1693:
1270:
1688:{\displaystyle {\begin{aligned}a_{{\overline {n}}|i}&=\sum _{k=1}^{n}{\frac {1}{(1+i)^{k}}}={\frac {1}{1+i}}\sum _{k=0}^{n-1}\left({\frac {1}{1+i}}\right)^{k}\\&={\frac {1}{1+i}}\left({\frac {1-(1+i)^{-n}}{1-(1+i)^{-1}}}\right)\quad \quad {\text{by using the equation for the sum of a geometric series}}\\&={\frac {1-(1+i)^{-n}}{1+i-1}}\\&={\frac {1-\left({\frac {1}{1+i}}\right)^{n}}{i}},\end{aligned}}}
3305:
3515:
that the annuitant lives to each future payment period. Valuation of life annuities also depends on the timing of payments just as with annuities certain, however life annuities may not be calculated with similar formulas because actuarial present value accounts for the probability of death at each
2789:
243:
Annuity due refers to a series of equal payments made at the same interval at the beginning of each period. Periods can be monthly, quarterly, semi-annually, annually, or any other defined period. Examples of annuity due payments include rentals, leases, and insurance payments, which are made to
3109:
151:– Registered products that are regulated by the SEC in the United States of America. They allow direct investment into various funds that are specially created for Variable annuities. Typically, the insurance company guarantees a certain death benefit or lifetime withdrawal benefits.
1928:
3425:
982:
159:– Annuities with payments linked to an index. Typically, the minimum payment will be 0% and the maximum will be predetermined. The performance of an index determines whether the minimum, the maximum or something in between is credited to the customer.
2649:
2158:
2305:
3995:
3084:
2940:
625:
of an annuity is the accumulated amount, including payments and interest, of a stream of payments made to an interest-bearing account. For an annuity-immediate, it is the value immediately after the n-th payment. The future value is given by:
2535:
1186:
2663:
3300:{\displaystyle \lim _{n\,\rightarrow \,\infty }{\text{PV}}(i,n,R)=\lim _{n\,\rightarrow \,\infty }R\times a_{{\overline {n}}|i}=\lim _{n\,\rightarrow \,\infty }R\times {\frac {1-\left(1+i\right)^{-n}}{i}}=\,{\frac {R}{i}}.}
1701:
Similarly, we can prove the formula for the future value. The payment made at the end of the last year would accumulate no interest and the payment made at the end of the first year would accumulate interest for a total of
1090:
61:
payments. Annuities can be classified by the frequency of payment dates. The payments (deposits) may be made weekly, monthly, quarterly, yearly, or at any other regular interval of time. Annuities may be calculated by
1275:
3614:
141:– These are annuities with fixed payments. If provided by an insurance company, the company guarantees a fixed return on the initial investment. In the United States, fixed annuities are not regulated by the
555:
858:
406:
712:
1712:
313:
of an annuity is the value of a stream of payments, discounted by the interest rate to account for the fact that payments are being made at various moments in the future. The present value is given in
3848:
3320:
872:
991:
of the loan, or the amount paid out by an interest-bearing account at the end of each period when the amount PV is invested at time zero, and the account becomes zero with the n-th withdrawal.
1258:
2543:
2016:
3489:
2413:
2166:
3697:
3644:
3879:
2951:
2807:
1942:
is an annuity whose payments are made at the beginning of each period. Deposits in savings, rent or lease payments, and insurance premiums are examples of annuities due.
2801:
payments is the sum of one annuity payment now and an ordinary annuity with one payment less, and also equal, with a time shift, to an ordinary annuity. Thus we have:
2424:
616:
3717:
3664:
3448:
2368:
2348:
2328:
778:
755:
735:
578:
472:
449:
429:
1101:
2784:{\displaystyle {\text{FV}}_{\text{due}}\left({\frac {0.09}{12}},7\times 12,\$ 100\right)=\$ 100\times {\ddot {s}}_{{\overline {84}}|0.0075}=\$ 11,730.01.}
1000:
4268:
4249:
3538:
2010:
Each annuity payment is allowed to compound for one extra period. Thus, the present and future values of an annuity-due can be calculated.
97:
are made at the end of payment periods, so that interest accrues between the issue of the annuity and the first payment. Payments of an
480:
2657:
The final value of a 7-year annuity-due with a nominal annual interest rate of 9% and monthly payments of $ 100 can be calculated by:
786:
323:
227:
If the payments are made at the end of the time periods, so that interest is accumulated before the payment, the annuity is called an
142:
632:
560:
In practice, often loans are stated per annum while interest is compounded and payments are made monthly. In this case, the interest
4220:
4186:
1923:{\displaystyle s_{{\overline {n}}|i}=1+(1+i)+(1+i)^{2}+\cdots +(1+i)^{n-1}=(1+i)^{n}a_{{\overline {n}}|i}={\frac {(1+i)^{n}-1}{i}}.}
987:
The rent is understood as either the amount paid at the end of each period in return for an amount PV borrowed at time zero, the
451:
is the per period interest rate. Present value is linear in the amount of payments, therefore the present value for payments, or
3420:{\displaystyle a_{{\overline {\infty }}|i}={\frac {1}{i}}{\text{ and }}{\ddot {a}}_{{\overline {\infty }}|i}={\frac {1}{d}},}
977:{\displaystyle {\text{PV}}\left({\frac {0.12}{12}},5\times 12,\$ 100\right)=\$ 100\times a_{{\overline {60}}|0.01}=\$ 4,495.50}
757:
is the per period interest rate. Future value is linear in the amount of payments, therefore the future value for payments, or
172:(usually after retirement). An annuity that begins payments as soon as the customer has paid, without a deferral period is an
3728:
2371:
4081:
Find the periodic payment of an accumulated value of $ 1,600,000, payable annually for 3 years at 9% compounded annually.
4107:
4102:
4038:− 1) 08695652174 × (−0.3424837676)÷ (−1304347826) = 2.2832251175 70000÷ 2.2832251175= $ 30658.3873 is the correct value
2794:
In Excel, the PV and FV functions take on optional fifth argument which selects from annuity-immediate or annuity-due.
4067:
Find the periodic payment of an accumulated value of $ 55,000, payable monthly for 3 years at 15% compounded monthly.
4042:
Find the periodic payment of an annuity due of $ 250,700, payable quarterly for 8 years at 5% compounded quarterly.
4289:
866:
The present value of a 5-year annuity with a nominal annual interest rate of 12% and monthly payments of $ 100 is:
4004:
Find the periodic payment of an annuity due of $ 70,000, payable annually for 3 years at 15% compounded annually.
4307:
4112:
31:
4124:
3504:
2644:{\displaystyle {\frac {1}{{\ddot {a}}_{{\overline {n}}|i}}}-{\frac {1}{{\ddot {s}}_{{\overline {n}}|i}}}=d.}
2153:{\displaystyle {\ddot {a}}_{{\overline {n|}}i}=(1+i)\times a_{{\overline {n|}}i}={\frac {1-(1+i)^{-n}}{d}},}
1215:
2300:{\displaystyle {\ddot {s}}_{{\overline {n|}}i}=(1+i)\times s_{{\overline {n|}}i}={\frac {(1+i)^{n}-1}{d}},}
155:
63:
129:
are guaranteed to be paid for a number of years and then become contingent on the annuitant being alive.
