The valuation of privately-held infrastructure equity investments

Abstract

This paper contributes a rigorous valuation framework to the debate on the benchmarking of privately-held infrastructure equity investments. We aim to achieve the following objectives: 1/ item Determine the most appropriate valuation framework for privately-held equity investments in infrastructure projects or entities; 2/ Design a methodology that can be readily applied given the current state of empirical knowledge and, going forward, at a minimum cost in terms of data collection; 3/ Derive some of the most relevant valuation and performance measures for long-term equity investors and regulators; 4/ Define a parsimonious data collection template that nevertheless allows meeting the three points above.

Full text

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20
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1−std Bounds
0 5 10 15 20 25 30
0
5
10
15
20
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Period
µt+τ(%)
BC IRR
Mean
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Mean
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0 5 10 15 20 25 30
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Equity Price
Period
Price (k$)
Mean
1−std Bounds
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Deals
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6
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10
11
12
Evolution of Prices
Deals
Price (k$)
Exact Mean
Observations
Filtered Mean
1−std Bounds
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/EBUB QPJOUT XF DBO VQEBUF UIF WBMVFT PG
UIF NFUBQBSBNFUFST FBDI UJNF XF PCTFSWF
OTVDDFTTFT BNPOHTU /OFX EBUB QPJOUT 
  *O TIPSU JG ɮ∼#FUB(ú,ġ),O∼
#JOPNJBM(ɮ,/)⇒ˆ
ɮ|O,/∼
#FUB(ú+O,ġ+/−O)5IF QPTUFSJPS EJTUSJCVUJPO PG ɮ
TVNNBSJTFT UIF TUBUF PG PVS LOPXMFEHF
CZ DPNCJOJOH JOGPSNBUJPO GSPN OFXMZ
BWBJMBCMF EBUB FYQSFTTFE UISPVHI
UIF MJLFMJIPPE GVODUJPO XJUI FY BOUF
JOGPSNBUJPO FYQSFTTFE UISPVHI UIF QSJPS
EJTUSJCVUJPO
&BDI UJNF OFX PCTFSWBUJPOT BSF NBEF /
QSPKFDUT OUSBOTJUJPOT GSPN TUBUF  UP TUBUF
 PVS LOPXMFEHF PG UIF NFUBQBSBNFUFST
PG UIF QSPCBCJMJUZ EJTUSJCVUJPO PG ɮ DBO CF
VQEBUFE BOE UIF BDDVSBDZ PG JUT EJTUSJCVUJPO
JNQSPWFE 5IF QPTUFSJPS EJTUSJCVUJPO PG
1S(ɮ)UIFO CFDPNFT B OFX QSJPS FBDI
UJNF OFX FNQJSJDBM PCTFSWBUJPOT CFDPNF
