4099:
20:
5424:
2053:(or a mortality table) is a mathematical construction that shows the number of people alive (based on the assumptions used to build the table) at a given age. In addition to the number of lives remaining at each age, a mortality table typically provides various probabilities associated with the development of these values.
1319:
The rate of discount equals the amount of interest earned during a one-year period, divided by the balance of money at the end of that period. By contrast, an annual effective rate of interest is calculated by dividing the amount of interest earned during a one-year period by the balance of money at
398:
Various proposals have been made to adopt a linear system, where all the notation would be on a single line without the use of superscripts or subscripts. Such a method would be useful for computing where representation of the halo system can be extremely difficult. However, a standard linear system
4141:
Notation directly above the basic symbol indicates when payments are made. Two dots indicates an annuity whose payments are made at the beginning of each year (an "annuity-due"); a horizontal line above the symbol indicates an annuity payable continuously (a "continuous annuity"); no mark above the
3902:
A life table generally shows the number of people alive at integral ages. If we need information regarding a fraction of a year, we must make assumptions with respect to the table, if not already implied by a mathematical formula underlying the table. A common assumption is that of a
Uniform
6022:
Notation directly above the basic symbol indicates the "type" of life insurance, whether payable at the end of the period or immediately. A horizontal line indicates life insurance payable immediately, whilst no mark above the symbol indicates payment is to be made at the end of the period
5942:
In the interest of simplicity the notation is limited and does not, for example, show whether the annuity is payable to a man or a woman (a fact that would typically be determined from the context, including whether the life table is based on male or female mortality rates).
1912:
6474:
5496:
When using these symbols, the rate of interest is not necessarily constant throughout the lifetime of the annuities. However, when the rate varies, the above formulas will no longer be valid; particular formulas can be developed for particular movements of the
1074:
This present value factor, or discount factor, is used to determine the amount of money that must be invested now in order to have a given amount of money in the future. For example, if you need 1 in one year, then the amount of money you should invest now is:
5218:
2039:
4575:
4348:
4992:
5207:
3406:
4887:
964:, interest to a debtor), more frequently than do effective rates. The result is more frequent compounding of interest income to the investor, (or interest expense to the debtor), when nominal rates are used.
951:
6613:
281:
3897:
4135:
Notation to the top-right indicates the frequency of payment (i.e., the number of annuity payments that will be made during each year). A lack of such notation means that payments are made annually.
1742:
1661:
5983:
Notation to the top-right indicates the timing of the payment of a death benefit. A lack of notation means payments are made at the end of the year of death. A figure in parentheses (for example
5045:
6297:
5429:
To understand the relationships shown above, consider that cash flows paid at a later time have a smaller present value than cash flows of the same total amount that are paid at earlier times.
1224:
1069:
356:
4698:
4426:
813:." It merely represents the number of interest conversions, or compounding times, per year. Semi-annual compounding, (or converting interest every six months), is frequently used in valuing
5857:
5101:
4088:
3610:
5946:
The
Actuarial Present Value of life contingent payments can be treated as the mathematical expectation of a present value random variable, or calculated through the current payment form.
2865:
3765:
5636:
5419:{\displaystyle a_{{\overline {n|}}i}<a_{{\overline {n|}}i}^{(m)}<{\overline {a}}_{{\overline {n|}}i}<{\ddot {a}}_{{\overline {n|}}i}^{(m)}<{\ddot {a}}_{{\overline {n|}}i}}
4645:
1317:
6133:
5713:
4202:
3172:
2404:
5587:
3031:
2641:
685:
733:
894:
5791:
1134:
773:
5506:
A life annuity is an annuity whose payments are contingent on the continuing life of the annuitant. The age of the annuitant is an important consideration in calculating the
3271:
2240:
6096:
5753:
2812:
1423:
1381:
956:
Effective and nominal rates of interest are not the same because interest paid in earlier measurement periods "earns" interest in later measurement periods; this is called
1923:
70:
Bar implies continuous – or paid at the moment of death; double dot implies paid at the beginning of the year; no mark implies paid at the end of the year;
3647:
3489:
2165:
of the table. Some mortality tables begin at an age greater than 0, in which case the radix is the number of people assumed to be alive at the youngest age in the table.
1100:
645:
5671:
4001:
3935:
2778:
1713:
1578:
1535:
1480:
807:
603:
501:
6014:
5546:
854:
465:
6055:
5491:
3967:
3802:
3061:
2926:
2888:
2738:
2704:
2344:
2270:
2213:
2185:
2159:
2131:
2081:
6229:
6187:
4728:
2581:
2517:
2484:
2456:
6208:
6166:
5977:
4129:
3693:
3535:
3452:
3221:
3107:
2972:
2670:
2316:
1734:
1692:
1501:
1444:
1339:
1248:
1157:
986:
430:
140:
5937:
5917:
5897:
5877:
5471:
5451:
5121:
4788:
4768:
4748:
4598:
4450:
4371:
4230:
3822:
3667:
3509:
3426:
3195:
3081:
2946:
2553:
2428:
2290:
2101:
1602:
1555:
569:
549:
529:
183:
160:
111:
91:
65:
45:
4462:
3824:. This is the expected number of complete years remaining to live (you may think of it as the expected number of birthdays that the person will celebrate).
4242:
4892:
4147:
5133:
3281:
5715:
indicates an annuity of 1 unit per year until the later death of member or death of spouse, to someone currently age 65 and spouse age 64.
5673:
indicates an annuity of 1 unit per year until the earlier death of member or death of spouse, to someone currently age 65 and spouse age 64
4796:
6016:) means the benefit is payable at the end of the period indicated (12 for monthly; 4 for quarterly; 2 for semi-annually; 365 for daily).
899:
6504:
tends to zero, so does this probability in the continuous case. The approximate force of mortality is this probability divided by Δ
191:
6493:
4138:
Notation to the bottom-right indicates the age of the person when the annuity starts and the period for which an annuity is paid.
1667:
6530:
3276:
These symbols may also be extended to multiple years, by inserting the number of years at the bottom left of the basic symbol.
1907:{\displaystyle \,(1+i)=\left(1+{\frac {i^{(m)}}{m}}\right)^{m}=e^{\delta }=\left(1-{\frac {d^{(m)}}{m}}\right)^{-m}=(1-d)^{-1}}
5755:
indicates an annuity of 1 unit per year payable 12 times a year (1/12 unit per month) until death to someone currently age 65
3830:
6679:
1160:
6469:{\displaystyle P_{\Delta x}(x)=P(x<X<x+\Delta \;x\mid \;X>x)={\frac {F_{X}(x+\Delta \;x)-F_{X}(x)}{(1-F_{X}(x))}}}
1610:
4999:
6674:
1169:
998:
286:
4650:
4378:
6684:
5801:
5050:
4009:
3543:
5793:
indicates an annuity of 1 unit per year payable at the start of each year until death to someone currently age 65
822:
2818:
5548:
indicates an annuity of 1 unit per year payable at the end of each year until death to someone currently age 65
605:
is the nominal rate of interest convertible semiannually. If the effective annual rate of interest is 12%, then
6651:
4107:
3701:
5594:
5589:
indicates an annuity of 1 unit per year payable for 10 years with payments being made at the end of each year
4730:
is added to the top-right corner, it represents the present value of an annuity whose payments occur each one
5638:
indicates an annuity of 1 unit per year for 10 years, or until death if earlier, to someone currently age 65
4605:
6624:
6272:
5507:
1253:
818:
433:
6103:
5678:
4163:
4142:
basic symbol indicates an annuity whose payments are made at the end of each year (an "annuity-immediate").
3115:
2349:
6639:
6634:
6251:
and is construed as an instantaneous rate of mortality at a certain age measured on an annualized basis.
5553:
2980:
2587:
650:
690:
508:
859:
5760:
1105:
738:
3938:
3229:
2218:
2034:{\displaystyle \,i>i^{(2)}>i^{(3)}>\cdots >\delta >\cdots >d^{(3)}>d^{(2)}>d}
6062:
6689:
5720:
2784:
1386:
1344:
4208:) represents the present value of an annuity-immediate, which is a series of unit payments at the
5124:
4146:
If the payments to be made under an annuity are independent of any life event, it is known as an
3617:
3459:
1078:
608:
384:
5643:
4098:
3972:
3906:
2757:
1697:
1562:
1506:
1451:
960:. That is, nominal rates of interest credit interest to an investor, (alternatively charge, or
778:
574:
472:
6629:
6144:
5986:
5523:
5514:
The age of the annuitant is placed at the bottom right of the symbol, without an "angle" mark.
1581:
957:
832:
443:
6032:
5476:
3944:
3779:
3038:
2903:
2873:
2710:
2676:
2321:
2247:
2190:
2170:
2136:
2108:
2058:
6213:
6171:
6019:
Notation to the bottom-right indicates the age of the person when the life insurance begins.
4706:
2559:
2495:
2462:
2434:
395:
before or after the main letter. Example notation using the halo system can be seen below.
6659:
6497:
4428:
represents the present value of an annuity-due, which is a series of unit payments at the
3771:
1320:
the beginning of the year. The present value (today) of a payment of 1 that is to be made
6192:
6150:
5961:
4113:
3672:
3514:
3431:
3200:
3086:
2951:
2649:
2295:
1718:
1676:
1485:
1428:
1323:
1232:
1141:
970:
414:
119:
6135:
indicates a life insurance benefit of 1 payable at the (mathematical) instant of death.
4570:{\displaystyle {\ddot {a}}_{{\overline {n|}}i}=1+v+\cdots +v^{n-1}={\frac {1-v^{n}}{d}}}
5955:
5922:
5902:
5882:
5862:
5456:
5436:
5106:
4773:
4753:
4733:
4583:
4435:
4356:
4215:
3807:
3652:
3494:
3411:
3180:
3066:
2931:
2538:
2413:
2275:
2086:
1587:
1540:
814:
554:
534:
514:
168:
145:
96:
76:
50:
30:
16:
Shorthand method to record math formulas that deal with interest rates and life tables
6668:
989:
4102:
Illustration of the payment streams represented by actuarial notation for annuities.
4155:
6098:
indicates a life insurance benefit of 1 payable at the end of the month of death.
4343:{\displaystyle \,a_{{\overline {n|}}i}=v+v^{2}+\cdots +v^{n}={\frac {1-v^{n}}{i}}}
19:
6057:
indicates a life insurance benefit of 1 payable at the end of the year of death.
