12
1
Life tables and Survival analysis
Life tables
Anousschka van der Meulen
Statistics Methods (201208)
Statistics Netherlands
The Hague/Heerlen, 2012
Explanation of symbols
.
data not available
*
provisional figure
**
revised provisional figure (but not definite)
x
publication prohibited (confidential figure)
–
nil
–
(between two figures) inclusive
0 (0.0)
less than half of unit concerned
empty cell
not applicable
2011–2012
2011 to 2012 inclusive
2011/2012
average for 2011 up to and including 2012
2011/’12
crop year, financial year, school year etc. beginning in 2011 and ending in 2012
2009/’10–
2011/’12
crop year, financial year, etc. 2009/’10 to 2011/’12 inclusive
Due to rounding, some totals may not correspond with the sum of the separate figures.
Publisher
Statistics Netherlands
Henri Faasdreef 312
2492 JP The Hague
Prepress
Statistics Netherlands
Grafimedia
Cover
Teldesign, Rotterdam
Information
Telephone +31 88 570 70 70
Telefax +31 70 337 59 94
Via contact form:
www.cbs.nl/information
60165201208 X-37
Where to order
E-mail: [email protected]
Telefax +31 45 570 62 68
Internet
www.cbs.nl
ISSN: 1876-0333
© Statistics Netherlands,
The Hague/Heerlen, 2012.
Reproduction is permitted,
provided Statistics Netherlands is quoted as source.
7DEOHRIFRQWHQWV
,QWURGXFWLRQWRWKHVXEWKHPH 3HULRGWDEOHE\DYHUDJHDJHRQ-DQXDU\
/LIHWDEOHV 3HULRGWDEOHE\DJHDWODVWELUWKGD\
&RQFOXVLRQ
5HIHUHQFHV ,QWURGXFWLRQWRWKHVXEWKHPH
*HQHUDOGHVFULSWLRQDQGUHDGLQJJXLGH
/LIHWDEOHVGHVFULEHWKHPRUWDOLW\DQGVXUYLYDOSDWWHUQVRIDSRSXODWLRQ2QWKHEDVLV
RI DQQXDO PRUWDOLW\ UDWLRV IRU HDFK DJH RU DJH JURXS WKHVH WDEOHV SURYLGH
LQIRUPDWLRQ RQ SDUDPHWHUV VXFK DV WKH QXPEHU RI VXUYLYRUV WKH QXPEHU RI GHDWKV
DQG WKHOLIH H[SHFWDQF\ $ OLIH WDEOH LV JHQHUDOO\ D WDEOH ZLWK ILYH FROXPQV 7KHVH
FROXPQVFRQWDLQWKHIROORZLQJYDULDEOHVLQVXFFHVVLRQDJHPRUWDOLW\UDWLRQXPEHU
RIVXUYLYRUVQXPEHURIGHDWKVDQGOLIHH[SHFWDQF\VHHWDEOH
7DEOH3DUWRIWKHSHULRGOLIHWDEOHE\DYHUDJHDJHRQ-DQXDU\PHQ
SHULRG
$JH
0RUWDOLW\UDWLR 1XPEHU
1XPEHURI
/LIHH[SHFWDQF\
RI
GHDWKV
H[
;
QT[
VXUYLYRUV
QG[
O[
$OLIHWDEOHFDQVKRZDUHDGHUKRZPDQ\SHRSOHRIDFHUWDLQDJHZLOOUHDFKWKHQH[W
DJH LQ D FHUWDLQ \HDU ,Q DGGLWLRQ WKH OLIH H[SHFWDQF\ WKH QXPEHU RI \HDUV RI OLIH
H[SHFWHGWRUHPDLQFDQEHFDOFXODWHGDWELUWKDQGDWYDULRXVDJHV$OLIHWDEOHFDQ
IRU H[DPSOH DOVR EH XVHGWR FRPSDUH WKH PRUWDOLW\ LQ D FHUWDLQ SDWLHQW SRSXODWLRQ
ZLWK WKH REVHUYHG PRUWDOLW\ LQ WKH HQWLUH SRSXODWLRQ )RU H[DPSOH LW FDQ EH
FDOFXODWHGZKHWKHUWKLVSDWLHQWSRSXODWLRQKDVDKLJKHURUORZHUPRUWDOLW\UDWHWKDQ
WKHHQWLUHSRSXODWLRQ,WLVSUREDEO\UDWKHUVXUSULVLQJWKDWZHDUHWDONLQJDERXWOLIH
H[SHFWDQF\ZKLOHDOOWKHSHRSOHLQTXHVWLRQKDYHDOUHDG\GLHG7KLVVKRXOGDFWXDOO\
EHXQGHUVWRRGDVµDYHUDJH¶LQVWHDGRIµH[SHFWDQF\¶
7KHUHDUHGLIIHUHQWW\SHVRIOLIHWDEOHV6WDWLVWLFV1HWKHUODQGVSXEOLVKHVWZRW\SHVRI
OLIHWDEOHVLQ6WDW/LQHSHULRGWDEOHVE\DYHUDJHDJHRQ-DQXDU\DQGSHULRGWDEOHV
E\DJHDWODVWELUWKGD\7KHVHDUHERWKWDEOHVWKDWGHVFULEHWKHPRUWDOLW\GHYHORSPHQW
LQDFHUWDLQSHULRGD\HDURUPXOWLSOH\HDUV7KHSHULRGWDEOHVE\DYHUDJHDJHRQ
-DQXDU\DUHGLVFXVVHGLQFKDSWHUDQGWKHSHULRGWDEOHVE\DJHDWODVWELUWKGD\LQ
FKDSWHU6HH9DQGHU0HXOHQDQG-DQVVHQ
/LIH WDEOHV KDYH EHHQ SXEOLVKHG VLQFH 7KH ILUVW 6WDWLVWLFV 1HWKHUODQGV
SXEOLFDWLRQ GDWHV IURP 6LQFH WKHQ YDULRXV W\SHV RI WDEOHV KDYH EHHQ
SXEOLVKHG 1RZDGD\V 6WDWLVWLFV 1HWKHUODQGV SXEOLVKHV LQ 6WDW/LQH OLIH WDEOHV SHU
ILYHFDOHQGDU\HDUVVWDUWLQJZLWKWKH\HDU
6FRSH
,QDGGLWLRQWRWKHSHULRGWDEOHVUHIHUUHGWRDERYHWKHUHDUHDOVRWKHVRFDOOHGFRKRUW
WDEOHV RU JHQHUDWLRQ WDEOHV 7KHVH GHVFULEH WKH PRUWDOLW\ LQ D FHUWDLQ ELUWK FRKRUW
3HULRGWDEOHVDUHSXEOLVKHGLQ6WDW/LQHEXWJHQHUDWLRQWDEOHVDUHQRWSXEOLVKHGKHUH
DW SUHVHQW ,Q WKH SDVW 6WDWLVWLFV 1HWKHUODQGV KDV SXEOLVKHG JHQHUDWLRQ WDEOHV LQ
SDSHUSXEOLFDWLRQV
7KH DERYHPHQWLRQHG WDEOHV DUH WDEOHV ZKHUH PHPEHUV RI WKH SRSXODWLRQ DUH RQO\
UHPRYHGIURPWKHWDEOHDVDUHVXOWRIGHDWK7KHUHDUHDOVRWDEOHVZKHUHPHPEHUVRI
WKH SRSXODWLRQ DUH ERWK HQWHUHG LQ DQG UHPRYHG IURP WKH WDEOH 7KH OLIH WDEOHV E\
PDULWDO VWDWXV DUH DQ H[DPSOH RI WKLV ,Q WKH SDVW 6WDWLVWLFV 1HWKHUODQGV KDV DOVR
SXEOLVKHGWKHVHWDEOHVLQSDSHUSXEOLFDWLRQV$WSUHVHQWWKH\DUHQRW\HWLQFOXGHGLQ
6WDW/LQH
$ FRKRUW WDEOH LV DFWXDOO\ WKH PRVW REYLRXV W\SH RI OLIH WDEOH 7KLV WDEOH
GHPRQVWUDWHV WKH VXUYLYDODQG PRUWDOLW\ SDWWHUQ IRU SHRSOH ZKR DUH DOO ERUQ LQWKH
VDPH\HDURULQWKHVDPHSHULRGRIIRUH[DPSOHILYH\HDUV7KHGUDZEDFNRIWKH
