Present Values an Accumulatons ANGUS S. MACDONALD Volume 3, pp. 1331 1336 In Encyclopea Of Actuaral Scence (ISBN -47-84676-3) Ete by Jozef L. Teugels an Bjørn Sunt John Wley & Sons, Lt, Chchester, 24
Present Values an Accumulatons Effectve Interest Money has a tme value; f we nvest $1 toay, we expect to get back more than $1 at some future tme as a rewar for lenng our money to someone else who wll use t prouctvely. Suppose that we nvest $1, an a year later we get back $(1 + ). The amount nveste s calle the prncpal, an we say that s the effectve rate of nterest per year. Evently, ths efnton epens on the tme unt we choose to use. In a rskless worl, whch may be well approxmate by the market for goo qualty government bons, wll be certan, but f the nvestment s rsky, s uncertan, an our expectaton at the outset to receve $(1 + ) can only be n the probablstc sense. We can regar the accumulaton of nveste money n ether a retrospectve or prospectve way. We may take a gven amount, $X say, to be nveste now an ask, as above, to what amount wll t accumulate after T years? Or, we may take a gven amount, $Y say, requre n T years tme (to meet some lablty perhaps) an ask, how much we shoul nvest now, so that the accumulaton n T years tme wll equal $Y? The latter quantty s calle the present value of $Y n T years tme. For example, f the effectve annual rate of nterest s per year, then we nee to nvest $1/(1 + ) now, n orer to receve $1 at the en of one year. In stanar actuaral notaton, 1/(1 + ) s enote v, an s calle the scount factor. It s mmeately clear that n a etermnstc settng, accumulatng an takng present values are nverse operatons. Although a tme unt must be ntrouce n the efnton of an v, money may be nveste over longer or shorter peros. Frst, conser an amount of $1 to be nveste for n complete years, at a rate per year effectve. Uner smple nterest, only the amount orgnally nveste attracts nterest payments each year, an after n years the accumulaton s $(1 + n). Uner compoun nterest, nterest s earne each year on the amount orgnally nveste an nterest alreay earne, an after n years the accumulaton s $(1 + ) n. Snce (1 + ) n (1 + n) ( >), an astute nvestor wll turn smple nterest nto compoun nterest just by wthrawng hs money each year an nvestng t afresh, f he s able to o so; therefore the use of smple nterest s unusual, an unless otherwse state, nterest s always compoun. Gven effectve nterest of per year, t s easly seen that $1 nveste for any length of tme T wll accumulate to $(1 + ) T. Ths gves us the rule for changng the tme unt; for example, f t was more convenent to use the month as tme unt, nterest of per year effectve woul be equvalent to nterest of j = (1 + ) 1/12 1 per month effectve, because (1 + j) 12 = 1 +. Changng Interest Rates an the Force of Interest The rate of nterest nee not be constant. To eal wth varable nterest rates n the greatest generalty, we efne the accumulaton factor A(t, s) to be the amount to whch $1 nveste at tme t wll accumulate by tme s>t. The corresponng scount factor s V(t,s), the amount that must be nveste at tme t to prouce $1 at tme s, an clearly V(t,s)= 1/A(t, s). The fact that nterest s compoun s expresse by the relaton A(t, s) = A(t, r)a(r, s) for t<r<s. (1) The force of nterest at tme t, enote (t),sefne as 1 A(,t) (t) = = log A(,t). (2) A(,t) t t The frst equalty gves an ornary fferental equaton for A(,t), whch wth bounary conton A(, ) = 1 has the followng soluton: ( t ) A(,t)= exp (s)s ( ( so V(,t) = exp t )) (s)s. (3) The specal case of constant nterest rates s now gven by settng (t) =, a constant, from whch we obtan the followng basc relatonshps: (1 + ) = e an = log(1 + ). (4)
2 Present Values an Accumulatons The theory of cash flows an ther accumulatons an present values has been put n a very general framework by Norberg [1]. Nomnal Interest In some cases, nterest may be expresse as an annual amount payable n equal nstalments urng the year; then the annual rate of nterest s calle nomnal. For example, uner a nomnal rate of nterest of 8% per year, payable quarterly, nterest payments of 2% of the prncpal woul be mae at the en of each quarter-year. A nomnal rate of per year payable m tmes urng the year s enote (m). Ths s equvalent to an effectve rate of nterest of (m) /m per 1/m year, an by the rule for changng tme unt, ths s equvalent to effectve nterest