BWBJMBCMF #BZFTJBO JOGFSFODF UIVT BMMPXT
TFRVFOUJBM MFBSOJOH BCPVU UIF FYQFDUFE
CFIBWJPVS PG JOGSBTUSVDUVSF QSPKFDU FRVJUZ
DBTI áPXT
5IF TBNF QSPDFTT JT VTFE UP FTUJNBUF ɮ
 &YBNQMF PG #BZFTJBO MFBSOJOH PG
EJWJEFOE TUBUF USBOTJUJPO QSPCBCJMJUJFT
*O UIJT TFDUJPO XF QSPWJEF BO JMMVTUSBUJPO
PG UIJT uMFBSOJOHu QSPDFTT GPMMPXJOH UIF
PCTFSWBUJPO PG UIF SFRVJSFE EBUB BOE IPX
GBTU UIF USVF QSPCBCJMJUJFT DIBSBDUFSJTJOH
EJWJEFOE DBTI áPXT NBZ CF LOPXO
8F CFHJO CZ BTTVNJOH UIBU UIF EJWJEFOE
QSPDFTT PG B HJWFO JOGSBTUSVDUVSF JOWFTUNFOU
JT DIBSBDUFSJTFE CZ UIF USVF USBOTJUJPO
"O &%)&$3JTL *OTUJUVUF 1VCMJDBUJPO 

5IF 7BMVBUJPO PG 1SJWBUFMZ)FME *OGSBTUSVDUVSF &RVJUZ *OWFTUNFOUT +BOVBSZ 
 &YQFDUFE %JWJEFOE .PEFM
5BCMF  5SVF QSJPS BOE QPTUFSJPS WBMVFT PG FYBNQMF USBOTJUJPO QSPCBCJMJUJFT BU UJNF U
ɮ ɮ
USVF WBMVF  
QSJPS WBMVF  
QPTUFSJPS WBMVF  
1SJPS WBSJBODF  
1PTUFSJPS WBSJBODF  
7BSJBODF SFEVDUJPO ∼ ∼
5BCMF  5SVF QSJPS BOE QPTUFSJPS WBMVFT PG FYBNQMF EJWJEFOE EJTUSJCVUJPO QBSBNFUFST BU UJNF U
&4$3 7BMVFT
NFBO Ȇ&4$3UTUE EFW ʤ&4$3UQSFDJTJPO
USVF WBMVF   
QSJPS WBMVF   
QPTUFSJPS   
-PHOPSNBM QBSBN &4$3 ∼-PH/(N,Q)
MPDBUJPO NTUE EFW ʤQSFDJTJPO Q
USVF WBMVF   
QSJPS WBMVF   
QPTUFSJPS   
.FUBQBSBN N∼/(Ȇ,ŋ)
NFBO ȆQSFDJTJPO ŋ
QSJPS WBMVF NQSJPS 
QPTUFSJPS  
.FUBQBSBN Q∼ƣ(B,C)
NFBO ȆQTUE EFW ʤQ
QSJPS QQSJPS 
QPTUFSJPS  
TIBQF B SBUF C
QSJPS WBMVF  
QPTUFSJPS  
 QSFDJTJPO JT UIF JOWFSTF PG UIF WBSJBODF
QSPCBCJMJUJFT BU TPNF UJNF UHJWFO JO UBCMF 
*O UIJT DBTF UIF USVF WBMVF PG ɮ =%
BOE UIBU PG ɮ =%JF JO UIJT UZQF
PG JOGSBTUSVDUVSF QSPKFDU JG B TUSJDUMZ QPTJUJWF
EJWJEFOE XBT QBJE BU UIF QSFWJPVT QFSJPE
UIFSF JT  DIBODF PG SFDFJWJOH B TUSJDUMZ
QPTJUJWF EJWJEFOE EVSJOH UIF DVSSFOU QFSJPE
BOE JG OP EJWJEFOE XBT QBJE BU UIF QSFWJPVT
QFSJPE UIFO UIF DIBODFT PG HFUUJOH QBJE
EVSJOH UIF DVSSFOU QFSJPE JT FWFO IJHIFS
BU  5IJT DPVME DPSSFTQPOE UP B NBUVSF
JOGSBTUSVDUVSF QSPKFDU QBZJOH QSFEJDUBCMF
DBTI áPXT BOE VOMJLFMZ UP FYQFSJFODF [FSP
EJWJEFOET GPS MPOH
5IF TBNF UBCMF BMTP SFQPSUT B QSJPS WBMVF GPS
ɮ BOE ɮ XIJDI CFGPSF PCTFSWJOH BOZ
EBUB XF TFU BU UIF DPOTFSWBUJWF FTUJNBUF
PG  GPS CPUI WBMVFT 4JODF UIF QSJPS
FYQFDUFE WBMVF PG ɮ ɮ JT . UIF CFUB
EJTUSJCVUJPO PG ɮ ɮ BMTP IBT NFBO PG
. 5IF QSJPS WBSJBODF PG CPUI QBSBNFUFST
JT TFU BU  XIJDI JT WFSZ DMPTF UP UIF
NBYJNVN WBSJBODF QPTTJCMF HJWFO UIF QSJPS
NFBO )FODF XF BTTVNF UIF WBMVFT PG