FRKRUW OLIH WDEOH LV WKDW GDWD DUH QHHGHG IRU WKH HQWLUH OLIH SURJUHVVLRQ RI D ELUWK
FRKRUWLQRWKHUZRUGVXSWRDQGLQFOXGLQJWKHGHDWKRIDOOLWVPHPEHUV+RZHYHU
WKHVH GDWD DUH RIWHQ QRW DYDLODEOH )RU UHFHQW ELUWK FRKRUWV WKLV LQIRUPDWLRQ LV QRW
DYDLODEOH DW DOO )RU H[DPSOH DW SUHVHQW IRU WKH FRKRUW µERUQ LQ WKH \HDU ¶
LQIRUPDWLRQ LV RQO\ DYDLODEOH DERXW WKH PRUWDOLW\ LQ WKH ILUVW IHZ \HDUV RI OLIH
,QGHHG RQ -DQXDU\ WKH ROGHVW PHPEHU RI WKLV FRKRUW ERUQ RQ -DQXDU\
ZDVRQO\MXVWVHYHQ\HDUVRIDJH,WLVSRVVLEOHWRGUDZXSWKHFRPSOHWHFRKRUW
OLIH WDEOH RQO\ IRU YHU\ ROG FRKRUWV WKDW KDYH FRPSOHWHO\ µGLHG RXW¶ WKH OLIH
H[SHFWDQF\FDQWKHQDOVREHFDOFXODWHGRQWKLVEDVLV7KLVH[DPSOHSURYLGHVDJRRG
LQGLFDWLRQ RI KRZ µH[SHFWDQF\¶ LQ D OLIH WDEOH VKRXOG EH LQWHUSUHWHG QDPHO\ DV DQ
µDYHUDJH¶$IWHUDOODFRKRUWZKRVHPHPEHUVDUHDOOGHDGFOHDUO\KDVQRUHPDLQLQJ
OLIHH[SHFWDQF\
7KHSHULRGWDEOHZDVGHYHORSHGWREHDEOHWRPDNHVWDWHPHQWVDERXWPRUWDOLW\UDWLRV
DQG WKH OLIH H[SHFWDQF\ IRU D SRSXODWLRQ EDVHG RQ UHFHQW GDWD 7KH REVHUYHG DJH
VSHFLILFPRUWDOLW\GDWDIRUDFHUWDLQFDOHQGDU\HDURUPXOWLSOHFDOHQGDU\HDUVWRJHWKHU
IRUPWKHEDVLVIRUDSHULRGWDEOH,WLVDVVXPHGWKDWWKLVDJHVSHFLILFPRUWDOLW\GDWD
ZLOO DOVR DSSO\ DV WKH SRSXODWLRQ E\ DJH LQ WKH REVHUYHG SHULRG UHDFKHV WKH DJHV
FRQFHUQHGLQWKHLUODWHUOLYHV,QWKLVZD\WKLVFUHDWHVZKDWLVNQRZQDVDV\QWKHWLF
ELUWKFRKRUW7KLVLVUHSUHVHQWHGLQWKH/H[LVGLDJUDPLQILJXUH$/H[LVGLDJUDP
KDVWKHDJHRQWKHYHUWLFDOD[LVDQGWKHSHULRGRQWKHKRUL]RQWDOD[LV%HFDXVHWKH
VDPH XQLW LV XVHG IRU ERWK RI WKHVH XVXDOO\ RQH \HDU WKH GLDJRQDOV UHSUHVHQW WKH
GLIIHUHQWFRKRUWV7KHFROXPQVVKRZREVHUYDWLRQVLQDFDOHQGDU\HDUDQGWKHURZV
REVHUYDWLRQVIRUDFHUWDLQDJH)RUPRUHLQIRUPDWLRQDERXWWKHLQWHUSUHWDWLRQRIWKH
/H[LVGLDJUDPSOHDVHUHIHUWR3UHVVDW
,Q ILJXUH WKH YHUWLFDO FROXPQ UHSUHVHQWV WKH REVHUYDWLRQV RI DOO DJHV LQ WKH
FDOHQGDU \HDU 7KHVH DUH REVHUYDWLRQV IURP GLIIHUHQW ELUWK FRKRUWV +HUH WKH
OLJKWJUH\GLDPRQGVKRZVWKHGDWDIRUFRKRUWWKHGDUNJUH\GLDPRQGWKHGDWD
IRUFRKRUWDQGWKHEODFNGLDPRQGWKHGDWDIRUFRKRUW,QWKHSHULRGOLIH
WDEOHIRUWKHDJHVSHFLILFPRUWDOLW\GDWDPRUWDOLW\UDWLRVDVREVHUYHGLQ
DUHDSSOLFDEOHIRUWKHDJHVSHFLILFPRUWDOLW\YLHZHGLQDV\QWKHWLFELUWKFRKRUWWKH
GLDPRQGVLQWKHGLDJRQDO7KLVLVUHSUHVHQWHGLQWKHILJXUHE\WKHKRUL]RQWDODUURZV
)RUWKHQHZERUQVLQ LW LV DVVXPHG WKDW DWIRUH[DPSOH \HDUV RI DJH WKH\
ZLOO EH H[SRVHG WR WKH PRUWDOLW\ UDWLRV H[SHULHQFHG E\ \HDUROGV LQ 7KLV
DSSOLHVLQWKHVDPHZD\IRUDOODJHV
7KLV DVVXPSWLRQ KDV LPSRUWDQW LPSOLFDWLRQV IRU WKH LQWHUSUHWDWLRQ RI WKH OLIH
H[SHFWDQF\LQWKHSHULRGOLIHWDEOH7KLVFRQFHUQVWKHQXPEHURIUHPDLQLQJ\HDUVRI
OLIHWKDWWKHVXUYLYRUVRIDFHUWDLQDJHFDQH[SHFWWROLYHRQDYHUDJHSURYLGHGWKDW
WKHDJHVSHFLILFPRUWDOLW\UDWLRVDVREVHUYHGLQWKH\HDURQZKLFKWKHWDEOHLVEDVHG
GR QRW FKDQJH GXULQJ WKH UHVW RI WKHLU OLYHV 7KH SUDFWLFDO YDOXH RI WKHVH WDEOHV LV
WKHUHIRUHGHWHUPLQHGE\KRZWUXHWKLVDVVXPSWLRQLV
)LJXUH /H[LV GLDJUDP ZLWK REVHUYDWLRQV LQ FDOHQGDU \HDU DQG WKH
K\SRWKHWLFDOFRKRUWWKDWLVIRUPHGRQWKLVEDVLV
7KH IRUPXODV WKDW DUH XVHG LQ WKLV UHSRUW DUH RQO\ LQWHQGHG IRU OLIH WDEOHV SHU DJH
\HDU DOVR NQRZQ DV FRPSOHWH OLIH WDEOHV ,Q DGGLWLRQ WR EHLQJ FDOFXODWHG SHU DJH
\HDUOLIHWDEOHVFDQDOVREHFDOFXODWHGIRUJURXSVRIDJH\HDUV7KHVHDUHDOVRNQRZQ
DVµDEULGJHG¶OLIHWDEOHVDQGDUHQRWGLVFXVVHGLQWKLVUHSRUW
3ODFHLQWKHVWDWLVWLFDOSURFHVV
7KH FRPSRVLWLRQ RI OLIH WDEOHV LV SUHFHGHG E\ D SURFHVV RI REVHUYDWLRQV DQG
WUHDWPHQWV 0RUWDOLW\ UDWLRV IRUP WKH EDVLV RI WKH OLIH WDEOH 0RUWDOLW\ UDWLRV DUH
FDOFXODWHG XVLQJ WKH SRSXODWLRQ DQG PRUWDOLW\ GDWD 6LQFH 6WDWLVWLFV
1HWKHUODQGVKDVKDGDFFHVVWRWKHSRSXODWLRQDQGPRUWDOLW\GDWDRIWKH1HWKHUODQGV
RULJLQDWLQJ IURP WKH 0XQLFLSDO 3HUVRQDO 5HFRUGV 'DWDEDVH *%$ %HIRUH 6WDWLVWLFV1HWKHUODQGVDOVRUHFHLYHGLQIRUPDWLRQDERXWWKHSRSXODWLRQDQGPRUWDOLW\
IURP WKH PXQLFLSDOLWLHV $W WKDW WLPH KRZHYHU WKH LQIRUPDWLRQ ZDV SURYLGHG DQG
SURFHVVHGLQDGLIIHUHQWZD\3ULQVDQG.XLMSHU
0RUWDOLW\ UDWLRV SOD\ IRU H[DPSOH DQ LPSRUWDQW UROH LQ WKH SRSXODWLRQ SURJQRVLV
IURP6WDWLVWLFV1HWKHUODQGV
'HILQLWLRQV
&RQFHSW
'HVFULSWLRQ
$GPLQLVWUDWLYHFRUUHFWLRQ
(DFKLQFOXVLRQLQDQGGHOHWLRQIURPWKHPXQLFLSDOGDWDEDVH
RWKHUWKDQDVDUHVXOWRIDELUWKGHDWKUHVLGHQFHGHSDUWXUHRU
PXQLFLSDOERUGHUFKDQJHUHJLVWHUHGDVVXFK
/LIHWDEOH
&RKRUWJHQHUDWLRQ
/LIHH[SHFWDQF\
5DGL[
0RUWDOLW\UDWLRRU
SUREDELOLW\
7DEOHWKDWLQGLFDWHVKRZPDQ\SHRSOHIURPDJURXSRI
WKRXVDQGQHZERUQVZLOOVWLOOEHDOLYHDWDODWHUDJH
$ELUWKFRKRUWRUFRKRUWIRUVKRUWFRQVLVWVRIWKHSHRSOHZKR
