1682:
1259:
1677:{\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}}}
3294:
3504:
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
2778:
232:
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
3098:
140:– 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.
1917:
3414:
971:
148:– 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.
2638:
2147:
2294:
3984:
3073:
2929:
614:
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:
2524:
1175:
2652:
3289:{\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}}.}
1690:
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
1079:
50:
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
1264:
3603:
130:– 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
544:
847:
395:
701:
1701:
302:
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
3837:
3309:
861:
980:
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.
1247:
2532:
2005:
3478:
2402:
2155:
3686:
3633:
3868:
2940:
2796:
1931:
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.
2790:
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:
2413:
605:
3706:
3653:
3437:
2357:
2337:
2317:
767:
744:
724:
567:
461:
438:
418:
1090:
2773:{\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.}
989:
3527:
1999:
Each annuity payment is allowed to compound for one extra period. Thus, the present and future values of an annuity-due can be calculated.
86:
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
469:
2646:
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:
775:
312:
216:
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
621:
549:
In practice, often loans are stated per annum while interest is compounded and payments are made monthly. In this case, the interest
1912:{\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}}.}
570:
976:
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
440:
is the per period interest rate. Present value is linear in the amount of payments, therefore the present value for payments, or
3409:{\displaystyle a_{{\overline {\infty }}|i}={\frac {1}{i}}{\text{ and }}{\ddot {a}}_{{\overline {\infty }}|i}={\frac {1}{d}},}
966:{\displaystyle {\text{PV}}\left({\frac {0.12}{12}},5\times 12,\$ 100\right)=\$ 100\times a_{{\overline {60}}|0.01}=\$ 4,495.50}
746:
is the per period interest rate. Future value is linear in the amount of payments, therefore the future value for payments, or
161:(usually after retirement). An annuity that begins payments as soon as the customer has paid, without a deferral period is an
4257:
4238:
3717:
4070:
Find the periodic payment of an accumulated value of $ 1,600,000, payable annually for 3 years at 9% compounded annually.
4027:− 1) 08695652174 × (−0.3424837676)÷ (−1304347826) = 2.2832251175 70000÷ 2.2832251175= $ 30658.3873 is the correct value
2783:
In Excel, the PV and FV functions take on optional fifth argument which selects from annuity-immediate or annuity-due.
131:
4056:
Find the periodic payment of an accumulated value of $ 55,000, payable monthly for 3 years at 15% compounded monthly.
4209:
4175:
4031:
Find the periodic payment of an annuity due of $ 250,700, payable quarterly for 8 years at 5% compounded quarterly.
855:
The present value of a 5-year annuity with a nominal annual interest rate of 12% and monthly payments of $ 100 is:
3993:
Find the periodic payment of an annuity due of $ 70,000, payable annually for 3 years at 15% compounded annually.
2360:
2633:{\displaystyle {\frac {1}{{\ddot {a}}_{{\overline {n}}|i}}}-{\frac {1}{{\ddot {s}}_{{\overline {n}}|i}}}=d.}
2142:{\displaystyle {\ddot {a}}_{{\overline {n|}}i}=(1+i)\times a_{{\overline {n|}}i}={\frac {1-(1+i)^{-n}}{d}},}
1204:
4096:
4091:
2289:{\displaystyle {\ddot {s}}_{{\overline {n|}}i}=(1+i)\times s_{{\overline {n|}}i}={\frac {(1+i)^{n}-1}{d}},}
118:
are guaranteed to be paid for a number of years and then become contingent on the annuitant being alive.
3442:
2366:
208:. Valuation of annuities certain may be calculated using formulas depending on the timing of payments.
4278:
3658:
4296:
4101:
3979:{\displaystyle R={\frac {A}{1+\left(1-\left(1+{\frac {j}{m}}\right)\right)^{-{\frac {(n-1)}{j/m}}}}}}
3303:
has a finite present value when there is a non-zero discount rate. The formulae for a perpetuity are
3611:
3068:{\displaystyle {\ddot {s}}_{{\overline {n|}}i}=s_{{\overline {n}}|i}(1+i)=s_{{\overline {n+1|}}i}-1}
2924:{\displaystyle {\ddot {a}}_{{\overline {n|}}i}=a_{{\overline {n}}|i}(1+i)=a_{{\overline {n-1|}}i}+1}
4272:
20:
4201:
4113:
3493:
303:
34:
is a series of payments made at equal intervals. Examples of annuities are regular deposits to a
2519:{\displaystyle {\ddot {s}}_{{\overline {n}}|i}=(1+i)^{n}\times {\ddot {a}}_{{\overline {n}}|i},}
144:
52:
4191:
8:
4267:
4194:
4133:
4118:
3843:
1189:-th payment must be discounted to the present by dividing by the interest, compounded by
181:
173:
90:
are made at the beginning of payment periods, so a payment is made immediately on issue.
576:
4301:
3691:
3638:
3422:
2342:
2322:
2302:
1170:{\displaystyle {\frac {1}{a_{{\overline {n}}|i}}}-{\frac {1}{s_{{\overline {n}}|i}}}=i}
752:
729:
709:
552:
446:
423:
403:
4253:
4234:
4205:
4171:
180:
of the future annuity payments. The valuation of an annuity entails concepts such as
98:
Annuities that provide payments that will be paid over a period known in advance are
58:
An annuity which provides for payments for the remainder of a person's lifetime is a
224:. Mortgage payments are annuity-immediate, interest is earned before being paid.
35:
4290:
185:
177:
39:
3711:
Also, this can be thought of as the present value of the remaining payments
4123:
3489:
1074:{\displaystyle s_{{\overline {n}}|i}=(1+i)^{n}\times a_{{\overline {n}}|i}}
189:
111:
59:
4271:
3501:
4192:
Jordan, Bradford D.; Ross, Stephen David; Westerfield, Randolph (2000).
4128:
3497:
3300:
63:
27:
3598:{\displaystyle {\frac {R}{i}}-(1+i)^{n}\left({\frac {R}{i}}-P\right).}
43:
3092:
is an annuity for which the payments continue forever. Observe that
2407:
The future and present values for annuities due are related since:
539:{\displaystyle {\text{PV}}(i,n,R)=R\times a_{{\overline {n}}|i}.}
200:
If the number of payments is known in advance, the annuity is an
47:
842:{\displaystyle {\text{FV}}(i,n,R)=R\times s_{{\overline {n}}|i}}
390:{\displaystyle a_{{\overline {n}}|i}={\frac {1-(1+i)^{-n}}{i}},}
696:{\displaystyle s_{{\overline {n}}|i}={\frac {(1+i)^{n}-1}{i}},}
114:, which is paid over the remaining lifetime of the annuitant.
