39:
352:
31:
514:), whereas previous writers had usually treated compound interest briefly in just one chapter in a mathematical textbook. Witt's book gave tables based on 10% (the maximum rate of interest allowable on loans) and other rates for different purposes, such as the valuation of property leases. Witt was a London mathematical practitioner and his book is notable for its clarity of expression, depth of insight, and accuracy of calculation, with 124 worked examples.
328:
70:
336:
570:
4230:
545:
In the 19th century, and possibly earlier, Persian merchants used a slightly modified linear Taylor approximation to the monthly payment formula that could be computed easily in their heads. In modern times, Albert
Einstein's supposed quote regarding compound interest rings true. "He who understands
287:
To help consumers compare retail financial products more fairly and easily, many countries require financial institutions to disclose the annual compound interest rate on deposits or advances on a comparable basis. The interest rate on an annual equivalent basis may be referred to variously in
3237:
Given a principal deposit and a recurring deposit, the total return of an investment can be calculated via the compound interest gained per unit of time. If required, the interest on additional non-recurring and recurring deposits can also be defined within the same formula (see below).
318:
and other terms. The effective annual rate is the total accumulated interest that would be payable up to the end of one year, divided by the principal sum. These rates are usually the annualised compound interest rate alongside charges other than interest, such as taxes and other fees.
408:
The interest on corporate bonds and government bonds is usually payable twice yearly. The amount of interest paid every six months is the disclosed interest rate divided by two and multiplied by the principal. The yearly compounded rate is higher than the disclosed
3907:
1762:
3196:
2563:
4126:
4017:
1231:
260:, where previously accumulated interest is not added to the principal amount of the current period. Compounded interest depends on the simple interest rate applied and the frequency at which the interest is compounded.
2041:
The interest on loans and mortgages that are amortizedâthat is, have a smooth monthly payment until the loan has been paid offâis often compounded monthly. The formula for payments is found from the following argument.
3585:
252:
accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower.
2868:
1995:
3727:
3014:
2142:
1315:
427:
is used to determine how to apply payments toward principal and interest. Interest generated on these loans is not added to the principal, but rather is paid off monthly as the payments are applied.
3470:
968:
885:
771:
1606:
3736:
1639:
3072:
2757:
2219:
2647:
272:
is the number of times per given unit of time the accumulated interest is capitalized, on a regular basis. The frequency could be yearly, half-yearly, quarterly, monthly, weekly, daily,
1421:
3063:
3911:
where C is each lump sum and k are non-monthly recurring deposits, respectively, and x and y are the differences in time between a new deposit and the total period t is modeling.
1893:
3382:
2453:
1520:
1071:
1475:
4244:
2796:
983:
When the number of compounding periods per year increases without limit, continuous compounding occurs, in which case the effective annual rate approaches an upper limit of
2605:
2460:
3384:
Adding all recurring deposits over the total period t, (i starts at 0 if deposits begin with the investment of principal; i starts at 1 if deposits begin the next month):
3227:
2409:
2363:
3629:
2955:
2894:
2017:
1118:
1350:
2676:
2930:
4022:
3921:
3325:
3303:
3281:
3259:
2696:
2383:
2310:
2288:
2266:
2244:
2068:
1098:
709:
689:
539:
279:
For example, monthly capitalization with interest expressed as an annual rate means that the compounding frequency is 12, with time periods measured in months.
1133:
4371:"Why Einstein Considered Compound Interest the Most Powerful Force in the Universe: Is the power of compound interest really the 8th Wonder of the World?"
232:
3918:
when the exact date and amount of each recurring deposit is not known, a formula that assumes a uniform recurring monthly deposit over the period, is:
4370:
1937:
1429:, the force of interest is a constant, and the accumulation function of compounding interest in terms of force of interest is a simple power of
902:
is used instead. The accumulation function shows what $ 1 grows to after any length of time. The accumulation function for compound interest is:
4274:
2959:
2073:
1245:
3636:
1359:
When the above formula is written in differential equation format, then the force of interest is simply the coefficient of amount of change:
905:
822:
4208:
714:
3479:
500:, stating that to find the number of years for an investment at compound interest to double, one should divide the interest rate into 72.
3387:
2147:
477:
2801:
2609:
510:, published in 1613, was a landmark in the history of compound interest. It was wholly devoted to the subject (previously called
225:
990:. Continuous compounding can be regarded as letting the compounding period become infinitesimally small, achieved by taking the
634:
1362:
17:
4197:"Interest Act, R.S.C., 1985, c. I-15, s. 6: Interest on Moneys Secured by Mortgage on Real Property or Hypothec on Immovables"
606:
4461:
4171:
3023:
1125:
3731:
If two or more types of deposits occur (either recurring or non-recurring), the compound value earned can be represented as
1100:
tends to infinity in continuous compounding, the continuous compound interest rate is referred to as the force of interest
1541:
4388:
1771:
613:
1480:
1018:
587:
521:
442:
are valued at ever-increasing frequency, until the limit is approached and the derivative is valued in continuous time.
218:
50:
1436:
2701:
653:
168:
4437:
3334:
2572:
620:
2036:
434:, to use continuous compounding. Continuous compounding in pricing these instruments is a natural consequence of
4204:
1525:
591:
602:
42:
The effect of earning 20% annual interest on an initial $ 1,000 investment and various compounding frequencies
4503:
2934:
3902:{\displaystyle {\text{Value}}=M{\frac {(1+r)^{t}-1}{r}}+P(1+r)^{t}+k{\frac {(1+r)^{t-x}-1}{r}}+C(1+r)^{t-y}}
2336:
A formula that is accurate to within a few percent can be found by noting that for typical U.S. note rates (
4166:
1757:{\displaystyle \left(1+{\frac {r_{1}}{n_{1}}}\right)^{n_{1}}=\left(1+{\frac {r_{2}}{n_{2}}}\right)^{n_{2}}}
795:
473:
160:
2909:
3191:{\displaystyle c\approx c_{0}\left(1+X+{\frac {1}{3}}X^{2}\right)=\$ 333.33(1+.675+.675^{2}/3)=\$ 608.96}
4508:
3588:
2414:
151:
4474:
60:
2904:
For a $ 120,000 mortgage with a term of 30 years and a note rate of 4.5%, payable monthly, we find:
2765:
486:
of about 1340. It gives the interest on 100 lire, for rates from 1% to 8%, for up to 20 years. The
206:
4288:
Lewin, C G (1970). "An Early Book on
Compound Interest - Richard Witt's Arithmeticall Questions".
