# Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.

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1 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 of the late Professor Biyami Schwarz Abstract Some attempts were made to evaluate the future value (FV) of the expected value ad the variace for various cash flows (CF). The motivatio stemmed from some recursive formulas. This method does ot apply directly to the evaluatio of preset values (PV). Oe ca get some estimates for the PV usig results about the FV. We will preset a direct approach to evaluate the PV of both factors for some CF. It will tur to be similar to that used to evaluate the FV. Furthermore it maes it possible to study the PV of these CF directly, ad may suggest a method to study some other CF as well. Subj. class : IE50; IE5 Keywords : Radom Rates of Iterest; Idepedet variables; Expected value; Variace; Auities; future value; preset value.. Itroductio A auity is a sequece of paymets C,..., C made i years i, i =,...,. If the paymets are made i the begiig of each year we have a auity due, ad if the paymets are made at the ed of each year we have a auity i arrear. Assume that the iterest i the year i, is i, ad that these iterests for i =,..., are idepedet radom variables with : E(+j i ) = +j ad Var(+j i ) = s for all i, i =,...,. (.) It is our goal to study the expected value ad the variace, for the preset value (PV) ad for the future value (FV) of the auity. For a series of yearly paymets,let PV() deote the preset value at the begiig of the first year of the paymets, ad let FV() deote the future value at the ed of the th year. There are five classical cases : The first is that of a sigle ivestmet (d) at the begiig of the first year, ad (a) at the ed of the th year. I each of the others we have a auity due i case (id) or a auity i arrear i case (ia) for i =, 3, 4, 5. The paymets i the various cases, ad the values of the FV() of ay auity, as a auity certai with fixed yearly rate of iterest, are give below.the defiite fuctios that express the formulas of the symbols are well ow [ e.g. (MS86)]. Electroic copy available at:

2 The cases are : (d) C =, ad C i = 0 for i =,..., ; FV() = ( + j ) (a) C =, ad C i = 0 for i =,..., ; FV() = (d) C i = for i =,..., ; FV() = s (a) ; FV () = (3d) C i = i for i =,..., ; FV() = ( Is ) (3a) ; FV() = ( Is ) (4d) C i = -i + for i =,..., ; FV() = ( Ds ) (4a) ; FV() = ( Ds ) (5d) C i = (+r) i- for i =,..., ; FV() = ( Cs ) r (5a) ; FV() = ( Cs ) r Recall that the symbol for (5d) is evaluated as s for a rate of iterest f that satisfies (+f) (+r) = (+j),ad that for (5a) is evaluated as. Future Values. s s /(+r) for the rate of iterest f. For the FV of the auities due, let S deote the radom value of the FV of a auity due of paymets evaluated at the ed of years, the S = C ( + j ). The followig equality holds: S ( S + C ) ( +j + for =,..., - (.) The radom variables ( S + C ) ad ( + j + ) remai idepedet for all. Let : E( S ) =, E( S ) = m for =,..., (.) The, for =,..., the followig hold i cases, 3, 4, 5 : = FV() as give above (.3) ad, Var( S ) = m - (.4) I particular the detailed evaluatio of FV() is well ow [e.g.(ms86)], so we have ow formulas for E( S ) =, ad i particular for E( S Oe derives the regressio formulas : ) =. + = ( + C ) ( + j for =,..., (.5) ad m + = ( m + C + C ) [( +j) + s )] for =,..., (.6) that lead to the value Var( S ), via successive iteratios. Electroic copy available at:

3 It turs out that oe eed ot pass through m, ad oe eed ot use regressio to evaluate Var( S ). I fact it is possible to get a defiite formula for S ). Var( To verify this poit oe observes that i the case (d) it is well ow that for all, =,..., the followig hold : E( S ) = ( + ad S ) = [( + ) + ] - ( + Var( ad, i the other cases we have : m = ( m + C + C ) [( + ) + ] - [( + C ) ( + ] m = [( + ) + ] ( m - ) + + /( + If we set V() = Var( S )/[( + ) + ], ad () = / [( + ) + ] the : V(+) = V() + (+) /( + Set S = 0,ad Var( S ) = 0,ad add (.8) for =,..., to get the equality : 0 0 V() = [ /( + ] A detailed calculatio to evaluate () (.9) () for all cases, except the first oe, was made i (Z0). Ufortuately there are some miscalculatios there. These were tae care of i (BMW), ad the author is grateful for the correctios. For the FV of the auities i arrear, let S deote the radom value of the FV of a auity arrear of paymets evaluated at the ed of years, the S = C. A similar approach to the oe tae for the FV of auities due ca be tae for the FV of auities i arrear ad a similar treatmet will lead to similar results. For the auities i arrear we replace (.) with the correspodig followig equalities : S = S ( C + for =,..., - (.0) E( S ) =, E( S ) = r for =,..., (.) The radom variables S ad ( + + are idepedet ad C + is a costat for all. I the case (a) we will get : E( S ) = ( + ad Var( S ) = [( + ) + ] - - ( + () We get similar equatios for all, =,..., -, for the cases, 3, 4, 5 : + = ( + C + (.) r + = r [( + ) + ] + ( + C + + Var( S ) = r = ( r - )[( + ) + ] + As above we deote i a similar way for all, =,..., - : C (.3) (.4) W() = Var( S )/ [( + ) + ], () = / [( + ) + ] (.5) The, W(+) = W() + () /[( + ) + ] Observe that =, Var( S ) = 0, ad r = to obtai the equality : A detailed evaluatio of W() = { /[( + ) + ] above i the remars as to the evaluatio of (). () (.6) () results i a way similar to the oe metioed

4 3. Iterlude. Our ext aim is to ivestigate the PV case. The PV() of a auity is the value of paymets at the begiig of the first year. A direct approach similar to the oe we used for the FV case, seems impossible sice i the regressio formula for the PV() the radom variables that arise are ot liearly idepedet. I sectio 4 we will suggest a differet treatmet of the auities that will eable the direct approach. We will the explore the PV i a way similar to the oe used for the FV. The differet way to study the auities will avoid the liear depedece. At this poit we wish to cosider the value of the quotiet of the FV by the PV. We observe that for ay give auity, whether due or i arrear, i ay of the five cases, the quotiet FV()/PV() will satisfy the followig equalities : E[FV()/PV()] = ( + ad, Var[FV()/PV()] = [( + ) + ] - ( + ad these values may be used to get estimates for the PV(), usig the FV(). It is importat to observe that the above relatios apply to the case of auities that cosists of a series of aual paymets made at the begiig of the first years, ad where FV() is the value at the ed of the th year ad PV() is the value at the begiig of the first year. 4. Preset Values. Deote by i the yearly discout factor for the ( i + ) th year. It follows that : - i + -i+. Let us deote : E(- i ) = - ad Var(- i ) = for all i, i =,...,. ( 4.) I geeral the relatio - = ( + ) - does ot hold. The problem that arises, whe tryig to follow a lie of thought similar to the oe used for the FV i the secod sectio, is that the radom variables ivolved are o loger idepedet. To overcome this difficulty, we cosider the auity bacwards : we will cosider all the above cases with C, for =,...,, ad set D = C -+, i each of the five cases. The auity we will cosider is the sequece of paymets D,..., D. I particular, the PV() of that auity is the value i the begiig of the (-+) th year of the last yearly paymets, D,..., D. The paymets i the various cases, ad the values of the PV() of ay auity as a auity certai with fixed yearly rate of discout,are give below.the defiite fuctios that express the formulas of the symbols are well ow [ e.g. (MS86) ]. The cases are : (d) D =, ad D i = 0 for i =,..., ; PV() = ( d ) - (a) D =, ad D i = 0 for i =,..., ; PV() = ( - ) (d) D i = for i =,..., ; PV() = a (a) ; PV() = a