6248:
4987:{\displaystyle {\ddot {a}}_{{\overline {n|}}i}^{(m)}={\frac {1-v^{n}}{d^{(m)}}}}
4151:
388:
377:
6244:
5202:{\displaystyle {\overline {a}}_{{\overline {n|}}i}={\frac {1-v^{n}}{\delta }}}
2050:
3223:) are living and dying, the relationship between these two probabilities is:
829:
frequently convert interest monthly. Following the above example again where
6254:
In a life table, we consider the probability of a person dying between age (
826:
392:
1102:. If you need 25 in 5 years the amount of money you should invest now is:
3903:
Distribution of Deaths (UDD) at each year of age. Under this assumption,
3401:{\displaystyle \,_{n}d_{x}=d_{x}+d_{x+1}+\cdots +d_{x+n-1}=l_{x}-l_{x+n}}
810:
373:
369:
4882:{\displaystyle a_{{\overline {n|}}i}^{(m)}={\frac {1-v^{n}}{i^{(m)}}}}
2083:
is the number of people alive, relative to an original cohort, at age
5493:, and is often omitted if the rate is clearly known from the context.
5212:
The present values of these annuities may be compared as follows:
4097:
961:
5899:
is the number of years of payments (or until death if earlier),
4232:
years (in other words: the value one period before the first of
946:{\displaystyle \,\left(1+{\frac {0.1139}{12}}\right)^{12}=1.12}
647:
represents the effective interest rate every six months. Since
5123:
increases without bound. The underlying annuity is known as a
4452:
years (in other words: the value at the time of the first of
3804:
is the curtate expectation of life for a person alive at age
1584:, is the limiting value of the nominal rate of interest when
3770:
Another statistic that can be obtained from a life table is
2161:: the number of people alive at age 0. This is known as the
276:{\displaystyle E(Z^{2})=E{\big (}(v^{k_{x}+1})^{2}{\big )},}
47:
is an assurance paying 1 on the insured event; lower-case
6608:{\displaystyle \mu \,(x)={\frac {F'_{X}(x)}{1-F_{X}(x)}}}
5453:
which represents the rate of interest may be replaced by
2103:. As age increases the number of people alive decreases.
1250:
can also be calculated from the following relationships:
188:
No fixed meaning, implies the second moment to calculate
67:
is an annuity paying 1 per annum at the appropriate time.
3892:{\displaystyle \,e_{x}=\sum _{t=1}^{\infty }\ _{t}p_{x}}
1446:
years in the future of an amount of 1 invested today.
6533:
6300:
6271:. In the continuous case, we could also consider the
6216:
6195:
6174:
6153:
6106:
6065:
6035:
5989:
5964:
5925:
5905:
5885:
5865:
5804:
5763:
5723:
5681:
5646:
5597:
5556:
5526:
5479:
5459:
5439:
5221:
5136:
5109:
5053:
5002:
4895:
4799:
4776:
4756:
4736:
4709:
4653:
4608:
4586:
4465:
4438:
4381:
4359:
4245:
4218:
4166:
4116:
4012:
3975:
3947:
3909:
3833:
3810:
3782:
3704:
3675:
3655:
3620:
3546:
3517:
3497:
3462:
3434:
3414:
3284:
3232:
3203:
3183:
3118:
3089:
3069:
3041:
2983:
2954:
2934:
2906:
2876:
2821:
2787:
2760:
2713:
2679:
2652:
2590:
2562:
2541:
2498:
2465:
2437:
2416:
2352:
2324:
2298:
2278:
2250:
2221:
2193:
2173:
2139:
2111:
2089:
2061:
1926:
1745:
1721:
1700:
1679:
1656:{\displaystyle \,\delta =\lim _{m\to \infty }i^{(m)}}
1613:
1590:
1565:
1543:
1509:
1488:
1454:
1431:
1389:
1347:
1326:
1256:
1235:
1172:
1144:
1108:
1081:
1001:
973:
902:
862:
835:
781:
741:
693:
653:
611:
577:
557:
537:
517:
475:
446:
417:
289:
194:
171:
148:
122:
99:
79:
53:
33:
4150:. Otherwise, in particular if payments end upon the
5040:{\displaystyle {\overline {a}}_{{\overline {n|}}i}}
3177:
Since the only possible alternatives from one age (
6607:
6468:
6223:
6202:
6181:
6160:
6127:
6090:
6049:
6008:
5971:
5931:
5911:
5891:
5871:
5851:
5785:
5747:
5707:
5665:
5630:
5581:
5540:
5485:
5465:
5445:
5418:
5201:
5115:
5095:
5039:
4986:
4881:
4782:
4762:
4742:
4722:
4692:
4639:
4592:
4569:
4444:
4420:
4373:in the denominator matches with 'i' in immediate)
4365:
4342:
4224:
4196:
4123:
4082:
3995:
3961:
3929:
3891:
3816:
3796:
3759:
3687:
3661:
3641:
3604:
3529:
3503:
3483:
3446:
3420:
3400:
3265:
3215:
3189:
3166:
3101:
3075:
3055:
3025:
2966:
2940:
2920:
2882:
2859:
2806:
2772:
2732:
2698:
2664:
2635:
2575:
2547:
2511:
2478:
2450:
2422:
2398:
2338:
2310:
2284:
2264:
2234:
2207:
2179:
2153:
2125:
2095:
2075:
2033:
1917:Their numerical value can be compared as follows:
1906:
1728:
1707:
1686:
1655:
1596:
1572:
1549:
1529:
1495:
1474:
1438:
1417:
1375:
1333:
1311:
1242:
1218:
1151:
1128:
1094:
1063:
980:
945:
888:
848:
801:
767:
727:
679:
639:
597:
563:
543:
523:
495:
459:
424:
350:
275:
177:
154:
134:
105:
85:
59:
39:
1219:{\displaystyle d={\frac {i}{1+i}}\approx i-i^{2}}
1064:{\displaystyle \,v={(1+i)}^{-1}\approx 1-i+i^{2}}
3491:is the probability of death between the ages of
2928:is the probability of death between the ages of
1622:
351:{\displaystyle v^{k_{x}+1}=e^{\delta (k_{x}+1)}}
4693:{\displaystyle {\ddot {s}}_{{\overline {n|}}i}}
4421:{\displaystyle {\ddot {a}}_{{\overline {n|}}i}}
3408:shows the number of people who die between age
440:. Thus if the annual interest rate is 12% then
372:to record mathematical formulas that deal with
4647:is the value at the time of the last payment,
551:times the effective rate of interest over one
5852:{\displaystyle a_{x:{\overline {n|}}i}^{(m)}}
5096:{\displaystyle \,a_{{\overline {n|}}i}^{(m)}}
4106:The basic symbol for the present value of an
4083:{\displaystyle \,l_{x+t}=(1-t)l_{x}+tl_{x+1}}
3605:{\displaystyle \,_{n}q_{x}={}_{n}d_{x}/l_{x}}
2187:is the limiting age of the mortality tables.
265:
222:
8:
6210:generally refers to net premiums per annum,
5979:. The following notation can then be added:
4600:in the denominator matches with 'd' in due)
4131:. The following notation can then be added:
2272:is the number of people who die between age
436:, which is the "true" rate of interest over
6262: + 1; this probability is called
2860:{\displaystyle d_{\omega -1}=l_{\omega -1}}
1482:, the nominal rate of discount convertible
6401:
6360:
6353:
6231:to special premiums, as a unique premium.