4413:
3203:
580:
482:
314:
302:
4196:
2388:
1236:
627:
296:
290:
273:
2339:
809:
is the overall length of time the interest is applied (expressed using the same time units as
4498:
3594:
2873:
2002:
1634:
To convert an interest rate from one compounding basis to another compounding basis, so that
1121:
1103:
899:
424:
308:
38:
1320:
4136:
2654:
1629:
488:
439:
431:
89:
8:
1524:
The force of interest is less than the annual effective interest rate, but more than the
991:
555:
110:
511:
4268:
4141:
3310:
3288:
3266:
3244:
2681:
2368:
2295:
2273:
2251:
2229:
2053:
1083:
978:
694:
674:
524:
126:
115:
2558:{\displaystyle c\approx {\frac {Pr}{1-e^{-nr}}}={\frac {P}{n}}{\frac {nr}{1-e^{-nr}}}}
817:
The total compound interest generated is the final value minus the initial principal:
351:
4493:
130:
84:
4355:
4351:
4324:
4297:
4239:
3473:
1353:
1006:
periods of continuous compounding can be expressed in terms of the initial amount
4146:
517:
457:
Compound interest when charged by lenders was once regarded as the worst kind of
420:
435:
4161:
3915:
898:
is simply a coefficient, it is often dropped for simplicity, and the resulting
559:
257:
189:
4328:
4301:
416:
are generally compounded semi-annually with monthly or more frequent payments.
30:
4487:
4234:
4156:
413:
4121:{\displaystyle r=\left({\frac {P'-\sum {M}/2}{P+\sum {M}/2}}\right)^{1/t}-1}
2698:
installments. In terms of these variables the approximation can be written
503:
493:
452:
4012:{\displaystyle r=\left({\frac {P'-P-\sum {M}}{P+\sum {M}/2}}\right)^{1/t}}
4176:
1226:{\displaystyle \delta _{t}={\frac {a'(t)}{a(t)}}={\frac {d}{dt}}\ln a(t)}
430:
It is sometimes mathematically simpler, for example, in the valuation of
184:
4342:
Milanfar, Peyman (1996). "A Persian Folk Method of
Figuring Interest".
3229:
so the approximation is an overestimate of about a sixth of a percent.
594: in this section. Unsourced material may be challenged and removed.
497:
466:
2678:
is the monthly payment required for a zeroâinterest loan paid off in
1538:
A way of modeling the force of inflation is with
Stoodley's formula:
1529:
462:
331:
Compound interest of 15% on initial $ 10,000 investment over 40 years
327:
120:
3580:{\displaystyle M'=M\sum _{i=0}^{t-1}(1+r)^{t}{\frac {1}{(1+r)^{i}}}}
569:
69:
4475:
http://moneychimp.com/features/portfolio_performance_calculator.htm
4315:
Lewin, C G (1981). "Compound
Interest in the Seventeenth Century".
4233: This article incorporates text from a publication now in the
4151:
2328:
PMT(interest_rate, number_payments, present_value, future_value, )
999:
340:
335:
249:
4477:"recommended by The Four Pillars of Investing and The Motley Fool"
2863:{\textstyle c\approx c_{0}\left(1+X+{\frac {X^{2}}{3}}\right)}
2030:
2385:=10â30 years), the monthly note rate is small compared to 1.
1990:{\displaystyle \delta =n\ln {\left(1+{\frac {r}{n}}\right)},}
458:
2019:
is the interest rate on a continuous compounding basis, and
4245:
CyclopĂŚdia, or an
Universal Dictionary of Arts and Sciences
1425:
For compound interest with a constant annual interest rate
2331:
3009:{\displaystyle c_{0}={\frac {\$ 120,000}{360}}=\$ 333.33}
2023:
is the stated interest rate with a compounding frequency
671:
The total accumulated value, including the principal sum
4261:
Francesco
Balducci Pegolotti, La Pratica della Mercatura
2137:{\displaystyle c={\frac {rP}{1-{\frac {1}{(1+r)^{n}}}}}}
1310:{\displaystyle a(t)=e^{\int _{0}^{t}\delta _{s}\,ds}\,,}
542:
in 1683 by studying a question about compound interest.
4462:"Using Compound Interest to Optimize Investment Spread"
3722:{\displaystyle P'=M{\frac {(1+r)^{t}-1}{r}}+P(1+r)^{t}}
963:{\displaystyle a(t)=\left(1+{\frac {r}{n}}\right)^{tn}}
880:{\displaystyle I=P\left(1+{\frac {r}{n}}\right)^{tn}-P}
2804:
2768:
2704:
355:
Inflation compounded over 40 years at different rates
4025:
3924:
3739:
3639:
3597:
3482:
3390:
3337:
3313:
3291:
3269:
3247:
3206:
3075:
3026:
2962:
2937:
2912:
2876:
2684:
2657:
2612:
2575:
2463:
2417:
2391:
2371:
2342:
2298:
2276:
2254:
2232:
2150:
2076:
2056:
2005:
1940:
1774:
1642:
1544:
1483:
1439:
1365:
1323:
1248:
1136:
1106:
1086:
1021:
908:
825:
766:{\displaystyle A=P\left(1+{\frac {r}{n}}\right)^{tn}}
717:
697:
677:
527:
3914:
A practical estimate for reverse calculation of the
2045:
1601:{\displaystyle \delta _{t}=p+{s \over {1+rse^{st}}}}
1124:a(t), the force of interest, or more generally the
345:$ 266,864 in total dividend payments over 40 years
4411:
4120:
4011:
3901:
3721:
3623:
3579:
3465:{\displaystyle M'=\sum _{i=0}^{t-1}{M(1+r)^{t-i}}}
3464:
3376:
3319:
3297:
3275:
3253:
3221:
3190:
3057:
3008:
2949:
2924:
2888:
2862:
2790:
2751:
2690:
2670:
2641:
2599:
2557:
2447:
2403:
2377:
2357:
2304:
2282:
2260:
2238:
2213:
2136:
2062:
2037:Mortgage calculator § Monthly payment formula
2011:
1989:
1887:
1756:
1600:
1514:
1469:
1415:
1344:
1309:
1225:
1112:
1092:
1065:
962:
879:
765:
703:
683:
533:
4485:
2752:{\textstyle c\approx c_{0}{\frac {Y}{1-e^{-Y}}}}
2214:{\displaystyle c={\frac {rP}{1-e^{-n\ln(1+r)}}}}
1918:is the interest rate with compounding frequency
1904:is the interest rate with compounding frequency
1352:, this can be viewed as a particular case of a
4248:(1st ed.). James and John Knapton, et al.
2899:
423:, not compound interest. With these loans, an
347:Dividends were not reinvested in this scenario
1126:logarithmic or continuously compounded return
226:
2567:which suggests defining auxiliary variables
4263:. Cambridge, Massachusetts. pp. 301â2.