3453:
2377:
581:
219:. Valuation of annuities certain may be calculated using formulas depending on the timing of payments.
3669:
3990:{\displaystyle R={\frac {A}{1+\left(1-\left(1+{\frac {j}{m}}\right)\right)^{-{\frac {(n-1)}{j/m}}}}}}
3314:
has a finite present value when there is a non-zero discount rate. The formulae for a perpetuity are
3622:
3079:{\displaystyle {\ddot {s}}_{{\overline {n|}}i}=s_{{\overline {n}}|i}(1+i)=s_{{\overline {n+1|}}i}-1}
2935:{\displaystyle {\ddot {a}}_{{\overline {n|}}i}=a_{{\overline {n}}|i}(1+i)=a_{{\overline {n-1|}}i}+1}
4283:
4278:
4144:
4129:
3854:
192:
184:
4212:
4312:
314:
45:
is a series of payments made at equal intervals. Examples of annuities are regular deposits to a
2530:{\displaystyle {\ddot {s}}_{{\overline {n}}|i}=(1+i)^{n}\times {\ddot {a}}_{{\overline {n}}|i},}
4264:
4245:
4216:
4182:
46:
4205:
1200:-th payment must be discounted to the present by dividing by the interest, compounded by
101:
are made at the beginning of payment periods, so a payment is made immediately on issue.
587:
3702:
3649:
3433:
2353:
2333:
2313:
1181:{\displaystyle {\frac {1}{a_{{\overline {n}}|i}}}-{\frac {1}{s_{{\overline {n}}|i}}}=i}
763:
740:
720:
563:
457:
434:
414:
4301:
196:
191:
of the future annuity payments. The valuation of an annuity entails concepts such as
188:
109:
Annuities that provide payments that will be paid over a period known in advance are
69:
An annuity which provides for payments for the remainder of a person's lifetime is a
50:
4134:
3500:
200:
122:
70:
235:. Mortgage payments are annuity-immediate, interest is earned before being paid.
3512:
4139:
3508:
3311:
74:
38:
3722:
Also, this can be thought of as the present value of the remaining payments
54:
1085:{\displaystyle s_{{\overline {n}}|i}=(1+i)^{n}\times a_{{\overline {n}}|i}}
4282:
17:
4203:
Jordan, Bradford D.; Ross, Stephen David; Westerfield, Randolph (2000).
58:
3609:{\displaystyle {\frac {R}{i}}-(1+i)^{n}\left({\frac {R}{i}}-P\right).}
3103:
is an annuity for which the payments continue forever. Observe that
2418:
The future and present values for annuities due are related since:
550:{\displaystyle {\text{PV}}(i,n,R)=R\times a_{{\overline {n}}|i}.}
211:
If the number of payments is known in advance, the annuity is an
853:{\displaystyle {\text{FV}}(i,n,R)=R\times s_{{\overline {n}}|i}}
401:{\displaystyle a_{{\overline {n}}|i}={\frac {1-(1+i)^{-n}}{i}},}
707:{\displaystyle s_{{\overline {n}}|i}={\frac {(1+i)^{n}-1}{i}},}
125:, which is paid over the remaining lifetime of the annuitant.
244:
cover services provided in the period following the payment.
3619:
Because the scheme is equivalent with borrowing the amount
3699:
of that borrowed amount in the bank to grow with interest
168:
An annuity that begins payments only after a period is a
3843:{\displaystyle R\left=R\times a_{{\overline {N-n}}|i}.}
1553:
by using the equation for the sum of a geometric series
4045:
R= 250,700/(1+〖(1-(1+((.05)/4) )〗^(-(32-1))/((.05)/4))
590:
4084:
R=1,600,000/((〖((1+((.09)/1) )〗^(3+1)-1)/((.09)/1)-1)
4070:
R=55,000/((〖((1+((.15)/12) )〗^(36+1)-1)/((.15)/12)-1)
3882:
3731:
3705:
3672:
3652:
3625:
3541:
3456:
3436:
3323:
3112:
3086:. The value one period after the time of the last of
2954:
2810:
2666:
2546:
2427:
2380:
2356:
2336:
2316:
2169:
2019:
1715:
1273:
1218:
1104:
1003:
875:
789:
766:
743:
723:
635:
566:
483:
460:
437:
417:
326:
117:
Annuities paid only under certain circumstances are
4181:. Mason, Ohio: Thomson South-Western. p. 230.
4010:= 70,000/(1+〖(1-(1+((.15)/1) )〗^(-(3-1))/((.15)/1))
4204:
4167:. Homewood, Illinois: Richard D. Irwin, Inc. p. 45
3989:
3842:
3711:
3691:
3658:
3638:
3608:
3483:
3442:
3419:
3299:
3078:
2934:
2783:
2643:
2529:
2407:
2362:
2342:
2322:
2299:
2152:
1922:
1687:
1252:
1180:
1084:
976:
852:
772:
749:
729:
706:
610:
572:
549:
466:
443:
423:
400:
4242:Mathematics of Investment and Credit, 5th Edition
3213:
3161:
3114:
73:. An annuity which continues indefinitely is a
4293:. Vol. II (9th ed.). pp. 72–89.
4244:. ACTEX Academic Series. ACTEX Publications.
994:Future and present values are related since:
85:Annuities may be classified in several ways.
8:
4060:R = S\,/((〖((1+(j/m) )〗^(n+1)-1)/(j/m)-1)
4057:Finding the Periodic Payment(R), Given S:
27:Series of payments made at equal intervals
3971:
3948:
3944:
3924:
3889:
3881:
3865:Formula for finding the periodic payment
3827:
3809:
3808:
3772:
3753:
3740:
3730:
3704:
3673:
3671:
3651:
3626:
3624:
3582:
3571:
3542:
3540:
3463:
3455:
3435:
3404:
3391:
3381:
3380:
3369:
3368:
3362:
3352:
3339:
3329:
3328:
3322:
3284:
3283:
3265:
3236:
3224:
3220:
3216:
3199:
3189:
3188:
3172:
3168:
3164:
3131:
3125:
3121:
3117:
3111:
3054:
3042:
3041:
3009:
2999:
2998:
2975:
2969:
2968:
2957:
2956:
2953:
2910:
2898:
2897:
2865:
2855:
2854:
2831:
2825:
2824:
2813:
2812:
2809:
2756:
2746:
2745:
2734:
2733:
2684:
2673:
2668:
2665:
2620:
2610:
2609:
2598:
2597:
2591:
2576:
2566:
2565:
2554:
2553:
2547:
2545:
2514:
2504:
2503:
2492:
2491:
2481:
2452:
2442:
2441:
2430:
2429:
2426:
2387:
2379:
2355:
2335:
2315:
2276:
2257:
2238:
2232:
2231:
2190:
2184:
2183:
2172:
2171:
2168:
2132:
2107:
2088:
2082:
2081:
2040:
2034:
2033:
2022:
2021:
2018:
1899:
1880:
1867:
1857:
1856:
1846:
1815:
1784:
1731:
1721:
1720:
1714:
1666:
1644:
1630:
1591:
1566:
1551:
1533:
1500:
1475:
1453:
1437:
1415:
1398:
1387:
1365:
1353:
1331:
1325:
1314:
1293:
1283:
1282:
1274:
1272:
1241:
1219:
1217:
1160:
1150:
1149:
1140:
1125:
1115:
1114:
1105:
1103:
1072:
1062:
1061:
1048:
1019:
1009:
1008:
1002:
949:
939:
938:
886:
876:
874:
840:
830:
829:
790:
788:
765:
742:
722:
683:
664:
651:
641:
640:
634:
600:
589:
565:
534:
524:
523:
484:
482:
459:
436:
416:
380:
355:
342:
332:
331:
325:
187:of an annuity entails calculation of the
3507:of the future life contingent payments.