of (1 + (m) /m) m 1 per year. Rates of Dscount Instea of supposng that nterest s always pa at the en of the year (or other tme unt), we can suppose that t s pa n avance, at the start of the year. Although ths s rarely encountere n practce, for obvous reasons, t s mportant n actuaral mathematcs. The effectve rate of scount per year, enote, sefneby = /(1 + ), an recevng ths n avance s clearly equvalent to recevng n arrears. We have the smple relaton = 1 v. Nomnal rates of scount (m) may also be efne, exactly as for nterest. Annutes Certan We often have to eal wth more than one payment, for example, we may be ntereste n the accumulaton of regular payments mae nto a bank account. Ths s smply one; both present values an accumulatons of multple payments can be foun by summng the present values or accumulatons of each nvual payment. An annuty s a seres of payments to be mae at efne tmes n the future. The smplest are level annutes, for example, of amount $1 per annum. The payments may be contngent on the occurrence or nonoccurrence of a future event for example, a penson s an annuty that s pa as long as the recpent survves but f they are guarantee regarless of events, the annuty s calle an annuty certan. Actuaral notaton extens to annutes certan as follows: A temporary annuty certan s one payable for a lmte term. The smplest example s a level annuty of $1 per year, payable at the en of each of the next n years. Its accumulaton at the en of n years s enote s n, an ts present value at the outset s enote a n.wehave s n = (1 + ) r = (1 + )n 1, (5) a n = r= n r=1 v r = 1 vn. (6) There are smple recursve relatonshps between accumulatons an present values of annutes certan of successve terms, such as s n+1 = 1 + (1 + )s n an a n+1 = v + va n, whch have very ntutve nterpretatons an can easly be verfe rectly. A perpetuty s an annuty wthout a lmte term. The present value of a perpetuty of $1 per year, payable n arrear, s enote a, an by takng the lmt n equaton (5) we have a = 1/.The accumulaton of a perpetuty s unefne. An annuty may be payable n avance nstea of n arrears, n whch case t s calle an annutyue. The actuaral symbols for accumulatons an present values are mofe by placng a par of ots over the s or a. For example, a temporary annuty-ue of $1 per year, payable yearly for n years woul have accumulaton s n after n years or present value ä n at outset; a perpetuty of $1 per year payable n avance woul have present value ä ; an so on. We have s n = n (1 + ) r = (1 + )n 1, (7) r=1 ä n = v r = 1 vn, (8) r= ä = 1. (9)
Present Values an Accumulatons 3 Annutes are commonly payable more frequently than annually, say m tmes per year. A level annuty of $1 per year, payable n arrears m tmes a year for n years has accumulaton enote s (m) n after n years an present value enote a (m) n at outset; the symbols for annutes-ue, perpetutes, an so on are mofe smlarly. We have s (m) n = (1 + )n 1, (1) (m) a (m) n = 1 vn, (m) (11) s (m) n = (1 + )n 1, (m) (12) ä (m) n = 1 vn (m). (13) Comparng, for example, equatons (5) an (1), we fn convenent relatonshps such as s (m) n = s (m) n. (14) In precomputer ays, when all calculatons nvolvng accumulatons an present values of annutes ha to be performe usng tables an logarthms, these relatonshps were useful. It was only necessary to tabulate s n or a n, an the ratos / (m) an / (m), at each annual rate of nterest neee, an all values of s (m) n an a (m) n coul be foun. In moern tmes ths trck s superfluous, snce, for example, s (m) n can be foun from frst prncples as the accumulaton of an annuty of $1/m, payable n arrears for nm tme unts at an effectve rate of nterest of (1 + ) 1/m 1per tme unt. Accorngly, the (m) an s (m) n notaton s ncreasngly of hstorcal nterest only. A few specal cases of nonlevel annutes arse often enough so that ther accumulatons an present values are nclue n the nternatonal actuaral notaton, namely, arthmetcally ncreasng annutes. An annuty payable annually for n years, of amount $t n the tth year, has accumulaton enote (Is) n an present value enote (Ia) n f payable n arrears, or (I s) n an (Iä) n f payable n avance. (Is) n = s n n, (15) (Ia) n = än nv n, (16) (Is) (m) n (I s) n = s n n, (17) (Iä) n = än nv n. (18) (an so on) s a val notaton for ncreasng annutes payable m tmes a year, but note that the payments are of amount $1/m urng the frst