3331:The compound interest for each deposit is:
2642:{\displaystyle c_{0}\equiv {\frac {P}{n}}.}
2031:Monthly amortized loan or mortgage payments
1929:
4368:
4273:: CS1 maint: location missing publisher (
2050:An exact formula for the monthly payment (
233:
219:
1416:{\displaystyle da(t)=\delta _{t}a(t)\,dt}
1406:
1303:
1294:
972:
654:Learn how and when to remove this message
282:
27:Compounding sum paid for the use of money
4341:
4238:
889:
350:
334:
326:
263:
37:
29:
4438:"Compound Interest Formula - Explained"
3058:{\displaystyle X={\frac {1}{2}}IT=.675}
2332:Approximate formula for monthly payment
666:
343:of 1.5% on initial $ 10,000 investment
14:
4486:
2316:
1120:. For any continuously differentiable
546:it earns it; he who doesn't pays it."
4317:Journal of the Institute of Actuaries
4314:
4290:Journal of the Institute of Actuaries
4287:
4258:
4172:Real versus nominal value (economics)
1080:As the number of compounding periods
256:Compound interest is contrasted with
4369:Schleckser, Jim (January 21, 2020).
4222:
2870:is valid to better than 1% provided
1623:
1128:, is a function of time as follows:
1075:
592:adding citations to reliable sources
563:
3232:
1888:{\displaystyle r_{2}=\left{n_{2}},}
24:
3213:
3182:
3137:
3000:
2979:
2352:
2325:function is used. The syntax is:
25:
4520:
4412:Investopedia Staff (2003-11-19).
2455:which yields the simplification:
2448:{\displaystyle \ln(1+r)\approx r}
2046:Exact formula for monthly payment
1515:{\displaystyle a(t)=e^{t\delta }}
1066:{\displaystyle P(t)=P_{0}e^{rt}.}
4228:
1470:{\displaystyle \delta =\ln(1+r)}
568:
276:, or not at all until maturity.
68:
4468:
4454:
4430:
4211:from the original on 2022-09-18
579:needs additional citations for
288:different markets as effective
4405:
4381:
4362:
4356:10.1080/0025570X.1996.11996479
4335:
4308:
4281:
4252:
4205:Department of Justice (Canada)
4189:
3884:
3871:
3838:
3825:
3807:
3794:
3767:
3754:
3710:
3697:
3670:
3657:
3618:
3606:
3565:
3552:
3537:
3524:
3446:
3433:
3365:
3352:
3176:
3143:
2791:{\textstyle X={\frac {1}{2}}Y}
2436:
2424:
2203:
2191:
2119:
2106:
1528:. It is the reciprocal of the
1526:annual effective discount rate
1493:
1487:
1464:
1452:
1403:
1397:
1378:
1372:
1333:
1327:
1258:
1252:
1239:of the accumulation function.
1220:
1214:
1181:
1175:
1167:
1161:
1031:
1025:
918:
912:
549:
461:and was severely condemned by
13:
1:
4182:
3377:{\displaystyle M'=M(1+r)^{t}}
2600:{\displaystyle Y\equiv nr=IT}
788:is the original principal sum
169:LindemannâWeierstrass theorem
4167:Rate of return on investment
3200:The exact payment amount is
803:is the compounding frequency
796:nominal annual interest rate
474:Francesco Balducci Pegolotti
7:
4389:"Compound Interest Formula"
4130:
3222:{\displaystyle c=\$ 608.02}
2900:Example of mortgage payment
2312:= number of payment periods
711:, is given by the formula:
560:Interest § Calculation
322:
10:
4525:
4242:, ed. (1728). "Interest".
3283:= rate of return (monthly)
2404:{\displaystyle r<<1}
2034:
1627:
976:
553:
478:table of compound interest
450:
446:
4442:www.thecalculatorsite.com
4329:10.1017/S0020268100040865
4302:10.1017/S002026810001636X
691:plus compounded interest
469:of many other countries.
4414:"Continuous Compounding"
2358:{\displaystyle I<8\%}
520:discovered the constant
472:The Florentine merchant
34:Effective interest rates
3624:{\displaystyle 1/(1+r)}
2950:{\displaystyle I=0.045}
2889:{\displaystyle X\leq 1}
2290:= monthly interest rate
2012:{\displaystyle \delta }
1930:continuously compounded
1113:{\displaystyle \delta }
508:Arithmeticall Questions
483:Pratica della mercatura
453:Interest § History
315:annual percentage yield
303:effective interest rate
4122:
4013:
3903:
3723:
3625:
3581:
3523:
3466:
3428:
3378:
3321:
3305:= monthly deposit, and
3299:
3277:
3255:
3223:
3192:
3059:
3010:
2951:
2926:
2890:
2864:
2792:
2753:
2692:
2672:
2643:
2601:
2559:
2449:
2405:
2379:
2359:
2306:
2284:
2262:
2240:
2215:
2138:
2064:
2013:
1991:
1889:
1758:
1602:
1516:
1471:
1417:
1346:
1345:{\displaystyle a(0)=1}
1311:
1237:logarithmic derivative
1227:
1114:
1094:
1067:
973:Continuous compounding
964:
881:
767:
705:
685:
535:
419:U.S. mortgages use an
404:
348:
332:
297:annual equivalent rate
291:annual percentage rate
283:Annual equivalent rate
58:mathematical constant
43:
35:
18:Continuous compounding
4259:Evans, Allan (1936).