DUHDOOERUQLQWKHVDPHSHULRG7KLVSRSXODWLRQFDQFKDQJH
DQQXDOO\GXHWRWKHLQIOX[DQGRXWIORZRIPLJUDQWVDQGGXHWR
GHDWK
7KHDYHUDJHQXPEHURI\HDUVWKDWSHRSOHRIDFHUWDLQDJH[
KDYHOLYHGIURPWKLVDJH[,WLVDQDYHUDJHYDOXHIRUWKHHQWLUH
SRSXODWLRQ,WLVULVN\WRPDNHVWDWHPHQWVDERXWLQGLYLGXDOVRID
FHUWDLQDJHXVLQJWKHOLIHWDEOHVGHVFULEHGKHUH$IWHUDOOJURXSV
RISHRSOHRIDFHUWDLQDJHFDQEHYHU\KHWHURJHQHRXVZKHQLW
FRPHVWRFHUWDLQDVSHFWVZKLFKLPSRUWDQWIRUPRUWDOLW\VXFKDV
KHDOWKDQGWKHULVNVWKDWSHRSOHUXQLQWKHLUGDLO\OLYHVIRU
H[DPSOHGULYLQJLQWUDIILFRURFFXSDWLRQDOULVNV7KHUHPDLQLQJ
OLIHH[SHFWDQF\FDQEHFDOFXODWHGIRUHDFKDJH
6L]HRIWKHSRSXODWLRQRI\HDUROGVRUWKHEHJLQQLQJ
SRSXODWLRQLQWKHWDEOHSRSXODWLRQ7KLVLVQRUPDOO\VHWDW
WKRXVDQGE\6WDWLVWLFV1HWKHUODQGV
7KHDYHUDJHSUREDELOLW\ RIGHDWKEHWZHHQWKHFXUUHQWDJHDQG
WKHQH[WDJH
6WULFWO\ VSHDNLQJ ZH FDQQRW WDON DERXW SUREDELOLWLHV ZLWK UHVSHFW WR OLIH WDEOHV JLYHQ WKH
KHWHURJHQHLW\ RI WKH JURXSV ,W ZRXOG EH EHWWHU WR WDON DERXW SURSRUWLRQV RU DYHUDJH
SUREDELOLWLHV+RZHYHUWRNHHSLQOLQH ZLWKQDWLRQDODQGLQWHUQDWLRQDOSXEOLFDWLRQVDQGIRU
WKHVDNHRIFRQYHQLHQFHZHZLOOXVHWKHFRQFHSWRIµSUREDELOLW\¶LQWKHUHVWRIWKLVUHSRUW
'HILQLWLRQRIDJH
µ$JH¶LQWKHOLIHWDEOHV
,QWKHOLIHWDEOHVWKDW6WDWLVWLFV1HWKHUODQGVSXEOLVKHVLQ6WDW/LQHWZRGLIIHUHQWDJHV
DUHXVHG
7KH DJH WKDW D SHUVRQ KDV DW D FHUWDLQ WLPH LV WKH WLPH GLIIHUHQFH EHWZHHQ WKDW
PRPHQW DQG WKH PRPHQW ZKHQ WKH SHUVRQ ZDV ERUQ ,W LV SRVVLEOH WR H[SUHVV WKLV
WLPH GLIIHUHQFH YHU\ SUHFLVHO\ LW FDQ EH DFFXUDWH GRZQ WR WKH QXPEHU RI KRXUV RU
PLQXWHV,QSUDFWLFHVXFKDFFXUDF\LVXVXDOO\QRWUHTXLUHGDQGDJHLVH[SUHVVHGLQ
\HDUVRUSDUWLDO\HDUV
,QWKHOLIHWDEOHVWKDW6WDWLVWLFV1HWKHUODQGVSXEOLVKHVDJHLVYLHZHGLQWZRGLIIHUHQW
ZD\V 7KH ILUVW LV WKH DJH WKDW D SHUVRQ KDV DFKLHYHG RQ KLV RU KHU PRVW UHFHQW
ELUWKGD\ZKLFKLVUHIHUUHGWRDVµDJHDWODVWELUWKGD\¶7KLVLVH[SUHVVHGDVDZKROH
QXPEHUDVLQRUGLQDU\ODQJXDJH:KHQWKHSHUVRQUHDFKHVKLVRUKHUQH[WELUWKGD\
WKHDJHLQFUHDVHVE\)RUH[DPSOHDSHUVRQERUQRQ'HFHPEHULV\HDUV
RIDJHEHWZHHQ'HFHPEHUDQG'HFHPEHU2Q'HFHPEHUWKLV
SHUVRQ WXUQV \HDUV ROG 7KH µDJH DW ODVW ELUWKGD\¶ LV D WUXQFDWHG QXPEHU 7KLV
PHDQV WKDW WKH GHFLPDO SDUW RI D QRQURXQGHG QXPEHU LV LJQRUHG $V D UHVXOW RQ
DYHUDJHWKLVDJHLVDKDOI\HDUORZHUWKDQWKHDJHWKHSHUVRQZRXOGKDYHLIWKHDJH
ZHUHPHDVXUHGPRUHDFFXUDWHO\LQWHUPVRIGD\VDQGKRXUV
7KHVHFRQGZD\DJHLVYLHZHGLQOLIHWDEOHVLVUHIHUUHGWRDVWKHµDJHRQ-DQXDU\¶
7KLVLVWKHDJHWKDWDSHUVRQKDVDWWKHVWDUWRIWKHQHZ\HDU$FKDUDFWHULVWLFRIWKLV
SDUWLFXODUGHILQLWLRQRIDJHLVWKDWDOOSHRSOHLQWKHSRSXODWLRQWXUQD\HDUROGHUDWWKH
VDPH WLPH RQ -DQXDU\ UHJDUGOHVV RI WKHLU DFWXDO ELUWKGD\ 7KH DJH GRHV QRW
FKDQJH GXULQJ WKH FDOHQGDU \HDU $VVXPLQJ WKDW WKH ELUWKGD\V DUH VSUHDG HYHQO\
WKURXJKRXWWKHFDOHQGDU\HDUWKHµDJHRQ-DQXDU\¶LVUHSUHVHQWHGE\DQXPEHUZLWK
RQHGHFLPDOHTXDOWRWKHVRFDOOHGKDOIDJH:HZLOOFODULI\WKLVXVLQJDQH[DPSOH
7KHDERYHPHQWLRQHGSHUVRQZLWKWKHGDWHRIELUWKRI'HFHPEHUWXUQHG
\HDUV RI DJH RQ -DQXDU\ 7KDW VDPH DJH LV UHDFKHG E\ VRPHRQH ZKR ZDV
ERUQIRUH[DPSOHLQ-DQXDU\2QDYHUDJHWKHSHRSOHLQWKHJHQHUDWLRQ
ZHUHERUQDURXQG-XO\RIWKDW\HDUZKLFKLVZK\WKH\ZRXOGEH\HDUVRIDJH
RQ -DQXDU\ 2Q -DQXDU\ WKH SHRSOH ERUQ LQ ZRXOG WXUQ \HDUVRIDJH2QWKHILUVWGD\RIWKHQHZ\HDUWKHµDJHRQ-DQXDU\¶LVFRUUHFWRQ
DYHUDJH%HFDXVHWKLVDJHGRHVQRWFKDQJHGXULQJWKHFDOHQGDU\HDUWKHµDJHDWODVW
ELUWKGD\¶MXVWOLNHWKHµDJHRQ-DQXDU\¶LVRQDYHUDJHDKDOI\HDUWRRORZGXULQJ
WKHFDOHQGDU\HDU
/LIHWDEOHVE\DJHRQ-DQXDU\
,QWKHOLIHWDEOHEDVHGRQµDJHRQ-DQXDU\¶WKHSHRSOHDUHJURXSHGE\ELUWK\HDU
$VSUHYLRXVO\VWDWHGWKLVDJHLVRQDYHUDJHDKDOI\HDUWRRORZGXULQJWKHFDOHQGDU
\HDU 7KLV LV YLVLEOH LQ WKH /H[LV GLDJUDP LQ ILJXUH 7KH FHQWURLG RI WKH VKDGHG
GLDPRQG OLHV RQ WKH KRUL]RQWDO OLQH IRU DJH 7KH YHUWLFDO OLQH WR WKH OHIW RI WKH
GLDPRQG VKRZV WKH DJH DW WKH VWDUW RI WKH QHZ \HDU 7KLV OLQH UXQV IURP DJH WR
DJHWKHUHIRUHDQDYHUDJHRI7KLVDJHLVVKRZQLQWKHOHIWFROXPQRIWKHOLIH
WDEOH
6LQFHLQWKHVKDGHGGLDPRQGWKHDYHUDJHDJHRISHRSOHZKRGLHGLVHTXDOWRWKH
TXHVWLRQPD\EHUDLVHGZK\WKLVDJHLVQRWXVHGLQWKHSUHVHQWDWLRQLQVWHDGRIWKHDJH
7KHUH FRXOG EH VRPHWKLQJ WR EH VDLG IRU WKLV DOEHLW WKDW LQ WKDW FDVH WKH
SUHVHQWHGDJHRIGHDWKZRXOGQRWEHUHDFKHGE\URXJKO\KDOIRIWKHSRSXODWLRQ
/LIHWDEOHE\DJHDWODVWELUWKGD\
$VVWDWHGDERYHWKHSUHVHQWHGDJHLQWKHOLIHWDEOHE\µDJHDWODVWELUWKGD\¶LVDOVRD
KDOI\HDUWRRORZ7KH/H[LVGLDJUDPLQILJXUHDOVRGHPRQVWUDWHVWKLV7KHFHQWURLG
RIWKHVKDGHGGLDPRQGOLHVRQWKHKRUL]RQWDOOLQHFRUUHVSRQGLQJWRDJH7KHDJH
DWODVWELUWKGD\WKDWFRUUHVSRQGVWRWKLVGLDPRQGLVDVVKRZQE\WKHKRUL]RQWDOOLQH