5 (3d) D i = -i+ for i =,..., ; PV() = ( Ia ) + ( ) a (3a) ; PV() = ( Ia ) + ( ) a (4d) D i = i for i =,..., ; PV() = ( Da ) (4a) ; PV () = ( Da ) (5d) D i = (+) -i for i =,..., ; PV() = (+) - ( Ca ) r (5a) ; PV() = (+) - ( Ca ) r The remars followig the list of cases i sectio, as for the evaluatio of the symbols i the 5 th case, apply i a similar way to the 5 th case above. let A deote the radom value of the PV() of a auity due, of a sequece of yearly paymets evaluated at the begiig of the first year, the A = D, ad we get : A ( - + D + for =,..., - (4.) A = with idepedet radom variables A ad ( - for all ad D + a costat. Let : E( A ) =, E( A ) = t for =,..., (4.3) The, for =,..., the followig hold i cases, 3, 4, 5 : = FV() as give above (4.4) ad, Var( A ) = t - (4.5) I particular, the detailed evaluatio of PV() is well ow [e.g.(ms86)], so we have ow formulas for E( A ) =, ad i particular for E( A Oe derives the regressio formulas : + = ( - +D + for =,..., ) =. (4.6) ad t + = t [( - ) + )] + D + + D for =,..., (4.7) that lead to the values of E( A ), ad Var( A ), via successive iteratios. It turs out that oe eed ot pass through t, ad oe eed ot use regressio to evaluate Var( A ). I fact it is possible to get a defiite formula for Var( A ). To verify this poit oe observes that i the case (d) it is well ow that : E( A ) = ( - - ad Var( A ) = [( - ) + ] - - ( - -) ad, i the other cases Oe ca derive the regressio formulas, for =,..., : t = [( - ) + ] ( t - ) + If we set M() = Var( A )/[( - ) + ], ad () = / [( - ) + ] the : M(+) = M() + () [( - ) + ] ad oticig that =, Var( A ) = 0, ad t = we obtai a similar equality :

6 M() = [ /[( - + ] A detailed calculatio to evaluate () (4.9) () for all cases, except the first oe, may be achieved alog the details as itroduced i (Z0) ad (BMW). A similar approach to the oe tae for the PV of auities due ca be tae for the PV of auities i arrear ad a similar treatmet will lead to similar results. Notice that it all starts by replacig (4.) with the correspodig equalities for this case : A = ( A + D + ) ( - + for =,..., - (4.0) E( A ) =, E( A ) = s for =,..., (4.) where A ad ( - + are idepedet radom variables ad D + is a costat for all. The, we get similar equatios for all, =,..., - : + = ( D + ) ( + (4.) s + = [( - ) + ] [s + D + + D ] (4.3) I the case (a) we will get : E( S ) = ( - ad Var( S ) = [( - ) + ] - ( - ad, i the other cases Oe ca derive the regressio formulas, for =,..., : Var( A ) = s = ( s - )[( - ) + ] + + /( ) (4.4) with otatios similar to the oes used for auities due, we get for =,..., - : N() = Var( A )/ [( - ) + ], () = / [( - ) + ] (4.5) The, N(+) = N() + (+)/(-) Set N(0) = 0,ad Var( A ) = 0,ad addig up for =,..., to get the equality : A detailed evaluatio of N() = [ /( - ) ] 0 () (4.6) () is similar to the oe used to evaluate (). Acowledgemet This research was supported by the Fud for the Promotio of Research at the Techhio Refereces (MS86) McCutcheo J. J., Scott W. F. 986 A Itroductio to the Mathematics of Fiace,Butterworth/Heiema, Lodo (Z0) Zas A. 00 Auities Uder Radom Rate of Iterest, IME 8, (00), - (BMW) Bureci K., Marciiu A., Wero A. Auities Uder Radom Rates of Iterest - revisited, IME 3, (003) (BD) Bedard D., Dufrese D. Pesio Fudig with Movig Average Rates of Retur, Scad. Act. J. (00) -7 (D) Dufrese D., Stability of pesio systems whe rates of retur are radom, IME 8,(989) 7-76 (DMW) Date, P., Mamo, R., Wag, I.C. Valuatio of cash flows uder radom rates of iterest: A liear algebraic approach IME (007) 4,

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