4123:
4014:
3904:
3724:
3626:
3582:
3497:
3467:
3402:
3379:
3322:
3300:
3278:
3256:
3224:
3193:
3060:
3011:
2952:
2927:
2891:
2865:
2793:
2754:
2693:
2673:
2671:{\displaystyle c_{0}}
2644:
2602:
2560:
2450:
2406:
2380:
2360:
2321:In spreadsheets, the
2307:
2285:
2263:
2241:
2216:
2139:
2065:
2014:
1992:
1890:
1759:
1603:
1517:
1472:
1418:
1347:
1312:
1228:
1122:accumulation function
1115:
1095:
1068:
965:
900:accumulation function
890:Accumulation function
882:
768:
706:
686:
536:
451:Further information:
440:financial derivatives
425:amortization schedule
354:
338:
330:
309:effective annual rate
270:compounding frequency
264:Compounding frequency
207:Schanuel's conjecture
41:
33:
4504:Mathematical finance
4344:Mathematics Magazine
4201:Justice Laws Website
4137:Credit card interest
4023:
3922:
3737:
3637:
3595:
3591:(common ratio :
3480:
3388:
3335:
3311:
3289:
3267:
3245:
3204:
3073:
3024:
2960:
2935:
2925:{\displaystyle T=30}
2910:
2874:
2802:
2766:
2702:
2682:
2655:
2610:
2573:
2461:
2415:
2389:
2369:
2340:
2296:
2274:
2252:
2230:
2148:
2074:
2054:
2003:
1938:
1772:
1640:
1630:Day count convention
1542:
1481:
1437:
1363:
1321:
1246:
1134:
1104:
1084:
1019:
906:
894:Since the principal
823:
715:
695:
675:
667:Periodic compounding
588:improve this article
525:
489:Summa de arithmetica
90:Exponential function
51:a series of articles
3589:closed-form formula
3261:= principal deposit
2317:Spreadsheet formula
1283:
1002:. The amount after
782:is the final amount
603:"Compound interest"
556:Time value of money
161:representations of
4142:Exponential growth
4118:
4009:
3899:
3719:
3621:
3577:
3462:
3374:
3317:
3295:
3273:
3251:
3219:
3188:
3055:
3006:
2947:
2922:
2886:
2860:
2788:
2749:
2688:
2668:
2639:
2597:
2555:
2445:
2401:
2375:
2355:
2302:
2280:
2258:
2236:
2211:
2134:
2060:
2009:
1987:
1885:
1754:
1598:
1512:
1467:
1413:
1342:
1307:
1269:
1223:
1110:
1090:
1063:
979:Logarithmic return
960:
877:
763:
701:
681:
531:
405:
349:
333:
44:
36:
4509:Actuarial science
4240:Chambers, Ephraim
4092:
3989:
3863:
3786:
3743:
3689:
3587:and applying the
3575:
3327:= time, in months
3320:{\displaystyle t}
3298:{\displaystyle M}
3276:{\displaystyle r}
3254:{\displaystyle P}
3117:
3041:
2995:
2853:
2783:
2747:
2691:{\displaystyle n}
2634:
2553:
2517:
2504:
2378:{\displaystyle T}
2305:{\displaystyle n}
2283:{\displaystyle r}
2261:{\displaystyle P}
2246:= monthly payment
2239:{\displaystyle c}
2209:
2132:
2129:
2063:{\displaystyle c}
1976:
1928:When interest is
1856:
1827:
1734:
1677:
1624:Compounding basis
1596:
1203:
1185:
1093:{\displaystyle n}
1076:Force of interest
944:
855:
813:, usually years).
747:
704:{\displaystyle I}
684:{\displaystyle P}
664:
663:
656:
638:
534:{\displaystyle e}
496:(1494) gives the
246:Compound interest
243:
242:
106:compound interest
85:Natural logarithm
16:(Redirected from
4516:
4478:
4472:
4466:
4465:
4458:
4452:
4451:
4449:
4448:
4434:
4428:
4427:
4425:
4424:
4409:
4403:
4402:
4400:
4399:
4385:
4379:
4378:
4366:
4360:
4359:
4339:
4333:
4332:
4312:
4306:
4305:
4285:
4279:
4278:
4272:
4264:
4256:
4250:
4249:
4232:
4231:
4226:
4220:
4219:
4217:
4216:
4193:
4127:
4125:
4124:
4119:
4111:
4110:
4106:
4097:
4093:
4091:
4087:
4082:
4067:
4063:
4058:
4047:
4038:
4018:
4016:
4015:
4010:
4008:
4007:
4003:
3994:
3990:
3988:
3984:
3979:
3964:
3963:
3946:
3937:
3908:
3906:
3905:
3900:
3898:
3897:
3864:
3859:
3852:
3851:
3823:
3815:
3814:
3787:
3782:
3775:
3774:
3752:
3744:
3741:
3728:
3726:
3725:
3720:
3718:
3717:
3690:
3685:
3678:
3677:
3655:
3647:
3630:
3628:
3627:
3622:
3605:
3586:
3584:
3583:
3578:
3576:
3574:
3573:
3572:
3547:
3545:
3544:
3522:
3511:
3490:
3474:geometric series
3472:Recognizing the
3471:
3469:
3468:
3463:
3461:
3460:
3459:
3427:
3416:
3398:
3383:
3381:
3380:
3375:
3373:
3372:
3345:
3326:
3324:
3323:
3318:
3304:
3302:
3301:
3296:
3282:
3280:
3279:
3274:
3260:
3258:
3257:
3252:
3233:Monthly deposits
3228:
3226:
3225:
3220:
3197:
3195:
3194:
3189:
3172:
3167:
3166:
3133:
3129:
3128:
3127:
3118:
3110:
3091:
3090:
3064:
3062:
3061:
3056:
3042:
3034:
3015:
3013:
3012:
3007:
2996:
2991:
2977:
2972:
2971:
2956:
2954:
2953:
2948:
2931:
2929:
2928:
2923:
2895:
2893:
2892:
2887:
2869:
2867:
2866:
2861:
2859:
2855:
2854:
2849:
2848:
2839:
2820:
2819:
2798:. The expansion
2797:
2795:
2794:
2789:
2784:
2776:
2758:
2756:
2755:
2750:
2748:
2746:
2745:
2744:
2722:
2720:
2719:
2697:
2695:
2694:
2689:
2677:
2675:
2674:
2669:
2667:
2666:
2648:
2646:
2645:
2640:
2635:
2627:
2622:
2621:
2606:
2604:
2603:
2598:
2564:
2562:
2561:
2556:
2554:
2552:
2551:
2550:
2528:
2520:
2518:
2510:
2505:
2503:
2502:
2501:
2479:
2471:
2454:
2452:
2451:
2446:
2410:
2408:
2407:
2402:
2384:
2382:
2381:
2376:
2364:
2362:
2361:
2356:
2311:
2309:
2308:
2303:
2289:
2287:
2286:
2281:
2267:
2265:
2264:
2259:
2245:
2243:
2242:
2237:
2220:
2218:
2217:
2212:
2210:
2208:
2207:
2206:
2166:
2158:
2144:or equivalently
2143:
2141:
2140:
2135:
2133:
2131:
2130:
2128:
2127:
2126:
2101:
2092:
2084:
2069:
2067:
2066:
2061:
2018:
2016:
2015:
2010:
1996:
1994:
1993:
1988:
1983:
1982:
1978:
1977:
1969:
1894:
1892:
1891:
1886:
1881:
1880:
1879:
1869:
1865:
1858:
1857:
1855:
1854:
1845:
1844:
1835:
1833:
1829:
1828:
1826:
1825:
1816:
1815:
1806:
1784:
1783:
1763:
1761:
1760:
1755:
1753:
1752:
1751:
1750:
1740:
1736:
1735:
1733:
1732:
1723:
1722:
1713:
1696:
1695:
1694:
1693:
1683:
1679:
1678:
1676:
1675:
1666:
1665:
1656:
1607:
1605:
1604:
1599:
1597:
1595:
1594:
1593:
1565:
1554:
1553:
1521:
1519:
1518:
1513:
1511:
1510:
1476:
1474:
1473:
1468:
1422:
1420:
1419:
1414:
1393:
1392:
1354:product integral
1351:
1349:
1348:
1343:
1316:
1314:
1313:
1308:
1302:
1301:
1293:
1292:
1282:
1277:
1232:
1230:
1229:
1224:
1204:
1202:
1191:
1186:
1184:
1170:
1160:
1151:
1146:
1145:
1119:
1117:
1116:
1111:
1099:
1097:
1096:
1091:
1072:
1070:
1069:
1064:
1059:
1058:
1046:
1045:
989:
969:
967:
966:
961:
959:
958:
950:
946:
945:
937:
886:
884:
883:
878:
870:
869:
861:
857:
856:
848:
772:
770:
769:
764:
762:
761:
753:
749:
748:
740:
710:
708:
707:
702:
690:
688:
687:
682:
659:
652:
648:
645:
639:
637:
596:
572:
564:
540:
538:
537:
532:
401:
395:
389:
383:
377:
371:
365:
359:
235:
228:
221:
164:
155:
144:
111:Euler's identity
72:
63:
46:
45:
21:
4524:
4523:
4519:
4518:
4517:
4515:
4514:
4513:
4484:
4483:
4482:
4481:
4473:
4469:
4460:
4459:
4455:
4446:
4444:
4436:
4435:
4431:
4422:
4420:
4410:
4406:
4397:
4395:
4387:
4386:
4382:
4367:
4363:
4340:
4336:
4313:
4309:
4286:
4282:
4266:
4265:
4257:
4253:
4229:
4227:
4223:
4214:
4212:
4195:
4194:
4190:
4185:
4147:Fisher equation
4133:
4102:
4098:
4083:
4078:
4068:
4059:
4054:
4040:
4039:
4037:
4033:
4032:
4024:
4021:
4020:
3999:
3995:
3980:
3975:
3965:
3959:
3939:
3938:
3936:
3932:
3931:
3923:
3920:
3919:
3887:
3883:
3841:
3837:
3824:
3822:
3810:
3806:
3770:
3766:
3753:
3751:
3740:
3738:
3735:
3734:
3713:
3709:
3673:
3669:
3656:
3654:
3640:
3638:
3635:
3634:
3601:
3596:
3593:
3592:
3568:
3564:
3551:
3546:
3540:
3536:
3512:
3501:
3483:
3481:
3478:
3477:
3449:
3445:
3429:
3417:
3406:
3391:
3389:
3386:
3385:
3368:
3364:
3338:
3336:
3333:
3332:
3312:
3309:
3308:
3290:
3287:
3286:
3268:
3265:
3264:
3246:
3243:
3242:
3235:
3205:
3202:
3201:
3168:
3162:
3158:
3123:
3119:
3109:
3096:
3092:
3086:
3082:
3074:
3071:
3070:
3033:
3025:
3022:
3021:
2978:
2976:
2967:
2963:
2961:
2958:
2957:
2936:
2933:
2932:
2911:
2908:
2907:
2902:
2875:
2872:
2871:
2844:
2840:
2838:
2825:
2821:
2815:
2811:
2803:
2800:
2799:
2775:
2767:
2764:
2763:
2737:
2733:
2726:
2721:
2715:
2711:
2703:
2700:
2699:
2683:
2680:
2679:
2662:
2658:
2656:
2653:
2652:
2626:
2617:
2613:
2611:
2608:
2607:
2574:
2571:
2570:
2540:
2536:
2529:
2521:
2519:
2509:
2491:
2487:
2480:
2472:
2470:
2462:
2459:
2458:
2416:
2413:
2412:
2390:
2387:
2386:
2370:
2367:
2366:
2341:
2338:
2337:
2334:
2329:
2319:
2297:
2294:
2293:
2275:
2272:
2271:
2253:
2250:
2249:
2231:
2228:
2227:
2178:
2174:
2167:
2159:
2157:
2149:
2146:
2145:
2122:
2118:
2105:
2100:
2093:
2085:
2083:
2075:
2072:
2071:
2055:
2052:
2051:
2048:
2039:
2033:
2004:
2001:
2000:
1968:
1961:
1957:
1956:
1939:
1936:
1935:
1924:
1917:
1910:
1903:
1875:
1871:
1870:
1850:
1846:
1840:
1836:
1834:
1821:
1817:
1811:
1807:
1805:
1798:
1794:
1793:
1792:
1788:
1779:
1775:
1773:
1770:
1769:
1746:
1742:
1741:
1728:
1724:
1718:
1714:
1712:
1705:
1701:
1700:
1689:
1685:
1684:
1671:
1667:
1661:
1657:
1655:
1648:
1644:
1643:
1641:
1638:
1637:
1632:
1626:
1620:are estimated.
1586:
1582:
1569:
1564:
1549:
1545:
1543:
1540:
1539:
1503:
1499:
1482:
1479:
1478:
1438:
1435:
1434:
1388:
1384:
1364:
1361:
1360:
1322:
1319:
1318:
1288:
1284:
1278:
1273:
1268:
1264:
1247:
1244:
1243:
1195:
1190:
1171:
1153:
1152:
1150:
1141:
1137:
1135:
1132:
1131:
1105:
1102:
1101:
1085:
1082:
1081:
1078:
1051:
1047:
1041:
1037:
1020:
1017:
1016:
1012:
984:
981:
975:
951:
936:
929:
925:
924:
907:
904:
903:
892:
862:
847:
840:
836:
835:
824:
821:
820:
754:
739:
732:
728:
727:
716:
713:
712:
696:
693:
692:
676:
673:
672:
669:
660:
649:
643:
640:
597:
595:
585:
573:
562:
552:
526:
523:
522:
518:Jacob Bernoulli
455:
449:
421:amortizing loan
403:
399:
397:
393:
391:
387:
385:
381:
379:
375:
373:
369:
367:
363:
361:
357:
346:
344:
325:
285:
266:
258:simple interest
239:
192:
162:
153:
142:
116:Euler's formula
59:
28:
23:
22:
15:
12:
11:
5:
4522:
4512:
4511:
4506:
4501:
4496:
4480:
4479:
4467:
4453:
4429:
4404:
4393:qrc.depaul.edu
4380:
4361:
4334:
4323:(3): 423â442.
4307:
4296:(1): 121â132.
4280:
4251:
4221:
4207:. 2002-12-31.
