1016:
1265:
896:
354:
667:
212:
588:
419:
787:
1120:
148:
1061:
62:
1153:
448:
249:
91:
911:
515:
1161:
802:
257:
487:
596:
153:
520:
1351:
455:
1278:
1346:
1341:
366:
486:; similarly, the existence of a strictly positive SDF is equivalent to the absence of arbitrage opportunities (see
678:
466:, when utility is separable and additive, though discounted by the risk-neutral rate), a "change of measure", "
1066:
107:
1024:
31:
28:. The name derives from the price of an asset being computable by "discounting" the future cash flow
25:
793:
21:
1129:
424:
225:
67:
1122:. Substituting this into the last expression and rearranging gives the following formula for the
1011:{\displaystyle 1=\operatorname {cov} ({\tilde {m}},{\tilde {R}})+E({\tilde {m}})E({\tilde {R}}).}
451:
1260:{\displaystyle E({\tilde {R}})-R_{f}=-R_{f}\operatorname {cov} ({\tilde {m}},{\tilde {R}}).}
483:
493:
219:
1335:
94:
1123:
463:
93:, and then taking the expectation. This definition is of fundamental importance in
891:{\displaystyle E({\tilde {m}}{\tilde {x}})=p,\quad E({\tilde {m}}{\tilde {R}})=1.}
349:{\displaystyle E({\tilde {m}}{\tilde {x}}_{i})=p_{i},{\text{for }}i=1,\ldots ,n.}
467:
902:
1270:
This shows that risk premiums are determined by covariances with any SDF.
459:
662:{\displaystyle E({\tilde {m}}{\tilde {R}}_{i})=1,\quad \forall i,}
359:
The stochastic discount factor is sometimes referred to as the
207:{\displaystyle {\tilde {x}}_{1},\ldots ,{\tilde {x}}_{n}}
590:
to denote the return, we can rewrite the definition as
583:{\displaystyle {\tilde {R}}_{i}={\tilde {x}}_{i}/p_{i}}
1164:
1132:
1069:
1027:
914:
805:
681:
599:
523:
496:
427:
369:
260:
228:
156:
110:
70:
34:
1259:
1147:
1114:
1055:
1010:
890:
781:
661:
582:
509:
454:. Other names sometimes used for the SDF are the "
442:
413:
348:
243:
206:
142:
85:
56:
414:{\displaystyle E({\tilde {m}}\,{\tilde {x}}_{i})}
450:can be interpreted as the kernel function in an
796:made up of the assets, then the SDF satisfies
482:The existence of an SDF is equivalent to the
8:
782:{\displaystyle E\left=0,\quad \forall i,j.}
1315:
1313:
1304:Asset Pricing and Portfolio Choice Theory
1240:
1239:
1225:
1224:
1209:
1193:
1172:
1171:
1163:
1134:
1133:
1131:
1106:
1097:
1077:
1076:
1068:
1047:
1029:
1028:
1026:
1021:Suppose there is a risk-free asset. Then
991:
990:
970:
969:
946:
945:
931:
930:
913:
868:
867:
856:
855:
825:
824:
813:
812:
804:
740:
729:
728:
718:
707:
706:
691:
690:
680:
631:
620:
619:
607:
606:
598:
574:
565:
559:
548:
547:
537:
526:
525:
522:
501:
495:
429:
428:
426:
402:
391:
390:
388:
377:
376:
368:
317:
308:
292:
281:
280:
268:
267:
259:
230:
229:
227:
198:
187:
186:
170:
159:
158:
155:
150:at the beginning of a period and payoffs
134:
115:
109:
72:
71:
69:
48:
37:
36:
33:
1324:. Princeton University Press. p. 9.
1297:
1295:
1293:
1289:
1115:{\displaystyle E({\tilde {m}})=1/R_{f}}
1126:of any asset or portfolio with return
7:
488:Fundamental theorem of asset pricing
222:), then SDF is any random variable
143:{\displaystyle p_{1},\ldots ,p_{n}}
1056:{\displaystyle {\tilde {R}}=R_{f}}
764:
650:
14:
901:By a simple standard identity on
490:). This being the case, then if
421:is written as an integral, then
57:{\displaystyle {\tilde {x}}_{i}}
18:stochastic discount factor (SDF)
848:
763:
649:
1251:
1245:
1230:
1221:
1183:
1177:
1168:
1139:
1088:
1082:
1073:
1034:
1002:
996:
987:
981:
975:
966:
957:
951:
936:
927:
879:
873:
861:
852:
836:
830:
818:
809:
746:
734:
712:
702:
696:
637:
625:
612:
603:
553:
531:
434:
408:
396:
382:
373:
298:
286:
273:
264:
235:
214:at the end of the period (all
192:
164:
77:
42:
1:
456:marginal rate of substitution
220:random (stochastic) variables
1148:{\displaystyle {\tilde {R}}}
443:{\displaystyle {\tilde {m}}}
244:{\displaystyle {\tilde {m}}}
86:{\displaystyle {\tilde {m}}}
104:assets with initial prices
1368:
1320:Cochrane, John H. (2001).