2942:. The value at the time of the first of
4156:
1698:which gives us the result as required.
1253:{\displaystyle {\frac {R}{(1+i)^{k}}}}
4211:. Boston: Irwin/McGraw-Hill. p.
4026:as, (1 ÷ 1.15)= 0.8695652174 2) find
3528:with interest, the amount owed after
3524:If an annuity is for repaying a debt
1204:terms. Hence the contribution of the
7:
3503:may be performed by calculating the
3646:to create a perpetuity with coupon
2350:is the per-term interest rate, and
3484:{\displaystyle d={\frac {i}{1+i}}}
3383:
3331:
3225:
3173:
3126:
2769:
2723:
2709:
2408:{\displaystyle d={\frac {i}{i+1}}}
1706: − 1) years. Therefore,
1192:Proof of annuity-immediate formula
962:
925:
911:
143:Securities and Exchange Commission
25:
4207:Fundamentals of corporate finance
3692:{\displaystyle {\frac {R}{i}}-P}
3491:is the effective discount rate.
1196:To calculate present value, the
4261:Theory of Interest, 3rd Edition
1550:
1549:
4179:Practical financial management
3963:
3951:
3828:
3769:
3756:
3639:{\displaystyle {\frac {R}{i}}}
3568:
3555:
3392:
3340:
3221:
3200:
3169:
3154:
3136:
3122:
3055:
3031:
3019:
3010:
2976:
2911:
2887:
2875:
2866:
2832:
2757:
2621:
2577:
2515:
2478:
2465:
2453:
2273:
2260:
2239:
2221:
2209:
2191:
2129:
2116:
2089:
2071:
2059:
2041:
1896:
1883:
1868:
1843:
1830:
1812:
1799:
1781:
1768:
1762:
1750:
1732:
1588:
1575:
1530:
1517:
1497:
1484:
1350:
1337:
1294:
1238:
1225:
1161:
1126:
1073:
1045:
1032:
1020:
950:
841:
813:
795:
680:
667:
652:
535:
507:
489:
377:
364:
343:
66:known as "annuity functions".
1:
4163:Kellison, Stephen G. (1970).
4022:Find PVOA factor as. 1) find
4240:Samuel A. Broverman (2010).
4108:Annuities under European law
4103:Annuities under American law
3822:
3386:
3334:
3194:
3060:
3004:
2981:
2916:
2860:
2837:
2751:
2615:
2571:
2509:
2447:
2244:
2196:
2094:
2046:
1862:
1726:
1288:
1155:
1120:
1067:
1014:
944:
835:
646:
529:
337:
737:is the number of terms and
431:is the number of terms and
4329:
3511:are used to calculate the
2372:effective rate of discount
127:Certain and life annuities
29:
4259:Stephen Kellison (2008).
4113:Annuities under Swiss law
3520:Amortization calculations
3450:is the interest rate and
4177:Lasher, William (2008).
2330:is the number of terms,
156:Equity-indexed annuities
121:. A common example is a
32:Annuity (disambiguation)
4290:Encyclopædia Britannica
4125:Amortization calculator
3505:actuarial present value
133:Variability of payments
105:Contingency of payments
4165:The Theory of Interest
4087:R = 1,600,000/3.573129
4073:R = 55,000/45.67944932
4048:R = 250,700/26.5692901
4013:R = 70,000/2.625708885
3991:
3844:
3713:
3693:
3660:
3640:
3610:
3485:
3444:
3421:
3301:
3080:
2936:
2785:
2645:
2531:
2409:
2364:
2344:
2324:
2301:
2154:
1976:———
1973:———
1970:———
1967:———
1924:
1689:
1409:
1330:
1254:
1182:
1086:
978:
854:
774:
751:
731:
708:
612:
574:
551:
468:
445:
425:
402:
278:———
275:———
272:———
269:———
64:mathematical functions
4263:. McGraw-Hill/Irwin.
3992:
3845:
3714:
3694:
3661:
3641:
3611:
3486:
3445:
3422:
3302:
3081:
2937:
2786:
2646:
2532:
2410:
2365:
2345:
2325:
2302:
2155:
1925:
1690:
1383:
1310:
1255:
1183:
1087:
979:
855:
775:
752:
732:
709:
613:
582:nominal interest rate
575:
552:
469:
446:
426:
403:
115:guaranteed annuities.