year, $2/m urng the secon year an so on, not the arthmetcally ncreasng sequence $1/m, $2/m, $3/m,... at ntervals of 1/m year. The notaton for the latter s (I (m) s) (m) n (an so on). In theory, annutes or other cash flows may be payable contnuously rather than scretely. In practce, ths s rarely encountere but t may be an aequate approxmaton to payments mae aly or weekly. In the nternatonal actuaral notaton, contnuous payment s ncate by a bar over the annuty symbol. For example, an annuty of $1 per year payable contnuously for n years has accumulaton s n an present value a n.we have s n = a n = a = (1 + ) n t t = e (n t) t = (1 + )n 1, (19) (1 + ) t t = e t t = 1 vn, (2) (1 + ) t t = e t t = 1.(21) Increasng contnuous annutes may have a rate of payment that ncreases contnuously, so that at tme t the rate of payment s $t peryear,orthat ncreases at screte tme ponts, for example, a rate of payment that s level at $t per year urng the tth year. The former s ncate by a bar that extens over the I, the latter by a bar that oes not. We have (Is) n = (r + 1) r= r+1 r (1 + ) n t t = s n n, (22)
4 Present Values an Accumulatons (Ia) n = (r + 1) r= r+1 r (1 + ) t t = än nv n, (23) (Is) n = t(1 + ) n t t = s n n, (24) (Ia) n = t(1 + ) t t = a n nv n. (25) Much of the above actuaral notaton serve to smplfy calculatons before wesprea computng power became avalable, an t s clear that t s now a trval task to calculate any of these present values an accumulatons (except possbly contnuous cash flows) wth a smple spreasheet; nee restrctons such as constant nterest rates an regular payments are no longer mportant. Only uner very partcular assumptons can any of the above actuaral formulae be aapte to nonconstant nterest rates [16]. For full treatments of the mathematcs of nterest rates, see [8, 9]. Accumulatons an Present Values Uner Uncertanty There may be uncertanty about the tmng an amount of future cash flows, an/or the rate of nterest at whch they may be accumulate or scounte. Probablstc moels have been evelope that attempt to moel each of these separately or n combnaton. Many of these moels are escrbe n etal n other artcles; here we just ncate some of the major lnes of evelopment. Note that when we amt uncertanty, present values an accumulatons are no longer equvalent, as they were n the etermnstc moel. For example, f a payment of $1 now wll accumulate to a ranom amount $X n a year, Jensen s nequalty (see Convexty) shows that E[1/X] = 1/E[X]. In fact, the only way to restore equalty s to conton on knowng X, n other wors, to remove all the uncertanty. Fnancal nsttutons are usually concerne wth managng future uncertanty, so both actuaral an fnancal mathematcs ten to stress present values much more than accumulatons. Lfe nsurance contracts efne payments that are contngent upon the eath or survval of one or more nvuals. The smplest nsurance contracts such as whole lfe nsurance guarantee to pay a fxe amount on eath, whle the smplest annutes guarantee a level amount throughout lfe. For smplcty, we wll suppose that cash flows are contnuous, an eath benefts are payable at the moment of eath. We can (a) represent the future lfetme of a person now age x by the ranom varable T x ; an (b) assume a fxe rate of nterest of per year effectve; an then the present value of $1 pa upon eath s the ranom varable v T x, an the present value of an annuty of $1 per annum, payable contnuously whle they lve, s the ranom varable a Tx. The prncple of equvalence states that two seres of contngent payments that have equal expecte present values can be equate n value; ths s just the law of large numbers (see Probablty Theory) apple to ranom present values. For example, n orer to fn the rate of premum P x that shoul be pa throughout lfe by the person now age x, we shoul solve E[v T x ] = P x E[a Tx ]. (26) In fact, these expecte values are entcal to the present values of contngent payments obtane by regarng the lfe table as a etermnstc moel of mortalty, an many of them are represente n the nternatonal actuaral notaton. For example, E[v T x ] = A x an E[a Tx ] = a x. Calculaton of these expecte present values requres a sutable lfe table (see Lfe Table; Lfe Insurance Mathematcs). In ths moel, expecte present values may be the bass of prcng an reservng n lfe nsurance an pensons, but the hgher moments an strbutons of the present values are of nterest for rsk management (see [15] for an early example, whch s an nterestng remner of just how racally the scope of actuaral scence has expane snce the avent of computers). For more on ths approach to lfe nsurance mathematcs, see [1, 2]. For more complcate contracts than lfe nsurance, such as sablty nsurance or ncome protecton nsurance, multple state moels were evelope an expecte present values of extremely general contngent payments were obtane as solutons of Thele s fferental equatons (see Lfe Insurance Mathematcs) [4, 5]. Ths evelopment reache ts logcal concluson when