  *U DBO FBTJMZ CF TIPXO UIBU
UIF WBSJBODF PG B SBOEPN WBSJBCMF
GPMMPXJOH B #FUB EJTUSJCVUJPO PG NFBO
ȆNVTU CF TUSJDUMZ JOGFSJPS UP Ȇ(−Ȇ)
ɮ BOE ɮ MBSHFMZ VOLOPXO BSPVOE UIFJS
PUIFSXJTF DPOTFSWBUJWF QSJPS NFBO FTUJNBUF
 "O &%)&$3JTL *OTUJUVUF 1VCMJDBUJPO

5IF 7BMVBUJPO PG 1SJWBUFMZ)FME *OGSBTUSVDUVSF &RVJUZ *OWFTUNFOUT  +BOVBSZ 
 &YQFDUFE %JWJEFOE .PEFM
'JHVSF  1SJPS BOE QPTUFSJPS EFOTJUJFT PG UIF #FUB EJTUSJCVUJPO PG ɮ PWFS  JUFSBUJPOT VTJOH  EBUB QPJOUT
0.80 0.85 0.90 0.95 1.00
0 10 20 30 40 50 60
true value
round 1
0.80 0.85 0.90 0.95 1.00
0 10 20 30 40 50 60
true value
round 2
0.80 0.85 0.90 0.95 1.00
0 10 20 30 40 50 60
true value
round 3
0.80 0.85 0.90 0.95 1.00
0 10 20 30 40 50 60
true value
round 4
0.80 0.85 0.90 0.95 1.00
0 10 20 30 40 50 60
true value
round 5
0.80 0.85 0.90 0.95 1.00
0 10 20 30 40 50 60
true value
round 6
0.80 0.85 0.90 0.95 1.00
0 10 20 30 40 50 60
true value
round 7
0.80 0.85 0.90 0.95 1.00
0 10 20 30 40 50 60
true value
round 8
0.80 0.85 0.90 0.95 1.00
0 10 20 30 40 50 60
true value
round 9
0.80 0.85 0.90 0.95 1.00
0 10 20 30 40 50 60
true value
round 10
0.80 0.85 0.90 0.95 1.00
0 10 20 30 40 50 60
true value
round 11
0.80 0.85 0.90 0.95 1.00
0 10 20 30 40 50 60
true value
round 12
"CPWF *O FBDI SPVOE PG OFX PCTFSWBUJPOT UIF QSJPS EFOTJUZ JT JOEJDBUFE JO CMBDL BOE UIF QPTUFSJPS JO PSBOHF /PUF UIBU FBDI QPTUFSJPS CFDPNFT UIF
QSJPS PG UIF OFYU SPVOE 5IF EPUUFE CMVF MJOF JOEJDBUFT UIF USVF WBMVF PG UIF QBSBNFUFS CFJOH FTUJNBUFE
#FMPX MFGU *O FBDI PCTFSWBUJPO SPVOE UIF HSFFO MJOF JOEJDBUFT UIF QSJPS BOE UIF QJOL EPU UIF QPTUFSJPS WBMVF
●
●●●●●●●●●●●
2 4 6 8 10 12
0.75 0.80 0.85 0.90 0.95 1.00
Posterior estimate of π11
observation rounds
probability of state transition
posterior estimates
prior estimates
true value
●
●
●
●
●
●●
●
●●●●
2 4 6 8 10 12
0.010 0.015 0.020 0.025 0.030
Change in standard deviation (learning) of π11
observation rounds
Standard deviation
"O &%)&$3JTL *OTUJUVUF 1VCMJDBUJPO 

5IF 7BMVBUJPO PG 1SJWBUFMZ)FME *OGSBTUSVDUVSF &RVJUZ *OWFTUNFOUT +BOVBSZ 
 &YQFDUFE %JWJEFOE .PEFM
'JHVSF  1SJPS BOE QPTUFSJPS EFOTJUJFT PG UIF #FUB EJTUSJCVUJPO PG ɮ PWFS  JUFSBUJPOT VTJOH  EBUB QPJOUT
0.80 0.85 0.90 0.95 1.00
0 10 20 30 40 50 60
true value
round 1
0.80 0.85 0.90 0.95 1.00
0 10 20 30 40 50 60
true value
round 2
0.80 0.85 0.90 0.95 1.00
0 10 20 30 40 50 60