4187:
4186:
4184:
4181:
4180:
4179:
4174:
4169:
4164:
4162:Rate of return
4159:
4154:
4149:
4144:
4139:
4132:
4129:
4117:
4114:
4109:
4105:
4101:
4096:
4090:
4086:
4081:
4077:
4074:
4071:
4066:
4062:
4057:
4053:
4050:
4046:
4043:
4036:
4031:
4028:
4006:
4002:
3998:
3993:
3987:
3983:
3978:
3974:
3971:
3968:
3962:
3958:
3955:
3952:
3949:
3945:
3942:
3935:
3930:
3927:
3916:rate of return
3896:
3893:
3890:
3886:
3882:
3879:
3876:
3873:
3870:
3867:
3862:
3858:
3855:
3850:
3847:
3844:
3840:
3836:
3833:
3830:
3827:
3821:
3818:
3813:
3809:
3805:
3802:
3799:
3796:
3793:
3790:
3785:
3781:
3778:
3773:
3769:
3765:
3762:
3759:
3756:
3750:
3747:
3716:
3712:
3708:
3705:
3702:
3699:
3696:
3693:
3688:
3684:
3681:
3676:
3672:
3668:
3665:
3662:
3659:
3653:
3650:
3646:
3643:
3620:
3617:
3614:
3611:
3608:
3604:
3600:
3571:
3567:
3563:
3560:
3557:
3554:
3550:
3543:
3539:
3535:
3532:
3529:
3526:
3521:
3518:
3515:
3510:
3507:
3504:
3500:
3496:
3493:
3489:
3486:
3458:
3455:
3452:
3448:
3444:
3441:
3438:
3435:
3432:
3426:
3423:
3420:
3415:
3412:
3409:
3405:
3401:
3397:
3394:
3371:
3367:
3363:
3360:
3357:
3354:
3351:
3348:
3344:
3341:
3329:
3328:
3316:
3306:
3294:
3284:
3272:
3262:
3250:
3234:
3231:
3218:
3215:
3212:
3209:
3187:
3184:
3181:
3178:
3175:
3171:
3165:
3161:
3157:
3154:
3151:
3148:
3145:
3142:
3139:
3136:
3132:
3126:
3122:
3116:
3113:
3108:
3105:
3102:
3099:
3095:
3089:
3085:
3081:
3078:
3054:
3051:
3048:
3045:
3040:
3037:
3032:
3029:
3005:
3002:
2999:
2994:
2990:
2987:
2984:
2981:
2975:
2970:
2966:
2946:
2943:
2940:
2921:
2918:
2915:
2901:
2898:
2885:
2882:
2879:
2858:
2852:
2847:
2843:
2837:
2834:
2831:
2828:
2824:
2818:
2814:
2810:
2807:
2787:
2782:
2779:
2774:
2771:
2743:
2740:
2736:
2732:
2729:
2725:
2718:
2714:
2710:
2707:
2687:
2665:
2661:
2638:
2633:
2630:
2625:
2620:
2616:
2596:
2593:
2590:
2587:
2584:
2581:
2578:
2549:
2546:
2543:
2539:
2535:
2532:
2527:
2524:
2516:
2513:
2508:
2500:
2497:
2494:
2490:
2486:
2483:
2478:
2475:
2469:
2466:
2444:
2441:
2438:
2435:
2432:
2429:
2426:
2423:
2420:
2400:
2397:
2394:
2374:
2354:
2351:
2348:
2345:
2333:
2330:
2327:
2318:
2315:
2314:
2313:
2301:
2291:
2279:
2269:
2257:
2247:
2235:
2205:
2202:
2199:
2196:
2193:
2190:
2187:
2184:
2181:
2177:
2173:
2170:
2165:
2162:
2156:
2153:
2125:
2121:
2117:
2114:
2111:
2108:
2104:
2099:
2096:
2091:
2088:
2082:
2079:
2059:
2047:
2044:
2032:
2029:
2008:
1986:
1981:
1975:
1972:
1967:
1964:
1960:
1955:
1952:
1949:
1946:
1943:
1922:
1915:
1908:
1901:
1884:
1878:
1874:
1868:
1864:
1861:
1853:
1849:
1843:
1839:
1832:
1824:
1820:
1814:
1810:
1804:
1801:
1797:
1791:
1787:
1782:
1778:
1749:
1745:
1739:
1731:
1727:
1721:
1717:
1711:
1708:
1704:
1699:
1692:
1688:
1682:
1674:
1670:
1664:
1660:
1654:
1651:
1647:
1625:
1622:
1592:
1589:
1585:
1581:
1578:
1575:
1572:
1568:
1563:
1560:
1557:
1552:
1548:
1509:
1506:
1502:
1498:
1495:
1492:
1489:
1486:
1466:
1463:
1460:
1457:
1454:
1451:
1448:
1445:
1442:
1412:
1409:
1405:
1402:
1399:
1396:
1391:
1387:
1383:
1380:
1377:
1374:
1371:
1368:
1341:
1338:
1335:
1332:
1329:
1326:
1306:
1300:
1297:
1291:
1287:
1281:
1276:
1272:
1267:
1263:
1260:
1257:
1254:
1251:
1222:
1219:
1216:
1213:
1210:
1207:
1201:
1198:
1194:
1189:
1183:
1180:
1177:
1174:
1169:
1166:
1163:
1159:
1156:
1149:
1144:
1140:
1109:
1089:
1077:
1074:
1062:
1057:
1054:
1050:
1044:
1040:
1036:
1033:
1030:
1027:
1024:
1010:
974:
971:
957:
954:
949:
943:
940:
935:
932:
928:
923:
920:
917:
914:
911:
891:
888:
876:
873:
868:
865:
860:
854:
851:
846:
843:
839:
834:
831:
828:
815:
814:
804:
798:
789:
783:
760:
757:
752:
746:
743:
738:
735:
731:
726:
723:
720:
700:
680:
668:
665:
662:
661:
576:
574:
567:
551:
548:
530:
448:
445:
444:
443:
428:
417:
414:mortgage loans
410:
398:
392:
386:
380:
374:
368:
362:
356:
324:
321:
284:
281:
265:
262:
241:
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238:
237:
230:
223:
215:
212:
211:
210:
209:
201:
200:
199:Related topics
196:
195:
194:
193:
190:Leonhard Euler
187:
179:
178:
174:
173:
172:
171:
166:
158:
146:
145:
138:
137:
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87:
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74:
73:
65:
64:
55:
54:
26:
9:
6:
4:
3:
2:
4521:
4510:
4507:
4505:
4502:
4500:
4497:
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4492:
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4476:
4471:
4463:
4457:
4443:
4439:
4433:
4419:
4415:
4408:
4394:
4390:
4384:
4376:
4372:
4365:
4357:
4353:
4349:
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4330:
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4303:
4299:
4295:
4291:
4284:
4276:
4270:
4262:
4255:
4247:
4246:
4241:
4236:
4235:public domain
4225:
4210:
4206:
4202:
4198:
4192:
4188:
4178:
4175:
4173:
4170:
4168:
4165:
4163:
4160:
4158:
4157:Interest rate
4155:
4153:
4150:
4148:
4145:
4143:
4140:
4138:
4135:
4134:
4128:
4115:
4112:
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3819:
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3811:
3803:
3800:
3797:
3791:
3788:
3783:
3779:
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3641:
