22:
621:
701:
461:
840:
228:
752:
523:
336:
395:
775:
264:
465:
399:
154:
94:
66:
43:
545:
73:
635:
558:
80:
113:
51:
62:
281:
47:
405:
871:
790:
174:
712:
549:
471:
294:
346:
87:
32:
36:
781:
760:
626:
which is the rate of change with time of the natural logarithm of the accumulation function.
268:
For various interest-accumulation protocols, the accumulation function is as follows (with
237:
8:
851:
533:
340:
288:
529:
865:
273:
150:
146:
21:
532:, as in the case of interest, the accumulation function is an
793:
763:
715:
696:{\displaystyle a(t)=e^{\int _{0}^{t}\delta _{u}\,du}}
638:
561:
474:
408:
349:
297:
240:
177:
616:{\displaystyle \delta _{t}={\frac {a'(t)}{a(t)}}\,}
834:
769:
746:
695:
615:
517:
455:
389:
330:
258:
222:
863:
552:, is a function of time defined as follows:
546:logarithmic or continuously compounded return
50:. Unsourced material may be challenged and
539:
141:expressing the ratio of the value at time
684:
612:
137:) is a function defined in terms of time
114:Learn how and when to remove this message
456:{\displaystyle a(t)=1+{\frac {td}{1-d}}}
864:
835:{\displaystyle r(t)=e^{\delta _{t}}-1}
48:adding citations to reliable sources
15:
223:{\displaystyle A(t)=A(0)\cdot a(t)}
13:
14:
883:
747:{\displaystyle a(t)=e^{t\delta }}
234:where the initial investment is
20:
518:{\displaystyle a(t)=(1-d)^{-t}}
331:{\displaystyle a(t)=1+t\cdot i}
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797:
725:
719:
648:
642:
606:
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592:
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503:
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478:
418:
412:
390:{\displaystyle a(t)=(1+i)^{t}}
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307:
301:
250:
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217:
211:
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196:
187:
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149:) and the initial investment (
1:
857:
164:(0)=1 and the value at time
7:
845:
10:
888:
528:In the case of a positive
770:{\displaystyle \delta }
540:Variable rate of return
63:"Accumulation function"
836:
782:annual percentage rate
771:
748:
697:
617:
519:
457:
391:
332:
260:
224:
837:
772:
749:
698:
618:
520:
458:
392:
333:
261:
259:{\displaystyle A(0).}
225:
128:accumulation function
872:Mathematical finance
791:
761:
713:
636:
559:
472:
406:
347:
295:
238:
175:
44:improve this article
852:Time value of money
673:
548:, sometimes called
534:increasing function
832:
767:
744:
693:
659:
613:
515:
453:
387:
328:
256:
220:
610:
550:force of interest
466:compound discount
451:
341:compound interest
153:). It is used in
124:
123:
116:
98:
879:
841:
839:
838:
833:
825:
824:
823:
822:
784:at any time is:
776:
774:
773:
768:
753:
751:
750:
745:
743:
742:
702:
700:
699:
694:
692:
691:
683:
682:
672:
667:
622:
620:
619:
614:
611:
609:
595:
585:
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571:
570:
524:
522:
521:
516:
514:
513:
462:
460:
459:
454:
452:
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431:
396:
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393:
388:
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385:
337:
335:
334:
329:
265:
263:
262:
257:
229:
227:
226:
221:
119:
112:
108:
105:
99:
97:
56:
24:
16:
887:
886:
882:
881:
880:
878:
877:
876:
862:
861:
860:
848:
818:
814:
813:
809:
792:
789:
788:
762:
759:
758:
735:
731:
714:
711:
710:
678:
674:
668:
663:
658:
654:
637:
634:
633:
596:
578:
577:
575:
566:
562:
560:
557:
556:
542:
506:
502:
473:
470:
469:
440:
432:
430:
407:
404:
403:
400:simple discount
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377:
348:
345:
344:
296:
293:
292:
289:simple interest
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236:
235:
176:
173:
172:
155:interest theory
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109:
103:
100:
57:
55:
41:
25:
12:
11:
5:
885:
875:
874:
859:
856:
855:
854:
847:
844:
843:
842:
831:
828:
821:
817:
812:
808:
805:
802:
799:
796:
780:The effective
766:
755:
754:
741:
738:
734:
730:
727:
724:
721:
718:
704:
703:
690:
687:
681:
677:
671:
666:
662:
657:
653:
650:
647:
644:
641:
624:
623:
608:
605:
602:
599:
594:
591:
588:
584:
581:
574:
569:
565:
541:
538:
530:rate of return
526:
525:
512:
509:
505:
501:
498:
495:
492:
489:
486:
483:
480:
477:
463:
449:
446:
443:
438:
435:
429:
426:
423:
420:
417:
414:
411:
397:
384:
380:
376:
373:
370:
367:
364:
361:
358:
355:
352:
338:
327:
324:
321:
318:
315:
312:
309:
306:
303:
300:
255:
252:
249:
246:
243:
232:
231:
219:
216:
213:
210:
207:
204:
201:
198:
195:
192:
189:
186:
183:
180:
122:
121:
28:
26:
19:
9:
6:
4:
3:
2:
884:
873:
870:
869:
867:
853:
850:
849:
829:
826:
819:
815:
810:
806:
800:
794:
787:
786:
785:
783:
778:
764:
757:for constant
739:
736:
732:
728:
722:
716:
709:
708:
707:
688:
685:
679:
675:
669:
664:
660:
655:
651:
645:
639:
632:
631:
630:
627:
603:
597:
589:
582:
579:
572:
567:
563:
555:
554:
553:
551:
547:
537:
535:
531:
510:
507:
499:
496:
493:
487:
481:
475:
467:
464:
447:
444:
441:
436:
433:
427:
424:
421:
415:
409:
401:
398:
382:
374:
371:
368:
362:
356:
350:
342:
339:
325:
322:
319:
316:
313:
310:
304:
298:
290:
287:
286:
285:
283:
282:discount rate
280:denoting the
279:
275:
274:interest rate
272:denoting the
271:
266:
253:
247:
241:
214:
208:
205:
199:
193:
190:
184:
178:
171:
170:
169:
168:is given by:
167:
163:
158:
156:
152:
151:present value
148:
144:
140:
136:
132:
129:
118:
115:
107:
104:December 2009
96:
93:
89:
86:
82:
79:
75:
72:
68:
65: –
64:
60:
59:Find sources:
53:
49:
45:
39:
38:
34:
29:This article
27:
23:
18:
17:
779:
756:
706:reducing to
705:
629:Conversely:
628:
625:
543:
527:
277:
269:
267:
233:
165:
161:
159:
147:future value
142:
138:
134:
130:
127:
125:
110:
101:
91:
84:
77:
70:
58:
42:Please help
30:
858:References
74:newspapers
827:−
816:δ
765:δ
740:δ
676:δ
661:∫
564:δ
508:−
497:−
445:−
323:⋅
206:⋅
31:does not
866:Category
846:See also
583:′
88:scholar
52:removed
37:sources
90:
83:
76:
69:
61:
160:Thus
95:JSTOR
81:books
544:The
276:and
126:The
67:news
35:any
33:cite
284:):
157:.
46:by
868::
777:.
536:.
468::
402::
343::
291::
830:1
820:t
811:e
807:=
804:)
801:t
798:(
795:r
737:t
733:e
729:=
726:)
723:t
720:(
717:a
689:u
686:d
680:u
670:t
665:0
656:e
652:=
649:)
646:t
643:(
640:a
607:)
604:t
601:(
598:a
593:)
590:t
587:(
580:a
573:=
568:t
511:t
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500:d
494:1
491:(
488:=
485:)
482:t
479:(
476:a
448:d
442:1
437:d
434:t
428:+
425:1
422:=
419:)
416:t
413:(
410:a
383:t
379:)
375:i
372:+
369:1
366:(
363:=
360:)
357:t
354:(
351:a
326:i
320:t
317:+
314:1
311:=
308:)
305:t
302:(
299:a
278:d
270:i
254:.
251:)
248:0
245:(
242:A
230:.
218:)
215:t
212:(
209:a
203:)
200:0
197:(
194:A
191:=
188:)
185:t
182:(
179:A
166:t
162:a
145:(
143:t
139:t
135:t
133:(
131:a
117:)
111:(
106:)
102:(
92:·
85:·
78:·
71:·
54:.
40:.
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