Present Values an Accumulatons 5 lfe hstores were formulate as countng processes, n whch settng the famlar expecte present values coul agan be erve [6] as well as computatonally tractable equatons for the hgher moments [13], an strbutons [3] of present values. All of classcal lfe nsurance mathematcs s generalze very elegantly usng countng processes [11, 12], an nterestng example of Jewell s avocacy that actuaral scence woul progress when moels were formulate n terms of the basc ranom events nstea of focusng on expecte values [7]. Alternatvely, or n aton, we may regar the nterest rates as ranom (see Interestrate Moelng), an evelop accumulatons an present values from that pont of vew. Uner sutable strbutonal assumptons, t may be possble to calculate or approxmate moments an strbutons of present values of smple contngent payments; for example, [14] assume that the force of nterest followe a seconorer autoregressve process, whle [17] assume that the rate of nterest was log-normal. The applcaton of such stochastc asset moels (see Asset Lablty Moelng) to actuaral problems has snce become extremely mportant, but the ervaton of explct expressons for moments or strbutons of expecte values an accumulatons s not common. Complex asset moels may be apple to complex moels of the entre nsurance company, an t woul be surprsng f analytcal results coul be foun; as a rule t s harly worthwhle to look for them, nstea, numercal methos such as Monte Carlo smulaton are use (see Stochastc Smulaton). References [1] Bowers, N.L., Gerber, H.U., Hckman, J.C., Jones, D.A. & Nesbtt, C.J. (1986). Actuaral Mathematcs, The Socety of Actuares, Itasca, IL. [2] Gerber, H.U. (199). Lfe Insurance Mathematcs, Sprnger-Verlag, Berln. [3] Hesselager, O. & Norberg, R. (1996). On probablty strbutons of present values n lfe nsurance, Insurance: Mathematcs & Economcs 18, 35 42. [4] Hoem, J.M. (1969). Markov chan moels n lfe nsurance, Blätter er Deutschen Gesellschaft für Verscherungsmathematk 9, 91 17. [5] Hoem, J.M. (1988). The versatlty of the Markov chan as a tool n the mathematcs of lfe nsurance, n Transactons of the 23r Internatonal Congress of Actuares, Helsnk, S, pp. 171 22. [6] Hoem, J.M. & Aalen, O.O. (1978). Actuaral values of payment streams, Scannavan Actuaral Journal 38 47. [7] Jewell, W.S. (198). Generalze moels of the nsurance busness (lfe an/or non-lfe nsurance), n Transactons of the 21st Internatonal Congress of Actuares, Zurch an Lausanne, S, pp. 87 141. [8] Kellson, S.G. (1991). The Theory of Interest, 2n Eton, Irwn, Burr Rge, IL. [9] McCutcheon, J.J. & Scott, W.F. (1986). An Introucton to the Mathematcs of Fnance, Henemann, Lonon. [1] Norberg, R. (199). Payment measures, nterest, an scountng. An axomatc approach wth applcatons to nsurance, Scannavan Actuaral Journal 14 33. [11] Norberg, R. (1991). Reserves n lfe an penson nsurance, Scannavan Actuaral Journal 3 24. [12] Norberg, R. (1992). Hattenorff s theorem an Thele s fferental equaton generalze, Scannavan Actuaral Journal 2 14. [13] Norberg, R. (1995). Dfferental equatons for moments of present values n lfe nsurance, Insurance: Mathematcs & Economcs 17, 171 18. [14] Pollar, J.H. (1971). On fluctuatng nterest rates, Bulletn e L Assocaton Royale es Actuares Belges 66, 68 97. [15] Pollar, A.H. & Pollar, J.H. (1969). A stochastc approach to actuaral functons, Journal of the Insttute of Actuares 95, 79 113. [16] Stooley, C.L. (1934). The effect of a fallng nterest rate on the values of certan actuaral functons, Transactons of the Faculty of Actuares 14, 137 175. [17] Waters, H.R. (1978). The moments an strbutons of actuaral functons, Journal of the Insttute of Actuares 15, 61 75. (See also Annutes; Interest-rate Moelng; Lfe Insurance Mathematcs) ANGUS S. MACDONALD