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3615:
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3602:
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3508:
3505:
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3498:
3494:
3491:
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3456:
3453:
3450:
3442:
3439:
3436:
3430:
3424:
3421:
3418:
3413:
3410:
3407:
3403:
3399:
3395:
3392:
3369:
3361:
3358:
3355:
3349:
3346:
3342:
3339:
3314:
3307:
3292:
3285:
3270:
3263:
3248:
3241:
3240:
3239:
3230:
3216:
3210:
3207:
3198:
3185:
3179:
3173:
3169:
3163:
3159:
3155:
3152:
3149:
3146:
3140:
3134:
3130:
3124:
3120:
3114:
3111:
3106:
3103:
3100:
3097:
3093:
3087:
3083:
3079:
3076:
3068:
3065:
3052:
3049:
3046:
3043:
3038:
3035:
3030:
3027:
3019:
3016:
3003:
2997:
2992:
2988:
2985:
2982:
2973:
2968:
2964:
2944:
2941:
2938:
2919:
2916:
2913:
2905:
2897:
2883:
2880:
2877:
2856:
2850:
2845:
2841:
2835:
2832:
2829:
2826:
2822:
2816:
2812:
2808:
2805:
2785:
2780:
2777:
2772:
2769:
2760:
2741:
2738:
2734:
2730:
2727:
2723:
2716:
2712:
2708:
2705:
2685:
2663:
2659:
2649:
2636:
2631:
2628:
2623:
2618:
2614:
2594:
2591:
2588:
2585:
2582:
2579:
2576:
2568:
2565:
2547:
2544:
2541:
2537:
2533:
2530:
2525:
2522:
2514:
2511:
2506:
2498:
2495:
2492:
2488:
2484:
2481:
2476:
2473:
2467:
2464:
2456:
2442:
2439:
2433:
2430:
2427:
2421:
2418:
2398:
2395:
2392:
2372:
2349:
2346:
2343:
2326:
2324:
2299:
2292:
2277:
2270:
2255:
2248:
2233:
2226:
2225:
2224:
2221:
2200:
2197:
2194:
2188:
2185:
2182:
2179:
2175:
2171:
2168:
2163:
2160:
2154:
2151:
2123:
2115:
2112:
2109:
2102:
2097:
2094:
2089:
2086:
2080:
2077:
2057:
2043:
2038:
2028:
2026:
2022:
2006:
1997:
1984:
1979:
1973:
1970:
1965:
1962:
1958:
1953:
1950:
1947:
1944:
1941:
1933:
1931:
1926:
1921:
1914:
1907:
1900:
1895:
1882:
1876:
1872:
1866:
1862:
1859:
1851:
1847:
1841:
1837:
1830:
1822:
1818:
1812:
1808:
1802:
1799:
1795:
1789:
1785:
1780:
1776:
1767:
1764:
1747:
1743:
1737:
1729:
1725:
1719:
1715:
1709:
1706:
1702:
1697:
1690:
1686:
1680:
1672:
1668:
1662:
1658:
1652:
1649:
1645:
1635:
1631:
1621:
1619:
1615:
1611:
1590:
1587:
1583:
1579:
1576:
1573:
1570:
1566:
1561:
1558:
1555:
1550:
1546:
1536:
1534:
1532:
1527:
1522:
1507:
1504:
1500:
1496:
1490:
1484:
1461:
1458:
1455:
1449:
1446:
1443:
1440:
1432:
1428:
1423:
1410:
1407:
1400:
1394:
1389:
1385:
1381:
1375:
1369:
1366:
1357:
1355:
1339:
1336:
1330:
1324:
1304:
1298:
1295:
1289:
1285:
1279:
1274:
1270:
1265:
1261:
1255:
1249:
1240:
1238:
1233:
1217:
1211:
1208:
1205:
1199:
1196:
1192:
1187:
1178:
1172:
1164:
1157:
1154:
1147:
1142:
1138:
1129:
1127:
1123:
1107:
1087:
1073:
1060:
1055:
1052:
1048:
1042:
1038:
1034:
1028:
1022:
1014:
1009:
1005:
1001:
997:
993:
987:
980:
970:
955:
952:
947:
941:
938:
933:
930:
926:
921:
915:
909:
901:
897:
887:
874:
871:
866:
863:
858:
852:
849:
844:
841:
837:
832:
829:
826:
818:
812:
808:
805:
802:
799:
797:
793:
790:
787:
784:
781:
778:
777:
776:
773:
758:
755:
750:
744:
741:
736:
733:
729:
724:
721:
718:
698:
678:
658:
655:
647:
636:
633:
629:
626:
622:
619:
615:
612:
608:
605: â
604:
600:
599:Find sources:
593:
589:
583:
582:
577:This section
575:
571:
566:
565:
561:
557:
547:
543:
541:
528:
519:
515:
513:
509:
505:
501:
499:
495:
491:
490:
485:
484:
479:
475:
470:
468:
464:
460:
454:
441:
437:
433:
429:
426:
422:
418:
415:
411:
407:
406:
353:
342:
337:
329:
320:
317:
316:
311:
310:
305:
304:
299:
298:
293:
292:
280:
277:
275:
271:
261:
259:
254:
251:
247:
236:
231:
229:
224:
222:
217:
216:
214:
213:
208:
205:
204:
203:
202:
198:
197:
191:
188:
186:
183:
182:
181:
180:
176:
175:
170:
167:
165:
159:
157:
156:is irrational
150:
149:
148:
147:
140:
139:
132:
128:
124:
123:
122:
119:
117:
114:
112:
109:
107:
104:
103:
102:
101:
97:
96:
91:
88:
86:
83:
82:
81:
80:
76:
75:
71:
67:
66:
62:
57:
56:
52:
48:
47:
40:
32:
19:
4499:Exponentials
4470:
4456:
4445:. Retrieved
4441:
4432:
4421:. Retrieved
4418:Investopedia
4417:
4407:
4396:. Retrieved
4392:
4383:
4374:
4364:
4347:
4343:
4337:
4320:
4316:
4310:
4293:
4289:
4283:
4260:
4254:
4243:
4224:
4213:. Retrieved
4200:
4191:
3913:
3910:
3733:
3730:
3633:
3330:
3236:
3199:
3069:
3066:
3020:
3018:which gives
3017:
2906:
2903:
2761:
2650:
2569:
2566:
2457:
2411:so that the
2335:
2322:
2320:
2222:
2049:
2040:
2024:
2020:
1998:
1934:
1927:
1919:
1912:
1905:
1898:
1896:
1768:
1765:
1636:
1633:
1617:
1613:
1609:
1537:
1530:
1523:
1430:
1426:
1424:
1358:
1242:Conversely:
1241:
1235:This is the
1234:
1130:
1079:
1015:
1007:
1003:
995:
985:
982:
895:
893:
819:
816:
810:
806:
800:
791:
785:
779:
774:
670:
650:
641:
631:
624:
617:
610:
598:
586:Please help
581:verification
578:
544:
516:
507:
504:Richard Witt
502:
494:Luca Pacioli
487:
481:
480:in his book
471:
456:
436:ItĂ´ calculus
313:
307:
301:
295:
289:
286:
278:
274:continuously
269:
267:
255:
245:
244:
125:exponential
105:
98:Applications
4177:Yield curve
2268:= principal
550:Calculation
476:provided a
467:common laws
432:derivatives
185:John Napier
152:proof that
4488:Categories
4447:2018-12-05
4423:2018-12-05
4398:2018-12-05
4350:(5): 376.