4279:Sprague, Thomas Bond
3880:
3861:Example calculations
3729:
3703:
3670:
3650:
3623:
3539:
3454:
3434:
3321:
3110:
2952:
2808:
2797:An annuity-due with
2664:
2544:
2425:
2378:
2354:
2334:
2314:
2167:
2017:
1713:
1271:
1264:to be 1, then:
1216:
1102:
1001:
873:
787:
764:
741:
721:
633:
588:
564:
481:
458:
435:
415:
324:
164:Deferral of payments
119:contingent annuities
30:For other uses, see
4145:Time value of money
4130:Fixed rate mortgage
3855:fixed rate mortgage
1260:. Just considering
611:{\textstyle i=I/12}
193:time value of money
3987:
3840:
3709:
3689:
3656:
3636:
3606:
3481:
3440:
3417:
3297:
3229:
3177:
3130:
3076:
2932:
2781:
2641:
2527:
2405:
2360:
2340:
2320:
2297:
2150:
1920:
1685:
1683:
1250:
1178:
1082:
974:
850:
770:
747:
727:
704:
608:
570:
547:
464:
441:
421:
398:
315:actuarial notation
217:guaranteed annuity
149:Variable annuities
89:Timing of payments
53:payments, monthly
4284:"Annuities"
4270:978-0-07-338244-9
4251:978-1-56698-767-7
4016:R = $ 26659.46724
3985:
3980:
3932:
3825:
3788:
3748:
3712:{\displaystyle i}
3681:
3659:{\displaystyle R}
3634:
3590:
3550:
3479:
3443:{\displaystyle i}
3412:
3389:
3377:
3365:
3360:
3337:
3292:
3278:
3212:
3197:
3160:
3134:
3113:
3063:
3007:
2984:
2965:
2919:
2863:
2840:
2821:
2754:
2742:
2692:
2676:
2671:
2630:
2618:
2606:
2586:
2574:
2562:
2512:
2500:
2450:
2438:
2403:
2363:{\displaystyle d}
2343:{\displaystyle i}
2323:{\displaystyle n}
2292:
2247:
2199:
2180:
2145:
2097:
2049:
2030:
2008:
2007:
1915:
1865:
1729:
1676:
1660:
1618:
1554:
1543:
1469:
1431:
1381:
1360:
1291:
1248:
1170:
1158:
1135:
1123:
1070:
1017:
947:
894:
879:
838:
793:
773:{\displaystyle R}
750:{\displaystyle i}
730:{\displaystyle n}
699:
649:
573:{\displaystyle I}
532:
487:
467:{\displaystyle R}
444:{\displaystyle i}
424:{\displaystyle n}
393:
340:
307:
306:
229:annuity-immediate
223:Annuity-immediate
174:immediate annuity
111:annuities certain
95:annuity-immediate
16:(Redirected from
4320:
4308:Finance theories
4294:
4286:
4274:
4255:
4227:
4226:
4210:
4200:
4194:
4192:
4174:
4168:
4161:
4090:R = $ 447,786.80
3996:
3994:
3993:
3988:
3986:
3984:
3983:
3982:
3981:
3979:
3975:
3966:
3949:
3943:
3939:
3938:
3934:
3933:
3925:
3890:
3849:
3847:
3846:
3841:
3836:
3835:
3831:
3826:
3821:
3810:
3794:
3790:
3789:
3784:
3783:
3782:
3754:
3749:
3741:
3718:
3716:
3715:
3710:
3698:
3696:
3695:
3690:
3682:
3674:
3665:
3663:
3662:
3657:
3645:
3643:
3642:
3637:
3635:
3627:
3615:
3613:
3612:
3607:
3602:
3598:
3591:
3583:
3576:
3575:
3551:
3543:
3490:
3488:
3487:
3482:
3480:
3478:
3464:
3449:
3447:
3446:
3441:
3426:
3424:
3423:
3418:
3413:
3405:
3400:
3399:
3395:
3390:
3382:
3379:
3378:
3370:
3366:
3363:
3361:
3353:
3348:
3347:
3343:
3338:
3330:
3306:
3304:
3303:
3298:
3293:
3285:
3279:
3274:
3273:
3272:
3264:
3260:
3237:
3228:
3208:
3207:
3203:
3198:
3190:
3176:
3135:
3132:
3129:
3085:
3083:
3082:
3077:
3069:
3068:
3064:
3059:
3058:
3043:
3018:
3017:
3013:
3008:
3000:
2990:
2989:
2985:
2980:
2979:
2970:
2967:
2966:
2958:
2941:
2939:
2938:
2933:
2925:
2924:
2920:
2915:
2914:
2899:
2874:
2873:
2869:
2864:
2856:
2846:
2845:
2841:
2836:
2835:
2826:
2823:
2822:
2814:
2790:
2788:
2787:
2782:
2765:
2764:
2760:
2755:
2747:
2744:
2743:
2735:
2719:
2715:
2693:
2685:
2678:
2677:
2674:
2672:
2669:
2650:
2648:
2647:
2642:
2631:
2629:
2628:
2624:
2619:
2611:
2608:
2607:
2599:
2592:
2587:
2585:
2584:
2580:
2575:
2567:
2564:
2563:
2555:
2548:
2536:
2534:
2533:
2528:
2523:
2522:
2518:
2513:
2505:
2502:
2501:
2493:
2486:
2485:
2461:
2460:
2456:
2451:
2443:
2440:
2439:
2431:
2414:
2412:
2411:
2406:
2404:
2402:
2388:
2369:
2367:
2366:
2361:
2349:
2347:
2346:
2341:
2329:
2327:
2326:
2321:
2306:
2304:
2303:
2298:
2293:
2288:
2281:
2280:
2258:
2253:
2252:
2248:
2243:
2242:
2233:
2205:
2204:
2200:
2195:
2194:
2185:
2182:
2181:
2173:
2159:
2157:
2156:
2151:
2146:
2141:
2140:
2139:
2108:
2103:
2102:
2098:
2093:
2092:
2083:
2055:
2054:
2050:
2045:
2044:
2035:
2032:
2031:
2023:
1945:
1944:
1929:
1927:
1926:
1921:
1916:
1911:
1904:
1903:
1881:
1876:
1875:
1871:
1866:
1858:
1851:
1850:
1826:
1825:
1789:
1788:
1740:
1739:
1735:
1730:
1722:
1694:
1692:
1691:
1686:
1684:
1677:
1672:
1671:
1670:
1665:
1661:
1659:
1645:
1631:
1623:
1619:
1617:
1600:
1599:
1598:
1567:
1559:
1555:
1552:
1548:
1544:
1542:
1541:
1540:
1509:
1508:
1507:
1476:
1470:
1468:
1454:
1446:
1442:
1441:
1436:
1432:
1430:
1416:
1408:
1397:
1382:
1380:
1366:
1361:
1359:
1358:
1357:
1332:
1329:
1324:
1302:
1301:
1297:
1292:
1284:
1259:
1257:
1256:
1251:
1249:
1247:
1246:
1245:
1220:
1187:
1185:
1184:
1179:
1171:
1169:
1168:
1164:
1159:
1151:
1141:
1136:
1134:
1133:
1129:
1124:
1116:
1106:
1091:
1089:
1088:
1083:
1081:
1080:
1076:
1071:
1063:
1053:
1052:
1028:
1027:
1023:
1018:
1010:
983:
981:
980:
975:
958:
957:
953:
948:
940:
921:
917:
895:
887:
880:
877:
859:
857:
856:
851:
849:
848:
844:
839:
831:
794:
791:
779:
777:
776:
771:
756:
754:
753:
748:
736:
734:
733:
728:
713:
711:
710:
705:
700:
695:
688:
687:
665:
660:
659:
655:
650:
642:
617:
615:
614:
609:
604:
579:
577:
576:
571:
556:
554:
553:
548:
543:
542:
538:
533:
525:
488:
485:
473:
471:
470:
465:
450:
448:
447:
442:
430:
428:
427:
422:
407:
405:
404:
399:
394:
389:
388:
387:
356:
351:
350:
346:
341:
333:
247:
246:
233:ordinary annuity
170:deferred annuity
21:
4328:
4327:
4323:
4322:
4321:
4319:
4318:
4317:
4298:
4297:
4277:
4271:
4258:
4252:
4239:
4236:
4231:
4230:
4223:
4202:
4201:
4197:
4189:
4176:
4175:
4171:
4162:
4158:
4153:
4121:
4099:
3967:
3950:
3917:
3913:
3906:
3902:
3901:
3894:
3878:
3877:
3863:
3811:
3804:
3768:
3755:
3739:
3735:
3727:
3726:
3701:
3700:
3668:
3667:
3648:
3647:
3621:
3620:
3581:
3577:
3567:
3537:
3536:
3522:
3497:
3468:
3452:
3451:
3432:
3431:
3367:
3364: and
3324:
3319:
3318:
3250:
3246:
3245:
3238:
3184:
3108:
3107:
3097:
3044:
3037:
2994:
2971:
2955:
2950:
2949:
2900:
2893:
2850:
2827:
2811:
2806:
2805:
2732:
2683:
2679:
2667:
2662:
2661:
2596:
2552:
2542:
2541:
2490:
2477:
2428:
2423:
2422:
2392:
2376:
2375:
2352:
2351:
2332:
2331:
2312:
2311:
2272:
2259:
2234:
2227:
2186:
2170:
2165:
2164:
2128:
2109:
2084:
2077:
2036:
2020:
2015:
2014:
1936:
1895:
1882:
1852:
1842:
1811:
1780:
1716:
1711:
1710:
1682:
1681:
1649:
1640:
1639:
1632:
1621:
1620:
1601:
1587:
1568:
1557:
1556:
1529:
1510:
1496:
1477:
1471:
1458:
1444:
1443:
1420:
1411:
1410:
1370:
1349:
1336:
1303:
1278:
1269:
1268:
1237:
1224:
1214:
1213:
1194:
1145:
1110:
1100:
1099:
1057:
1044:
1004:
999:
998:
934:
885:
881:
871:
870:
825:
785:
784:
762:
761:
739:
738:
719:
718:
679:
666:
636:
631:
630:
586:
585:
580:is stated as a
562:
561:
519:
479:
478:
456:
455:
433:
432:
413:
412:
376:
357:
327:
322:
321:
241:
225:
213:annuity certain
209:
207:Annuity-certain
182:
166:
139:Fixed annuities
135:
107:
93:Payments of an
91:
83:
47:savings account
35:
28:
23:
22:
15:
12:
11:
5:
4326:
4324:
4316:
4315:
4310:
4300:
4299:
4296:
4295:
4275:
4269:
4256:
4250:
4235:
4232:
4229:
4228:
4221:
4195:
4187:
4169:
4155:
4154:
4152:
4149:
4148:
4147:
4142:
4137:
4132:
4127:
4120:
4117:
4116:
4115:
4110:
4105:
4098:
4095:
4094:
4093:
4092:
4091:
4088:
4085:
4079:
4078:
4077:
4076:R = $ 1,204.04
4074:
4071:
4055:
4054:
4053:
4052:
4051:R = $ 9,435.71
4049:
4046:
4020:
4019:
4018:
4017:
4014:
4011:
3998:
3997:
3978:
3974:
3970:
3965:
3962:
3959:
3956:
3953:
3947:
3942:
3937:
3931:
3928:
3923:
3920:
3916:
3912:
3909:
3905:
3900:
3897:
3893:
3888:
3885:
3862:
3859:
3851:
3850:
3839:
3834:
3830:
3824:
3820:
3817:
3814:
3807:
3803:
3800:
3797:
3793:
3787:
3781:
3778:
3775:
3771:
3767:
3764:
3761:
3758:
3752:
3747:
3744:
3738:
3734:
3708:
3688:
3685:
3680:
3677:
3666:, and putting
3655:
3633:
3630:
3617:
3616:
3605:
3601:
3597:
3594:
3589:
3586:
3580:
3574:
3570:
3566:
3563:
3560:
3557:
3554:
3549:
3546:
3521:
3518:
3501:life annuities
3496:
3495:Life annuities
3493:
3477:
3474:
3471:
3467:
3462:
3459:
3439:
3428:
3427:
3416:
3411:
3408:
3403:
3398:
3394:
3388:
3385:
3376:
3373:
3359:
3356:
3351:
3346:
3342:
3336:
3333:
3327:
3308:
3307:
3296:
3291:
3288:
3282:
3277:
3271:
3268:
3263:
3259:
3256:
3253:
3249:
3244:
3241:
3235:
3232:
3227:
3223:
3219:
3215:
3211:
3206:
3202:
3196:
3193:
3187:
3183:
3180:
3175:
3171:
3167:
3163:
3159:
3156:
3153:
3150:
3147:
3144:
3141:
3138:
3128:
3124:
3120:
3116:
3096:
3093:
3092:
3091:
3090:payments of 1.
3075:
3072:
3067:
3062:
3057:
3053:
3050:
3047:
3040:
3036:
3033:
3030:
3027:
3024:
3021:
3016:
3012:
3006:
3003:
2997:
2993:
2988:
2983:
2978:
2974:
2964:
2961:
2947:
2946:payments of 1.