233:
cover services provided in the period following the payment.
3608:
Because the scheme is equivalent with borrowing the amount
3688:
of that borrowed amount in the bank to grow with interest
157:
An annuity that begins payments only after a period is a
3832:{\displaystyle R\left=R\times a_{{\overline {N-n}}|i}.}
1542:
by using the equation for the sum of a geometric series
4034:
R= 250,700/(1+〖(1-(1+((.05)/4) )〗^(-(32-1))/((.05)/4))
579:
4073:
R=1,600,000/((〖((1+((.09)/1) )〗^(3+1)-1)/((.09)/1)-1)
4059:
R=55,000/((〖((1+((.15)/12) )〗^(36+1)-1)/((.15)/12)-1)
3871:
3720:
3694:
3661:
3641:
3614:
3530:
3445:
3425:
3312:
3101:
3075:. The value one period after the time of the last of
2943:
2799:
2655:
2535:
2416:
2369:
2345:
2325:
2305:
2158:
2008:
1704:
1262:
1207:
1180:
1093:
992:
864:
778:
755:
732:
712:
624:
555:
472:
449:
426:
406:
315:
106:
Annuities paid only under certain circumstances are
4228:
4170:. Mason, Ohio: Thomson South-Western. p. 230.
3999:= 70,000/(1+〖(1-(1+((.15)/1) )〗^(-(3-1))/((.15)/1))
4193:
4156:. Homewood, Illinois: Richard D. Irwin, Inc. p. 45
3978:
3831:
3700:
3680:
3647:
3627:
3597:
3472:
3431:
3408:
3288:
3067:
2923:
2772:
2632:
2518:
2396:
2351:
2331:
2311:
2288:
2141:
1911:
1676:
1241:
1169:
1073:
965:
841:
761:
738:
718:
695:
599:
561:
538:
455:
432:
412:
389:
4231:Mathematics of Investment and Credit, 5th Edition
4288:
4247:
4185:
4159:
3202:
3150:
3103:
4165:
62:. An annuity which continues indefinitely is a
4282:. Vol. II (9th ed.). pp. 72–89.
4233:. ACTEX Academic Series. ACTEX Publications.
983:Future and present values are related since:
74:Annuities may be classified in several ways.
3508:
4049:R = S\,/((〖((1+(j/m) )〗^(n+1)-1)/(j/m)-1)
4046:Finding the Periodic Payment(R), Given S:
121:
93:
16:Series of payments made at equal intervals
3854:Formula for finding the periodic payment
3272:
3213:
3209:
3161:
3157:
3114:
3110:
176:of an annuity entails calculation of the
3496:of the future life contingent payments.
2931:. The value at the time of the first of
4266:
3849:
1687:which gives us the result as required.
152:
4289:
1242:{\displaystyle {\frac {R}{(1+i)^{k}}}}
4200:. Boston: Irwin/McGraw-Hill. p.
4015:as, (1 ÷ 1.15)= 0.8695652174 2) find
3517:with interest, the amount owed after
3513:If an annuity is for repaying a debt
1193:terms. Hence the contribution of the
77:
3492:may be performed by calculating the
211:
3635:to create a perpetuity with coupon
2339:is the per-term interest rate, and
13:
3473:{\displaystyle d={\frac {i}{1+i}}}
3372:
3320:
3214:
3162:
3115:
2758:
2712:
2698:
2397:{\displaystyle d={\frac {i}{i+1}}}
1695: − 1) years. Therefore,
1181:Proof of annuity-immediate formula
951:
914:
900:
195:
132:Securities and Exchange Commission
14:
4313:
4196:Fundamentals of corporate finance
3483:
4222:
4085:
3681:{\displaystyle {\frac {R}{i}}-P}
3480:is the effective discount rate.
1185:To calculate present value, the
4250:Theory of Interest, 3rd Edition
1539:
1538:
4168:Practical financial management
4146:
3952:
3940:
3817:
3758:
3745:
3628:{\displaystyle {\frac {R}{i}}}
3557:
3544:
3381:
3329:
3210:
3189:
3158:
3143:
3125:
3111:
3044:
3020:
3008:
2999:
2965:
2900:
2876:
2864:
2855:
2821:
2746:
2610:
2566:
2504:
2467:
2454:
2442:
2262:
2249:
2228:
2210:
2198:
2180:
2118:
2105:
2078:
2060:
2048:
2030:
1922:
1885:
1872:
1857:
1832:
1819:
1801:
1788:
1770:
1757:
1751:
1739:
1721:
1577:
1564:
1519:
1506:
1486:
1473:
1339:
1326:
1283:
1227:
1214:
1150:
1115:
1062:
1034:
1021:
1009:
939:
830:
802:
784:
669:
656:
641:
524:
496:
478:
366:
353:
332:
227:
55:known as "annuity functions".
1:
4152:Kellison, Stephen G. (1970).
4139:
4011:Find PVOA factor as. 1) find
3083:
4229:Samuel A. Broverman (2010).
4097:Annuities under European law
4092:Annuities under American law
3811:
3375:
3323:
3183:
3049:
2993:
2970:
2905:
2849:
2826:
2740:
2604:
2560:
2498:
2436:
2233:
2185:
2083:
2035:
1851:
1715:
1277:
1144:
1109:
1056:
1003:
933:
824:
635:
518:
326:
168:
7:
4107:
726:is the number of terms and
420:is the number of terms and
10:
4318:
3500:are used to calculate the
2361:effective rate of discount
116:Certain and life annuities
18:
4248:Stephen Kellison (2008).
4102:Annuities under Swiss law
3509:Amortization calculations
3439:is the interest rate and
4166:Lasher, William (2008).
2319:is the number of terms,
145:Equity-indexed annuities
110:. A common example is a
69:
21:Annuity (disambiguation)
4279:Encyclopædia Britannica
4114:Amortization calculator
3494:actuarial present value
122:Variability of payments
94:Contingency of payments
4154:The Theory of Interest
4076:R = 1,600,000/3.573129
4062:R = 55,000/45.67944932
4037:R = 250,700/26.5692901
4002:R = 70,000/2.625708885
3980:
3833:
3702:
3682:
3649:
3629:
3599:
3474:
3433:
3410:
3290:
3069:
2925:
2774:
2634:
2520:
2398:
2353:
2333:
2313:
2290:
2143:
1965:———
1962:———
1959:———
1956:———
1913:
1678:
1398:
1319:
1243:
1171:
1075:
967:
843:
763:
740:
720:
697:
601:
563:
540:
457:
434:
414:
391:
267:———
264:———
261:———
258:———
53:mathematical functions
4252:. McGraw-Hill/Irwin.