true value
round 3
0.80 0.85 0.90 0.95 1.00
0 10 20 30 40 50 60
true value
round 4
0.80 0.85 0.90 0.95 1.00
0 10 20 30 40 50 60
true value
round 5
0.80 0.85 0.90 0.95 1.00
0 10 20 30 40 50 60
true value
round 6
0.80 0.85 0.90 0.95 1.00
0 10 20 30 40 50 60
true value
round 7
0.80 0.85 0.90 0.95 1.00
0 10 20 30 40 50 60
true value
round 8
0.80 0.85 0.90 0.95 1.00
0 10 20 30 40 50 60
true value
round 9
0.80 0.85 0.90 0.95 1.00
0 10 20 30 40 50 60
true value
round 10
0.80 0.85 0.90 0.95 1.00
0 10 20 30 40 50 60
true value
round 11
0.80 0.85 0.90 0.95 1.00
0 10 20 30 40 50 60
true value
round 12
●●
●●●●
●●●●●●
2 4 6 8 10 12
0.80 0.85 0.90 0.95 1.00
Posterior estimate of π01
observation rounds
probability of state transition
posterior estimates
prior estimates
true value
●
●
●
●
●●●
●●
●
●●
2 4 6 8 10 12
0.010 0.015 0.020
Change in standard deviation (learning) of π01
observation rounds
Standard deviation
 "O &%)&$3JTL *OTUJUVUF 1VCMJDBUJPO

5IF 7BMVBUJPO PG 1SJWBUFMZ)FME *OGSBTUSVDUVSF &RVJUZ *OWFTUNFOUT  +BOVBSZ 
 &YQFDUFE %JWJEFOE .PEFM
'SPN UIFTF QSJPS NFBO BOE WBSJBODF PG UIF
UXP ɮQBSBNFUFST XIJDI BSF TJNQMZ BE IPD
HVFTTFT PO PVS QBSU XF DBO EFSJWF UIF QSJPS
WBMVFT PG UIF NFUBQBSBNFUFST úBOE ġPG
UIFJS #FUB EJTUSJCVUJPO GSPN UIF EFàOJUJPO PG
UIF NFBO BOE WBSJBODF PG #FUBEJTUSJCVUFE
WBSJBCMFT BT TIPXT JO BQQFOEJY 
/FYU XF HFOFSBUF  SBOEPN TBNQMFT
PG PCTFSWBUJPOT UIBU GPMMPX UIF USVF
CJOPNJBM EJTUSJCVUJPO PG UIF PCTFSWBCMF
EBUB UIF OVNCFS PG TUSJDUMZ QPTJUJWF
EJWJEFOET PCTFSWFE HJWFO UIF QSFWJPVT
TUBUF GPS B TBNQMF PG /= QSPKFDUT
VTJOH B DPNQVUBUJPO TPGUXBSF 3 
BOE VQEBUF UIF WBMVFT PG UIF NFUB
QBSBNFUFST BDDPSEJOH UP UIF SVMF EFTDSJCFE
JO FRVBUJPO 
5IF OFX PS QPTUFSJPS WBMVFT PG úBOE ġ
DBO CF VTFE UP DPNQVUF UIF QPTUFSJPS NFBO
BOE WBSJBODF PG FBDI ɮQBSBNFUFS *O UIJT
FYBNQMF XF SFQFBU UIJT QSPDFEVSF  UJNFT
UP TIPX UIF FYUFOU BOE QBDF PG UIF MFBSOJOH
QSPDFTT BGGPSEFE CZ #BZFTJBO JOGFSFODF
'JHVSFT  BOE  JMMVTUSBUF UIJT VQEBUJOH
QSPDFTT CZ XIJDI FBDI QPTUFSJPS WBMVF
CFDPNFT B OFX QSJPS FBDI UJNF OFX EBUB
DBO CF PCTFSWFE 5IF NFBO PS FYQFDUFE
WBMVF PG FBDI ɮSBQJEMZ DPOWFSHFT UPXBSET
JUT USVF WBMVF BOE UIF WBSJBODF PG FBDI
QBSBNFUFS ɮSBQJEMZ EFDSFBTFT JOEJDBUJOH
UIBU UIFJS WBMVFT BSF LOPXO XJUI NPSF BOE
NPSF DFSUBJOUZ
'JOBMMZ FTUJNBUFT PG ɮ BOE ɮ DBO CF
VTFE UP DPNQVUF UIF QSPCBCJMJUZ PG SFDFJWJOH