2931:
2928:
2923:
2918:
2913:
2909:
2906:
2903:
2896:
2892:
2889:
2886:
2883:
2880:
2877:
2872:
2868:
2862:
2859:
2853:
2849:
2844:
2839:
2834:
2830:
2820:
2817:
2792:
2791:
2780:
2777:
2774:
2771:
2768:
2763:
2759:
2753:
2750:
2741:
2738:
2731:
2728:
2725:
2722:
2718:
2714:
2711:
2708:
2705:
2702:
2699:
2696:
2691:
2688:
2682:
2652:
2651:
2640:
2637:
2634:
2627:
2623:
2617:
2614:
2605:
2602:
2595:
2590:
2583:
2579:
2573:
2570:
2561:
2558:
2551:
2538:
2537:
2526:
2521:
2517:
2511:
2508:
2499:
2496:
2489:
2484:
2480:
2476:
2473:
2470:
2467:
2464:
2459:
2455:
2449:
2446:
2437:
2434:
2401:
2398:
2395:
2391:
2386:
2383:
2359:
2339:
2319:
2308:
2307:
2296:
2291:
2287:
2284:
2279:
2275:
2271:
2268:
2265:
2262:
2256:
2251:
2246:
2241:
2237:
2230:
2226:
2223:
2220:
2217:
2214:
2211:
2208:
2203:
2198:
2193:
2189:
2179:
2176:
2161:
2160:
2149:
2144:
2138:
2135:
2131:
2127:
2124:
2121:
2118:
2115:
2112:
2106:
2101:
2096:
2091:
2087:
2080:
2076:
2073:
2070:
2067:
2064:
2061:
2058:
2053:
2048:
2043:
2039:
2029:
2026:
2006:
2005:
2002:
1997:
1991:
1988:
1985:
1981:
1980:
1977:
1974:
1971:
1968:
1964:
1963:
1960:
1958:
1955:
1952:
1949:
1935:
1932:
1931:
1930:
1919:
1914:
1910:
1907:
1902:
1898:
1894:
1891:
1888:
1885:
1879:
1874:
1870:
1864:
1861:
1855:
1849:
1845:
1841:
1838:
1835:
1832:
1829:
1824:
1821:
1818:
1814:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1787:
1783:
1779:
1776:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1743:
1738:
1734:
1728:
1725:
1719:
1696:
1695:
1680:
1675:
1669:
1664:
1658:
1655:
1652:
1648:
1643:
1638:
1635:
1629:
1626:
1624:
1622:
1616:
1613:
1610:
1607:
1604:
1597:
1594:
1590:
1586:
1583:
1580:
1577:
1574:
1571:
1565:
1562:
1560:
1558:
1547:
1539:
1536:
1532:
1528:
1525:
1522:
1519:
1516:
1513:
1506:
1503:
1499:
1495:
1492:
1489:
1486:
1483:
1480:
1474:
1467:
1464:
1461:
1457:
1452:
1449:
1447:
1445:
1440:
1435:
1429:
1426:
1423:
1419:
1414:
1407:
1404:
1401:
1396:
1393:
1390:
1386:
1379:
1376:
1373:
1369:
1364:
1356:
1352:
1348:
1345:
1342:
1339:
1335:
1328:
1323:
1320:
1317:
1313:
1309:
1306:
1304:
1300:
1296:
1290:
1287:
1281:
1277:
1276:
1244:
1240:
1236:
1233:
1230:
1227:
1223:
1193:
1190:
1189:
1188:
1177:
1174:
1167:
1163:
1157:
1154:
1148:
1144:
1139:
1132:
1128:
1122:
1119:
1113:
1109:
1093:
1092:
1079:
1075:
1069:
1066:
1060:
1056:
1051:
1047:
1043:
1040:
1037:
1034:
1031:
1026:
1022:
1016:
1013:
1007:
985:
984:
973:
970:
967:
964:
961:
956:
952:
946:
943:
937:
933:
930:
927:
924:
920:
916:
913:
910:
907:
904:
901:
898:
893:
890:
884:
861:
860:
847:
843:
837:
834:
828:
824:
821:
818:
815:
812:
809:
806:
803:
800:
797:
769:
746:
726:
715:
714:
703:
698:
694:
691:
686:
682:
678:
675:
672:
669:
663:
658:
654:
648:
645:
639:
607:
603:
599:
596:
593:
569:
558:
557:
546:
541:
537:
531:
528:
522:
518:
515:
512:
509:
506:
503:
500:
497:
494:
491:
463:
440:
420:
409:
408:
397:
392:
386:
383:
379:
375:
372:
369:
366:
363:
360:
354:
349:
345:
339:
336:
330:
305:
304:
301:
296:
293:
290:
287:
283:
282:
279:
276:
273:
270:
266:
265:
262:
259:
256:
253:
250:
240:
237:
224:
221:
208:
205:
181:
178:
165:
162:
161:
160:
152:
146:
134:
131:
106:
103:
90:
87:
82:
79:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4325:
4314:
4311:
4309:
4306:
4305:
4303:
4292:
4291:
4285:
4280:
4276:
4272:
4266:
4262:
4257:
4253:
4247:
4243:
4238:
4237:
4234:Other sources
4233:
4224:
4222:0-07-231289-0
4218:
4214:
4209:
4208:
4199:
4196:
4190:
4188:0-324-42262-8
4184:
4180:
4173:
4170:
4166:
4160:
4157:
4150:
4146:
4143:
4141:
4138:
4136:
4133:
4131:
4128:
4126:
4123:
4122:
4118:
4114:
4111:
4109:
4106:
4104:
4101:
4100:
4097:Legal regimes
4096:
4089:
4086:
4083:
4082:
4080:
4075:
4072:
4069:
4068:
4066:
4065:
4064:
4061:
4058:
4050:
4047:
4044:
4043:
4041:
4040:
4039:
4037:
4033:
4029:
4025:
4015:
4012:
4009:
4006:
4005:
4003:
4002:
4001:
3976:
3972:
3968:
3960:
3957:
3954:
3945:
3940:
3935:
3929:
3926:
3921:
3918:
3914:
3910:
3907:
3903:
3898:
3895:
3891:
3886:
3883:
3876:
3875:
3874:
3872:
3868:
3860:
3858:
3856:
3837:
3832:
3818:
3815:
3812:
3805:
3801:
3798:
3795:
3791:
3785:
3779:
3776:
3773:
3765:
3762:
3759:
3750:
3745:
3742:
3736:
3732:
3725:
3724:
3723:
3720:
3706:
3686:
3683:
3678:
3675:
3653:
3631:
3628:
3603:
3599:
3595:
3592:
3587:
3584:
3578:
3572:
3564:
3561:
3558:
3552:
3547:
3544:
3535:
3534:
3533:
3531:
3527:
3519:
3517:
3514:
3510:
3506:
3502:
3499:Valuation of
3494:
3492:
3475:
3472:
3469:
3465:
3460:
3457:
3437:
3414:
3409:
3406:
3401:
3396:
3374:
3371:
3357:
3354:
3349:
3344:
3325:
3317:
3316:
3315:
3313:
3294:
3289:
3286:
3280:
3275:
3269:
3266:
3261:
3257:
3254:
3251:
3247:
3242:
3239:
3233:
3230:
3217:
3209:
3204:
3191:
3185:
3181:
3178:
3165:
3157:
3151:
3148:
3145:
3142:
3139:
3118:
3106:
3105:
3104:
3102:
3094:
3089:
3073:
3070:
3065:
3051:
3048:
3045:
3038:
3034:
3028:
3025:
3022:
3014:
3001:
2995:
2991:
2986:
2972:
2962:
2959:
2948:
2945:
2929:
2926:
2921:
2907:
2904:
2901:
2894:
2890:
2884:
2881:
2878:
2870:
2857:
2851:
2847:
2842:
2828:
2818:
2815:
2804:
2803:
2802:
2800:
2795:
2778:
2775:
2772:
2766:
2761:
2748:
2739:
2736:
2729:
2726:
2720:
2716:
2712:
2706:
2703:
2700:
2697:
2694:
2689:
2686:
2680:
2660:
2659:
2658:
2656:
2638:
2635:
2632:
2625:
2612:
2603:
2600:
2593:
2588:
2581:
2568:
2559:
2556:
2549:
2540:
2539:
2524:
2519:
2506:
2497:
2494:
2487:
2482:
2474:
2471:
2468:
2462:
2457:
2444:
2435:
2432:
2421:
2420:
2419:
2416:
2399:
2396:
2393:
2389:
2384:
2381:
2373:
2357:
2337:
2317:
2294:
2289:
2285:
2282:
2277:
2269:
2266:
2263:
2254:
2249:
2235:
2228:
2224:
2218:
2215:
2212:
2206:
2201:
2187:
2177:
2174:
2163:
2162:
2147:
2142:
2136:
2133:
2125:
2122:
2119:
2113:
2110:
2104:
2099:
2085:
2078:
2074:
2068:
2065:
2062:
2056:
2051:
2037:
2027:
2024:
2013:
2012:
2011:
2003:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1982:
1978:
1975:
1972:
1969:
1966:
1965:
1961:
1959:
1956:
1953:
1950:
1947:
1946:
1943:
1941:
1933:
1917:
1912:
1908:
1905:
1900:
1892:
1889:
1886:
1877:
1872:
1859:
1853:
1847:
1839:
1836:
1833:
1827:
1822:
1819:
1816:
1808:
1805:
1802:
1796:
1793:
1790:
1785:
1777:
1774:
1771:
1765:
1759:
1756:
1753:
1747:
1744:
1741:
1736:
1723:
1717:
1709:
1708:
1707:
1705:
1699:
1678:
1673:
1667:
1662:
1656:
1653:
1650:
1646:
1641:
1636:
1633:
1627:
1625:
1614:
1611:
1608:
1605:
1602:
1595:
1592:
1584:
1581:
1578:
1572:
1569:
1563:
1561:
1545:
1537:
1534:
1526:
1523:
1520:
1514:
1511:
1504:
1501:
1493:
1490:
1487:
1481:
1478:
1472:
1465:
1462:
1459:
1455:
1450:
1448:
1438:
1433:
1427:
1424:
1421:
1417:
1412:
1405:
1402:
1399:
1394:
1391:
1388:
1384:
1377:
1374:
1371:
1367:
1362:
1354:
1346:
1343:
1340:
1333:
1326:
1321:
1318:
1315:
1311:
1307:
1305:
1298:
1285:
1279:
1267:
1266:
1265:
1263:
1242:
1234:
1231:
1228:
1221:
1211:
1207:
1203:
1199:
1191:
1175:
1172:
1165:
1152:
1146:
1142:
1137:
1130:
1117:
1111:
1107:
1098:
1097:
1096:
1077:
1064:
1058:
1054:
1049:
1041:
1038:
1035:
1029:
1024:
1011:
1005:
997:
996:
995:
992:
990:
971:
968:
965:
959:
954:
941:
935:
931:
928:
922:
918:
914:
908:
905:
902:
899:
896:
891:
888:
882:
869:
868:
867:
865:
845:
832:
826:
822:
819:
816:
810:
807:
804:
801:
798:
783:
782:
781:
767:
760:
744:
724:
701:
696:
692:
689:
684:
676:
673:
670:
661:
656:
643:
637:
629:
628:
627:
624:
619:
605:
601:
597:
594:
591:
583:
567:
544:
539:
526:
520:
516:
513:
510:
504:
501:
498:
495:
492:
477:
476:
475:
461:
454:
438:
418:
395:
390:
384:
381:
373:
370:
367:
361:
358:
352:
347:
334:
328:
320:
319:
318:
316:
312:
311:present value
302:
300:
297:
294:
291:
288:
285:
284:
280:
277:
274:
271:
268:
267:
263:
260:
257:
254:
251:
249:
248:
245:
238:
236:
234:
230:
222:
220:
218:
214:
206:
204:
202:
198:
197:interest rate
194:
190:
189:present value
186:
179:
177:
175:
171:
163:
158:
157:
153:
150:
147:
144:
140:
137:
136:
132:
130:
128:
124:
120:
116:
112:
104:
102:
100:
96:
88:
86:
80:
78:
76:
72:
67:
65:
60:
57:payments and
56:
52:
51:home mortgage
48:
44:
40:
33:
19:
4288:
4260:
4241:
4206:
4198:
4178:
4172:
4164:
4159:
4135:Life annuity
4062:
4059:
4056:
4035:
4031:
4027:
4023:
4021:
4007:
3999:
3870:
3866:
3864:
3852:
3721:
3618:
3532:payments is
3529:
3525:
3523:
3498:
3429:
3310:Therefore a
3309:
3100:
3098:
3087:
2943:
2798:
2796:
2793:
2654:
2653:
2417:
2309:
2009:
1999:
1993:
1939:
1937:
1703:
1700:
1697:
1261:
1209:
1208:-th payment
1205:
1201:
1197:
1195:
1094:
993:
988:
986:
863:
862:
758:
716:
623:future value
622:
620:
559:
452:
410:
310:
308:
298:
242:
232:
228:
226:
216:
212:
210:
201:future value
183:
173:
169:
167:
154:
148:
138:
126:
123:life annuity
118:
114:
110:
108:
98:
94:
92:
84:
71:life annuity
68:
42:
36:
3513:probability
3509:Life tables
1940:annuity-due
1934:Annuity-due
239:Annuity Due
99:annuity-due
4302:Categories
4151:References
4140:Perpetuity
4063:Examples:
4000:Examples:
3312:perpetuity
3101:perpetuity
3095:Perpetuity
75:perpetuity
49:, monthly
39:investment
4313:Annuities
3958:−
3946:−
3911:−
3853:See also
3823:¯
3816:−
3802:×
3777:−
3751:−
3684:−
3593:−
3553:−
3387:¯
3384:∞
3375:¨
3335:¯
3332:∞
3267:−
3243:−
3234:×
3226:∞
3222:→
3195:¯
3182:×
3174:∞
3170:→
3127:∞
3123:→
3071:−
3061:¯
3005:¯
2982:¯
2963:¨
2917:¯
2905:−
2861:¯
2838:¯
2819:¨
2770:$
2752:¯
2740:¨
2730:×
2724:$
2710:$
2701:×
2616:¯
2604:¨
2589:−
2572:¯
2560:¨
2510:¯
2498:¨
2488:×
2448:¯
2436:¨
2374:given by
2283:−
2245:¯
2225:×
2197:¯
2178:¨
2134:−
2114:−
2095:¯
2075:×
2047:¯
2028:¨
1962:payments
1906:−
1863:¯
1820:−
1794:⋯
1727:¯
1637:−
1612:−
1593:−
1573:−
1535:−
1515:−
1502:−
1482:−
1403:−
1385:∑
1312:∑
1289:¯
1212:would be
1156:¯
1138:−
1121:¯
1068:¯
1055:×
1015:¯
989:principal
963:$
945:¯
932:×
926:$
912:$
903:×
836:¯
823:×
690:−
647:¯
530:¯
517:×
382:−
362:−
338:¯
264:payments
185:Valuation
180:Valuation
55:insurance
18:Annuities
4281:(1878).
4119:See also
4034:− 1) ÷ (
3869:, given
2655:Example:
2004:periods
1979:—
864:Example:
303:periods
281:—
2779:730.01.
2370:is the
59:pension
43:annuity
4267:
4248:
4219:
4185:
3430:where
2762:0.0075
2310:where
972:495.50
717:where
584:, and
411:where
199:, and
3516:age.
231:, or
81:Types
41:, an
4265:ISBN
4246:ISBN
4217:ISBN
4183:ISBN
2687:0.09
1996:− 1
1990:...
1954:...
1095:and
955:0.01
889:0.12
780:is:
759:rent
621:The
474:is:
453:rent
317:by:
309:The
295:...
