ACTUARIAL RESEARCH CLEARING HOUSE 1990 VOL. INTEREST, AMORTIZATION AND SIMPLICITY by Thomas M. Zavist, A.S.A. 37
Iterest m Amortizatio ad Simplicity Cosider simple iterest for a momet. Suppose you have a effective iterest rate of 1% per year, ad you wat to brig a quatity forward with iterest for three moths. Obviously, you multiply by 1.03. Simple iterest seems so easy, but what if you have to go backward with iterest for three moths istead? Do you multiply by 0.97 or divide by 1.037 Suppose you have a mid-year adjustmet. Do you multiply by 1.06, make the adjustmet, ad the multiply by 1.06 agai to get to the ed of the year? You will ed up with 0.36% too much iterest for the year. Perhaps you divide by 1.06 ad multiply by 1.1 after the adjustmet, i order to ed up with exactly 1% aually. Does that mea that the quatity i the first example should have bee divided by 1.09 ad multiplied by 1.1, istead of Just multiplied by 1.037 Simple iterest is ot so simple after all. I fact, simple iterest is very complicated. It is much more cofusig to compute tha compoud iterest, because with compoud iterest, every step is defied precisely, ad o cofusio ever arises. Moreover, compoud iterest describes how iterest behaves i the real world much better tha simple iterest does. Amortizatio schedules are aalogous to iterest schedules i may ways, ad a liear amortizatio schedule has all the problems of simple iterest ad the some. O the oe had, bases with the same amortizatio period but differet startig dates caot be combied, so detailed records of prior bases must be maitaied from year to year. O the other had, bases with differig sigs ca produce peculiar ad udesirable effects whe aggregated. Imagie that you have a $50,000 gai oe year ad a $50,000 loss the ext. Suppose you eglect iterest, as accoutats are wot to do, ad amortize each base over te years. After oe year, $5,000 of the gai has bee realized, but the urealized portio of the gai, coupled with the loss, produces a urealized et loss of $5,000. Because the aual amouts of amortizatio of the two bases cacel oe aother, the urealized et loss of $5,000 remais o the books udisturbed for ie more years, the vaishes i the course of the sigle fial year. Why should a accoutat with a urealized et loss of $5,000 sit aroud twiddlig his thumbs for ie full years ad suddely realize all of it i the eleveth year? To do so is othig less tha silly. The startig date of a urealized gai or loss beig amortized is of o more relevace tha the year prited o a dollar bill that is earig iterest i a bak. Each dollar of urealized gai or loss should be treated equal. The oly way to aggregate several gais ad losses, ad always amortize them smoothly, is to amortize a fixed proportio of the urealized balace every period, which meas usig a expoetial amortizatio schedule istead of a liear amortizatio schedule. 38
Recall that the force of mortality is the egative of the derivative (with respect to time) of the logarithm of a expected populatio. By aalogy, let the force of amortizatio.(t) at time t be equal to the egative of the derivative (with respect to time) of the logarithm of the balace b(t) of a base. The outstadig balace b(t) of a base amortized liearly from time t = 0 to time t = is give by A_tJ b(t) - b(o), so the force of amortizatio is give by d.(t) = - - - d [ log b(t) log ~ + log b(0) - log A -t J -t d v 1 log a = - -tl A_tl,_tl T] With simple iterest, the force of iterest varies periodically, but with liear amortizatio, the force of amortizatio diverges to ifiity at time t =, iasmuch as the deomiator i the last expressio vaishes at time t =. Not oly does the force of amortizatio diverge to ifiity, but the uweighted mea force of amortizatio betwee time t = 0 ad time t = is ifiite as well ad is give by i ~(t) I - -- d log 0 0 -t J log Y& log = 39
Fortuately, the mea force of amortizatio weighted by the outstadig balace is fiite ad is give by I i_t i v -t.(t) b(t) --b(0) v 0 0 0 i i I -t[ b(0) a b(t) s 0 0 -'] 0 ~ 1 - exp(- ) ] -t[ D~~ ~~ 1 - exp(-~) I - (I - ~) ] ~ 1-1 - + By cotrast, if the force of amortizatio = is costat, the b(t) = exp(-.t) b(0). Thus, a costat force of amortizatio produces a expoetial amortizatio schedule. Sice the effective rate of iterest i is give by l+i = exp(~), defie the effective rate of amortizatio to be m such that l-m = exp(-~). Therefore, the outstadig balace is give by t b(t) = (l-m) b(0) : (l-m) b(t-1), ad the paymet made at the ed of each year to pay iterest ad amortize the pricipal is give by (m+i) b(t-l). 40
By holdig m fixed, a chage of iterest rate chages oly the yearly paymet but ot the schedule of uamortized balaces, thereby makig a chage i iterest rate easy to implemet. Furthermore, the effective rate of amortizatio is approximated by Usig m ~ / = 0. to approximate the effective rate of amortizatio i the $50,000 example above, examie the urealized et loss uder the traditioal liear amortizatio schedule as compared to a expoetial amortizatio schedule, as follows: Urealized Net Loss Year Liear Amortizatio o $ (50,000) 1 5,000 5,000 3 5,000 4 5,000 5 5,000 6 5,000 7 5,000 8 5,000 9 5,000 10 5,000 11 0 1 0 13 0 14 0 15 0 Urealized Net Loss Expoetial Amortizatio $ (50,000) I0,000 8,000 6,400 5,10 4,096 3.77.61.097 1.678 1,34 1,074 859 687 550 440 Besides producig a smoother amortizatio schedule, the expoetial amortizatio schedule allows all the bases, which have the same fprce of amortizatio, to be combied ad multiplied each year by the simple factor of (l-m). Cosequetly, like bases occurrig at differet dates ca be combied ito a sigle base, ad the record keepig becomes simpler. Beware that the expoetially amortized bases are ever amortized fully. Because like bases ca be combied, however, the umber of bases is limited. I other words, the expoetially amortized bases do ot proliferate like rabbits, despite beig immortal. Expoetial amortizatio is ot very useful for mortgages, but it has practical applicatios for the pesio actuary, especially with regard to the actuarial value of assets, fudig stadard accout, ad FASB expese ad disclosure. It may apply to the isurace busiess, but I am ot familiar with how isurace actuaries use amortizatio. Of course, there are legal obstacles to overcome. 41
Compoud iterest is a improvemet upo simple iterest that makes the calculatio of iterest simpler. Likewise, expoetial amortizatio is a improvemet upo liear amortizatio that makes the calculatio of amortizatio simpler. Most importat, however, a expoetial amortizatio schedule avoids the peculiar effects produced by the iteractio of several bases amortized liearly. 4