3981:
3834:
3703:
3683:
3650:
3630:
3600:
3475:
3434:
3411:
3291:
3070:
2926:
2775:
2635:
2521:
2399:
2354:
2334:
2314:
2291:
2144:
1914:
1679:
1372:
1299:
1244:
1172:
1076:
968:
844:
764:
741:
721:
698:
602:
571:nominal interest rate
564:
541:
458:
435:
415:
392:
104:guaranteed annuities.
4268:Sprague, Thomas Bond
3869:
3850:Example calculations
3718:
3692:
3659:
3639:
3612:
3528:
3443:
3423:
3310:
3099:
2941:
2797:
2786:An annuity-due with
2653:
2533:
2414:
2367:
2343:
2323:
2303:
2156:
2006:
1702:
1260:
1253:to be 1, then:
1205:
1091:
990:
862:
776:
753:
730:
710:
622:
577:
553:
470:
447:
424:
404:
313:
153:Deferral of payments
108:contingent annuities
19:For other uses, see
4134:Time value of money
4119:Fixed rate mortgage
3844:fixed rate mortgage
1249:. Just considering
600:{\textstyle i=I/12}
182:time value of money
3976:
3829:
3698:
3678:
3645:
3625:
3595:
3470:
3429:
3406:
3286:
3218:
3166:
3119:
3065:
2921:
2770:
2630:
2516:
2394:
2349:
2329:
2309:
2286:
2139:
1909:
1674:
1672:
1239:
1167:
1071:
963:
839:
759:
736:
716:
693:
597:
559:
536:
453:
430:
410:
387:
304:actuarial notation
206:guaranteed annuity
138:Variable annuities
78:Timing of payments
42:payments, monthly
4273:"Annuities"
4259:978-0-07-338244-9
4240:978-1-56698-767-7
4005:R = $ 26659.46724
3974:
3969:
3921:
3814:
3777:
3737:
3701:{\displaystyle i}
3670:
3648:{\displaystyle R}
3623:
3579:
3539:
3468:
3432:{\displaystyle i}
3401:
3378:
3366:
3354:
3349:
3326:
3281:
3267:
3201:
3186:
3149:
3123:
3102:
3052:
2996:
2973:
2954:
2908:
2852:
2829:
2810:
2743:
2731:
2681:
2665:
2660:
2619:
2607:
2595:
2575:
2563:
2551:
2501:
2489:
2439:
2427:
2392:
2352:{\displaystyle d}
2332:{\displaystyle i}
2312:{\displaystyle n}
2281:
2236:
2188:
2169:
2134:
2086:
2038:
2019:
1997:
1996:
1904:
1854:
1718:
1665:
1649:
1607:
1543:
1532:
1458:
1420:
1370:
1349:
1280:
1237:
1159:
1147:
1124:
1112:
1059:
1006:
936:
883:
868:
827:
782:
762:{\displaystyle R}
739:{\displaystyle i}
719:{\displaystyle n}
688:
638:
562:{\displaystyle I}
521:
476:
456:{\displaystyle R}
433:{\displaystyle i}
413:{\displaystyle n}
382:
329:
296:
295:
218:annuity-immediate
212:Annuity-immediate
163:immediate annuity
100:annuities certain
84:annuity-immediate
4309:
4297:Finance theories
4283:
4275:
4263:
4244:
4216:
4215:
4199:
4189:
4183:
4181:
4163:
4157:
4150:
4079:R = $ 447,786.80
3985:
3983:
3982:
3977:
3975:
3973:
3972:
3971:
3970:
3968:
3964:
3955:
3938:
3932:
3928:
3927:
3923:
3922:
3914:
3879:
3838:
3836:
3835:
3830:
3825:
3824:
3820:
3815:
3810:
3799:
3783:
3779:
3778:
3773:
3772:
3771:
3743:
3738:
3730:
3707:
3705:
3704:
3699:
3687:
3685:
3684:
3679:
3671:
3663:
3654:
3652:
3651:
3646:
3634:
3632:
3631:
3626:
3624:
3616:
3604:
3602:
3601:
3596:
3591:
3587:
3580:
3572:
3565:
3564:
3540:
3532:
3479:
3477:
3476:
3471:
3469:
3467:
3453:
3438:
3436:
3435:
3430:
3415:
3413:
3412:
3407:
3402:
3394:
3389:
3388:
3384:
3379:
3371:
3368:
3367:
3359:
3355:
3352:
3350:
3342:
3337:
3336:
3332:
3327:
3319:
3295:
3293:
3292:
3287:
3282:
3274:
3268:
3263:
3262:
3261:
3253:
3249:
3226:
3217:
3197:
3196:
3192:
3187:
3179:
3165:
3124:
3121:
3118:
3074:
3072:
3071:
3066:
3058:
3057:
3053:
3048:
3047:
3032:
3007:
3006:
3002:
2997:
2989:
2979:
2978:
2974:
2969:
2968:
2959:
2956:
2955:
2947:
2930:
2928:
2927:
2922:
2914:
2913:
2909:
2904:
2903:
2888:
2863:
2862:
2858:
2853:
2845:
2835:
2834:
2830:
2825:
2824:
2815:
2812:
2811:
2803:
2779:
2777:
2776:
2771:
2754:
2753:
2749:
2744:
2736:
2733:
2732:
2724:
2708:
2704:
2682:
2674:
2667:
2666:
2663:
2661:
2658:
2639:
2637:
2636:
2631:
2620:
2618:
2617:
2613:
2608:
2600:
2597:
2596:
2588:
2581:
2576:
2574:
2573:
2569:
2564:
2556:
2553:
2552:
2544:
2537:
2525:
2523:
2522:
2517:
2512:
2511:
2507:
2502:
2494:
2491:
2490:
2482:
2475:
2474:
2450:
2449:
2445:
2440:
2432:
2429:
2428:
2420:
2403:
2401:
2400:
2395:
2393:
2391:
2377:
2358:
2356:
2355:
2350:
2338:
2336:
2335:
2330:
2318:
2316:
2315:
2310:
2295:
2293:
2292:
2287:
2282:
2277:
2270:
2269:
2247:
2242:
2241:
2237:
2232:
2231:
2222:
2194:
2193:
2189:
2184:
2183:
2174:
2171:
2170:
2162:
2148:
2146:
2145:
2140:
2135:
2130:
2129:
2128:
2097:
2092:
2091:
2087:
2082:
2081:
2072:
2044:
2043:
2039:
2034:
2033:
2024:
2021:
2020:
2012:
1934:
1933:
1918:
1916:
1915:
1910:
1905:
1900:
1893:
1892:
1870:
1865:
1864:
1860:
1855:
1847:
1840:
1839:
1815:
1814:
1778:
1777:
1729:
1728:
1724:
1719:
1711:
1683:
1681:
1680:
1675:
1673:
1666:
1661:
1660:
1659:
1654:
1650:
1648:
1634:
1620:
1612:
1608:
1606:
1589:
1588:
1587:
1556:
1548:
1544:
1541:
1537:
1533:
1531:
1530:
1529:
1498:
1497:
1496:
1465:
1459:
1457:
1443:
1435:
1431:
1430:
1425:
1421:
1419:
1405:
1397:
1386:
1371:
1369:
1355:
1350:
1348:
1347:
1346:
1321:
1318:
1313:
1291:
1290:
1286:
1281:
1273:
1248:
1246:
1245:
1240:
1238:
1236:
1235:
1234:
1209:
1176:
1174:
1173:
1168:
1160:
1158:
1157:
1153:
1148:
1140:
1130:
1125:
1123:
1122:
1118:
1113:
1105:
1095:
1080:
1078:
1077:
1072:
1070:
1069:
1065:
1060:
1052:
1042:
1041:
1017:
1016:
1012:
1007:
999:
972:
970:
969:
964:
947:
946:
942:
937:
929:
910:
906:
884:
876:
869:
866:
848:
846:
845:
840:
838:
837:
833:
828:
820:
783:
780:
768:
766:
765:
760:
745:
743:
742:
737:
725:
723:
722:
717:
702:
700:
699:
694:
689:
684:
677:
676:
654:
649:
648:
644:
639:
631:
606:
604:
603:
598:
593:
568:
566:
565:
560:
545:
543:
542:
537:
532:
531:
527:
522:
514:
477:
474:
462:
460:
459:
454:
439:
437:
436:
431:
419:
417:
416:
411:
396:
394:
393:
388:
383:
378:
377:
376:
345:
340:
339:
335:
330:
322:
236:
235:
222:ordinary annuity
159:deferred annuity
4317:
4316:
4312:
4311:
4310:
4308:
4307:
4306:
4287:
4286:
4260:
4241:
4225:
4220:
4219:
4212:
4190:
4186:
4178:
4164:
4160:
4151:
4147:
4142:
4110:
4088:
3960:
3956:
3939:
3937:
3933:
3913:
3906:
3902:
3895:
3891:
3890:
3883:
3878:
3870:
3867:
3866:
3852:
3816:
3800:
3798:
3797:
3793:
3761:
3757:
3744:
3742:
3729:
3728:
3724:
3719:
3716:
3715:
3693:
3690:
3689:
3662:
3660:
3657:
3656:
3640:
3637:
3636:
3615:
3613:
3610:
3609:
3571:
3570:
3566:
3560:
3556:
3531:
3529:
3526:
3525:
3511:
3486:
3457:
3452:
3444:
3441:
3440:
3424:
3421:
3420:
3393:
3380:
3370:
3369:
3358:
3357:
3356:
3353: and
3351:
3341:
3328:
3318:
3317:
3313:
3311:
3308:
3307:
3273:
3254:
3239:
3235:
3234:
3227:
3225:
3205:
3188:
3178:
3177:
3173:
3153:
3120:
3106:
3100:
3097:
3096:
3086:
3043:
3033:
3031:
3030:
3026:
2998:
2988:
2987:
2983:
2964:
2960:
2958:
2957:
2946:
2945:
2944:
2942:
2939:
2938:
2899:
2889:
2887:
2886:
2882:
2854:
2844:
2843:
2839:
2820:
2816:
2814:
2813:
2802:
2801:
2800:
2798:
2795:
2794:
2745:
2735:
2734:
2723:
2722:
2721:
2673:
2672:
2668:
2662:
2657:
2656:
2654:
2651:
2650:
2609:
2599:
2598:
2587:
2586:
2585:
2580:
2565:
2555:
2554:
2543:
2542:
2541:
2536:
2534:
2531:
2530:
2503:
2493:
2492:
2481:
2480:
2479:
2470:
2466:
2441:
2431:
2430:
2419:
2418:
2417:
2415:
2412:
2411:
2381:
2376:
2368:
2365:
2364:
2344:
2341:
2340:
2324:
2321:
2320:
2304:
2301:
2300:
2265:
2261:
2248:
2246:
2227:
2223:
2221:
2220:
2216:
2179:
2175:
2173:
2172:
2161:
2160:
2159:
2157:
2154:
2153:
2121:
2117:
2098:
2096:
2077:
2073:
2071:
2070:
2066:
2029:
2025:
2023:
2022:
2011:
2010:
2009:
2007:
2004:
2003:
1925:
1888:
1884:
1871:
1869:
1856:
1846:
1845:
1841:
1835:
1831:
1804:
1800:
1773:
1769:
1720:
1710:
1709:
1705:
1703:
1700:
1699:
1671:
1670:
1655:
1638:
1633:
1629:
1628:
1621:
1619:
1610:
1609:
1590:
1580:
1576:
1557:
1555:
1546:
1545:
1540:
1522:
1518:
1499:
1489:
1485:
1466:
1464:
1460:
1447:
1442:
1433:
1432:
1426:
1409:
1404:
1400:
1399:
1387:
1376:
1359:
1354:
1342:
1338:
1325:
1320:
1314:
1303:
1292:
1282:
1272:
1271:
1267:
1263:
1261:
1258:
1257:
1230:
1226:
1213:
1208:
1206:
1203:
1202:
1183:
1149:
1139:
1138:
1134:
1129:
1114:
1104:
1103:
1099:
1094:
1092:
1089:
1088:
1061:
1051:
1050:
1046:
1037:
1033:
1008:
998:
997:
993:
991:
988:
987:
938:
928:
927:
923:
875:
874:
870:
865:
863:
860:
859:
829:
819:
818:
814:
779:
777:
774:
773:
754:
751:
750:
731:
728:
727:
711:
708:
707:
672:
668:
655:
653:
640:
630:
629:
625:
623:
620:
619:
589:
578:
575:
574:
569:is stated as a
554:
551:
550:
523:
513:
512:
508:
473:
471:
468:
467:
448:
445:
444:
425:
422:
421:
405:
402:
401:
369:
365:
346:
344:
331:
321:
320:
316:
314:
311:
310:
230:
214:
202:annuity certain
198:
196:Annuity-certain
171:
155:
128:Fixed annuities
124:
96:
82:Payments of an
80:
72:
36:savings account
24:
17:
12:
11:
5:
4315:
4305:
4304:
4299:
4285:
4284:
4264:
4258:
4245:
4239:
4224:
4221:
4218:
4217:
4210:
4184:
4176:
4158:
4144:
4143:
4141:
4138:
4137:
4136:
4131:
4126:
4121:
4116:
4109:
4106:
4105:
4104:
4099:
4094:
4087:
4084:
4083:
4082:
4081:
4080:
4077:
4074:
4068:
4067:
4066:
4065:R = $ 1,204.04
4063:
4060:
4044:
4043:
4042:
4041:
4040:R = $ 9,435.71
4038:
4035:
4009:
4008:
4007:
4006:
4003:
4000:
3987:
3986:
3967:
3963:
3959:
3954:
3951:
3948:
3945:
3942:
3936:
3931:
3926:
3920:
3917:
3912:
3909:
3905:
3901:
3898:
3894:
3889:
3886:
3882:
3877:
3874:
3851:
3848:
3840:
3839:
3828:
3823:
3819:
3813:
3809:
3806:
3803:
3796:
3792:
3789:
3786:
3782:
3776:
3770:
3767:
3764:
3760:
3756:
3753:
3750:
3747:
3741:
3736:
3733:
3727:
3723:
3697:
3677:
3674:
3669:
3666:
3655:, and putting
3644:
3622:
3619:
3606:
3605:
3594:
3590:
3586:
3583:
3578:
3575:
3569:
3563:
3559:
3555:
3552:
3549:
3546:
3543:
3538:
3535:
3510:
3507:
3490:life annuities
3485:
3484:Life annuities
3482:
3466:
3463:
3460:
3456:
3451:
3448:
3428:
3417:
3416:
3405:
3400:
3397:
3392:
3387:
3383:
3377:
3374:
3365:
3362:
3348:
3345:
3340:
3335:
3331:
3325:
3322:
3316:
3297:
3296:
3285:
3280:
3277:
3271:
3266:
3260:
3257:
3252:
3248:
3245:
3242:
3238:
3233:
3230:
3224:
3221:
3216:
3212:
3208:
3204:
3200:
3195:
3191:
3185:
3182:
3176:
3172:
3169:
3164:
3160:
3156:
3152:
3148:
3145:
3142:
3139:
3136:
3133:
3130:
3127:
3117:
3113:
3109:
3105:
3085:
3082:
3081:
3080:
3079:payments of 1.
3064:
3061:
3056:
3051:
3046:
3042:
3039:
3036:
3029:
3025:
3022:
3019:
3016:
3013:
3010:
3005:
3001:
2995:
2992:
2986:
2982:
2977:
2972:
2967:
2963:
2953:
2950:
2936:
2935:payments of 1.
2920:
2917:
2912:
2907:
2902:
2898:
2895:
2892:
2885:
2881:
2878:
2875:
2872:
2869:
2866:
2861:
2857:
2851:
2848:
2842:
2838:
2833:
2828:
2823:
2819:
2809:
2806:
2781:
2780:
2769:
2766:
2763:
2760:
2757:
2752:
2748:
2742:
2739:
2730:
2727:
2720:
2717:
2714:
2711:
2707:
2703:
2700:
2697:
2694:
2691:
2688:
2685:
2680:
2677:
2671:
2641:
2640:
2629:
2626:
2623:
2616:
2612:
2606:
2603:
2594:
2591:
2584:
2579:
2572:
2568:
2562:
2559:
2550:
2547:
2540:
2527:
2526:
2515:
2510:
2506:
2500:
2497:
2488:
2485:
2478:
2473:
2469:
2465:
2462:
2459:
2456:
2453:
2448:
2444:
2438:
2435:
2426:
2423:
2390:
2387:
2384:
2380:
2375:
2372:
2348:
2328:
2308:
2297:
2296:
2285:
2280:
2276:
2273:
2268:
2264:
2260:
2257:
2254:
2251:
2245:
2240:
2235:
2230:
2226:
2219:
2215:
2212:
2209:
2206:
2203:
2200:
2197:
2192:
2187:
2182:
2178:
2168:
2165:
2150:
2149:
2138:
2133:
2127:
2124:
2120:
2116:
2113:
2110:
2107:
2104:
2101:
2095:
2090:
2085:
2080:
2076:
2069:
2065:
2062:
2059:
2056:
2053:
2050:
2047:
2042:
2037:
2032:
2028:
2018:
2015:
1995:
1994:
1991:
1986:
1980:
1977:
1974:
1970:
1969:
1966:
1963:
1960:
1957:
1953:
1952:
1949:
1947:
1944:
1941:
1938:
1924:
1921:
1920:
1919:
1908:
1903:
1899:
1896:
1891:
1887:
1883:
1880:
1877:
1874:
1868:
1863:
1859:
1853:
1850:
1844:
1838:
1834:
1830:
1827:
1824:
1821:
1818:
1813:
1810:
1807:
1803:
1799:
1796:
1793:
1790:
1787:
1784:
1781:
1776:
1772:
1768:
1765:
1762:
1759:
1756:
1753:
1750:
1747:
1744:
1741:
1738:
1735:
1732:
1727:
1723:
1717:
1714:
1708:
1685:
1684:
1669:
1664:
1658:
1653:
1647:
1644:
1641:
1637:
1632:
1627:
1624:
1618:
1615:
1613:
1611:
1605:
1602:
1599:
1596:
1593:
1586:
1583:
1579:
1575:
1572:
1569:
1566:
1563:
1560:
1554:
1551:
1549:
1547:
1536:
1528:
1525:
1521:
1517:
1514:
1511:
1508:
1505:
1502:
1495:
1492:
1488:
1484:
1481:
1478:
1475:
1472:
1469:
1463:
1456:
1453:
1450:
1446:
1441:
1438:
1436:
1434:
1429:
1424:
1418:
1415:
1412:
1408:
1403:
1396:
1393:
1390:
1385:
1382:
1379:
1375:
1368:
1365:
1362:
1358:
1353:
1345:
1341:
1337:
1334:
1331:
1328:
1324:
1317:
1312:
1309:
1306:
1302:
1298:
1295:
1293:
1289:
1285:
1279:
1276:
1270:
1266:
1265:
1233:
1229:
1225:
1222:
1219:
1216:
1212:
1182:
1179:
1178:
1177:
1166:
1163:
1156:
1152:
1146:
1143:
1137:
1133:
1128:
1121:
1117:
1111:
1108:
1102:
1098:
1082:
1081:
1068:
1064:
1058:
1055:
1049:
1045:
1040:
1036:
1032:
1029:
1026:
1023:
1020:
1015:
1011:
1005:
1002:
996:
974:
973:
962:
959:
956:
953:
950:
945:
941:
935:
932:
926:
922:
919:
916:
913:
909:
905:
902:
899:
896:
893:
890:
887:
882:
879:
873:
850:
849:
836:
832:
826:
823:
817:
813:
810:
807:
804:
801:
798:
795:
792:
789:
786:
758:
735:
715:
704:
703:
692:
687:
683:
680:
675:
671:
667:
664:
661:
658:
652:
647:
643:
637:
634:
628:
596:
592:
588:
585:
582:
558:
547:
546:
535:
530:
526:
520:
517:
511:
507:
504:
501:
498:
495:
492:
489:
486:
483:
480:
452:
429:
409:
398:
397:
386:
381:
375:
372:
368:
364:
361:
358:
355:
352:
349:
343:
338:
334:
328:
325:
319:
294:
293:
290:
285:
282:
279:
276:
272:
271:
268:
265:
262:
259:
255:
254:
251:
248:
245:
242:
239:
229:
226:
213:
210:
197:
194:
170:
167:
154:
151:
150:
149:
141:
135:
123:
120:
95:
92:
79:
76:
71:
68:
15:
9:
6:
4:
3:
2:
4314:
4303:
4300:
4298:
4295:
4294:
4292:
4281:
4280:
4274:
4269:
4265:
4261:
4255:
4251:
4246:
4242:
4236:
4232:
4227:
4226:
4223:Other sources
4213:
4211:0-07-231289-0
4207:
4203:
4198:
4197:
4188:
4179:
4177:0-324-42262-8
4173:
4169:
4162:
4155:
4149:
4145:
4135:
4132:
4130:
4127:
4125:
4122:
4120:
4117:
4115:
4112:
4111:
4103:
4100:
4098:
4095:
4093:
4090:
4089:
4086:Legal regimes
4078:
4075:
4072:
4071:
4069:
4064:
4061:
4058:
4057:
4055:
4054:
4053:
4050:
4047:
4039:
4036:
4033:
4032:
4030:
4029:
4028:
4026:
4022:
4018:
4014:
4004:
4001:
3998:
3995:
3994:
3992:
3991:
3990:
3965:
3961:
3957:
3949:
3946:
3943:
3934:
3929:
3924:
3918:
3915:
3910:
3907:
3903:
3899:
3896:
3892:
3887:
3884:
3880:
3875:
3872:
3865:
3864:
3863:
3861:
3857:
3847:
3845:
3826:
3821:
3807:
3804:
3801:
3794:
3790:
3787:
3784:
3780:
3774:
3768:
3765:
3762:
3754:
3751:
3748:
3739:
3734:
3731:
3725:
3721:
3714:
3713:
3712:
3709:
3695:
3675:
3672:
3667:
3664:
3642:
3620:
3617:
3592:
3588:
3584:
3581:
3576:
3573:
3567:
3561:
3553:
3550:
3547:
3541:
3536:
3533:
3524:
3523:
3522:
3520:
3516:
3506:
3503:
3499:
3495:
3491:
3488:Valuation of
3481:
3464:
3461:
3458:
3454:
3449:
3446:
3426:
3403:
3398:
3395:
3390:
3385:
3363:
3360:
3346:
3343:
3338:
3333:
3314:
3306:
3305:
3304:
3302:
3283:
3278:
3275:
3269:
3264:
3258:
3255:
3250:
3246:
3243:
3240:
3236:
3231:
3228:
3222:
3219:
3206:
3198:
3193:
3180:
3174:
3170:
3167:
3154:
3146:
3140:
3137:
3134:
3131:
3128:
3107:
3095:
3094:
3093:
3091:
3078:
3062:
3059:
3054:
3040:
3037:
3034:
3027:
3023:
3017:
3014:
3011:
3003:
2990:
2984:
2980:
2975:
2961:
2951:
2948:
2937:
2934:
2918:
2915:
2910:
2896:
2893:
2890:
2883:
2879:
2873:
2870:
2867:
2859:
2846:
2840:
2836:
2831:
2817:
2807:
2804:
2793:
2792:
2791:
2789:
2784:
2767:
2764:
2761:
2755:
2750:
2737:
2728:
2725:
2718:
2715:
2709:
2705:
2701:
2695:
2692:
2689:
2686:
2683:
2678:
2675:
2669:
2649:
2648:
2647:
2645:
2627:
2624:
2621:
2614:
2601:
2592:
2589:
2582:
2577:
2570:
2557:
2548:
2545:
2538:
2529:
2528:
2513:
2508:
2495:
2486:
2483:
2476:
2471:
2463:
2460:
2457:
2451:
2446:
2433:
2424:
2421:
2410:
2409:
2408:
2405:
2388:
2385:
2382:
2378:
2373:
2370:
2362:
2346:
2326:
2306:
2283:
2278:
2274:
2271:
2266:
2258:
2255:
2252:
2243:
2238:
2224:
2217:
2213:
2207:
2204:
2201:
2195:
2190:
2176:
2166:
2163:
2152:
2151:
2136:
2131:
2125:
2122:
2114:
2111:
2108:
2102:
2099:
2093:
2088:
2074:
2067:
2063:
2057:
2054:
2051:
2045:
2040:
2026:
2016:
2013:
2002:
2001:
2000:
1992:
1990:
1987:
1984:
1981:
1978:
1975:
1972:
1971:
1967:
1964:
1961:
1958:
1955:
1954:
1950:
1948:
1945:
1942:
1939:
1936:
1935:
1932:
1930:
1906:
1901:
1897:
1894:
1889:
1881:
1878:
1875:
1866:
1861:
1848:
1842:
1836:
1828:
1825:
1822:
1816:
1811:
1808:
1805:
1797:
1794:
1791:
1785:
1782:
1779:
1774:
1766:
1763:
1760:
1754:
1748:
1745:
1742:
1736:
1733:
1730:
1725:
1712:
1706:
1698:
1697:
1696:
1694:
1688:
1667:
1662:
1656:
1651:
1645:
1642:
1639:
1635:
1630:
1625:
1622:
1616:
1614:
1603:
1600:
1597:
1594:
1591:
1584:
1581:
1573:
1570:
1567:
1561:
1558:
1552:
1550:
1534:
1526:
1523:
1515:
1512:
1509:
1503:
1500:
1493:
1490:
1482:
1479:
1476:
1470:
1467:
1461:
1454:
1451:
1448:
1444:
1439:
1437:
1427:
1422:
1416:
1413:
1410:
1406:
1401:
1394:
1391:
1388:
1383:
1380:
1377:
1373:
1366:
1363:
1360:
1356:
1351:
1343:
1335:
1332:
1329:
1322:
1315:
1310:
1307:
1304:
1300:
1296:
1294:
1287:
1274:
1268:
1256:
1255:
1254:
1252:
1231:
1223:
1220:
1217:
1210:
1200:
1196:
1192:
1188:
1164:
1161:
1154:
1141:
1135:
1131:
1126:
1119:
1106:
1100:
1096:
1087:
1086:
1085:
1066:
1053:
1047:
1043:
1038:
1030:
1027:
1024:
1018:
1013:
1000:
994:
986:
985:
984:
981:
979:
960:
957:
954:
948:
943:
930:
924:
920:
917:
911:
907:
903:
897:
894:
891:
888:
885:
880:
877:
871:
858:
857:
856:
854:
834:
821:
815:
811:
808:
805:
799:
796:
793:
790:
787:
772:
771:
770:
756:
749:
733:
713:
690:
685:
681:
678:
673:
665:
662:
659:
650:
645:
632:
626:
618:
617:
616:
613:
608:
594:
590:
586:
583:
580:
572:
556:
533:
528:
515:
509:
505:
502:
499:
493:
490:
487:
484:
481:
466:
465:
464:
450:
443:
427:
407:
384:
379:
373:
370:
362:
359:
356:
350:
347:
341:
336:
323:
317:
309:
308:
307:
305:
301:
300:present value
291:
289:
286:
283:
280:
277:
274:
273:
269:
266:
263:
260:
257:
256:
252:
249:
246:
243:
240:
238:
237:
234:
225:
223:
219:
209:
207:
203:
193:
191:
187:
186:interest rate
183:
179:
178:present value
175:
166:
164:
160:
147:
146:
142:
139:
136:
133:
129:
126:
125:
119:
117:
113:
109:
105:
101:
91:
89:
85:
75:
67:
65:
61:
56:
54:
49:
46:payments and
45:
41:
40:home mortgage
37:
33:
29:
22:
4277:
4249:
4230:
4195:
4187:
4167:
4161:
4153:
4148:
4124:Life annuity
4051:
4048:
4045:
4024:
4020:
4016:
4012:
4010:
3996:
3988:
3859:
3855:
3853:
3841:
3710:
3607:
3521:payments is
3518:
3514:
3512:
3487:
3418:
3299:Therefore a
3298:
3089:
3087:
3076:
2932:
2787:
2785:
2782:
2643:
2642:
2406:
2298:
1998:
1988:
1982:
1928:
1926:
1692:
1689:
1686:
1250:
1198:
1197:-th payment
1194:
1190:
1186:
1184:
1083:
982:
977:
975:
852:
851:
747:
705:
612:future value
611:
609:
548:
441:
399:
299:
297:
287:
231:
221:
217:
215:
205:
201:
199:
190:future value
172:
162:
158:
156:
143:
137:
127:
115:
112:life annuity
107:
103:
99:
97:
87:
83:
81:
73:
60:life annuity
57:
31:
25:
3502:probability
3498:Life tables
1929:annuity-due
1923:Annuity-due
228:Annuity Due
88:annuity-due
4291:Categories
4140:References
4129:Perpetuity
4052:Examples:
3989:Examples:
3301:perpetuity
3090:perpetuity
3084:Perpetuity
64:perpetuity
38:, monthly
28:investment
4302:Annuities
3947:−
3935:−
3900:−
3842:See also
3812:¯
3805:−
3791:×
3766:−
3740:−
3673:−
3582:−
3542:−
3376:¯
3373:∞
3364:¨
3324:¯
3321:∞
3256:−
3232:−
3223:×
3215:∞
3211:→
3184:¯
3171:×
3163:∞
3159:→
3116:∞
3112:→
3060:−
3050:¯
2994:¯
2971:¯
2952:¨
2906:¯
2894:−
2850:¯
2827:¯
2808:¨
2759:$
2741:¯
2729:¨
2719:×
2713:$
2699:$
2690:×
2605:¯
2593:¨
2578:−
2561:¯
2549:¨
2499:¯
2487:¨
2477:×
2437:¯
2425:¨
2363:given by
2272:−
2234:¯
2214:×
2186:¯
2167:¨
2123:−
2103:−
2084:¯
2064:×
2036:¯
2017:¨
1951:payments
1895:−
1852:¯
1809:−
1783:⋯
1716:¯
1626:−
1601:−
1582:−
1562:−
1524:−
1504:−
1491:−
1471:−
1392:−
1374:∑
1301:∑
1278:¯
1201:would be
1145:¯
1127:−
1110:¯
1057:¯
1044:×
1004:¯
978:principal
952:$
934:¯
921:×
915:$
901:$
892:×
825:¯
812:×
679:−
636:¯
519:¯
506:×
371:−
351:−
327:¯
253:payments
174:Valuation
169:Valuation
44:insurance
4270:(1878).
4108:See also
4023:− 1) ÷ (
3858:, given
2644:Example:
1993:periods
1968:—
853:Example:
292:periods
270:—
2768:730.01.
2359:is the
48:pension
32:annuity
4256:
4237:
4208:
4174:
3419:where
2751:0.0075
2299:where
961:495.50
706:where
573:, and
400:where
188:, and
3505:age.
220:, or
70:Types
30:, an
4254:ISBN
4235:ISBN
4206:ISBN
4172:ISBN
2676:0.09
1985:− 1
1979:...
1943:...
1084:and
944:0.01
878:0.12
769:is:
748:rent
610:The
463:is:
442:rent
306:by:
298:The
284:...
247:...
4202:175
4019:× (
3203:lim
3151:lim
3104:lim
2716:100
2702:100
2664:due
1927:An
918:100
904:100
204:or
102:or
26:In
4293::
4276:.
4204:.
3862::
3846:.
3708:.
3122:PV
3088:A
2762:11
2738:84
2693:12
2679:12
2659:FV
2404:.