B QPTJUJWF EJWJEFOE BU UJNF UGPMMPXJOH
FRVBUJPO  HJWFO B WBMVF PG QU−XIJDI
IBT UP IBWF CFFO DPNQVUFE àSTU PS DBO CF
UIF JOJUJBM DPOEJUJPO XIFO UIF QSPKFDU CFHJOT
FH Q= XJUI DFSUBJOUZ BT TIPXO PO
àHVSF 
*U TIPVME CF OPUFE UIBU NPTU PG UIF
uMFBSOJOHu PDDVST EVSJOH UIF àSTU GFX SPVOET
PG PCTFSWBUJPOT EVSJOH XIJDI UIF FYQFDUFE
WBMVF PG QBSBNFUFST ɮ BOE ɮ SBQJEMZ
DPOWFSHF UPXBSET UIFJS USVF WBMVF BOE UIFJS
WBSJBODF EFDSFBTFT FRVBMMZ SBQJEMZ #FDBVTF
XF TUBSUFE XJUI B IJHI MFWFM PG WBSJBODF
GPS FBDI WBMVF PG ɮ UIF MFBSOJOH QSPDFTT
GSPN PCTFSWJOH FWFO UIF àSTU GFX SPVOET
PG EBUB JT FYUFOTJWF BOE WBSJBODF SFEVDUJPO
WFSZ TJHOJàDBOU 5IFTF SFTVMUT TVHHFTU UIBU
PCTFSWJOH UIF àSTU àWF SPVOET PG  QSPKFDUT
JT TVGàDJFOU UP BDIJFWF QBSBNFUFS FTUJNBUFT
UIBU BSF WFSZ DMPTF UP UIFJS USVF WBMVFT
BTTVNJOH UIBU UIF OBUVSF PG UIF VOEFSMZJOH
QSPDFTT EPFT OPU DIBOHF
*U NVTU BMTP CF TUSFTTFE UIBU MFBSOJOH DPVME
UBLF QMBDF NPSF SBQJEMZ JG UIF QSJPS XBT NPSF
uJOGPSNFEu JF UIF QSPEVDU PG B SPCVTU FY
BOUF NPEFM PG QSPKFDU DBTI áPXT JOTUFBE PG
UIF BE IPD JOUVJUJPOT UIBU XFSF VTFE IFSF 
  8F QSPQPTF TVDI B NPEFM JO UIF
DBTF PG OPOSFDPVSTF JOGSBTUSVDUVSF
QSPKFDU àOBODF VTJOH B QPXFSGVM
DBMJCSBUJPO UPPM SFMZJOH PO EFCU TFSWJDF
DPWFS SBUJPT JO #MBOD#SVEF FU BM
  DPOEJUJPOBM %JWJEFOE 1BZNFOU
%JTUSJCVUJPO
'PMMPXJOH UIF FYQSFTTJPO PG UIF FYQFDUFE
WBMVF PG &4$3UHJWFO JO FRVBUJPO  PODF
UIF QSPCBCJMJUZ QU+ʲPG CFJOH JO UIF TUSJDUMZ
QPTJUJWF EJWJEFOE QBZNFOU TUBUF BU UJNF
U+ʲJT LOPXO PS DBMJCSBUFE BT CFTU BT
DVSSFOU JOGPSNBUJPO BMMPXT UIF EJTUSJCVUJPO
PG EJWJEFOET JO UIF QBZNFOU TUBUF TUJMM OFFET
UP CF FTUJNBUFE
"O &%)&$3JTL *OTUJUVUF 1VCMJDBUJPO 

5IF 7BMVBUJPO PG 1SJWBUFMZ)FME *OGSBTUSVDUVSF &RVJUZ *OWFTUNFOUT +BOVBSZ 
 &YQFDUFE %JWJEFOE .PEFM
'JHVSF  1SJPS BOE QPTUFSJPS FTUJNBUFT PG UIF QSPCBCJMJUZ PG PCTFSWJOH B QPTJUJWF EJWJEFOE BU UJNF U
●
●
●
●●●
●●
●
●●
●
2 4 6 8 10 12
0.75 0.80 0.85 0.90 0.95 1.00
Mean estimate of the probability of observing a positive dividend, pt
observation rounds
probability
posterior estimates
prior estimates
true value
$POEJUJPOBM PO CFJOH JO UIF QPTJUJWF
EJWJEFOE TUBUF &4$3U+ʲ∈],+∞[ *G XF
DBO HJWF &4$3U+ʲB GVODUJPOBM GPSN UIFO JU
DBO CF DBMJCSBUFE VTJOH BO JOGPSNFE QSJPS BT
XF TVHHFTU CFMPX BOE MBUFS VQEBUFE XIFO
OFX EBUB DBO CF PCTFSWFE $POWFSTFMZ
JG OP GVODUJPOBM GPSN JT BTTVNFE GPS UIF
EJTUSJCVUJPO PG &4$3U+ʲ|1U+ʲ= OVNFSJDBM