531:times a year, and is numerically equal to
6587:
6557:
6550:
6537:
6532:
6445:
6415:
6383:
6376:
6305:
6299:
6217:
6215:
6196:
6194:
6175:
6173:
6154:
6152:
6119:
6109:
6107:
6105:
6076:
6071:
6066:
6064:
6041:
6036:
6034:
5994:
5988:
5965:
5963:
5924:
5904:
5884:
5864:
5837:
5822:
5816:
5809:
5803:
5777:
5766:
5765:
5762:
5733:
5728:
5722:
5686:
5680:
5651:
5645:
5615:
5609:
5602:
5596:
5567:
5561:
5555:
5532:
5527:
5525:
5478:
5458:
5438:
5400:
5394:
5393:
5382:
5381:
5365:
5350:
5344:
5343:
5332:
5331:
5311:
5305:
5304:
5294:
5278:
5263:
5257:
5256:
5233:
5227:
5226:
5220:
5187:
5174:
5155:
5149:
5148:
5138:
5135:
5108:
5081:
5066:
5060:
5059:
5054:
5052:
5021:
5015:
5014:
5004:
5001:
4970:
4959:
4946:
4931:
4916:
4910:
4909:
4898:
4897:
4894:
4865:
4854:
4841:
4826:
4811:
4805:
4804:
4798:
4775:
4755:
4735:
4710:
4708:
4674:
4668:
4667:
4656:
4655:
4652:
4621:
4615:
4614:
4609:
4607:
4585:
4555:
4542:
4527:
4486:
4480:
4479:
4468:
4467:
4464:
4437:
4402:
4396:
4395:
4384:
4383:
4380:
4358:
4328:
4315:
4306:
4287:
4258:
4252:
4251:
4246:
4244:
4217:
4178:
4172:
4171:
4165:
4117:
4115:
4068:
4052:
4018:
4013:
4011:
3981:
3976:
3974:
3953:
3948:
3946:
3915:
3910:
3908:
3883:
3873:
3863:
3852:
3839:
3834:
3832:
3809:
3788:
3783:
3781:
3760:{\displaystyle \,_{n}p_{x}=l_{x+n}/l_{x}}
3751:
3742:
3730:
3717:
3707:
3706:
3703:
3674:
3654:
3633:
3623:
3622:
3619:
3596:
3587:
3581:
3571:
3569:
3559:
3549:
3548:
3545:
3516:
3496:
3475:
3465:
3464:
3461:
3433:
3413:
3386:
3373:
3348:
3323:
3310:
3297:
3287:
3286:
3283:
3251:
3238:
3233:
3231:
3202:
3182:
3158:
3149:
3137:
3124:
3119:
3117:
3088:
3068:
3047:
3042:
3040:
3017:
3008:
3002:
2989:
2984:
2982:
2953:
2933:
2912:
2907:
2905:
2875:
2845:
2826:
2820:
2792:
2786:
2759:
2718:
2712:
2684:
2678:
2651:
2621:
2608:
2595:
2589:
2567:
2561:
2540:
2503:
2497:
2470:
2464:
2442:
2436:
2415:
2384:
2371:
2358:
2353:
2351:
2330:
2325:
2323:
2297:
2277:
2256:
2251:
2249:
2222:
2220:
2199:
2194:
2192:
2172:
2145:
2140:
2138:
2117:
2112:
2110:
2088:
2067:
2062:
2060:
2013:
1994:
1957:
1938:
1927:
1925:
1895:
1867:
1845:
1839:
1818:
1805:
1783:
1777:
1746:
1744:
1722:
1720:
1701:
1699:
1680:
1678:
1641:
1625:
1614:
1612:
1589:
1566:
1564:
1542:
1515:
1510:
1508:
1489:
1487:
1460:
1455:
1453:
1432:
1430:
1409:
1392:
1390:
1388:
1367:
1350:
1348:
1346:
1327:
1325:
1300:
1283:
1257:
1255:
1236:
1234:
1210:
1179:
1171:
1145:
1143:
1120:
1109:
1107:
1082:
1080:
1055:
1027:
1010:
1002:
1000:
974:
972:
931:
916:
903:
901:
868:
863:
861:
836:
834:
787:
782:
780:
747:
742:
740:
711:
699:
694:
692:
665:
654:
652:
629:
617:
612:
610:
583:
578:
576:
556:
536:
516:
481:
476:
474:
447:
445:
418:
416:
331:
320:
299:
294:
288:
264:
263:
257:
239:
234:
221:
220:
205:
193:
170:
147:
121:
98:
78:
52:
32:
5919:is the number of payments per year, and
5631:{\displaystyle a_{65:{\overline {10|}}}}
4456:payments). This value is obtained from:
4236:payments). This value is obtained from:
2407:
18:
4640:{\displaystyle \,s_{{\overline {n|}}i}}
6660:International Actuarial Notation suite
6512:tend to zero, we get the function for
1425:for the future (or accumulated) value
1312:{\displaystyle \,(1-d)=v={(1+i)}^{-1}}
6654:Journal of the Institute of Actuaries
6247:and other social scientists call the
6128:{\displaystyle \,{\overline {A}}_{x}}
5708:{\displaystyle a_{\overline {65:64}}}
4197:{\displaystyle a_{{\overline {n|}}i}}
3167:{\displaystyle \,p_{x}=l_{x+1}/l_{x}}
2399:{\displaystyle \,d_{x}=l_{x}-l_{x+1}}
7:
6275:that a person who has attained age (
2346:may be calculated using the formula
5582:{\displaystyle a_{\overline {10|}}}
3649:is the probability that a life age
3063:is the probability that a life age
3026:{\displaystyle \,q_{x}=d_{x}/l_{x}}
2636:{\displaystyle d_{x}=l_{x}-l_{x+1}}
1383:. This is analogous to the formula
992:of 1 to be paid one year from now:
680:{\displaystyle \,(1.0583)^{2}=1.12}
6398:
6350:
6306:
3864:
1632:
728:{\displaystyle \,i^{(2)}/2=0.0583}
358:implying double force of interest.
14:
1673:The general relationship between
889:{\displaystyle \,i^{(12)}=0.1139}
6494:cumulative distribution function
5786:{\displaystyle {\ddot {a}}_{65}}
1129:{\displaystyle \,25\times v^{5}}
768:{\displaystyle \,i^{(2)}=0.1166}
6496:of the continuous age-at-death
4770:years, and each payment is one
3266:{\displaystyle \,p_{x}+q_{x}=1}
2235:{\displaystyle \,n\geq \omega }
775:. The appearing in the symbol
368:is a shorthand method to allow
6599:
6593:
6572:
6566:
6544:
6538:
6460:
6457:
6451:
6432:
6427:
6421:
6405:
6389:
6370:
6329:
6320:
6314:
6091:{\displaystyle \,A_{x}^{(12)}}
6083:
6077:
6001:
5995:
5844:
5838:
5823:
5740:
5734:
5616:
5568:
5401:
5372:
5366:
5351:
5312:
5285:
5279:
5264:
5234:
5156:
5088:
5082:
5067:
5022:
4977:
4971:
4938:
4932:
4917:
4872:
4866:
4833:
4827:
4812:
4717:
4711:
4675:
4622:
4487:
4403:
4259:
4179:
4045:
4033:
2020:
2014:
2001:
1995:
1964:
1958:
1945:
1939:
1892:
1879:
1852:
1846:
1790:
1784:
1759:
1747:
1648:
1642:
1629:
1537:. Discount is converted on an
1522:
1516:
1503:times a year, is analogous to
1467:
1461:
1405:
1393:
1363:
1351:
1296:
1284:
1270:
1258:
1161:annual effective discount rate
1023:
1011:
875:
869:
794:
788:
754:
748:
706:
700:
662:
655:
624:
618:
590:
584:
488:
482:
387:, where symbols are placed as
343:
324:
254:
227:
211:
198:
129:
123:
1:
5879:is the age of the annuitant,
5748:{\displaystyle a_{65}^{(12)}}
4750:th of a year for a period of
2807:{\displaystyle l_{\omega -1}}
1418:{\displaystyle \,{(1+i)}^{n}}
1376:{\displaystyle \,{(1-d)}^{n}}
23:Example of actuarial symbol.
6114:
5828:
5699:
5621:
5573:
5406:
5356:
5317:
5299:
5269:
5239:
5161:
5143:
5072:
5027:
5009:
4922:
4817:
4700:the value one period later.
4680:
4627:
4492:
4408:
4264:
4184:
823:monetary financial liability
383:Traditional notation uses a
3642:{\displaystyle \,_{n}p_{x}}
3484:{\displaystyle \,_{n}q_{x}}
1095:{\displaystyle \,1\times v}
640:{\displaystyle \,i^{(2)}/2}
6706:
2133:is the starting point for
1666:In this case, interest is
825:instruments, whereas home
5666:{\displaystyle a_{65:64}}
5047:is the limiting value of
4154:'s death, it is called a
3996:{\displaystyle \,l_{x+1}}
3930:{\displaystyle \,l_{x+t}}
2773:{\displaystyle \omega -1}
1708:{\displaystyle \,\delta }
1604:increases without bound:
1573:{\displaystyle \,\delta }
1530:{\displaystyle \,i^{(m)}}
1475:{\displaystyle \,d^{(m)}}
802:{\displaystyle \,i^{(m)}}
598:{\displaystyle \,i^{(2)}}
496:{\displaystyle \,i^{(m)}}
6652:1949 description in the
6279:) will die between age (
6009:{\displaystyle A^{(12)}}
5541:{\displaystyle \,a_{65}}
1668:convertible continuously
849:{\displaystyle \,i=0.12}
571:of a year. For example,
460:{\displaystyle \,i=0.12}
6625:Actuarial present value
6273:conditional probability
6050:{\displaystyle \,A_{x}}
5954:The basic symbol for a
5508:actuarial present value
5486:{\displaystyle \delta }
3962:{\displaystyle \,l_{x}}
3797:{\displaystyle \,e_{x}}
3056:{\displaystyle \,p_{x}}
2921:{\displaystyle \,q_{x}}
2883:{\displaystyle \omega }
2733:{\displaystyle d_{x+1}}
2699:{\displaystyle l_{x+1}}
2339:{\displaystyle \,d_{x}}
2265:{\displaystyle \,d_{x}}
2208:{\displaystyle \,l_{n}}
2180:{\displaystyle \omega }
2154:{\displaystyle \,l_{x}}
2126:{\displaystyle \,l_{0}}
2076:{\displaystyle \,l_{x}}
1341:years in the future is
819:fixed income securities
434:effective interest rate
6640:Mathematics of finance
6635:Annual percentage rate
6609:
6470:
6225:
6224:{\displaystyle \,\pi }
6204:
6183:
6182:{\displaystyle \,\pi }
6162:
6129:
6092:
6051:
6010:
5973:
5939:is the interest rate.