258:...
4213:175
4030:× (
3214:lim
3162:lim
3115:lim
2727:100
2713:100
2675:due
1938:An
929:100
915:100
215:or
113:or
37:In
4304::
4287:.
4215:.
3873::
3857:.
3719:.
3133:PV
3099:A
2773:11
2749:84
2704:12
2690:12
2670:FV
2415:.
1987:1
1984:0
1957:↓
1951:↓
1948:↓
942:60
906:12
892:12
878:PV
792:FV
618:.
606:12
486:PV
292:2
289:1
286:0
261:↓
255:↓
252:↓
203:.
195:,
176:.
77:.
4273:.
4254:.
4225:.
4193:.
4191:.
4036:r
4032:r
4028:r
4024:r
4008:R
3977:m
3973:/
3969:j
3964:)
3961:1
3955:n
3952:(
3941:)
3936:)
3930:m
3927:j
3922:+
3919:1
3915:(
3908:1
3904:(
3899:+
3896:1
3892:A
3887:=
3884:R
3871:A
3867:R
3838:.
3833:i
3829:|
3819:n
3813:N
3806:a
3799:R
3796:=
3792:]
3786:i
3780:N
3774:n
3770:)
3766:1
3763:+
3760:i
3757:(
3746:i
3743:1
3737:[
3733:R
3707:i
3687:P
3679:i
3676:R
3654:R
3632:i
3629:R
3604:.
3600:)
3596:P
3588:i
3585:R
3579:(
3573:n
3569:)
3565:i
3562:+
3559:1
3556:(
3548:i
3545:R
3530:n
3526:P
3476:i
3473:+
3470:1
3466:i
3461:=
3458:d
3438:i
3415:,
3410:d
3407:1
3402:=
3397:i
3393:|
3372:a
3358:i
3355:1
3350:=
3345:i
3341:|
3326:a
3295:.
3290:i
3287:R
3281:=
3276:i
3270:n
3262:)
3258:i
3255:+
3252:1
3248:(
3240:1
3231:R
3218:n
3210:=
3205:i
3201:|
3192:n
3186:a
3179:R
3166:n
3158:=
3155:)
3152:R
3149:,
3146:n
3143:,
3140:i
3137:(
3119:n
3088:n
3074:1
3066:i
3056:|
3052:1
3049:+
3046:n
3039:s
3035:=
3032:)
3029:i
3026:+
3023:1
3020:(
3015:i
3011:|
3002:n
2996:s
2992:=
2987:i
2977:|
2973:n
2960:s
2944:n
2930:1
2927:+
2922:i
2912:|
2908:1
2902:n
2895:a
2891:=
2888:)
2885:i
2882:+
2879:1
2876:(
2871:i
2867:|
2858:n
2852:a
2848:=
2843:i
2833:|
2829:n
2816:a
2799:n
2776:,
2767:=
2758:|
2737:s
2721:=
2717:)
2707:,
2698:7
2695:,
2681:(
2639:.
2636:d
2633:=
2626:i
2622:|
2613:n
2601:s
2594:1
2582:i
2578:|
2569:n
2557:a
2550:1
2525:,
2520:i
2516:|
2507:n
2495:a
2483:n
2479:)
2475:i
2472:+
2469:1
2466:(
2463:=
2458:i
2454:|
2445:n
2433:s
2400:1
2397:+
2394:i
2390:i
2385:=
2382:d
2358:d
2338:i
2318:n
2295:,
2290:d
2286:1
2278:n
2274:)
2270:i
2267:+
2264:1
2261:(
2255:=
2250:i
2240:|
2236:n
2229:s
2222:)
2219:i
2216:+
2213:1
2210:(
2207:=
2202:i
2192:|
2188:n
2175:s
2148:,
2143:d
2137:n
2130:)
2126:i
2123:+
2120:1
2117:(
2111:1
2105:=
2100:i
2090:|
2086:n
2079:a
2072:)
2069:i
2066:+
2063:1
2060:(
2057:=
2052:i
2042:|
2038:n
2025:a
2000:n
1994:n
1918:.
1913:i
1909:1
1901:n
1897:)
1893:i
1890:+
1887:1
1884:(
1878:=
1873:i
1869:|
1860:n
1854:a
1848:n
1844:)
1840:i
1837:+
1834:1
1831:(
1828:=
1823:1
1817:n
1813:)
1809:i
1806:+
1803:1
1800:(
1797:+
1791:+
1786:2
1782:)
1778:i
1775:+
1772:1
1769:(
1766:+
1763:)
1760:i
1757:+
1754:1
1751:(
1748:+
1745:1
1742:=
1737:i
1733:|
1724:n
1718:s
1704:n
1702:(
1679:,
1674:i
1668:n
1663:)
1657:i
1654:+
1651:1
1647:1
1642:(
1634:1
1628:=
1615:1
1609:i
1606:+
1603:1
1596:n
1589:)
1585:i
1582:+
1579:1
1576:(
1570:1
1564:=
1546:)
1538:1
1531:)
1527:i
1524:+
1521:1
1518:(
1512:1
1505:n
1498:)
1494:i
1491:+
1488:1
1485:(
1479:1
1473:(
1466:i
1463:+
1460:1
1456:1
1451:=
1439:k
1434:)
1428:i
1425:+
1422:1
1418:1
1413:(
1406:1
1400:n
1395:0
1392:=
1389:k
1378:i
1375:+
1372:1
1368:1
1363:=
1355:k
1351:)
1347:i
1344:+
1341:1
1338:(
1334:1
1327:n
1322:1
1319:=
1316:k
1308:=
1299:i
1295:|
1286:n
1280:a
1262:R
1243:k
1239:)
1235:i
1232:+
1229:1
1226:(
1222:R
1210:R
1206:k
1202:k
1198:k
1176:i
1173:=
1166:i
1162:|
1153:n
1147:s
1143:1
1131:i
1127:|
1118:n
1112:a
1108:1
1078:i
1074:|
1065:n
1059:a
1050:n
1046:)
1042:i
1039:+
1036:1
1033:(
1030:=
1025:i
1021:|
1012:n
1006:s
969:,
966:4
960:=
951:|
936:a
923:=
919:)
909:,
900:5
897:,
883:(
846:i
842:|
833:n
827:s
820:R
817:=
814:)
811:R
808:,
805:n
802:,
799:i
796:(
768:R
745:i
725:n
702:,
697:i
693:1
685:n
681:)
677:i
674:+
671:1
668:(
662:=
657:i
653:|
644:n
638:s
602:/
598:I
595:=
592:i
568:I
545:.
540:i
536:|
527:n
521:a
514:R
511:=
508:)
505:R
502:,
499:n
496:,
493:i
490:(
462:R
439:i
419:n
396:,
391:i
385:n
378:)
374:i
371:+
368:1
365:(
359:1
353:=
348:i
344:|
335:n
329:a
299:n
145:.
34:.
20:)
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