DWWKHERWWRPRIWKHGLDPRQG
7KHVDPHDVLQWKHOLIHWDEOHE\DJHRQ-DQXDU\DGHFLVLRQZDVPDGHWREDVHWKH
SUHVHQWDWLRQRQWKHDJHWKDWWKHSHRSOHKDYHDWWKHWLPHWKDWWKHLUOLIHOLQHUHDFKHVWKH
GLDPRQG 7KH DJH WKDW WKH\ KDYH DW WKDW PRPHQW LV D KDOI \HDU ORZHU WKDQ WKH
DYHUDJHDJHRIGHDWKLQWKHGLDPRQG
µ$JH¶LQWKHREVHUYHGSRSXODWLRQRIWKH1HWKHUODQGV
7KHGHILQLWLRQRIDJHWKDW6WDWLVWLFV1HWKHUODQGVXVHVLQGHVFULELQJWKHSRSXODWLRQRI
WKH1HWKHUODQGVDQGDOVRWKHPRUWDOLW\LQWKLVSRSXODWLRQGHYLDWHVIURPWKHDJHVWKDW
DUH XVHG LQ WKH OLIH WDEOHV 6WDWLVWLFV 1HWKHUODQGV SXEOLVKHV WKH SRSXODWLRQ RI WKH
1HWKHUODQGVE\DJHLQ6WDW/LQHXQGHUWKHWLWOHµ3RSXODWLRQRQ-DQXDU\DJHJHQGHU
PDULWDO VWDWXV¶ ,Q WKLV SXEOLFDWLRQ WKH DJH LV QRW SUHVHQWHG DV D KDOI DJH EXW DV D
ZKROHDJH+HUHDJHLVXQGHUVWRRGDVWKHDJHUHDFKHGDWWKHODVWELUWKGD\EHIRUHWKH
QHZ\HDU:HSURYLGHDQH[DPSOHWRFODULI\WKLV$FFRUGLQJWRWKHµ3RSXODWLRQRQ
-DQXDU\ DJH JHQGHU PDULWDO VWDWXV¶ D PDQ ERUQ LQ LV \HDUV ROG RQ -DQXDU\$IWHUDOOWKLVSHUVRQ¶VODVWELUWKGD\ZDVLQDQGDWWKDWWLPHKH
FHOHEUDWHGKLV ELUWKGD\7RFRPSDUHDFFRUGLQJWRWKHOLIHWDEOHVIRUWKLV
PDQ¶VDJHLVDKDOI\HDUKLJKHUQDPHO\
WK
7RPDNHWKLVHYHQPRUHFRPSOLFDWHGIRUWKHSXEOLFDWLRQLQ6WDW/LQHRIWKHPRUWDOLW\
E\DJHWKHDJHLVFDOFXODWHGDVWKHGLIIHUHQFHEHWZHHQWKH\HDURIGHDWKDQGWKH\HDU
RI ELUWK 7KHUHIRUH VRPHRQH ERUQ LQ DQG ZKR GLHV LQ LV \HDUV ROG
DFFRUGLQJWRWKLVGHILQLWLRQWKHUHIRUHDKDOI\HDUROGHUWKDQWKLVSHUVRQ¶VDJHLQWKH
OLIHWDEOHIRU7KLVDJHLVUHIHUUHGWRDVWKHµDJHRQ'HFHPEHU¶$IWHUDOODW
WKHHQGRIDFDOHQGDU\HDUHYHU\RQHZLOOKDYHDOUHDG\FHOHEUDWHGWKHLUELUWKGD\LQ
WKDW\HDU
3OHDVHQRWH7KHUHLVDPXWXDOGHSHQGHQF\EHWZHHQDJHFRKRUWDQGFDOHQGDU\HDU
)RUDFHUWDLQFDOHQGDU\HDUDQGJLYHQELUWK\HDUWKHDJHLVIL[HG
*HQHUDOQRWDWLRQ
7KHIROORZLQJQRWDWLRQVDUHXVHGIRUWKHSDUDPHWHUVLQWKHOLIHWDEOH
T PRUWDOLW\ UDWLR RU PRUWDOLW\ SUREDELOLW\ 7KLV LV WKH SUREDELOLW\ RI GHDWK
EHWZHHQDJH[DQGWKHQH[WDJH[QLQFDOHQGDU\HDUW7KHVXEVFULSWQLQGLFDWHV
WKHOHQJWKRIWKHDJHLQWHUYDO
Q
[ W
O WKHQXPEHURIVXUYLYRUVDWDJH[
[
G WKHQXPEHURISHRSOHWKDWGLHDWDJH[
[
7KHIROORZLQJQRWDWLRQVDSSO\IRUWKHSDUDPHWHUVLQWKHSRSXODWLRQ
' QXPEHURIREVHUYHGGHDWKVE\DJH[LQFDOHQGDU\HDUW
Q
[W
1 SRSXODWLRQRQ-DQXDU\ZLWKDJH[LQFDOHQGDU\HDUW
[W
' QXPEHURIGHDWKVE\DJH[RQODVWELUWKGD\IRUELUWKFRKRUWF
[F
' QXPEHU RI GHDWKV E\ DJH [ DW ODVW ELUWKGD\ IRU ELUWK FRKRUW F DQG FDOHQGDU
\HDUW
[WF
%
0,
QXPEHURISHRSOHLQELUWKFRKRUWFZKRFHOHEUDWHELUWKGD\[LQFDOHQGDU\HDUW
[WF
PLJUDWLRQEDODQFHRISHRSOHLQELUWKFRKRUWFZLWKDJH[LQFDOHQGDU\HDUW
[WF
)RUWKHVDNHRIVLPSOLFLW\ZKHUHYHUSRVVLEOHWKHUHVWRIWKLVUHSRUWZLOOEHEDVHGRQ
PRUWDOLW\ UDWLRV E\ VHSDUDWH DJH \HDU 7KLV PHDQV WKDW WKH VXEVFULSW ZLWK WKH DJH
LQWHUYDO Q ZLOO EH RPLWWHG IURP WKH IRUPXODV Q 6OLJKWO\ GLIIHUHQW IRUPXODV
DSSO\ IRU WKH PRUWDOLW\ LQ WKH FDOHQGDU \HDU RI ELUWK \HDUROGV E\ DJH RQ -DQXDU\DQGWKHKLJKHVWDJHJURXSDJH7KLVZLOOEHH[SODLQHGLQVHFWLRQ
/LIHWDEOHV
6KRUWGHVFULSWLRQ
7DEOH JLYHV DQ H[DPSOH RI SDUW RI D OLIH WDEOH 7KH GLIIHUHQW FROXPQV LQ WKH OLIH
WDEOH FRQWDLQ WKH IROORZLQJ YDULDEOHV LQ WKLV RUGHU DJH PRUWDOLW\ UDWLR QXPEHU RI
VXUYLYRUV QXPEHU RI GHDWKV DQG WKH OLIH H[SHFWDQF\ 6WDWLVWLFV 1HWKHUODQGV
FDOFXODWHV WKH PRUWDOLW\ UDWLRV IRU HDFK W\SH RI OLIH WDEOH LQ D GLIIHUHQW ZD\ 7KH
FDOFXODWLRQ RI WKHVH UDWHV LV GHVFULEHG LQ VHSDUDWH FKDSWHUV VHH FKDSWHUV DQG 7KHRWKHUFROXPQVLQWKHOLIHWDEOHFDQEHFDOFXODWHGEDVHGRQWKHPRUWDOLW\UDWLRV
7KLVLVGRQHLQWKHVDPHZD\IRUDOOWKHWDEOHVDQGLVH[SODLQHGLQVHFWLRQ
$SSOLFDELOLW\
$OLIHWDEOHFDQEHXVHGIRUH[DPSOHWRFRPSDUHWKHPRUWDOLW\LQDFHUWDLQSDWLHQW
SRSXODWLRQZLWKWKHPRUWDOLW\LQWKHZKROHSRSXODWLRQ7KLVFDQEHGRQHE\DSSO\LQJ
WKHPRUWDOLW\UDWLRVREVHUYHGLQWKHZKROHSRSXODWLRQWRWKHSDWLHQWSRSXODWLRQ7KH
GHDWK QXPEHUV FDOFXODWHG LQ WKLV ZD\ FDQ EH FRPSDUHG ZLWK WKH DFWXDO REVHUYHG
PRUWDOLW\LQWKHSDWLHQWSRSXODWLRQ
'HWDLOHGGHVFULSWLRQ
&DOFXODWLRQRIWKHQXPEHURIVXUYLYRUVDQGGHDWKVLQWKHWDEOHSRSXODWLRQ
7KHFDOFXODWLRQRIWKHFROXPQµVXUYLYRUVLQWKHWDEOHSRSXODWLRQ¶EHJLQVZLWKWKH
VHOHFWLRQ RI WKH VL]H RI WKH K\SRWKHWLFDO FRKRUW DOVR FDOOHG WKH UDGL[ O 7KH
VHOHFWLRQ RI WKLV UDGL[ LV DUELWUDU\ ,Q SUDFWLFH URXQG QXPEHUV DUH XVHG EHWZHHQ DQG PLOOLRQ 6WDWLVWLFV 1HWKHUODQGV XVHV D UDGL[ RI SHRSOH DJHG 7KLV
JHQHUDOO\ FRUUHVSRQGV WR WKH QXPEHU RI ER\V DQG JLUOV ERUQ DQQXDOO\ LQ WKH
1HWKHUODQGV
7KHQXPEHURISHRSOHOLYLQJDWDJH[QO LQWKHOLIHWDEOHLVFDOFXODWHGXVLQJWKH
QXPEHURISHRSOHOLYLQJDWDJH[O PLQXVWKHSHRSOHZKRKDYHGLHGEHWZHHQDJH[
DQG[Q G [Q
[
Q
[