TBNQMJOH UFDIOJRVFT FH .BSLPW $IBJO
.POUF $BSMP DBO CF VTFE UP VQEBUF PVS
LOPXMFEHF PG UIF EJTUSJCVUJPO PG EJWJEFOET
*O XIBU GPMMPXT XF TIPX IPX XF DBO
FBTJMZ UP EFSJWF UIF NFUBQBSBNFUFST PG
B EJTUSJCVUJPO PG EJWJEFOET UIBU GPMMPX B
MPHOPSNBM QSPDFTT "HBJO UIJT BTTVNQUJPO
JT OPU OFDFTTBSZ CVU BMMPXT VT UP QSFTFOU B
TJNQMF JMMVTUSBUJWF FYBNQMF
.PSFPWFS BT XF TIPX JO #MBOD#SVEF
C UIFSF BSF HPPE SFBTPOT UP
CFMJFWF UIBU DBTI áPXT BSF JOEFFE MPH
OPSNBMMZ EJTUSJCVUFE BU MFBTU JO UIF DBTF PG
JOGSBTUSVDUVSF QSPKFDU àOBODF
  *O #MBOD#SVEF C XF TIPX
FNQJSJDBMMZ UIBU UIF EFCU TFSWJDF
DPWFS SBUJP %4$3 JO OPOSFDPVSTF
JOGSBTUSVDUVSF QSPKFDU àOBODF JT MPH
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●
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●●●●
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●
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0246810
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0246810
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0 5 10 15 20 25 30
0
2
4
6
8
10
Base Case & Expected Dividend
Period
Dividend (k$)
Base Case
Expected
0 5 10 15 20 25 30
0
20
40
60
80
100
Dividend Volatility
Period
Volatility (%)
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SJHIU QBOFM TIPXT UIF USBOTJUJPO QSPCBCJMJUJFT GSPN QBZNFOU TUBUF UP CPUI QBZNFOU BOE OPOQBZNFOU TUBUFT 5IF OPOQBZNFOU TUBUF JT EFOPUFE CZ 
BOE UIF QBZNFOU TUBUF JT EFOPUFE CZ 
0 5 10 15 20 25 30
0
20
40
60
80
100
Transition Probabilities from Non−Payment State
Period
Probability (%)
0→1
0→0
0 5 10 15 20 25 30
0
20
40
60
80
100
Transition Probabilities from Payment State
Period
Probability (%)
1→1
1→0
8F TUBSU XJUI UIF BTTVNFE EJWJEFOE
DIBSBDUFSJTUJDT EFQJDUFE JO àHVSFT  UP
 BOE BTTVNF UXFOUZ PCTFSWFE QSJDFT
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UIFO àMUFS PWFS UIFTF PCTFSWBUJPOT POF CZ
POF UP GPSN FYQFDUBUJPOT SFHBSEJOH GVUVSF
FYQFDUFE SFUVSOT
*O UIF MFGU QBOFM àHVSF  TIPXT UIF
BTTVNFE SJTL GSFF GPSXBSE DVSWF SG
U+ʲ XIJDI
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DVSWF
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SFUVSOT ǮU+ʲ PCUBJOFE BGUFS àMUFSJOH PWFS
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JOJUJBMMZ EFDSFBTF BT UIF EJWJEFOE WPMBUJMJUZ
EFDSFBTFT CVU UIFO JODSFBTF OFBS UIF MPBOT
UBJM BT EJWJEFOE WPMBUJMJUZ SJTFT $IBOHFT