5933:
5913:
5893:
5873:
5853:
5787:
5749:
5709:
5667:
5632:
5583:
5542:
5487:
5467:
5447:
5420:
5203:
5117:
5097:
5041:
4988:
4883:
4784:
4764:
4744:
4724:
4694:
4641:
4594:
4571:
4446:
4422:
4367:
4344:
4226:
4198:
4125:
4103:
4084:
3997:
3963:
3931:
3893:
3868:
3818:
3798:
3761:
3689:
3663:
3643:
3606:
3531:
3505:
3485:
3448:
3422:
3402:
3267:
3217:
3191:
3168:
3103:
3077:
3057:
3027:
2968:
2942:
2922:
2884:
2861:
2808:
2774:
2734:
2700:
2666:
2637:
2577:
2549:
2513:
2480:
2452:
2424:
2400:
2340:
2312:
2286:
2266:
2236:
2209:
2181:
2155:
2127:
2097:
2077:
2035:
1908:
1730:
1709:
1688:
1657:
1598:
1574:
1551:
1531:
1497:
1476:
1440:
1419:
1377:
1335:
1313:
1244:
1220:
1153:
1130:
1096:
1065:
982:
947:
890:
850:
803:
769:
729:
681:
641:
599:
565:
545:
525:
497:
461:
426:
362:
352:
277:
179:
156:
136:
107:
93:-year-old person, for
87:
61:
41:
6680:Mathematical notation
6610:
6471:
6226:
6205:
6184:
6163:
6143:The basic symbol for
6130:
6093:
6052:
6011:
5974:
5934:
5914:
5894:
5874:
5854:
5788:
5750:
5710:
5668:
5633:
5584:
5543:
5488:
5468:
5448:
5421:
5204:
5118:
5098:
5042:
4989:
4884:
4785:
4765:
4745:
4725:
4723:{\displaystyle \,(m)}
4695:
4642:
4595:
4572:
4447:
4423:
4368:
4345:
4227:
4199:
4126:
4101:
4085:
3998:
3964:
3932:
3894:
3848:
3819:
3799:
3762:
3690:
3664:
3644:
3607:
3532:
3506:
3486:
3449:
3423:
3403:
3268:
3218:
3192:
3169:
3104:
3078:
3058:
3028:
2969:
2943:
2923:
2885:
2862:
2809:
2775:
2735:
2701:
2667:
2638:
2578:
2576:{\displaystyle l_{x}}
2550:
2514:
2512:{\displaystyle l_{0}}
2481:
2479:{\displaystyle d_{x}}
2453:
2451:{\displaystyle l_{x}}
2425:
2401:
2341:
2313:
2287:
2267:
2237:
2210:
2182:
2156:
2128:
2098:
2078:
2036:
1909:
1731:
1710:
1689:
1658:
1599:
1575:
1552:
1532:
1498:
1477:
1441:
1420:
1378:
1336:
1314:
1245:
1221:
1154:
1131:
1097:
1066:
983:
948:
891:
851:
804:
770:
730:
682:
642:
600:
566:
546:
526:
509:nominal interest rate
498:
462:
427:
353:
278:
180:
157:
137:
108:
88:
62:
42:
22:
6531:
6298:
6214:
6193:
6172:
6151:
6104:
6063:
6033:
5987:
5962:
5923:
5903:
5883:
5863:
5802:
5761:
5721:
5679:
5644:
5595:
5554:
5524:
5477:
5457:
5437:
5219:
5134:
5107:
5051:
5000:
4893:
4797:
4774:
4754:
4734:
4707:
4651:
4606:
4584:
4463:
4436:
4379:
4357:
4243:
4216:
4164:
4114:
4010:
3973:
3945:
3939:linear interpolation
3907:
3831:
3808:
3780:
3702:
3673:
3669:will survive to age
3653:
3618:
3544:
3515:
3495:
3460:
3432:
3412:
3282:
3230:
3201:
3181:
3116:
3087:
3083:will survive to age
3067:
3039:
2981:
2952:
2932:
2904:
2874:
2819:
2785:
2758:
2711:
2677:
2650:
2588:
2560:
2539:
2496:
2463:
2435:
2414:
2350:
2322:
2296:
2276:
2248:
2219:
2191:
2171:
2137:
2109:
2087:
2059:
1924:
1743:
1719:
1698:
1677:
1611:
1588:
1563:
1541:
1507:
1486:
1452:
1429:
1387:
1345:
1324:
1254:
1233:
1170:
1142:
1106:
1079:
999:
971:
900:
860:
833:
779:
739:
691:
651:
609:
575:
555:
535:
515:
473:
444:
415:
287:
192:
169:
146:
120:
97:
77:
51:
31:
6675:Applied mathematics
6565:
6203:{\displaystyle \,P}
6161:{\displaystyle \,P}
6087:
5972:{\displaystyle \,A}
5848:
5744:
5376:
5289:
5092:
4942:
4837:
4124:{\displaystyle \,a}
3688:{\displaystyle x+n}
3530:{\displaystyle x+n}
3447:{\displaystyle x+n}
3216:{\displaystyle x+1}
3102:{\displaystyle x+1}
2967:{\displaystyle x+1}
2665:{\displaystyle x+1}
2311:{\displaystyle x+1}
1729:{\displaystyle \,d}
1687:{\displaystyle \,i}
1496:{\displaystyle \,m}
1439:{\displaystyle \,n}
1334:{\displaystyle \,n}
1243:{\displaystyle \,d}
1152:{\displaystyle \,d}
981:{\displaystyle \,v}
425:{\displaystyle \,i}
399:has yet to emerge.
135:{\displaystyle (x)}
6605:
6553:
6514:force of mortality
6466:
6241:force of mortality
6235:Force of mortality
6221:
6200:
6179:
6158:
6125:
6088:
6067:
6047:
6006:
5969:
5929:
5909:
5889:
5869:
5849:
5805:
5783:
5745:
5724:
5705:
5663:
5628:
5579:
5538:
5483:
5463:
5443:
5416:
5330:
5252:
5199:
5125:continuous annuity
5113:
5093:
5055:
5037:
4984:
4896:
4879:
4800:
4780:
4760:
4740:
4720:
4690:
4637:
4590:
4567:
4442:
4418:
4363:
4340:
4222:
4194:
4121:
4104:
4080:
3993:
3959:
3927:
3889:
3814:
3794:
3757:
3685:
3659:
3639:
3602:
3527:
3501:
3481:
3444:
3418:
3398:
3263:
3213:
3187:
3164:
3099:
3073:
3053:
3023:
2964:
2938:
2918:
2880:
2857:
2804:
2770:
2730:
2696:
2662:
2633:
2573:
2545:
2509:
2476:
2448:
2420:
2396:
2336:
2308:
2282:
2262:
2232:
2205:
2177:
2151:
2123:
2093:
2073:
2031:
1904:
1726:
1705:
1684:
1653:
1636:
1594:
1570:
1547:
1527:
1493:
1472:
1436:
1415:
1373:
1331:
1309:
1240:
1216:
1149:
1126:
1092:
1061:
978:
943:
886:
846:
799:
765:
725:
677:
637:
595:
561:
541:
521:
493:
457:
422:
366:Actuarial notation
363:
348:
273:
175:
152:
132:
103:
83:
57:
37:
6685:Actuarial science
6630:Actuarial science
6603:
6464:
6239:Among actuaries,
6117:
5932:{\displaystyle i}
5912:{\displaystyle m}
5892:{\displaystyle n}
5872:{\displaystyle x}
5831:
5774:
5702:
5624:
5576:
5466:{\displaystyle d}
5446:{\displaystyle i}
5409:
5390:
5359:
5340:
5320:
5302:
5272:
5242:
5197:
5164:
5146:
5116:{\displaystyle m}
5075:
5030:
5012:
4982:
4925:
4906:
4877:
4820:
4783:{\displaystyle m}
4763:{\displaystyle n}
4743:{\displaystyle m}
4683:
4664:
4630:
4593:{\displaystyle d}
4565:
4495:
4476:
4445:{\displaystyle n}
4432:of each year for
4411:
4392:
4366:{\displaystyle i}
4338:
4267:
4225:{\displaystyle n}
4212:of each year for
4187:
3872:
3817:{\displaystyle x}
3662:{\displaystyle x}
3504:{\displaystyle x}
3421:{\displaystyle x}
3190:{\displaystyle x}
3076:{\displaystyle x}
2941:{\displaystyle x}
2899:
2898:
2548:{\displaystyle x}
2423:{\displaystyle x}
2285:{\displaystyle x}
2096:{\displaystyle x}
1860:
1798:
1621:
1597:{\displaystyle m}
1582:force of interest
1550:{\displaystyle m}
1195:
958:compound interest
924:
564:{\displaystyle m}
544:{\displaystyle m}
524:{\displaystyle m}
178:{\displaystyle m}
155:{\displaystyle n}
106:{\displaystyle n}
86:{\displaystyle x}
60:{\displaystyle a}
40:{\displaystyle A}
6697:
6614:
6612:
6611:
6606:
6604:
6602:
6592:
6591:
6575:
6561:
6551:
6475:
6473:
6472:
6467:
6465:
6463:
6450:
6449:
6430:
6420:
6419:
6388:
6387:
6377:
6313:
6312:
6230:
6228:
6227:
6222:
6209:
6207:
6206:
6201:
6188:
6186:
6185:
6180:
6167:
6165:
6164:
6159:
6134:
6132:
6131:
6126:
6124:
6123:
6118:
6110:
6097:
6095:
6094:
6089:
6086:
6075:
6056:
6054:
6053:
6048:
6046:
6045:
6015:
6013:
6012:
6007:
6005:
6004:
5978:
5976:
5975:
5970:
5938:
5936:
5935:
5930:
5918:
5916:
5915:
5910:
5898:
5896:
5895:
5890:
5878:
5876:
5875:
5870:
5858:
5856:
5855:
5850:
5847:
5836:
5832:
5827:
5826:
5817:
5792:
5790:
5789:
5784:
5782:
5781:
5776:
5775:
5767:
5754:
5752:
5751:
5746:
5743:
5732:
5714:
5712:
5711:
5706:
5704:
5703:
5698:
5687:
5672:
5670:
5669:
5664:
5662:
5661:
5637:
5635:
5634:
5629:
5627:
5626:
5625:
5620:
5619:
5610:
5588:
5586:
5585:
5580:
5578:
5577:
5572:
5571:
5562:
5547:
5545:
5544:
5539:
5537:
5536:
5492:
5490:
5489:
5484:
5472:
5470:
5469:
5464:
5452:
5450:
5449:
5444:
5425:
5423:
5422:
5417:
5415:
5414:
5410:
5405:
5404:
5395:
5392:
5391:
5383:
5375:
5364:
5360:
5355:
5354:
5345:
5342:
5341:
5333:
5326:
5325:
5321:
5316:
5315:
5306:
5303:
5295:
5288:
5277:
5273:
5268:
5267:
5258:
5248:
5247:
5243:
5238:
5237:
5228:
5208:
5206:
5205:
5200:
5198:
5193:
5192:
5191:
5175:
5170:
5169:
5165:
5160:
5159:
5150:
5147:
5139:
5122:
5120:
5119:
5114:
5102:
5100:
5099:
5094:
5091:
5080:
5076:
5071:
5070:
5061:
5046:
5044:
5043:
5038:
5036:
5035:
5031:
5026:
5025:
5016:
5013:
5005:
4993:
4991:
4990:
4985:
4983:
4981:
4980:
4965:
4964:
4963:
4947:
4941:
4930:
4926:
4921:
4920:
4911:
4908:
4907:
4899:
4888:
4886:
4885:
4880:
4878:
4876:
4875:
4860:
4859:
4858:
4842:
4836:
4825:
4821:
4816:
4815:
4806:
4789:
4787:
4786:
4781:
4769:
4767:
4766:
4761:
4749:
4747:
4746:
4741:
4729:
4727:
4726:
4721:
4699:
4697:
4696:
4691:
4689:
4688:
4684:
4679:
4678:
4669:
4666:
4665:
4657:
4646:
4644:
4643:
4638:
4636:
4635:
4631:
4626:
4625:
4616:
4599:
4597:
4596:
4591:
4576:
4574:
4573:
4568:
4566:
4561:
4560:
4559:
4543:
4538:
4537:
4501:
4500:
4496:
4491:
4490:
4481:
4478:
4477:
4469:
4451:
4449:
4448:
4443:
4427:
4425:
4424:
4419:
4417:
4416:
4412:
4407:
4406:
4397:
4394:
4393:
4385:
4372:
4370:
4369:
4364:
4349:
4347:
4346:
4341:
4339:
4334:
4333:
4332:
4316:
4311:
4310:
4292:
4291:
4273:
4272:
4268:
4263:
4262:
4253:
4231:
4229:
4228:
4223:
4203:
4201:
4200:
4195:
4193:
4192:
4188:
4183:
4182:
4173:
4130:
4128:
4127:
4122:
4089:
4087:
4086:
4081:
4079:
4078:
4057:
4056:
4029:
4028:
4002:
4000:
3999:
3994:
3992:
3991:
3968:
3966:
3965:
3960:
3958:
3957:
3936:
3934:
3933:
3928:
3926:
3925:
3898:
3896:
3895:
3890:
3888:
3887:
3878:
3877:
3870:
3867:
3862:
3844:
3843:
3823:
3821:
3820:
3815:
3803:
3801:
3800:
3795:
3793:
3792:
3766:
3764:
3763:
3758:
3756:
3755:
3746:
3741:
3740:
3722:
3721:
3712:
3711:
3694:
3692:
3691:
3686:
3668:
3666:
3665:
3660:
3648:
3646:
3645:
3640:
3638:
3637:
3628:
3627:
3611:
3609:
3608:
3603:
3601:
3600:
3591:
3586:
3585:
3576:
3575:
3570:
3564:
3563:
3554:
3553:
3536:
3534:
3533:
3528:
3510:
3508:
3507:
3502:
3490:
3488:
3487:
3482:
3480:
3479:
3470:
3469:
3453:
3451:
3450:
3445:
3427:
3425:
3424:
3419:
3407:
3405:
3404:
3399:
3397:
3396:
3378:
3377:
3365:
3364:
3334:
3333:
3315:
3314:
3302:
3301:
3292:
3291:
3272:
3270:
3269:
3264:
3256:
3255:
3243:
3242:
3222:
3220:
3219:
3214:
3196:
3194:
3193:
3188:
3173:
3171:
3170:
3165:
3163:
3162:
3153:
3148:
3147:
3129:
3128:
3108:
3106:
3105:
3100:
3082:
3080:
3079:
3074:
3062:
3060:
3059:
3054:
3052:
3051:
3032:
3030:
3029:
3024:
3022:
3021:
3012:
3007:
3006:
2994:
2993:
2973:
2971:
2970:
2965:
2947:
2945:
2944:
2939:
2927:
2925:
2924:
2919:
2917:
2916:
2889:
2887:
2886:
2881:
2866:
2864:
2863:
2858:
2856:
2855:
2837:
2836:
2813:
2811:
2810:
2805:
2803:
2802:
2779:
2777:
2776:
2771:
2739:
2737:
2736:
2731:
2729:
2728:
2705:
2703:
2702:
2697:
2695:
2694:
2671:
2669:
2668:
2663:
2642:
2640:
2639:
2634:
2632:
2631:
2613:
2612:
2600:
2599:
2582:
2580:
2579:
2574:
2572:
2571:
2554:
2552:
2551:
2546:
2518:
2516:
2515:
2510:
2508:
2507:
2485:
2483:
2482:
2477:
2475:
2474:
2457:
2455:
2454:
2449:
2447:
2446:
2429:
2427:
2426:
2421:
2408:
2405:
2403:
2402:
2397:
2395:
2394:
2376:
2375:
2363:
2362:
2345:
2343:
2342:
2337:
2335:
2334:
2317:
2315:
2314:
2309:
2291:
2289:
2288:
2283:
2271:
2269:
2268:
2263:
2261:
2260:
2241:
2239:
2238:
2233:
2215:is zero for all
2214:
2212:
2211:
2206:
2204:
2203:
2186:
2184:
2183:
2178:
2160:
2158:
2157:
2152:
2150:
2149:
2132:
2130:
2129:
2124:
2122:
2121:
2102:
2100:
2099:
2094:
2082:
2080:
2079:
2074:
2072:
2071:
2040:
2038:
2037:
2032:
2024:
2023:
2005:
2004:
1968:
1967:
1949:
1948:
1913:
1911:
1910:
1905:
1903:
1902:
1875:
1874:
1866:
1862:
1861:
1856:
1855:
1840:
1823:
1822:
1810:
1809:
1804:
1800:
1799:
1794:
1793:
1778:
1735:
1733:
1732:
1727:
1714:
1712:
1711:
1706:
1693:
1691:
1690:
1685:
1662:
1660:
1659:
1654:
1652:
1651:
1635:
1603:
1601:
1600:
1595:
1579:
1577:
1576:
1571:
1556:
1554:
1553:
1548:
1536:
1534:
1533:
1528:
1526:
1525:
1502:
1500:
1499:
1494:
1481:
1479:
1478:
1473:
1471:
1470:
1445:
1443:
1442:
1437:
1424:
1422:
1421:
1416:
1414:
1413:
1408:
1382:
1380:
1379:
1374:
1372:
1371:
1366:
1340:
1338:
1337:
1332:
1318:
1316:
1315:
1310:
1308:
1307:
1299:
1249:
1247:
1246:
1241:
1225:
1223:
1222:
1217:
1215:
1214:
1196:
1194:
1180:
1158:
1156:
1155:
1150:
1135:
1133:
1132:
1127:
1125:
1124:
1101:
1099:
1098:
1093:
1070:
1068:
1067:
1062:
1060:
1059:
1035:
1034:
1026:
987:
985:
984:
979:
952:
950:
949:
944:
936:
935:
930:
926:
925:
917:
895:
893:
892:
887:
879:
878:
855:
853:
852:
847:
808:
806:
805:
800:
798:
797:
774:
772:
771:
766:
758:
757:
734:
732:
731:
726:
715:
710:
709:
686:
684:
683:
678:
670:
669:
646:
644:
643:
638:
633:
628:
627:
604:
602:
601:
596:
594:
593:
570:
568:
567:
562:
550:
548:
547:
542:
530:
528:
527:
522:
502:
500:
499:
494:
492:
491:
466:
464:
463:
458:
431:
429:
428:
423:
403:Example notation
357:
355:
354:
349:
347:
346:
336:
335:
312:
311:
304:
303:
282:
280:
279:
274:
269:
268:
262:
261:
252:
251:
244:
243:
226:
225:
210:
209:
184:
182:
181:
176:
161:
159:
158:
153:
141:
139:
138:
133:
112:
110:
109:
104:
92:
90:
89:
84:
66:
64:
63:
58:
46:
44:
43:
38:
6705:
6704:
6700:
6699:
6698:
6696:
6695:
6694:
6665:
6664:
6648:
6621:
6583:
6576:
6552:
6529:
6528:
6498:random variable
6487:
6441:
6431:
6411:
6379:
6378:
6301:
6296:
6295:
6270:
6243:refers to what
6237:
6212:
6211:
6191:
6190:
6170:
6169:
6149:
6148:
6141:
6108:
6102:
6101:
6061:
6060:
6037:
6031:
6030:
5990:
5985:
5984:
5960:
5959:
5952:
5921:
5920:
5901:
5900:
5881:
5880:
5861:
5860:
5818:
5800:
5799:
5796:or in general:
5764:
5759:
5758:
5719:
5718:
5688:
5682:
5677:
5676:
5647:
5642:
5641:
5611:
5598:
5593:
5592:
5563:
5557:
5552:
5551:
5528:
5522:
5521:
5510:of an annuity.
5504:
5475:
5474:
5455:
5454:
5435:
5434:
5396:
5380:
5346:
5307:
5293:
5259:
5229:
5222:
5217:
5216:
5183:
5176:
5151:
5137:
5132:
5131:
5105:
5104:
5062:
5049:
5048:
5017:
5003:
4998:
4997:
4966:
4955:
4948:
4912:
4891:
4890:
4861:
4850:
4843:
4807:
4795:
4794:
4772:
4771:
4752:
4751:
4732:
4731:
4705:
4704:
4670:
4654:
4649:
4648:
4617:
4610:
4604:
4603:
4582:
4581:
4551:
4544:
4523:
4482:
4466:
4461:
4460:
4434:
4433:
4398:
4382:
4377:
4376:
4355:
4354:
4324:
4317:
4302:
4283:
4254:
4247:
4241:
4240:
4214:
4213:
4174:
4167:
4162:
4161:
4148:annuity-certain
4112:
4111:
4096:
4064:
4048:
4014:
4008:
4007:
3977:
3971:
3970:
3949:
3943:
3942:
3911:
3905:
3904:
3879:
3869:
3835:
3829:
3828:
3806:
3805:
3784:
3778:
3777:
3772:life expectancy
3747:
3726:
3713:
3705:
3700:
3699:
3671:
3670:
3651:
3650:
3629:
3621:
3616:
3615:
3592:
3577:
3568:
3555:
3547:
3542:
3541:
3513:
3512:
3493:
3492:
3471:
3463:
3458:
3457:
3430:
3429:
3410:
3409:
3382:
3369:
3344:
3319:
3306:
3293:
3285:
3280:
3279:
3247:
3234:
3228:
3227:
3199:
3198:
3197:) to the next (
3179:
3178:
3154:
3133:
3120:
3114:
3113:
3085:
3084:
3065:
3064:
3043:
3037:
3036:
3013:
2998:
2985:
2979:
2978:
2950:
2949:
2930:
2929:
2908:
2902:
2901:
2872:
2871:
2841:
2822:
2817:
2816:
2788:
2783:
2782:
2756:
2755:
2714:
2709:
2708:
2680:
2675:
2674:
2648:
2647:
2617:
2604:
2591:
2586:
2585:
2563:
2558:
2557:
2537:
2536:
2499:
2494:
2493:
