value at the begg of the frst year of the paymets, ad FV() deotes the future value at the ed of the th year.. Future Value For the FV of the autes due, let S deote the radom value of the FV of a auty due of paymets evaluated at the ed of years,the S = C (+j ). The followg equalty holds: S = (S +C )(+j ) for =,..., - (.) + + The radom varables ( S + C ) ad (+ j + ) rema depedet for all. Let : E(S ) = µ, E(S ) = m for =,..., (.) µ = FV() as the certa case (.3)
3 3 ad Var( S ) = m - µ (.4) Detaled evaluato of FV() the varous s ow, e.g. McCutcheo ad Scott (986) where specfc formulas for E( S )= µ, ad for E( S )= µ are gve. Oe derves the recursve formulas : ad m + = ( m + µ C + µ + = ( µ + C ) ( +j) for =,..., (.5) C ) [( + j ) + s )] for =,..., (.6) that lead to the value Var( S ), va successve teratos. We wll obta formulas to evaluate Var( S ) the varous cases. I case (d) t s well ow that for all, =,..., the followg hold : ad Var( ) [ ] S = (+ j) + s ( + j) E( S ) = (+ j ) ad, the other cases we have : m - µ =( m +µ C + C )[(+ j ) +s ]-[( µ + C )(+ j )] + + m+- µ =[(+ j) + s ](m - µ )+ µ + +s /(+ j) (.7) Set V()=Var(S )/ [(+ j) + s ] ad µ () = µ /[(+ j ) +s ] the: V(+)=V()+ µ (+)s /( + j ) (.8) Set S = 0,ad Var( S 0 0 ) = 0,ad add (.8) for =,..., to get: V() = [s /(+ j) ] µ () = (.9) Formulas evaluatg µ () these cases, are gve Zas (00). Some = errors were otced ad corrected by Burec, Marcu ad Wero (003)). For the FV of autes pad arrear smlar expressos may be derved.
4 4 3. Iterlude. Our ext am s to vestgate the PV case. The PV() of a auty s the value of aual paymets at the begg of the frst year. A drect approach smlar to the oe we used for the FV case, seems mpossble sce the recursve formulas for the PV() the radom varables that arse are ot learly depedet. I secto 4 we suggest a dfferet approach to the PV of autes. The reader may compare t to that used by Dufrese (989)). We proceed to explore the PV a way smlar to the oe used for the FV. At ths pot we pomt out that estmates for the value of the PV may be easly derved from the value of FV cosderrg the quotet of FV by PV. We observe that the quotet FV()/PV() satsfes the followg equaltes : E[FV()/PV()]=(+ j ) ad Var[FV()/PV()]=[(+ j ) +s ] -(+ j ) ad these values may be used to get estmates for the PV(), usg the FV(). It s mportat to observe that the above relatos apply to the case of autes that cossts of a seres of aual paymets made durg the frst years, ad where FV() s ther value at the ed of the th year ad PV() s ther value at the begg of the frst year. 4. Preset Value Deote by d the yearly dscout factor for the( + ) th year. It follows tha: - -d =(+ j -+). Let us deote : E(- d )=- d ad Var(- d )= z for all, =,...,. ( 4.) I geeral the relato - - d = (+ j) does ot hold.
5 5 The problem that arses, whe tryg to follow a le of thought smlar to the oe used for the FV the secod secto, s that the radom varables volved are o loger depedet. To overcome ths dffculty we cosder the auty bacwards. We wll cosder all the above cases wth C, for =,...,, ad set D = C -+ each of the fve cases. The auty we cosder s the sequece of paymets D,..., D. I partcular, the PV() of that auty s the value the begg of the (-+) th year of the last yearly paymets, D,..., D. The formulas for the values of PV() as autes certa at a fxed yearly rate of dscout are dscussed McCutcheo ad Scott (986). The classcal cases are: (d) D = ad D = 0 for =,..., PV() = ( d ) - (a) PV() = ( - d ) (d) D = for =,..., PV() = a (a) PV() = (3d) D = -+ for =,..., (3a) (4d) D = for =,..., (4a) a PV() = (Ia) + ( - )a PV() = (Ia) + ( - )a PV() = (Da) PV() = (Da) (5d) D = (+r) - - r for =,..., PV() = (+r) (Ca) (5a) - r PV() = (+r) (Ca) The PV() (5d) s evaluated as a ad the PV() (5a) s evaluated as - a(+r) both at the rate of terest f so that (+f)(+r) = (+j).
6 6 Let A deote the radom value of the PV() of a auty due, of a sequece of yearly paymets evaluated at the begg of the frst year,the : A = D, ad A = A ( - d )+ D, for =,..., - (4.) wth depedet radom varables A ad ( - d ) for all ad D + a costat. Let : E(A ) = θ ad E(A ) = t for =,..., (4.3) The, for =,..., the followg hold cases, 3, 4, 5 : θ = FV() as gve above ad, Var( A ) = t - θ (4.4) The detaled evaluato of PV() s ow e.g. McCutcheo abd Scott (986), so there are ow formulas for E( Oe derves the two recursve formulas : A ) = θ, ad partcular for E( A ) = θ. θ+ = θ (- d ) + D for =,..., (4.5) + t = t [(- d) + z )]+θ D ( - d)+ D for =,..., (4.6) that lead to the values of E( The formula we get for Var( A ), ad Var( A ), va successve teratos. A ) replaces the recursve teratos to evaluate t. To verfy ths pot oe observes that the case (d) t s well ow that : - ad E(A ) = (- d) Var(A )= [(- d) + z ] - (- d) - (-) ad, the other cases Oe ca derve the recursve formulas, for =,..., : t - θ = [(- d) + z ](t - θ ) + θ z (4.7) + + Set,ad M() = Var(A )/[(- d) + z ] θ()= θ /[(- d) + z ] the : M(+) = M() + θ ()z /[(- d) + z ] (4.8) ad otcg that σ =, Var( A ) = 0, ad t = we obta a smlar equalty:
7 7 M() = [z /[(- d) + z ] - = θ() (4.9) A detaled calculato to evaluate - = θ () for all cases, except the frst oe, may be acheved alog the detals as troduced by Zas (00) ad Burec, Marcu ad Wero (003). A smlar approach for the PV of autes due ca be used for that of autes arrear ad leads to smlar results. Startg by replacg (4.) wth the equaltes : A = (A + D ) ( - d ), for =,..., - (4.0) We observe that costat for all. E( A ) = σ, E( A ) = s for =,..., (4.) A ad (- d + ) are depedet radom varables ad D + s
8 8 Refereces Bedard D., Dufrese D. (00) Peso Fudg wth Movg Average Rates of Retur, Scad. Act. J. -7 Burec K., Marcu A., Wero A. (003) Autes Uder Radom Rates of Iterest revsted, IME 3, Dufrese D. (989) Stablty of peso systems whe rates of retur are radom, IME 8, 7-76 McCutcheo J. J., Scott W. F. (986) A Itroducto to the Mathematcs of Face,Butterworth/Heema, Lodo Zas A. (00) Autes Uder Radom Rate of Iterest, IME 8, Address Departmet of Mathematcs, Teco, I.I.T, Hafa, 3000, Israel e-mal : azas@techux.techo..ac.l Fax : Acowledgemet Ths research was supported by the Fud for the Promoto of Research at the Techho Footote I memory of the late Professor Byam Schwarz
Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.
Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 3000, Haifa, Israel I memory
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