4215:2024-08-14
4183:References
2365:and terms
2035:See also:
1628:See also:
977:See also:
614:newspapers
554:See also:
498:Rule of 72
121:half-lives
77:Properties
4269:cite book
4113:−
4076:∑
4052:∑
4049:−
3973:∑
3957:∑
3954:−
3948:−
3892:−
3854:−
3846:−
3777:−
3680:−
3517:−
3499:∑
3454:−
3422:−
3404:∑
3214:$
3183:$
3138:$
3080:≈
3001:$
2980:$
2881:≤
2809:≈
2739:−
2731:−
2709:≈
2624:≡
2580:≡
2542:−
2534:−
2493:−
2485:−
2468:≈
2440:≈
2422:
2353:%
2189:
2180:−
2172:−
2098:−
2007:δ
1954:
1942:δ
1860:−
1547:δ
1508:δ
1450:
1441:δ
1386:δ
1286:δ
1271:∫
1209:
1139:δ
1108:δ
872:−
644:June 2019
512:anatocism
463:Roman law
412:Canadian
141:Defining
4494:Interest
4209:Archived
4152:Interest
4131:See also
4045:′
3944:′
3645:′
3488:′
3396:′
3343:′
3067:so that
2396:<<
1533:-folding
1158:′
1000:infinity
998:goes to
506:'s book
465:and the
438:, where
402: 1%
396: 2%
390: 3%
384: 4%
378: 5%
372: 6%
366: 7%
360: 8%
341:dividend
323:Examples
294:(EAPR),
250:interest
49:Part of
4237::
2223:where:
1535:time.
1317:(Since
794:is the
775:where:
628:scholar
447:History
339:Annual
300:(AER),
3217:608.02
3186:608.96
3141:333.33
3004:333.33
1999:where
1932:, use
1911:, and
1897:where
1608:where
630:
623:
616:
609:
601:
400:
394:
388:
382:
376:
370:
364:
358:
177:People
127:growth
53:on the
3742:Value
2945:0.045
2651:Here
2323:PMT()
2070:) is
992:limit
635:JSTOR
621:books
459:usury
409:rate.
131:decay
4275:link
3160:.675
3153:.675
3053:.675
2762:Let
2347:<
1766:use
1616:and
1013:as:
607:news
558:and
268:The
129:and
4375:Inc
4352:doi
4325:doi
4321:108
4298:doi
4019:or
3631:):
2993:360
2989:000
2983:120
1477:or
1356:.)
994:as
988:â 1
590:by
492:of
248:is
4490::
4440:.
4416:.
4391:.
4373:.
4348:69
4346:.
4319:.
4294:96
4292:.
4271:}}
4267:{{
4203:.
4199:.
3476::
2920:30
2896:.
2759:.
2419:ln
2186:ln
2027:.
1951:ln
1925:.
1612:,
1447:ln
1433::
1206:ln
312:,
306:,
4464:.
4450:.
4426:.
4401:.
4377:.
4358:.
4354::
4331:.
4327::
4304:.
4300::
4277:)
4218:.
4116:1
4108:t
4104:/
4100:1
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4089:2
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4073:+
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4035:(
4030:=
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3934:(
3929:=
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3211:=
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3180:=
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3174:3
3170:/
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3135:=
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3098:1
3094:(
3088:0
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3077:c
3050:=
3047:T
3044:I
3039:2
3036:1
3031:=
3028:X
2998:=
2986:,
2974:=
2969:0
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2914:T
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2637:.
2632:n
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2589:=
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2507:=
2499:r
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2116:r
2113:+
2110:1
2107:(
2103:1
2095:1
2090:P
2087:r
2081:=
2078:c
2058:c
2025:n
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1985:,
1980:)
1974:n
1971:r
1966:+
1963:1
1959:(
1948:n
1945:=
1923:2
1920:n
1916:2
1913:r
1909:1
1906:n
1902:1
1899:r
1883:,
1877:2
1873:n
1867:]
1863:1
1852:2
1848:n
1842:1
1838:n
1831:)
1823:1
1819:n
1813:1
1809:r
1803:+
1800:1
1796:(
1790:[
1786:=
1781:2
1777:r
1748:2
1744:n
1738:)
1730:2
1726:n
1720:2
1716:r
1710:+
1707:1
1703:(
1698:=
1691:1
1687:n
1681:)
1673:1
1669:n
1663:1
1659:r
1653:+
1650:1
1646:(
1618:s
1614:r
1610:p
1591:t
1588:s
1584:e
1580:s
1577:r
1574:+
1571:1
1567:s
1562:+
1559:p
1556:=
1551:t
1531:e
1505:t
1501:e
1497:=
1494:)
1491:t
1488:(
1485:a
1465:)
1462:r
1459:+
1456:1
1453:(
1444:=
1431:e
1427:r
1411:t
1408:d
1404:)
1401:t
1398:(
1395:a
1390:t
1382:=
1379:)
1376:t
1373:(
1370:a
1367:d
1340:1
1337:=
1334:)
1331:0
1328:(
1325:a
1305:,
1299:s
1296:d
1290:s
1280:t
1275:0
1266:e
1262:=
1259:)
1256:t
1253:(
1250:a
1221:)
1218:t
1215:(
1212:a
1200:t
1197:d
1193:d
1188:=
1182:)
1179:t
1176:(
1173:a
1168:)
1165:t
1162:(
1155:a
1148:=
1143:t
1088:n
1061:.
1056:t
1053:r
1049:e
1043:0
1039:P
1035:=
1032:)
1029:t
1026:(
1023:P
1011:0
1008:P
1004:t
996:n
986:e
956:n
953:t
948:)
942:n
939:r
934:+
931:1
927:(
922:=
919:)
916:t
913:(
910:a
896:P
875:P
867:n
864:t
859:)
853:n
850:r
845:+
842:1
838:(
833:P
830:=
827:I
811:r
807:t
801:n
792:r
786:P
780:A
759:n
756:t
751:)
745:n
742:r
737:+
734:1
730:(
725:P
722:=
719:A
699:I
679:P
657:)
651:(
646:)
642:(
632:¡
625:¡
618:¡
611:¡
584:.
529:e
234:e
227:t
220:v
163:e
154:e
143:e
61:e
20:)
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