2466:
2461:
2460:
2438:
2433:
2432:
2412:
2411:
2380:
2367:
2354:
2348:
2347:
2326:
2320:
2319:
2294:
2293:
2274:
2273:
2252:
2246:
2245:
2217:
2216:
2195:
2189:
2188:
2169:
2168:
2141:
2135:
2134:
2113:
2107:
2106:
2085:
2084:
2063:
2057:
2056:
2047:
2009:
1990:
1953:
1934:
1922:
1921:
1891:
1841:
1832:
1828:
1827:
1814:
1779:
1770:
1766:
1765:
1741:
1740:
1717:
1716:
1696:
1695:
1675:
1674:
1637:
1609:
1608:
1586:
1585:
1561:
1560:
1539:
1538:
1511:
1505:
1504:
1484:
1483:
1456:
1450:
1449:
1427:
1426:
1391:
1385:
1384:
1349:
1343:
1342:
1322:
1321:
1282:
1252:
1251:
1231:
1230:
1206:
1184:
1168:
1167:
1140:
1139:
1116:
1104:
1103:
1077:
1076:
1051:
1009:
997:
996:
988:represents the
969:
968:
909:
905:
904:
898:
897:
864:
858:
857:
831:
830:
783:
777:
776:
743:
737:
736:
695:
689:
688:
661:
649:
648:
613:
607:
606:
579:
573:
572:
553:
552:
533:
532:
513:
512:
503:(pronounced "i
477:
471:
470:
442:
441:
413:
412:
410:
405:
361:
327:
316:
295:
290:
285:
284:
253:
235:
230:
201:
190:
189:
167:
166:
144:
143:
118:
117:
95:
94:
75:
74:
49:
48:
29:
28:
17:
12:
11:
5:
6703:
6701:
6693:
6692:
6687:
6682:
6677:
6667:
6666:
6663:
6662:
6657:
6647:
6646:External links
6644:
6643:
6642:
6637:
6632:
6627:
6620:
6617:
6616:
6615:
6601:
6598:
6595:
6590:
6586:
6582:
6579:
6574:
6571:
6568:
6564:
6560:
6556:
6549:
6546:
6543:
6540:
6536:
6483:
6477:
6476:
6462:
6459:
6456:
6453:
6448:
6444:
6440:
6437:
6434:
6429:
6426:
6423:
6418:
6414:
6410:
6407:
6404:
6400:
6397:
6394:
6391:
6386:
6382:
6375:
6372:
6369:
6366:
6363:
6359:
6356:
6352:
6349:
6346:
6343:
6340:
6337:
6334:
6331:
6328:
6325:
6322:
6319:
6316:
6311:
6308:
6304:
6287: + Δ
6266:
6236:
6233:
6220:
6199:
6178:
6157:
6140:
6137:
6122:
6116:
6113:
6085:
6082:
6079:
6074:
6070:
6044:
6040:
6025:
6024:
6020:
6017:
6003:
6000:
5997:
5993:
5968:
5956:life insurance
5951:
5950:Life insurance
5948:
5928:
5908:
5888:
5868:
5846:
5843:
5840:
5835:
5830:
5825:
5821:
5815:
5812:
5808:
5780:
5773:
5770:
5742:
5739:
5736:
5731:
5727:
5701:
5697:
5694:
5691:
5685:
5660:
5657:
5654:
5650:
5623:
5618:
5614:
5608:
5605:
5601:
5575:
5570:
5566:
5560:
5535:
5531:
5516:
5515:
5503:
5502:Life annuities
5500:
5499:
5498:
5494:
5482:
5462:
5442:
5433:The subscript
5427:
5426:
5413:
5408:
5403:
5399:
5389:
5386:
5379:
5374:
5371:
5368:
5363:
5358:
5353:
5349:
5339:
5336:
5329:
5324:
5319:
5314:
5310:
5301:
5298:
5292:
5287:
5284:
5281:
5276:
5271:
5266:
5262:
5255:
5251:
5246:
5241:
5236:
5232:
5225:
5210:
5209:
5196:
5190:
5186:
5182:
5179:
5173:
5168:
5163:
5158:
5154:
5145:
5142:
5112:
5090:
5087:
5084:
5079:
5074:
5069:
5065:
5058:
5034:
5029:
5024:
5020:
5011:
5008:
4995:
4994:
4979:
4976:
4973:
4969:
4962:
4958:
4954:
4951:
4945:
4940:
4937:
4934:
4929:
4924:
4919:
4915:
4905:
4902:
4874:
4871:
4868:
4864:
4857:
4853:
4849:
4846:
4840:
4835:
4832:
4829:
4824:
4819:
4814:
4810:
4803:
4790:th of a unit.
4779:
4759:
4739:
4719:
4716:
4713:
4703:If the symbol
4687:
4682:
4677:
4673:
4663:
4660:
4634:
4629:
4624:
4620:
4613:
4589:
4578:
4577:
4564:
4558:
4554:
4550:
4547:
4541:
4536:
4533:
4530:
4526:
4522:
4519:
4516:
4513:
4510:
4507:
4504:
4499:
4494:
4489:
4485:
4475:
4472:
4441:
4415:
4410:
4405:
4401:
4391:
4388:
4362:
4351:
4350:
4337:
4331:
4327:
4323:
4320:
4314:
4309:
4305:
4301:
4298:
4295:
4290:
4286:
4282:
4279:
4276:
4271:
4266:
4261:
4257:
4250:
4221:
4206:a-angle-n at i
4191:
4186:
4181:
4177:
4170:
4144:
4143:
4139:
4136:
4120:
4095:
4092:
4091:
4090:
4077:
4074:
4071:
4067:
4063:
4060:
4055:
4051:
4047:
4044:
4041:
4038:
4035:
4032:
4027:
4024:
4021:
4017:
3990:
3987:
3984:
3980:
3956:
3952:
3924:
3921:
3918:
3914:
3900:
3899:
3886:
3882:
3876:
3866:
3861:
3858:
3855:
3851:
3847:
3842:
3838:
3813:
3791:
3787:
3768:
3767:
3754:
3750:
3745:
3739:
3736:
3733:
3729:
3725:
3720:
3716:
3710:
3684:
3681:
3678:
3658:
3636:
3632:
3626:
3613:
3612:
3599:
3595:
3590:
3584:
3580:
3574:
3567:
3562:
3558:
3552:
3526:
3523:
3520:
3500:
3478:
3474:
3468:
3443:
3440:
3437:
3417:
3395:
3392:
3389:
3385:
3381:
3376:
3372:
3368:
3363:
3360:
3357:
3354:
3351:
3347:
3343:
3340:
3337:
3332:
3329:
3326:
3322:
3318:
3313:
3309:
3305:
3300:
3296:
3290:
3274:
3273:
3262:
3259:
3254:
3250:
3246:
3241:
3237:
3212:
3209:
3206:
3186:
3175:
3174:
3161:
3157:
3152:
3146:
3143:
3140:
3136:
3132:
3127:
3123:
3098:
3095:
3092:
3072:
3050:
3046:
3034:
3033:
3020:
3016:
3011:
3005:
3001:
2997:
2992:
2988:
2963:
2960:
2957:
2937:
2915:
2911:
2897:
2896:
2893:
2890:
2879:
2868:
2867:
2854:
2851:
2848:
2844:
2840:
2835:
2832:
2829:
2825:
2814:
2801:
2798:
2795:
2791:
2780:
2769:
2766:
2763:
2752:
2751:
2748:
2745:
2741:
2740:
2727:
2724:
2721:
2717:
2706:
2693:
2690:
2687:
2683:
2672:
2661:
2658:
2655:
2644:
2643:
2630:
2627:
2624:
2620:
2616:
2611:
2607:
2603:
2598:
2594:
2583:
2570:
2566:
2555:
2544:
2533:
2532:
2529:
2526:
2522:
2521:
2519:
2506:
2502:
2491:
2487:
2486:
2473:
2469:
2458:
2445:
2441:
2430:
2419:
2393:
2390:
2387:
2383:
2379:
2374:
2370:
2366:
2361:
2357:
2333:
2329:
2307:
2304:
2301:
2281:
2259:
2255:
2231:
2228:
2225:
2202:
2198:
2176:
2148:
2144:
2120:
2116:
2092:
2070:
2066:
2046:
2043:
2042:
2041:
2030:
2027:
2022:
2019:
2016:
2012:
2008:
2003:
2000:
1997:
1993:
1989:
1986:
1983:
1980:
1977:
1974:
1971:
1966:
1963:
1960:
1956:
1952:
1947:
1944:
1941:
1937:
1933:
1930:
1915:
1914:
1901:
1898:
1894:
1890:
1887:
1884:
1881:
1878:
1873:
1870:
1865:
1859:
1854:
1851:
1848:
1844:
1838:
1835:
1831:
1826:
1821:
1817:
1813:
1808:
1803:
1797:
1792:
1789:
1786:
1782:
1776:
1773:
1769:
1764:
1761:
1758:
1755:
1752:
1749:
1725:
1704:
1683:
1664:
1663:
1650:
1647:
1644:
1640:
1634:
1631:
1628:
1624:
1620:
1617:
1593:
1569:
1546:
1524:
1521:
1518:
1514:
1492:
1469:
1466:
1463:
1459:
1435:
1412:
1407:
1404:
1401:
1398:
1395:
1370:
1365:
1362:
1359:
1356:
1353:
1330:
1306:
1303:
1298:
1295:
1292:
1289:
1286:
1281:
1278:
1275:
1272:
1269:
1266:
1263:
1260:
1239:
1227:
1226:
1213:
1209:
1205:
1202:
1199:
1193:
1190:
1187:
1183:
1178:
1175:
1148:
1123:
1119:
1115:
1112:
1091:
1088:
1085:
1072:
1071:
1058:
1054:
1050:
1047:
1044:
1041:
1038:
1033:
1030:
1025:
1022:
1019:
1016:
1013:
1008:
1005:
977:
942:
939:
934:
929:
923:
920:
915:
912:
908:
885:
882:
877:
874:
871:
867:
845:
842:
839:
821:) and similar
796:
793:
790:
786:
764:
761:
756:
753:
750:
746:
724:
721:
718:
714:
708:
705:
702:
698:
676:
673:
668:
664:
660:
657:
636:
632:
626:
623:
620:
616:
592:
589:
586:
582:
560:
540:
520:
490:
487:
484:
480:
456:
453:
450:
432:is the annual
421:
409:
408:Interest rates
406:
404:
401:
374:interest rates
360:
359:
345:
342:
339:
334:
330:
326:
323:
319:
315:
310:
307:
302:
298:
293:
272:
267:
260:
256:
250:
247:
242:
238:
233:
229:
224:
219:
216:
213:
208:
204:
200:
197:
186:
174:
163:
151:
131:
128:
125:
114:
102:
82:
71:
68:
56:
36:
27:An upper-case
24:
15:
13:
10:
9:
6:
4:
3:
2:
6702:
6691:
6688:
6686:
6683:
6681:
6678:
6676:
6673:
6672:
6670:
6661:
6658:
6656:
6655:
6650:
6649:
6645:
6641:
6638:
6636:
6633:
6631:
6628:
6626:
6623:
6622:
6618:
6596:
6588:
6584:
6580:
6577:
6569:
6562:
6558:
6554:
6547:
6541:
6534:
6527:
6526:
6525:
6523:
6519:
6516:, denoted as
6515:
6511:
6508:. If we let Δ
6507:
6503:
6499:
6495:
6491:
6486:
6482:
6454:
6446:
6442:
6438:
6435:
6424:
6416:
6412:
6408:
6402:
6395:
6392:
6384:
6380:
6373:
6367:
6364:
6361:
6357:
6354:
6347:
6344:
6341:
6338:
6335:
6332:
6326:
6323:
6317:
6309:
6302:
6294:
6293:
6292:
6290:
6286:
6282:
6278:
6274:
6269:
6265:
6261:
6257:
6252:
6250:
6246:
6242:
6234:
6232:
6218:
6197:
6176:
6155:
6146:
6138:
6136:
6120:
6111:
6099:
6080:
6072:
6068:
6058:
6042:
6038:
6028:
6027:For example:
6021:
6018:
5998:
5991:
5982:
5981:
5980:
5966:
5957:
5949:
5947:
5944:
5940:
5926:
5906:
5886:
5866:
5841:
5833:
5819:
5813:
5810:
5806:
5797:
5794:
5778:
5771:
5768:
5756:
5737:
5729:
5725:
5716:
5695:
5692:
5689:
5683:
5674:
5658:
5655:
5652:
5648:
5639:
5612:
5606:
5603:
5599:
5590:
5564:
5558:
5549:
5533:
5529:
5519:
5518:For example:
5513:
5512:
5511:
5509:
5501:
5495:
5480:
5460:
5440:
5432:
5431:
5430:
5411:
5397:
5387:
5384:
5377:
5369:
5361:
5347:
5337:
5334:
5327:
5322:
5308:
5296:
5290:
5282:
5274:
5260:
5253:
5249:
5244:
5230:
5223:
5215:
5214:
5213:
5194:
5188:
5184:
5180:
5177:
5171:
5166:
5152:
5140:
5130:
5129:
5128:
5126:
5110:
5085:
5077:
5063:
5056:
5032:
5018:
5006:
4974:
4967:
4960:
4956:
4952:
4949:
4943:
4935:
4927:
4913:
4903:
4900:
4869:
4862:
4855:
4851:
4847:
4844:
4838:
4830:
4822:
4808:
4801:
4793:
4792:
4791:
4777:
4757:
4737:
4714:
4701:
4685:
4671:
4661:
4658:
4632:
4618:
4611:
4601:
4587:
4562:
4556:
4552:
4548:
4545:
4539:
4534:
4531:
4528:
4524:
4520:
4517:
4514:
4511:
4508:
4505:
4502:
4497:
4483:
4473:
4470:
4459:
4458:
4457:
4455:
4439:
4431:
4413:
4399:
4389:
4386:
4374:
4360:
4335:
4329:
4325:
4321:
4318:
4312:
4307:
4303:
4299:
4296:
4293:
4288:
4284:
4280:
4277:
4274:
4269:
4255:
4248:
4239:
4238:
4237:
4235:
4219:
4211:
4207:
4189:
4175:
4168:
4159:
4157:
4153:
4149:
4140:
4137:
4134:
4133:
4132:
4118:
4109:
4100:
4093:
4075:
4072:
4069:
4065:
4061:
4058:
4053:
4049:
4042:
4039:
4036:
4030:
4025:
4022:
4019:
4015:
4006:
4005:
4004:
3988:
3985:
3982:
3978:
3954:
3950:
3940:
3922:
3919:
3916:
3912:
3884:
3880:
3874:
3859:
3856:
3853:
3849:
3845:
3840:
3836:
3827:
3826:
3825:
3811:
3789:
3785:
3775:
3773:
3752:
3748:
3743:
3737:
3734:
3731:
3727:
3723:
3718:
3714:
3708:
3698:
3697:
3696:
3682:
3679:
3676:
3656:
3634:
3630:
3624:
3597:
3593:
3588:
3582:
3578:
3572:
3565:
3560:
3556:
3550:
3540:
3539:
3538:
3524:
3521:
3518:
3498:
3476:
3472:
3466:
3455:
3441:
3438:
3435:
3415:
3393:
3390:
3387:
3383:
3379:
3374:
3370:
3366:
3361:
3358:
3355:
3352:
3349:
3345:
3341:
3338:
3335:
3330:
3327:
3324:
3320:
3316:
3311:
3307:
3303:
3298:
3294:
3288:
3277:
3260:
3257:
3252:
3248:
3244:
3239:
3235:
3226:
3225:
3224:
3210:
3207:
3204:
3184:
3159:
3155:
3150:
3144:
3141:
3138:
3134:
3130:
3125:
3121:
3112:
3111:
3110:
3096:
3093:
3090:
3070:
3048:
3044:
3018:
3014:
3009:
3003:
2999:
2995:
2990:
2986:
2977:
2976:
2975:
2961:
2958:
2955:
2935:
2913:
2909:
2894:
2891:
2877:
2870:
2869:
2852:
2849:
2846:
2842:
2838:
2833:
2830:
2827:
2823:
2815:
2799:
2796:
2793:
2789:
2781:
2767:
2764:
2761:
2754:
2753:
2749:
2746:
2743:
2742:
2725:
2722:
2719:
2715:
2707:
2691:
2688:
2685:
2681:
2673:
2659:
2656:
2653:
2646:
2645:
2628:
2625:
2622:
2618:
2614:
2609:
2605:
2601:
2596:
2592:
2584:
2568:
2564:
2556:
2542:
2535:
2534:
2530:
2527:
2524:
2523:
2520:
2504:
2500:
2492:
2489:
2488:
2471:
2467:
2459:
2443:
2439:
2431:
2417:
2410:
2409:
2406:
2391:
2388:
2385:
2381:
2377:
2372:
2368:
2364:
2359:
2355:
2331:
2327:
2305:
2302:
2299:
2279:
2257:
2253:
2243:
2229:
2226:
2223:
2200:
2196:
2174:
2166:
2164:
2146:
2142:
2118:
2114:
2104:
2090:
2068:
2064:
2054:
2052:
2044:
2028:
2025:
2017:
2010:
2006:
1998:
1991:
1987:
1984:
1981:
1978:
1975:
1972:
1969:
1961:
1954:
1950:
1942:
1935:
1931:
1928:
1920:
1919:
1918:
1899:
1896:
1888:
1885:
1882:
1876:
1871:
1868:
1863:
1857:
1849:
1842:
1836:
1833:
1829:
1824:
1819:
1815:
1811:
1806:
1801:
1795:
1787:
1780:
1774:
1771:
1767:
1762:
1756:
1753:
1750:
1739:
1738:
1737:
1723:
1702:
1681:
1671:
1669:
1645:
1638:
1626:
1618:
1615:
1607:
1606:
1605:
1591:
1583:
1567:
1558:
1544:
1519:
1512:
1490:
1464:
1457:
1447:
1433:
1410:
1402:
1399:
1396:
1368:
1360:
1357:
1354:
1328:
1304:
1301:
1293:
1290:
1287:
1279:
1276:
1273:
1267:
1264:
1261:
1237:
1229:The value of
1211:
1207:
1203:
1200:
1197:
1191:
1188:
1185:
1181:
1176:
1173:
1166:
1165:
1164:
1162:
1146:
1137:
1121:
1117:
1113:
1110:
1089:
1086:
1083:
1056:
1052:
1048:
1045:
1042:
1039:
1036:
1031:
1028:
1020:
1017:
1014:
1006:
1003:
995:
994:
993:
991:
990:present value
975:
965:
963:
959:
954:
940:
937:
932:
927:
921:
918:
913:
910:
906:
883:
880:
872:
865:
843:
840:
837:
828:
824:
820:
816:
812:
791:
784:
762:
759:
751:
744:
722:
719:
716:
712:
703:
696:
674:
671:
666:
658:
634:
630:
621:
614:
587:
580:
558:
538:
518:
510:
506:
485:
478:
468:
454:
451:
448:
439:
435:
419:
407:
402:
400:
396:
394:
390:
386:
381:
379:
375:
371:
367:
340:
337:
332:
328:
321:
317:
313:
308:
305:
300:
296:
291:
270:
258:
248:
245:
240:
236:
231:
217:
214:
206:
202:
195:
187:
172:
164:
149:
126:
115:
100:
80:
72:
69:
54:
34:
26:
25:
21:
6653:
6521:
6517:
6513:
6509:
6505:
6501:
6489:
6484:
6480:
6478:
6288:
6284:
6280:
6276:
6267:
6263:
6259:
6255:
6253:
6240:
6238:
6142:
6100:
6059:
6029:
6026:
5953:
5945:
5941:
5798:
5795:
5757:
5717:
5675:
5640:
5591:
5550:
5520:
5517:
5505:
5428:
5211:
4996:
4702:
4602:
4579:
4453:
4429:
4375:
4352:
4233:
4209:
4205:
4160:
4156:life annuity
4145:
4105:
3901:
3776:
3769:
3614:
3456:
3278:
3275:
3176:
3035:
2900:
2244:
2167:
2162:
2105:
2055:
2048:
1916:
1672:
1665:
1559:
1448:
1228:
1138:
1073:
966:
955:
511:convertible
504:
469:
437:
411:
397:
382:
365:
364:
142:dies within
6283:) and age (
6249:hazard rate
4152:beneficiary
2045:Life tables
1557:-ly basis.
967:The symbol
809:is not an "
507:m") is the
389:superscript
385:halo system
378:life tables
6690:Demography
6669:Categories
6258:) and age
6245:economists
6023:indicated.
2051:life table
856:, we have
817:(see also
735:and hence
687:, we have
283:but often
165:deferred (
6581:−
6535:μ
6500:, X. As Δ
6492:) is the
6439:−
6409:−
6399:Δ
6358:∣
6351:Δ
6307:Δ
6219:π
6177:π
6115:¯
5829:¯
5772:¨
5700:¯
5622:¯
5574:¯
5481:δ
5407:¯
5388:¨
5357:¯
5338:¨
5318:¯
5300:¯
5270:¯
5240:¯
5195:δ
5181:−
5162:¯
5144:¯
5073:¯
5028:¯
5010:¯
4953:−
4923:¯
4904:¨
4848:−
4818:¯
4681:¯
4662:¨
4628:¯
4549:−
4532:−
4518:⋯
4493:¯
4474:¨
4430:beginning
4409:¯
4390:¨
4322:−
4297:⋯
4265:¯
4185:¯
4094:Annuities
4040:−
3865:∞
3850:∑
3380:−
3359:−
3339:⋯
2878:ω
2850:−
2847:ω
2831:−
2828:ω
2797:−
2794:ω
2765:−
2762:ω
2615:−
2378:−
2230:ω
2227:≥
2175:ω
1985:⋯
1979:δ
1973:⋯
1897:−
1886:−
1869:−
1837:−
1820:δ
1703:δ
1633:∞
1630:→
1616:δ
1568:δ
1358:−
1302:−
1265:−
1204:−
1198:≈
1114:×
1087:×
1043:−
1037:≈
1029:−
827:mortgages
393:subscript
370:actuaries
322:δ
6619:See also
6563:′
5859:, where
3941:between
3511:and age
3428:and age
2948:and age
2292:and age
811:exponent
116:paid if
6145:premium
6139:Premium
4108:annuity
4003:. i.e.