1306:. Oxford University Press.
64:by the stochastic factor
1279:Hansen–Jagannathan bound
363:as, if the expectation
1302:Kerry E. Back (2010).
1261:
1149:
1116:
1057:
1012:
892:
783:
663:
584:
517:is positive, by using
511:
444:
415:
350:
245:
208:
144:
87:
58:
1262:
1150:
1117:
1058:
1013:
893:
784:
664:
585:
512:
510:{\displaystyle p_{i}}
445:
416:
351:
246:
209:
145:
88:
59:
1352:Mathematical finance
1162:
1130:
1067:
1025:
912:
803:
792:Also, if there is a
679:
597:
521:
494:
425:
367:
258:
226:
154:
108:
68:
32:
26:mathematical finance
1347:Financial economics
1342:Stochastic calculus
472:state-price density
22:financial economics
16:The concept of the
1257:
1145:
1112:
1053:
1008:
888:
779:
672:and this implies
659:
580:
507:
452:integral transform
440:
411:
346:
241:
204:
140:
83:
54:
1248:
1233:
1180:
1142:
1085:
1037:
999:
978:
954:
939:
876:
864:
833:
821:
737:
715:
699:
628:
615:
556:
534:
437:
399:
385:
320:
289:
276:
238:
195:
167:
80:
45:
1359:
1326:
1325:
1317:
1308:
1307:
1299:
1266:
1264:
1263:
1258:
1250:
1249:
1241:
1235:
1234:
1226:
1214:
1213:
1198:
1197:
1182:
1181:
1173:
1154:
1152:
1151:
1146:
1144:
1143:
1135:
1121:
1119:
1118:
1113:
1111:
1110:
1101:
1087:
1086:
1078:
1062:
1060:
1059:
1054:
1052:
1051:
1039:
1038:
1030:
1017:
1015:
1014:
1009:
1001:
1000:
992:
980:
979:
971:
956:
955:
947:
941:
940:
932:
897:
895:
894:
889:
878:
877:
869:
866:
865:
857:
835:
834:
826:
823:
822:
814:
788:
786:
785:
780:
753:
749:
745:
744:
739:
738:
730:
723:
722:
717:
716:
708:
701:
700:
692:
668:
666:
665:
660:
636:
635:
630:
629:
621:
617:
616:
608:
589:
587:
586:
581:
579:
578:
569:
564:
563:
558:
557:
549:
542:
541:
536:
535:
527:
516:
514:
513:
508:
506:
505:
484:law of one price
470:deflator" or a "
458:" (the ratio of
449:
447:
446:
441:
439:
438:
430:
420:
418:
417:
412:
407:
406:
401:
400:
392:
387:
386:
378:
355:
353:
352:
347:
321:
318:
313:
312:
297:
296:
291:
290:
282:
278:
277:
269:
250:
248:
247:
242:
240:
239:
231:
213:
211:
210:
205:
203:
202:
197:
196:
188:
175:
174:
169:
168:
160:
149:
147:
146:
141:
139:
138:
120:
119:
92:
90:
89:
84:
82:
81:
73:
63:
61:
60:
55:
53:
52:
47:
46:
38:
1367:
1366:
1362:
1361:
1360:
1358:
1357:
1356:
1332:
1331:
1330:
1329:
1319:
1318:
1311:
1301:
1300:
1291:
1286:
1276:
1205:
1189:
1160:
1159:
1128:
1127:
1102:
1065:
1064:
1043:
1023:
1022:
910:
909:
801:
800:
727:
705:
689:
685:
677:
676:
618:
595:
594:
570:
546:
524:
519:
518:
497:
492:
491:
480:
423:
422:
389:
365:
364:
304:
279:
256:
255:
224:
223:
185:
157:
152:
151:
130:
111:
106:
105:
66:
65:
35:
30:
29:
12:
11:
5:
1365:
1363:
1355:
1354:
1349:
1344:
1334:
1333:
1328:
1327:
1309:
1288:
1287:
1285:
1282:
1275:
1272:
1268:
1267:
1256:
1253:
1247:
1244:
1238:
1232:
1229:
1223:
1220:
1217:
1212:
1208:
1204:
1201:
1196:
1192:
1188:
1185:
1179:
1176:
1170:
1167:
1141:
1138:
1109:
1105:
1100:
1096:
1093:
1090:
1084:
1081:
1075:
1072:
1050:
1046:
1042:
1036:
1033:
1019:
1018:
1007:
1004:
998:
995:
989:
986:
983:
977:
974:
968:
965:
962:
959:
953:
950:
944:
938:
935:
929:
926:
923:
920:
917:
899:
898:
887:
884:
881:
875:
872:
863:
860:
854:
851:
847:
844:
841:
838:
832:
829:
820:
817:
811:
808:
790:
789:
778:
775:
772:
769:
766:
762:
759:
756:
752:
748:
743:
736:
733:
726:
721:
714:
711:
704:
698:
695:
688:
684:
670:
669:
658:
655:
652:
648:
645:
642:
639:
634:
627:
624:
614:
611:
605:
602:
577:
573:
568:
562:
555:
552:
545:
540:
533:
530:
504:
500:
479:
476:
436:
433:
410:
405:
398:
395:
384:
381:
375:
372:
361:pricing kernel
357:
356:
345:
342:
339:
336:
333:
330:
327:
324:
316:
311:
307:
303:
300:
295:
288:
285:
275:
272:
266:
263:
237:
234:
201:
194:
191:
184:
181:
178:
173:
166:
163:
137:
133:
129:
126:
123:
118:
114:
79:
76:
51:
44:
41:
13:
10:
9:
6:
4:
3:
2:
1364:
1353:
1350:
1348:
1345:
1343:
1340:
1339:
1337:
1323:
1322:Asset Pricing
1316:
1314:
1310:
1305:
1298:
1296:
1294:
1290:
1283:
1281:
1280:
1273:
1271:
1254:
1242:
1236:
1227:
1218:
1215:
1210:
1206:
1202:
1199:
1194:
1190:
1186:
1174:
1165:
1158:
1157:
1156:
1136:
1125:
1107:
1103:
1098:
1094:
1091:
1079:
1070:
1048:
1044:
1040:
1031:
1005:
993:
984:
972:
963:
960:
948:
942:
933:
924:
921:
918:
915:
908:
907:
906:
904:
885:
882:
870:
858:
849:
845:
842:
839:
827:
815:
806:
799:
798:
797:
795:
776:
773:
770:
767:
760:
757:
754:
750:
741:
731:
724:
719:
709:
693:
686:
682:
675:
674:
673:
656:
653:
646:
643:
640:
632:
622:
609:
600:
593:
592:
591:
575:
571:
566:
560:
550:
543:
538:
528:
502:
498:
489:
485:
477:
475:
473:
469:
465:
461:
457:
453:
431:
403:
393:
379:
370:
362:
343:
340:
337:
334:
331:
328:
325:
322:
314:
309:
305:
301:
293:
283:
270:
261:
254:
253:
252:
232:
221:
217:
199:
189:
182:
179:
176:
171:
161:
135:
131:
127:
124:
121:
116:
112:
103:
100:If there are
98:
96:
95:asset pricing
74:
49:
39:
27:
23:
19:
1321:
1303:
1277:
1269:
1124:risk premium
1020:
900:
791:
671:
481:
471:
360:
358:
215:
101:
99:
17:
15:
903:covariances
468:state-price
251:satisfying
20:is used in
1336:Categories
1284:References
905:, we have
478:Properties
1246:~
1231:~
1219:
1203:−
1187:−
1178:~
1140:~
1083:~
1035:~
997:~
976:~
952:~
937:~
925:
874:~
862:~
831:~
819:~
794:portfolio
765:∀
735:~
725:−
713:~
697:~
651:∀
626:~
613:~
554:~
532:~
435:~
397:~
383:~
335:…
319:for
287:~
274:~
236:~
193:~
180:…
165:~
125:…
78:~
43:~
1274:See also
1063:implies
460:utility
464:states
218:s are
24:and
1216:cov
922:cov
474:".
462:of
97:.
1338::
1312:^
1292:^
1155::
886:1.
1255:.
1252:)
1243:R
1237:,
1228:m
1222:(
1211:f
1207:R
1200:=
1195:f
1191:R
1184:)
1175:R
1169:(
1166:E
1137:R
1108:f
1104:R
1099:/
1095:1
1092:=
1089:)
1080:m
1074:(
1071:E
1049:f
1045:R
1041:=
1032:R
1006:.
1003:)
994:R
988:(
985:E
982:)
973:m
967:(
964:E
961:+
958:)
949:R
943:,
934:m
928:(
919:=
916:1
883:=
880:)
871:R
859:m
853:(
850:E
846:,
843:p
840:=
837:)
828:x
816:m
810:(
807:E
777:.
774:j
771:,
768:i
761:,
758:0
755:=
751:]
747:)
742:j
732:R
720:i
710:R
703:(
694:m
687:[
683:E
657:,
654:i
647:,
644:1
641:=
638:)
633:i
623:R
610:m
604:(
601:E
576:i
572:p
567:/
561:i
551:x
544:=
539:i
529:R
503:i
499:p
432:m
409:)
404:i
394:x
380:m
374:(
371:E
344:.
341:n
338:,
332:,
329:1
326:=
323:i
315:,
310:i
306:p
302:=
299:)
294:i
284:x
271:m
265:(
262:E
233:m
216:x
200:n
190:x
183:,
177:,
172:1
162:x
136:n
132:p
128:,
122:,
117:1
113:p
102:n
75:m
50:i
40:x
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