O [ + Q = O [ − Q G [ 7KHQXPEHURISHRSOHZKRKDYHGLHGEHWZHHQDJH[DQG[QLQWKLVIRUPXOD G LV
FDOFXODWHGE\PXOWLSO\LQJWKHPRUWDOLW\SUREDELOLW\EHWZHHQDJH[DQG[Q T E\
WKHQXPEHURISHRSOHVWLOOOLYLQJDWWKHVWDUWRIWKHLQWHUYDOO Q
Q
Q G [ =Q T[
⋅ O [ [
[
[
6WDUWLQJDWDJHWKHUDGL[RILVVXEVWLWXWHGILUVW$IWHUWKLVWKHQXPEHURI
SHRSOH ZKR KDYH GLHG EHWZHHQ WKH DJHV RI DQG \HDUV FDQ EH FDOFXODWHG E\
PXOWLSO\LQJWKHPRUWDOLW\UDWLRDWDJHE\WKHUDGL[RIIRUPXOD7KLV
SURGXFHVWKHQXPEHURISHRSOHZKRKDYHGLHGZKLFKVXEVWLWXWLQJLQIRUPXOD
PXVWEHVXEWUDFWHGIURPWKHUDGL[WRREWDLQWKHQXPEHURISHRSOHOLYLQJDWDJH
\HDUV,QWKLVZD\IRUHDFKQH[WKLJKHUDJHLQWKHOLIHWDEOHWKHQXPEHURISHRSOH
ZKR KDYH GLHG DQG WKHQ WKH QXPEHU RI SHRSOH OLYLQJ FDQ EH FDOFXODWHG XQWLO WKH
HQWLUHSRSXODWLRQRISHRSOHKDVGLHG
&DOFXODWLRQRIWKHOLIHH[SHFWDQF\
8VLQJWKHYDULDEOHVRISHRSOHOLYLQJDQGSHRSOHZKRKDYHGLHGFDOFXODWHGLQVHFWLRQ
WKHOLIHH[SHFWDQF\FDQDOVREHFDOFXODWHG
7KH OLIH H[SHFWDQF\ LQGLFDWHV WKH DYHUDJH QXPEHU RI \HDUV RI OLIH UHPDLQLQJ DW D
FHUWDLQDJH)RUWKHFDOFXODWLRQRIWKLVWKHWRWDOQXPEHURI\HDUVRIOLIHUHPDLQLQJ
IURP D FHUWDLQ DJH [ LV GHWHUPLQHG DOVR FDOOHG 7 IRU WKH WDEOH SRSXODWLRQ
'LYLGLQJWKHWRWDOQXPEHURI\HDUVRIOLIHUHPDLQLQJ7 E\WKHQXPEHURISHRSOH
VWLOO OLYLQJ DW WKLV DJH O UHVXOWVLQWKHDYHUDJH QXPEHU RI \HDUVRIOLIHUHPDLQLQJ
H [
[
[
[
H [ = 7[ O [ )RU[ WKLVIRUPXODERLOVGRZQWRWKHXVXDOPDQQHURIFDOFXODWLQJWKHDYHUDJHLQ
SUREDELOLW\WKHRU\QDPHO\E\GLYLGLQJWKHVXPRIWKHDJHVRQZKLFKWKHPHPEHUV
RIWKHOLIHWDEOHSRSXODWLRQGLHE\WKHUDGL[KHUH7KLVVDPHPHWKRGFDQEH
DSSOLHGPXWDWLVPXWDQGLVWRKLJKHUDJHV
)RUWKHFDOFXODWLRQRIWKHWRWDOQXPEHURI\HDUVRIOLIHUHPDLQLQJ7 WKHQXPEHURI
\HDUVRIOLIHDWWDLQHGSHUDJHDJHJURXS / PXVWEHGHWHUPLQHGILUVW7KHQXPEHU
RI\HDUVRIOLIHDWWDLQHGEHWZHHQWZRVXEVHTXHQWDJHVLVGHWHUPLQHGE\WKHQXPEHU
RISHRSOHVWLOOOLYLQJDW[QDQGWKHDYHUDJHQXPEHURI\HDUVOLYHGE\WKRVHZKRGLH
EHWZHHQDJH[DQG[Q7KHQXPEHURISHRSOHVWLOOOLYLQJDWDJH[QDOOFRQWULEXWHQ
\HDUVWRWKHQXPEHURI\HDUVRIOLIHDWWDLQHG)RUWKHQXPEHURISHRSOHZKRGLHLQ
WKHDJHLQWHUYDOLWLVDVVXPHGWKDWWKH\GLHRQDYHUDJHKDOIZD\WKURXJKWKHLQWHUYDO
,QEULHI
[
Q
/[ = Q ⋅ O[+ Q + Q⋅Q G [ Q
6LQFH G O ±O
Q
[
[
[Q
WKLVIRUPXODFDQEHUHZULWWHQDV
/[ = Q ⋅ O[ + O[+ Q Q
[
8VXDOO\WKHOHQJWKRIWKHDJHLQWHUYDOQLVRQH\HDUH[FHSWIRUDJHLQWKHSHULRG
WDEOHE\DJHRQ-DQXDU\WKHUHWKHDJHLQWHUYDOLVDKDOI\HDU7KHWLPHLQWHUYDOLV
DOVRGLIIHUHQWIRUWKHKLJKHVWDJH+HUHLQWKHRU\WKHLQWHUYDOLVLQILQLWH
6WDWLVWLFV1HWKHUODQGVXVHVWKHHVWLPDWLRQIURP:XQVFKDQG7HUPRWHIRUWKH
FDOFXODWLRQRIWKHQXPEHURI\HDUVRIOLIHDWWDLQHGIURPWKHEHJLQQLQJRIWKHKLJKHVW
DJHJURXS7KHQXPEHURI\HDUVRIOLIHIURPWKHKLJKHVWDJH[ / KHUH’VWDQGV
IRUDQLQILQLWHDJHLQWHUYDOFDQEHHVWLPDWHGDVIROORZV
’
∞
/ [ + = O [ ⋅ ∞ T [ (
)
+HUH ∞ T LVFDOFXODWHGXVLQJ
[
[
∞T [
= T [ + ( − T [ + )
7KH PRUWDOLW\ UDWLR T WKDW LV VXEVWLWXWHG KHUH LV WKH DQQXDO PRUWDOLW\ UDWLR DV
FDOFXODWHGXVLQJIRUPXOD
[
7KHFDOFXODWLRQLQIRUPXODLVDFWXDOO\MXVWDFDOFXODWLRQZLWKDFRQVWDQWDQQXDO
PRUWDOLW\ UDWLR IRU DOO DJHV LQ WKH KLJKHVW DJH JURXS 7KH KLJKHVW DJH PXVW EH
VHOHFWHG LQ VXFK D ZD\ WKDW WKH VL]H RI WKH WDEOH SRSXODWLRQ DW WKDW DJH LV VPDOO LQ
FRPSDULVRQWRWKHUDGL[WRHQVXUHWKDWWKHLPSDFWRILVOLPLWHGRQWKHYDOXHRI
WKHOLIHH[SHFWDQF\H [ ò«
[
)RUWKHFDOFXODWLRQRIWKHOLIHH[SHFWDQF\DWDFHUWDLQDJHWKHVXPRIDOO\HDUVRIOLIH
DWWDLQHGIURPWKDWDJHLVQHHGHG7KLVLVWKHQGLYLGHGE\WKHQXPEHURISHRSOHVWLOO
OLYLQJ DW DJH [ 7KH FDOFXODWLRQ IRU WKH OLIH H[SHFWDQF\ H LV DV IROORZV ZKHQ
ZULWWHQDVDIRUPXOD
[
H[ =
O[
∑
∞
D=[ Q
/D 7KLVIRUPXODFDQDOVREHZULWWHQLQUHFXUVLYHIRUP
H H O O [
[
[
[
,IWKHOLIHH[SHFWDQF\IRUHDFKDJHLVGHVLUHGLWLVWKHUHIRUHFRQYHQLHQWWRVWDUWZLWK
WKHVXPPDWLRQDWWKHERWWRPRIWKHOLIHWDEOHIRUWKHKLJKHVWDJHDQGWKHQWRPRYH
XSZDUGVLHWR\RXQJHUDJHV
7KLVIRUPXODFDQEHDSSOLHGWRERWKWKHOLIHWDEOHE\DYHUDJHDJHRQ-DQXDU\DQG
WKHOLIHWDEOHE\DJHDWODVWELUWKGD\,QWKHOLIHWDEOHE\DYHUDJHDJHRQ-DQXDU\
Q LVXVHGLQWKHFDOFXODWLRQRIWKHQXPEHURISHUVRQ\HDUVIRUWKHDJHRIWR
([DPSOH
7RLOOXVWUDWHWKHQXPEHURIGHDWKVDQGVXUYLYRUVIURPWDEOHLVFDOFXODWHG$WDJH
WKHPRUWDOLW\UDWLRLVDQGWKHQXPEHURIVXUYLYRUVLV6XEVWLWXWLRQLQ
IRUPXOD WKHQ SURGXFHV WKH QXPEHU RI GHDWKV EHIRUH UHDFKLQJ WKH DJH RI \HDUV G 8VLQJWKLVWKHQXPEHURIVXUYLYRUVDWDJH
FDQEHFDOFXODWHGO ± 7KLVQXPEHURIVXUYLYRUVFDQWKHQ
EH PXOWLSOLHG E\ WKH PRUWDOLW\ UDWLR DW DJH WR FDOFXODWH WKH QXPEHU RI GHDWKV
EHIRUHUHDFKLQJWKHDJHRI\HDUV
&KDUDFWHULVWLFV
'HDWKLVDJHGHSHQGHQW,QJHQHUDOWKHKLJKHUWKHDYHUDJHDJHRIDSRSXODWLRQWKH
KLJKHUWKHQXPEHURIGHDWKVSHURIWKLVSRSXODWLRQ/LIHWDEOHVGRQRWKDYHWKLV