1159:is the
185:years).
6479:where
6291:) as:
4204:(read
3871:
1580:, the
919:0.1139
896:since
884:0.1139
763:0.1166
723:0.0583
659:1.0583
438:a year
162:years;
113:years;
5497:rate.
5103:when
3937:is a
2163:radix
962:debit
815:bonds
505:upper
6365:>
6342:<
6336:<
6168:or
5378:<
5328:<
5291:<
5250:<
3969:and
2750:...
2747:...
2744:...
2531:...
2528:...
2525:...
2026:>
2007:>
1988:>
1982:>
1976:>
1970:>
1951:>
1932:>
1736:is:
1715:and
941:1.12
844:0.12
675:1.12
455:0.12
376:and
73:for
6524:):
6147:is
5958:is
5473:or
4210:end
4110:is
1623:lim
391:or
6671::
6189:.
6081:12
5999:12
5779:65
5738:12
5730:65
5696:64
5690:65
5659:64
5653:65
5613:10
5604:65
5565:10
5534:65
5127:.
4889:,
4158:.
3774:.
3695:.
3537:.
3454:.
3109:.
2974:.
2895:0
2892:0
2490:0
2318:.
2242:.
2049:A
1694:,
1670:.
1163::
1136:.
1111:25
953:.
933:12
922:12
873:12
467:.
380:.
6600:)
6597:x
6594:(
6589:X
6585:F
6578:1
6573:)
6570:x
6567:(
6559:X
6555:F
6548:=
6545:)
6542:x
6539:(
6522:x
6520:(
6518:μ
6510:x
6506:x
6502:x
6490:x
6488:(
6485:X
6481:F
6461:)
6458:)
6455:x
6452:(
6447:X
6443:F
6436:1
6433:(
6428:)
6425:x
6422:(
6417:X
6413:F
6406:)
6403:x
6396:+
6393:x
6390:(
6385:X
6381:F
6374:=
6371:)
6368:x
6362:X
6355:x
6348:+
6345:x
6339:X
6333:x
6330:(
6327:P
6324:=
6321:)
6318:x
6315:(
6310:x
6303:P
6289:x
6285:x
6281:x
6277:x
6268:x
6264:q
6260:x
6256:x
6198:P
6156:P
6121:x
6112:A
6084:)
6078:(
6073:x
6069:A
6043:x
6039:A
6002:)
5996:(
5992:A
5967:A
5927:i
5907:m
5887:n
5867:x
5845:)
5842:m
5839:(
5834:i
5824:|
5820:n
5814::
5811:x
5807:a
5769:a
5741:)
5735:(
5726:a
5693::
5684:a
5656::
5649:a
5617:|
5607::
5600:a
5569:|
5559:a
5530:a
5461:d
5441:i
5412:i
5402:|
5398:n
5385:a
5373:)
5370:m
5367:(
5362:i
5352:|
5348:n
5335:a
5323:i
5313:|
5309:n
5297:a
5286:)
5283:m
5280:(
5275:i
5265:|
5261:n
5254:a
5245:i
5235:|
5231:n
5224:a
5189:n
5185:v
5178:1
5172:=
5167:i
5157:|
5153:n
5141:a
5111:m
5089:)
5086:m
5083:(
5078:i
5068:|
5064:n
5057:a
5033:i
5023:|
5019:n
5007:a
4978:)
4975:m
4972:(
4968:d
4961:n
4957:v
4950:1
4944:=
4939:)
4936:m
4933:(
4928:i
4918:|
4914:n
4901:a
4873:)
4870:m
4867:(
4863:i
4856:n
4852:v
4845:1
4839:=
4834:)
4831:m
4828:(
4823:i
4813:|
4809:n
4802:a
4778:m
4758:n
4738:m
4718:)
4715:m
4712:(
4686:i
4676:|
4672:n
4659:s
4633:i
4623:|
4619:n
4612:s
4588:d
4580:(
4563:d
4557:n
4553:v
4546:1
4540:=
4535:1
4529:n
4525:v
4521:+
4515:+
4512:v
4509:+
4506:1
4503:=
4498:i
4488:|
4484:n
4471:a
4454:n
4440:n
4414:i
4404:|
4400:n
4387:a
4361:i
4353:(
4336:i
4330:n
4326:v
4319:1
4313:=
4308:n
4304:v
4300:+
4294:+
4289:2
4285:v
4281:+
4278:v
4275:=
4270:i
4260:|
4256:n
4249:a
4234:n
4220:n
4190:i
4180:|
4176:n
4169:a
4119:a
4076:1
4073:+
4070:x
4066:l
4062:t
4059:+
4054:x
4050:l
4046:)
4043:t
4037:1
4034:(
4031:=
4026:t
4023:+
4020:x
4016:l
3989:1
3986:+
3983:x
3979:l
3955:x
3951:l
3923:t
3920:+
3917:x
3913:l
3885:x
3881:p
3875:t
3860:1
3857:=
3854:t
3846:=
3841:x
3837:e
3812:x
3790:x
3786:e
3753:x
3749:l
3744:/
3738:n
3735:+
3732:x
3728:l
3724:=
3719:x
3715:p
3709:n
3683:n
3680:+
3677:x
3657:x
3635:x
3631:p
3625:n
3598:x
3594:l
3589:/
3583:x
3579:d
3573:n
3566:=
3561:x
3557:q
3551:n
3525:n
3522:+
3519:x
3499:x
3477:x
3473:q
3467:n
3442:n
3439:+
3436:x
3416:x
3394:n
3391:+
3388:x
3384:l
3375:x
3371:l
3367:=
3362:1
3356:n
3353:+
3350:x
3346:d
3342:+
3336:+
3331:1
3328:+
3325:x
3321:d
3317:+
3312:x
3308:d
3304:=
3299:x
3295:d
3289:n
3261:1
3258:=
3253:x
3249:q
3245:+
3240:x
3236:p
3211:1
3208:+
3205:x
3185:x
3160:x
3156:l
3151:/
3145:1
3142:+
3139:x
3135:l
3131:=
3126:x
3122:p
3097:1
3094:+
3091:x
3071:x
3049:x
3045:p
3019:x
3015:l
3010:/
3004:x
3000:d
2996:=
2991:x
2987:q
2962:1
2959:+
2956:x
2936:x
2914:x
2910:q
2853:1
2843:l
2839:=
2834:1
2824:d
2800:1
2790:l
2768:1
2726:1
2723:+
2720:x
2716:d
2692:1
2689:+
2686:x
2682:l
2660:1
2657:+
2654:x
2629:1
2626:+
2623:x
2619:l
2610:x
2606:l
2602:=
2597:x
2593:d
2569:x
2565:l
2543:x
2505:0
2501:l
2472:x
2468:d
2444:x
2440:l
2418:x
2392:1
2389:+
2386:x
2382:l
2373:x
2369:l
2365:=
2360:x
2356:d
2332:x
2328:d
2306:1
2303:+
2300:x
2280:x
2258:x
2254:d
2224:n
2201:n
2197:l
2147:x
2143:l
2119:0
2115:l
2091:x
2069:x
2065:l
2029:d
2021:)
2018:2
2015:(
2011:d
2002:)
1999:3
1996:(
1992:d
1965:)
1962:3
1959:(
1955:i
1946:)
1943:2
1940:(
1936:i
1929:i
1900:1
1893:)
1889:d
1883:1
1880:(
1877:=
1872:m
1864:)
1858:m
1853:)
1850:m
1847:(
1843:d
1834:1
1830:(
1825:=
1816:e
1812:=
1807:m
1802:)
1796:m
1791:)
1788:m
1785:(
1781:i
1775:+
1772:1
1768:(
1763:=
1760:)
1757:i
1754:+
1751:1
1748:(
1724:d
1682:i
1649:)
1646:m
1643:(
1639:i
1627:m
1619:=
1592:m
1545:m
1523:)
1520:m
1517:(
1513:i
1491:m
1468:)
1465:m
1462:(
1458:d
1434:n
1411:n
1406:)
1403:i
1400:+
1397:1
1394:(
1369:n
1364:)
1361:d
1355:1
1352:(
1329:n
1305:1
1297:)
1294:i
1291:+
1288:1
1285:(
1280:=
1277:v
1274:=
1271:)
1268:d
1262:1
1259:(
1238:d
1212:2
1208:i
1201:i
1192:i
1189:+
1186:1
1182:i
1177:=
1174:d
1147:d
1122:5
1118:v
1090:v
1084:1
1057:2
1053:i
1049:+
1046:i
1040:1
1032:1
1024:)
1021:i
1018:+
1015:1
1012:(
1007:=
1004:v
976:v
938:=
928:)
914:+
911:1
907:(
881:=
876:)
870:(
866:i
841:=
838:i
795:)
792:m
789:(
785:i
760:=
755:)
752:2
749:(
745:i
720:=
717:2
713:/
707:)
704:2
701:(
697:i
672:=
667:2
663:)
656:(
635:2
631:/
625:)
622:2
619:(
615:i
591:)
588:2
585:(
581:i
559:m
539:m
519:m
489:)
486:m
483:(
479:i
452:=
449:i
420:i
344:)
341:1
338:+
333:x
329:k
325:(
318:e
314:=
309:1
306:+
301:x
297:k
292:v
271:,
266:)
259:2
255:)
249:1
246:+
241:x
237:k
232:v
228:(
223:(
218:E
215:=
212:)
207:2
203:Z
199:(
196:E
173:m
150:n
130:)
127:x
124:(
101:n
81:x
55:a
35:A
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