1976:1
1973:0
1946:↓
1940:↓
1937:↓
931:60
895:12
881:12
867:PV
781:FV
607:.
595:12
475:PV
281:2
278:1
275:0
250:↓
244:↓
241:↓
192:.
184:,
165:.
66:.
4262:.
4243:.
4214:.
4182:.
4180:.
4025:r
4021:r
4017:r
4013:r
3997:R
3966:m
3962:/
3958:j
3953:)
3950:1
3944:n
3941:(
3930:)
3925:)
3919:m
3916:j
3911:+
3908:1
3904:(
3897:1
3893:(
3888:+
3885:1
3881:A
3876:=
3873:R
3860:A
3856:R
3827:.
3822:i
3818:|
3808:n
3802:N
3795:a
3788:R
3785:=
3781:]
3775:i
3769:N
3763:n
3759:)
3755:1
3752:+
3749:i
3746:(
3735:i
3732:1
3726:[
3722:R
3696:i
3676:P
3668:i
3665:R
3643:R
3621:i
3618:R
3593:.
3589:)
3585:P
3577:i
3574:R
3568:(
3562:n
3558:)
3554:i
3551:+
3548:1
3545:(
3537:i
3534:R
3519:n
3515:P
3465:i
3462:+
3459:1
3455:i
3450:=
3447:d
3427:i
3404:,
3399:d
3396:1
3391:=
3386:i
3382:|
3361:a
3347:i
3344:1
3339:=
3334:i
3330:|
3315:a
3284:.
3279:i
3276:R
3270:=
3265:i
3259:n
3251:)
3247:i
3244:+
3241:1
3237:(
3229:1
3220:R
3207:n
3199:=
3194:i
3190:|
3181:n
3175:a
3168:R
3155:n
3147:=
3144:)
3141:R
3138:,
3135:n
3132:,
3129:i
3126:(
3108:n
3077:n
3063:1
3055:i
3045:|
3041:1
3038:+
3035:n
3028:s
3024:=
3021:)
3018:i
3015:+
3012:1
3009:(
3004:i
3000:|
2991:n
2985:s
2981:=
2976:i
2966:|
2962:n
2949:s
2933:n
2919:1
2916:+
2911:i
2901:|
2897:1
2891:n
2884:a
2880:=
2877:)
2874:i
2871:+
2868:1
2865:(
2860:i
2856:|
2847:n
2841:a
2837:=
2832:i
2822:|
2818:n
2805:a
2788:n
2765:,
2756:=
2747:|
2726:s
2710:=
2706:)
2696:,
2687:7
2684:,
2670:(
2628:.
2625:d
2622:=
2615:i
2611:|
2602:n
2590:s
2583:1
2571:i
2567:|
2558:n
2546:a
2539:1
2514:,
2509:i
2505:|
2496:n
2484:a
2472:n
2468:)
2464:i
2461:+
2458:1
2455:(
2452:=
2447:i
2443:|
2434:n
2422:s
2389:1
2386:+
2383:i
2379:i
2374:=
2371:d
2347:d
2327:i
2307:n
2284:,
2279:d
2275:1
2267:n
2263:)
2259:i
2256:+
2253:1
2250:(
2244:=
2239:i
2229:|
2225:n
2218:s
2211:)
2208:i
2205:+
2202:1
2199:(
2196:=
2191:i
2181:|
2177:n
2164:s
2137:,
2132:d
2126:n
2119:)
2115:i
2112:+
2109:1
2106:(
2100:1
2094:=
2089:i
2079:|
2075:n
2068:a
2061:)
2058:i
2055:+
2052:1
2049:(
2046:=
2041:i
2031:|
2027:n
2014:a
1989:n
1983:n
1907:.
1902:i
1898:1
1890:n
1886:)
1882:i
1879:+
1876:1
1873:(
1867:=
1862:i
1858:|
1849:n
1843:a
1837:n
1833:)
1829:i
1826:+
1823:1
1820:(
1817:=
1812:1
1806:n
1802:)
1798:i
1795:+
1792:1
1789:(
1786:+
1780:+
1775:2
1771:)
1767:i
1764:+
1761:1
1758:(
1755:+
1752:)
1749:i
1746:+
1743:1
1740:(
1737:+
1734:1
1731:=
1726:i
1722:|
1713:n
1707:s
1693:n
1691:(
1668:,
1663:i
1657:n
1652:)
1646:i
1643:+
1640:1
1636:1
1631:(
1623:1
1617:=
1604:1
1598:i
1595:+
1592:1
1585:n
1578:)
1574:i
1571:+
1568:1
1565:(
1559:1
1553:=
1535:)
1527:1
1520:)
1516:i
1513:+
1510:1
1507:(
1501:1
1494:n
1487:)
1483:i
1480:+
1477:1
1474:(
1468:1
1462:(
1455:i
1452:+
1449:1
1445:1
1440:=
1428:k
1423:)
1417:i
1414:+
1411:1
1407:1
1402:(
1395:1
1389:n
1384:0
1381:=
1378:k
1367:i
1364:+
1361:1
1357:1
1352:=
1344:k
1340:)
1336:i
1333:+
1330:1
1327:(
1323:1
1316:n
1311:1
1308:=
1305:k
1297:=
1288:i
1284:|
1275:n
1269:a
1251:R
1232:k
1228:)
1224:i
1221:+
1218:1
1215:(
1211:R
1199:R
1195:k
1191:k
1187:k
1165:i
1162:=
1155:i
1151:|
1142:n
1136:s
1132:1
1120:i
1116:|
1107:n
1101:a
1097:1
1067:i
1063:|
1054:n
1048:a
1039:n
1035:)
1031:i
1028:+
1025:1
1022:(
1019:=
1014:i
1010:|
1001:n
995:s
958:,
955:4
949:=
940:|
925:a
912:=
908:)
898:,
889:5
886:,
872:(
835:i
831:|
822:n
816:s
809:R
806:=
803:)
800:R
797:,
794:n
791:,
788:i
785:(
757:R
734:i
714:n
691:,
686:i
682:1
674:n
670:)
666:i
663:+
660:1
657:(
651:=
646:i
642:|
633:n
627:s
591:/
587:I
584:=
581:i
557:I
534:.
529:i
525:|
516:n
510:a
503:R
500:=
497:)
494:R
491:,
488:n
485:,
482:i
479:(
451:R
428:i
408:n
385:,
380:i
374:n
367:)
363:i
360:+
357:1
354:(
348:1
342:=
337:i
333:|
324:n
318:a
288:n
134:.
23:.
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