GHSHQGHQF\ ,Q WKLV UHVSHFW WDEOHV DUH FRPSDUDEOH ZLWK RWKHU VWDQGDUGLVDWLRQ
PHWKRGV&RQVHTXHQWO\OLIHWDEOHVFDQEHXVHGZKHQFRPSDULQJGHDWKVLQGLIIHUHQW
SRSXODWLRQV VXFK DV E\ JHQGHU UHJLRQ FRXQWULHV SURYLQFHV HWF EDFNJURXQG
SDWLHQWSRSXODWLRQVRUWLPHWKH1HWKHUODQGVLQDQG
4XDOLW\LQGLFDWRUV
,QWKH6WDWLVWLFV1HWKHUODQGVOLIHWDEOHVQRFRQILGHQFHLQWHUYDOVDUHSURYLGHG7KHVH
WDEOHV DUH FDOFXODWHG EDVHG RQ WKH GHDWKV LQ WKH HQWLUH 'XWFK SRSXODWLRQ 7KHUH LV
WKHUHIRUHQRVDPSOHVHOHFWLRQZLWKWKHDFFRPSDQ\LQJHUURUPDUJLQV
3HULRGWDEOHE\DYHUDJHDJHRQ-DQXDU\
6KRUWGHVFULSWLRQ
7KHSHULRGWDEOHE\DYHUDJHDJHRQ-DQXDU\XVHVPRUWDOLW\UDWLRVIRUDYHUDJHDJH
RQ-DQXDU\7KHVHPRUWDOLW\UDWLRVE\DYHUDJHDJHRQ-DQXDU\LQWXUQPDNHXVH
RIWKHQXPEHURIGHDWKVE\µDJHRQ'HFHPEHU¶)RUH[DPSOHWKHPRUWDOLW\UDWLR
E\DYHUDJHDJHRQ-DQXDU\IRU\HDUVXVHVWKHQXPEHURIGHDWKVE\DJH\HDUV
RQ'HFHPEHUVHHWKHOLJKWJUH\GLDPRQGLQILJXUH,QWKLVFDVHIRUWKHSHRSOH
ZKR GLH WKH QXPEHU RI ZKROH \HDUVLV GHWHUPLQHG ZKLFK RQ 'HFHPEHU RI WKH
UHIHUHQFH\HDUKDVHODSVHGVLQFHWKHELUWKGD\RIWKHSHUVRQ7KLVµDJH¶FDQHDVLO\EH
FDOFXODWHGDVµ\HDURIGHDWKPLQXV\HDURIELUWK¶LQWKHH[DPSOHLQILJXUH
7KHSHULRGWDEOHE\DYHUDJHDJHRQ-DQXDU\FDQEHLGHQWLILHGE\WKHKDOIDJHLQWKH
DJHFROXPQ$QH[DPSOHRIWKLVWDEOHLVSUHVHQWHGLQWDEOH
)LJXUH /H[LV GLDJUDP ZLWK WKH REVHUYDWLRQ DUHD IRU WKH SHULRG WDEOH E\ DJH RQ
-DQXDU\ OLJKW JUH\ GLDPRQG DQG IRU WKH SHULRG WDEOH E\ DJH DW ODVW ELUWKGD\
GDUNJUH\GLDPRQG
$SSOLFDELOLW\
7KHVHPRUWDOLW\UDWLRVDUHXVHGIRUH[DPSOHIRUSURJQRVLVSXUSRVHVEHFDXVHWKH\
DUHXVXDOO\EDVHGRQWKHVLWXDWLRQRQ-DQXDU\RIDFHUWDLQSURJQRVLV\HDU
'HWDLOHGGHVFULSWLRQ
&DOFXODWLRQRIWKHPRUWDOLW\UDWLRV
7KH IRUPXOD IRU WKH PRUWDOLW\ UDWLR WKDW LV XVHG E\ 6WDWLVWLFV 1HWKHUODQGV FDQ EH
GHULYHGIURPWKHIRUPXODVWKDWDUHSUHVHQWHGIRUH[DPSOHLQ3UHVWRQHWDO
:KHQGHULYLQJWKHIRUPXODXVHGE\6WDWLVWLFV1HWKHUODQGVLWLVDVVXPHGWKDWQ DV
VWDWHGLQVHFWLRQ DQG D +HUHQ LVWKHOHQJWK RI WKH DJHLQWHUYDOLQ WKLV
FDVHRQHFDOHQGDU\HDUDQG D LVWKHDYHUDJHQXPEHURISHUVRQ\HDUVOLYHGLQWKH
LQWHUYDO RI WKRVH ZKR GLHG LQ WKH LQWHUYDO ,I LW LV DVVXPHG WKDW WKH GHDWKV DUH
GLVWULEXWHG HYHQO\ WKURXJKRXW WKH \HDU WKLV LV D JRRG HVWLPDWH RI WKH QXPEHU RI
SHUVRQ\HDUV6WDWLVWLFV1HWKHUODQGVFDOFXODWHVWKHPRUWDOLW\UDWLRE\DYHUDJHDJHRQ
-DQXDU\DVIROORZV
Q
[
Q
T
[
+ W
='
[
+W
[(1
[
[ W
+1
[
+ W + +' [ + W
)] 7KHFDOFXODWLRQRIWKLVPRUWDOLW\UDWLRLPSOLFLWO\WDNHVDFFRXQWRIWKHLQIOX[WRDQG
RXWIORZ IURP WKH SRSXODWLRQ GXH WR PLJUDWLRQ 7KLV LV DFKLHYHG EHFDXVH WKH
SRSXODWLRQLVLQFOXGHGERWKDWWKHVWDUWDQGWKHHQGRIWKH\HDU
3OHDVHQRWH$WILUVWJODQFHIRUPXODVHHPVWRGHYLDWHIURP+RZHYHULQ
YLHZRIWKHIDFWWKDWLQDOLIHWDEOHWKHUHIRUHLQIRUPXODQHLWKHULPPLJUDWLRQ
QRU HPLJUDWLRQ RFFXU LW LV WUXH WKDW O O ± G ZKLFK PHDQV WKDW PXWDWLV
PXWDQGLVWKHGHQRPLQDWRURIFDQEHZULWWHQDVO $IWHUDOOWKHOHWWHUOLQWKH
OLIHWDEOHSRSXODWLRQLVFRPSDUDEOHZLWKWKH1LQWKHREVHUYHGSRSXODWLRQ
[
[
[
[
7KHVXEVFULSWWLQGLFDWHVWKHFDOHQGDU\HDU[WKHDJHFRUUHVSRQGLQJWRWKHSRSXODWLRQ
RI WKH 1HWKHUODQGV RQ -DQXDU\ $V SUHYLRXVO\ VWDWHG WKH DJH WKDW LV XVHG LQ WKH
GHVFULSWLRQRIWKHGHDWKVLQWKH\HDUWLVDZKROH\HDUKLJKHU7KLVLVWKHUHDVRQIRU
WKHWHUP' 7KHDJHLQWKHOLIHWDEOHLVDKDOI\HDUKLJKHUZKLFKLVH[SUHVVHGLQ
WKHWHUPT
[W
[W
7KHPRUWDOLW\UDWLRFDOFXODWHGLQWKLVPDQQHULVDPRUWDOLW\UDWLRIRUµKDOIDJHV¶7KLV
LVVKRZQLQDQH[DPSOHLQILJXUH,QWKHH[DPSOHWKHQHFHVVDU\GDWDDUHVKRZQIRU
WKHFDOFXODWLRQ RI WKH PRUWDOLW\ UDWLR IRU \HDUROGV LQ T
)RU WKLV
FDOFXODWLRQZHXVHWKHQXPEHURISHRSOHZKRGLHGWKDWUHDFKHGWKHDJH RI\HDUV
GXULQJWKHFDOHQGDU\HDU'
7KLVLVUHODWHGWRWKHSRSXODWLRQWKDWZDVVWLOO
\HDUROGRQ-DQXDU\DQGZKLFKZDVDJHDWWKHHQGRIWKH\HDU $JHDFFRUGLQJWRWKHOLIHWDEOHLQRWKHUZRUGVWKHDYHUDJHDJHRQ-DQXDU\
7KHDJHWKDWWKHSHRSOHUHDFKHGRUZRXOGKDYHUHDFKHGRQWKHLUELUWKGD\LQµDJHRQ
'HFHPEHU¶
$JH WKDW SHRSOH ZRXOG KDYH UHDFKHG RQ WKHLU ELUWKGD\ LQ DQG WKH DJH WKDW SHRSOH
UHDFKHGRUZRXOGKDYHUHDFKHGRQWKHLUELUWKGD\LQµDJHRQ-DQXDU\¶UHVSHFWLYHO\
)LJXUH/H[LVGLDJUDPIRUWKHFDOFXODWLRQRIPRUWDOLW\UDWLRT
, 2002
7KH PRUWDOLW\ UDWLR WKDW LV FDOFXODWHG IRU WKHVH \HDUROGV LV WKHUHIRUH WKH
SUREDELOLW\RIGHDWKIURPWKHDYHUDJHDJHRI\HDUVEHIRUHUHDFKLQJWKHDYHUDJH
DJHRI\HDUV7KHGHQRPLQDWRULQWKLVH[DPSOHLVWKHSRSXODWLRQRI\HDUROGV
RQ -DQXDU\ SOXV KDOI RI WKH QHW QXPEHU RI PLJUDQWV DQG DGPLQLVWUDWLYH
FRUUHFWLRQV LQ WKH FDOHQGDU \HDU ,W LV DVVXPHG KHUH WKDW WKH PLJUDWLRQ DQG
FRUUHFWLRQVDUHHYHQO\GLVWULEXWHGWKURXJKRXWWKH\HDU
)RUPXODGRHVQRWDSSO\IRUWKH\RXQJHVWDJHFDWHJRU\LQWKHWDEOH$IWHUDOO
WKHUHLVQRVWDUWLQJSRSXODWLRQRQ-DQXDU\)RU\HDUROGVWKHPRUWDOLW\UDWLRLV
FDOFXODWHGDVIROORZV
T
W
='
W
(1
W
+
+'
W
)
7KLVPHDQVWKDWWKHPRUWDOLW\UDWLRIRU\HDUROGVLQLVEDVHGRQWKHQXPEHURI
SHRSOHZKRGLHGLQZKRZRXOGKDYHKDGWKHDJHRI\HDUVRQ'HFHPEHULQ
WKDW VDPH \HDU LH ZKR ZHUH ERUQ LQ GLYLGHG E\ WKH VWDUWLQJ SRSXODWLRQ
ILJXUH 7KH VWDUWLQJ SRSXODWLRQ IRUWKH \HDUROGV LV GHWHUPLQHG E\ DGGLQJ WKH
QXPEHU\HDUROGVZKRGLHGLQ' WRWKHQXPEHURI\HDUROGVSUHVHQWLQ
,I ZH DVVXPH WKDW FKLOGUHQ DUH ERUQ
WKH SRSXODWLRQ RQ -DQXDU\ 1
HYHQO\GLVWULEXWHGWKURXJKRXWDFDOHQGDU\HDUWKHQWKH\ZLOOEHRQDYHUDJHDKDOI
\HDUROGRQWKHODVWGD\RIWKDWFDOHQGDU\HDU7KHPRUWDOLW\UDWLRIRU\HDUROGVLV
WKHUHIRUHWKHSUREDELOLW\RIG\LQJEHIRUHWKHDYHUDJHDJHRIDKDOI\HDU7KLVPHDQV
WKDWWKH\HDUROGVDUHWKHH[FHSWLRQWRWKHDVVXPSWLRQLQWKLVDUWLFOHWKDWQ IRU
WKHPQ DSSOLHVLQVWHDG7KLVFDQEHZULWWHQDV T W
W
W
)LJXUH/H[LVGLDJUDPIRUWKHFDOFXODWLRQRIPRUWDOLW\UDWLRT
, 2002
7KHDQQXDOPRUWDOLW\UDWLRIRUWKHKLJKHVWDJHJURXSFDQEHFDOFXODWHGE\DGGLQJXS
WKH YDOXHV IRU WKH KLJKHVW DJH JURXSV DQG VXEVWLWXWLQJ WKHP LQ IRUPXOD )RU
\HDUROGVDQGROGHULQ\HDUWWKLVFDQEHIRUH[DPSOH
T
+ W
=
∑' +
∞
[ W
[
=
∞
 ∞
⋅  ∑ 1 + ∑ 1
=
 =
[ W
[
[
+
[ W
+
∞

+ ∑ ' +  =

[ W
[
7KLVYDOXHLVQHHGHGWRFDOFXODWHWKHOLIHH[SHFWDQF\2IWHQWLPHVKRZHYHUWKHOLIH
WDEOHRQO\QRWHVWKHWRWDOPRUWDOLW\UDWLRIRUWKHKLJKHVWDJHJURXSIRUH[DPSOHIRU
\HDUROGVDQGROGHUT ,QGHHGWKHSUREDELOLW\RIG\LQJEHIRUHUHDFKLQJ
WKHLQILQLWHHQGRIWKLVDJHJURXSLV
([DPSOH
7KH H[DPSOH IURP ILJXUH ZLOO EH HODERUDWHG KHUH XVLQJ DQ H[DPSOH ZLWK ILJXUHV
EDVHGRQWKHDFWXDOREVHUYHGYDOXHV)RUWKHPRUWDOLW\UDWLRIRUWKHDJHRIò
IRUPHQZDVFDOFXODWHGEDVHGRQWKHIROORZLQJ'
1
1
6XEVWLWXWLRQLQIRUPXODSURGXFHVDPRUWDOLW\UDWLRRI
T
= [ ⋅ (++)] = 7KLV LV WKH SUREDELOLW\ RI D ò\HDUROG ER\ G\LQJ EHIRUH UHDFKLQJ WKH DJH RI ò
\HDUV ,Q RWKHU ZRUGV LW LV WKH SUREDELOLW\ RI D ER\ G\LQJ LQ WKH \HDU WKDW KH
FHOHEUDWHVKLVVHFRQGELUWKGD\
&KDUDFWHULVWLFV
7KH PRUWDOLW\ UDWLR DVFDOFXODWHGIRUWKHSHULRGWDEOHE\ DYHUDJH DJH RQ -DQXDU\
DOZD\VFRQFHUQVRQHSHULRGLQILJXUHRQHFRKRUWDQGWZRDJHV
DQG \HDUV RI DJH ,Q WKH LQWHUQDWLRQDO OLWHUDWXUH WKLV PRUWDOLW\ SUREDELOLW\ LV
WKHUHIRUHNQRZQDVWKHFRKRUWVSHFLILFSHULRGSUREDELOLW\
&RKRUWVSHFLILF SHULRG SUREDELOLWLHV DUH XVHG IRU VXFK SXUSRVHV DV SURJQRVHV
EHFDXVHWKHVHJHQHUDOO\DVVXPHWKHVLWXDWLRQRQ-DQXDU\RIFHUWDLQSURJQRVLV\HDU
4XDOLW\LQGLFDWRUV
7KHPRUWDOLW\UDWLRVIRUWKLVWDEOHFRQFHUQLQHDFKFDVHWZRDJHVLQDVLQJOHFDOHQGDU
\HDU7KHDJHUDQJHIRUDPRUWDOLW\UDWLRDWRQHVSHFLILFDJHWKHUHIRUHFRYHUVDWRWDO
RIDJH\HDUVVHHDOVRILJXUH7KLVFDQEHDGUDZEDFNRIWKLVFDOFXODWLRQPHWKRG
EHFDXVHLWPHDQVWKDWWKHJURXSLVQRWYHU\KRPRJHQRXVZLWKUHVSHFWWRWKHDJHRI
GHDWK
3HULRGWDEOHE\DJHDWODVWELUWKGD\
6KRUWGHVFULSWLRQ
7KH SHULRG WDEOH EDVHG RQ PRUWDOLW\ GDWD E\ DJH DW ODVW ELUWKGD\ LV DOVR NQRZQ DV
µSHULRGWDEOHE\DJHDWODVWELUWKGD\¶7KLVSHULRGWDEOHXVHVZKROHDJHVDQGPDNHV
XVHRIPRUWDOLW\UDWLRVLQZKROH\HDUVRQWKHGD\RIGHDWK,IIRUH[DPSOHDSHUVRQ
GLHVRQ$SULORIDFHUWDLQ\HDUDQGZRXOGKDYHWXUQHG\HDUVRIDJHRQ$SULO
WKHQWKHDJHDWODVWELUWKGD\DWWKHWLPHRIGHDWKLV\HDUVRIDJHDFFRUGLQJWRWKLV
GHILQLWLRQ7KLVSHULRGWDEOHXVHVPRUWDOLW\GDWDRYHUDSHULRGRIWZRFDOHQGDU\HDUV
VHH WKH GDUN JUH\ GLDPRQG ILJXUH LQ WKLV FDVH WKH SHULRG LV $Q
H[DPSOHRIWKLVWDEOHLVSUHVHQWHGLQWDEOH
7DEOH 3DUW RI WKH OLIH WDEOH E\ DJH DW ODVW ELUWKGD\ PHQ SHULRG $JH
PRUWDOLW\UDWLR
QXPEHURI
QXPEHURI
OLIHH[SHFWDQF\
[
QT[
VXUYLYRUVO[
GHDWKVQG[
H[
$SSOLFDELOLW\
7KHSHULRGOLIHWDEOHVE\DJHDWODVWELUWKGD\DUHXVHGIRUH[DPSOHE\DFWXDULHVLQ
WKHLQVXUDQFHVHFWRUWRGHWHUPLQHLQVXUDQFHSUHPLXPV
'HWDLOHGGHVFULSWLRQ
&DOFXODWLRQRIWKHPRUWDOLW\UDWLRV
7KHPRUWDOLW\UDWLRVIRUWKHZKROHDJHVDUHGHWHUPLQHGXVLQJWKHQXPEHURISHRSOH
LQ D FHUWDLQ FRKRUW ZKR FHOHEUDWH WKHLU ELUWKGD\ LQ D FHUWDLQ FDOHQGDU \HDU DQG WKH
QXPEHU RI WKHP ZKR KDYH GLHG EHIRUH WKH\ UHDFK WKHLU QH[W ELUWKGD\ LQ WKH QH[W
FDOHQGDU \HDU 0RUWDOLW\ GDWD IURP WZR VXFFHVVLYH FDOHQGDU \HDUV W DQG W DUH
WKHUHIRUHQHHGHGIRUWKHFDOFXODWLRQ)RUWKLVUHDVRQWKHSHULRGRIWZR\HDUVLVQRWHG
DERYHWKHOLIHWDEOHE\DJHDWODVWELUWKGD\VXFKDVVHHDOVRWKHGDUNJUH\
GLDPRQGLQILJXUH
7KH QXPEHU RI GHDWKV IRU DJH DW ODVW ELUWKGD\ [ DQG FRKRUW F DQG WKHUHIRUH
DXWRPDWLFDOO\IRUWKHWZRFDOHQGDU\HDUVWDQGWLVFDOFXODWHGIURPZKDWLVNQRZQ
DVGRXEO\FODVVLILHGPRUWDOLW\GDWDLQRWKHUZRUGVPRUWDOLW\GDWDE\DJHSHULRGDQG
FRKRUWWKHWZREODFNWULDQJOHVLQILJXUH
' [ F = ' [ W F + ' [ W + F ZKHUH
' WKHQXPEHURIGHDWKVE\DJH[RQODVWELUWKGD\IRUELUWKFRKRUWF
'
'
[F
WKHQXPEHURIGHDWKVE\DJH[RQODVWELUWKGD\IRUELUWKFRKRUWFDQGFDOHQGDU
\HDUW
[WF
QXPEHURIGHDWKVE\DJH[RQODVWELUWKGD\IRUELUWKFRKRUWFDQGFDOHQGDU
\HDUW
[WF
,QWKHH[DPSOHILJXUH'
F
'
'
)LJXUH/H[LVGLDJUDPIRUFDOFXODWLRQRIPRUWDOLW\UDWLRT
, c=2000
7KHQXPEHURISHRSOHLQFRKRUWFWKDWFHOHEUDWHELUWKGD\[LQFDOHQGDU\HDUW% FDQEHDSSUR[LPDWHGXVLQJIRUPXOD7KHFDOFXODWLRQWDNHVH[SOLFLWDFFRXQWRI
PLJUDWLRQ
[WF
%[ W F = 1 [ W + + '[ W F − 0, [ W F + 0, [ W + F ZKHUH
%
QXPEHURISHRSOHLQELUWKFRKRUWFWKDWFHOHEUDWHVELUWKGD\[LQFDOHQGDU\HDU
W
[WF
1
0,
0,
SRSXODWLRQRQ-DQXDU\ZLWKDJH[LQFDOHQGDU\HDUW
[W
PLJUDWLRQEDODQFHRISHRSOHLQELUWKFRKRUWFZLWKDJH[LQFDOHQGDU\HDUW
[WF
W
[ WF
PLJUDWLRQEDODQFHRISHRSOHLQELUWKFRKRUWFZLWKDJH[LQFDOHQGDU\HDU
7KLV PHDQV WKDW GRXEO\ FODVVLILHG GDWD DUH DOVR QHHGHG IRU PLJUDWLRQ ,Q WKLV
IRUPXOD LW LV DVVXPHG WKDW WKH KDOI RI WKH QHW PLJUDWLRQ IRU DJH [ DQG FRKRUW F LV
FRXQWHGLQWKHVWDUWLQJSRSXODWLRQ
7KHPRUWDOLW\UDWLRIRUDJHDWODVWELUWKGD\IRU\HDUWDQGWLVWKHQDVIROORZV
T[ F =
1 [ W+
'[ W F + '[ W + F
+ '[ W F − 0, [ W F + 0, [ W + F
7KHPLJUDWLRQEDODQFHDVXVHGLQWKLVFDOFXODWLRQLVWKHEDODQFHRILPPLJUDWLRQDQG
HPLJUDWLRQ LQFOXGLQJ DGPLQLVWUDWLYH FRUUHFWLRQV $GPLQLVWUDWLYH FRUUHFWLRQV DUH
IRUPHGE\WKHHQWULHVDQGGHOHWLRQVIURPWKH0XQLFLSDO3HUVRQDO5HFRUGV'DWDEDVH
RWKHU WKDQ GXH WR ELUWK GHDWK UHVLGHQFH GHSDUWXUH RU PXQLFLSDO ERUGHU FKDQJH
$GPLQLVWUDWLYH GHOHWLRQV DUH XVXDOO\ WKH UHVXOW RI QRQUHSRUWHG HPLJUDWLRQ
DGPLQLVWUDWLYHHQWULHVRIWKHVXEVHTXHQWLPPLJUDWLRQ
([DPSOH
$QH[DPSOHRISDUWRIDOLIHWDEOHE\DJHDWODVWELUWKGD\LVVKRZQLQWDEOH
&KDUDFWHULVWLFV
8QOLNH IRU WKH PRUWDOLW\ UDWLRV E\ KDOI DJHV ZKHUH WKH SUREDELOLWLHV IRU D SHULRG
FDOHQGDU\HDUDUHFDOFXODWHGDVSHFLILFDJHLVFRQVLGHUHGIRUWKHPRUWDOLW\UDWLRVE\
DJH DW ODVW ELUWKGD\ &RQVHTXHQWO\ WKHVH UDWHV DUH DOVR FDOOHG DJHVSHFLILF FRKRUW
SUREDELOLWLHV
$JHVSHFLILFFRKRUWSUREDELOLWLHVDUHXVHGDVWKHEDVLVIRUWKHVRFDOOHGµ$*WDEOHV¶
LH WDEOHV IRU DFWXDULDO SXUSRVHV $* LV WKH 'XWFK DEEUHYLDWLRQ IRU WKH $FWXDULDO
$VVRFLDWLRQRIWKH 1HWKHUODQGV7KHVH $*WDEOHV JHQHUDOO\ FRYHU D SHULRG RIILYH
FDOHQGDU\HDUV,QFLGHQWDOO\LQDFWXDULDODSSOLFDWLRQVWKHPRUWDOLW\UDWLRVDUHWUHDWHG
LQ VXFK D ZD\ WKDW WKHUH LV D VHDPOHVV WUDQVLWLRQ EHWZHHQ WKHP $ ZHOONQRZQ
WHFKQLTXHIRUWKLVLVWKH0DNHKDPURXQGLQJWHFKQLTXH
4XDOLW\LQGLFDWRUV
7KHPRUWDOLW\UDWLRVIRUWKLVWDEOHFRQFHUQLQHDFKFDVHWZRFDOHQGDU\HDUVIRURQH
DJHVHHILJXUH7KHUHDUHWKHUHIRUHWZRFDOHQGDU\HDUVEHWZHHQWKHILUVWDQGWKH
ODVW GHDWK LQ RQH DJH \HDU $V D UHVXOW WKH JURXS RI SHRSOH ZKR GLHG LV QRW
KRPRJHQRXVZLWKUHVSHFWWRWKHGDWHRIGHDWK7KLVFRXOGEHDGLVDGYDQWDJHRIWKLV
OLIHWDEOH
&RQFOXVLRQ
2QWKHRUHWLFDOJURXQGVWKHUHLVQRPDMRUSUHIHUHQFHIRURQHRIWKHWZRW\SHVRIOLIH
WDEOHV DV GHVFULEHG LQ FKDSWHUV DQG ,I WKH GHYHORSPHQW RI GHDWKV RYHU WLPH LV
YHU\ VWHDG\ VXFK DV LQ WKH SRSXODWLRQ RI WKH 1HWKHUODQGV ERWK DSSURDFKHV ZLOO
UHVXOWLQ D XVDEOH OLIH WDEOH )URP D QXPEHUV SHUVSHFWLYH LWLV DOVR QRW SRVVLEOHWR
IRUPXODWHKDUGFULWHULD7KHRUHWLFDOO\LWFDQEHDUJXHGWKDWIRUH[DPSOHWKHQXPEHU
RIVXUYLYRUVDJHGDFFRUGLQJWRRQHWDEOHIDOOVEHWZHHQWKHQXPEHURIDQG
\HDUROGVXUYLYRUVDFFRUGLQJWRWKHRWKHUDQGFRQYHUVHO\WKDWWKHQXPEHURI\HDU
ROGVXUYLYRUVIDOOVEHWZHHQWKHQXPEHUVRIDQG\HDUROGV+RZHYHUWKHVH
FRQGLWLRQV DUH GLIILFXOW WR LPSOHPHQW LQ SUDFWLFH EHFDXVH WKH OLIH WDEOHV GR QRW
UHSUHVHQWWKHVDPHSHULRGV$OLIHWDEOHE\µDJHRQ-DQXDU\¶FRQFHUQVRQHRUPRUH
ZKROHFDOHQGDU\HDUVVXFKDVZKLOHDOLIHWDEOHE\µDJHDWODVWELUWKGD\¶
FRQFHUQVVSOLWFDOHQGDU\HDUVVXFKDV
5HIHUHQFHV
3UHVVDW5/RFDWLRQLQ7LPH,Q%RJXH'-(($UULDJH*:5XPVH\
DQG93UDFKXDEPRKHGV5HDGLQJV,Q3RSXODWLRQ5HVHDUFK0HWKRGRORJ\
9RO%DVLFWRROVSS6RFLDO'HYHORSPHQW&HQWHU&KLFDJR
3UHVWRQ 6+ +HXYHOLQH3 DQG *XLOORW 0 'HPRJUDSK\0HDVXULQJ DQG
PRGHOLQJSRSXODWLRQSURFHVVHV%ODFNZHOOSXEOLVKHUV2[IRUG8.0DOGHQ
86$
3ULQV .HHV DQG +DQV .XLMSHU %HYRONLQJVVWDWLVWLHNHQ RQGHU KHW
SHUVRRQVNDDUWHQVWHOVHO HQ KHW *%$VWHOVHO RYHUHHQNRPVWHQ HQ YHUVFKLOOHQ
%HYRONLQJVWUHQGVILUVWTXDUWHU
9DQ GHU 0HXOHQ $ DQG ) -DQVVHQ $FKWHUJURQGHQ HQ EHUHNHQLQJVZLM]HQ
YDQ&%6RYHUOHYLQJVWDIHOV%HYRONLQJVWUHQGVWKLUGTXDUWHU
:XQVFK *- DQG 7HUPRWH 0* ,QWURGXFWLRQ WR GHPRJUDSKLF DQDO\VLV
SULQFLSOHVDQGPHWKRGV3OHQXP3UHVV1HZ<RUN
9HUVLRQKLVWRU\
9HUVLRQ
'DWH
'HVFULSWLRQ
$XWKRUV
5HYLHZHUV
)LUVW'XWFKYHUVLRQ
$QRXVFKNDYDQGHU0HXOHQ %RE/RGGHU
.HHV3ULQV
(
)LUVW(QJOLVKYHUVLRQ $QRXVFKNDYDQGHU0HXOHQ 'XWFKYHUVLRQ2YHUOHYLQJVWDIHOVHQORQJLWXGLQDOHDQDO\VH2YHUOHYLQJVWDIHOV
(QJOLVKYHUVLRQ/LIHWDEOHVDQG6XUYLYDODQDO\VLV/LIHWDEOHV