JO FYQFDUFE EJWJEFOE QBZPVUT BMTP JNQBDU
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'JHVSF  "TTVNFE EJWJEFOE EJTUSJCVUJPO 5IF MFGU QBOFM TIPXT UIF QSPCBCJMJUJFT PG CFJOH JO QBZNFOU OPOQBZNFOU TUBUFT BOE UIF SJHIU QBOFM
TIPXT UIF FYQFDUFE MPTT JO FBDI QFSJPE BT B QFSDFOUBHF PG UIBU QFSJPET CBTF DBTF EJWJEFOE
0 5 10 15 20 25 30
0
20
40
60
80
100
Lockup and Payment Probabilities
Period
Probability (%)
Lockup
Payment
0 5 10 15 20 25 30
0
5
10
15
20
Expected Period Losses
Period
Expected Loss (%)
'JHVSF  5IF SJTLGSFF GPSXBSE DVSWF SG
U+ʲ VTFE JO UIF FYBNQMF JT TIPXO JO UIF MFGU QBOFM BOE àMUFSFE FTUJNBUFT PG FYQFDUFE FYDFTT SFUVSOT ǮU+ʲ
BGUFS PCTFSWJOH  USBOTBDUJPOT BSF TIPXO JO UIF SJHIU QBOFM
0 5 10 15 20 25 30
0
1
2
3
4
5
Risk Free Rate
Period
rt+τ
f(%)
0 5 10 15 20 25 30
−5
0
5
10
15
20
25
Expected Excess Returns
Period
λt+τ(%)
Prior
Filtered Mean
1−std Bounds
FYQFDUFE SFUVSOT BOE FYQMBJO GPS FYBNQMF
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FYDFTT SFUVSOT BSF DPNCJOFE UP DPNQVUF
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4JOHMF QFSJPE SBUFT BSF PCUBJOFE CZ BEEJOH
UIF TJOHMF QFSJPE SJTLGSFF SBUFT GPSXBSE SJTL
GSFF SBUFT UP UIF FYQFDUFE FYDFTT SFUVSOT
BOE NVMUJ QFSJPE FYQFDUFE SFUVSOT BSF
PCUBJOFE GSPN UIF TJOHMF QFSJPE FYQFDUFE
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BGUFS  USBOTBDUJPOT BSF TIPXO JO UIF SJHIU QBOFM 5IFTF FYQFDUFE SFUVSOT BSF CBTFE PO JOGPSNBUJPO BWBJMBCMF BU UJNF U JF UIF PCTFSWFE 
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0 5 10 15 20 25 30
−5
0
5
10
15
20
25
Expected Single Period Returns
Period
rt+τ(%)
Mean Return
1−std Bounds
0 5 10 15 20 25 30
0
5
10
15
20
Expected Multi Period Returns (Discount Rates)
Period
µt+τ(%)
BC IRR
Mean
1−std Bounds
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0 5 10 15 20 25 30
0.0
0.2
0.4
0.6
0.8
1.0
Stochastic Discount Factor
Period
mt+τ
Mean
1−std Bounds
0 5 10 15 20 25 30
0
5
10
15
Equity Price
Period
Price (k$)
Mean
1−std Bounds
5IF TUPDIBTUJD EJTDPVOU GBDUPS NBHOJàFT UIF
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0 5 10 15 20 25 30
0
20
40
60
80
100
Cash Yield
Period
Cash Yield (%)
0 5 10 15 20 25 30
0
5
10
15
20
Duration
Period
Duration (year)
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20 40 60 80 100
0
5
10
15
Price vs Volatility
Avg Dividend Volatility (%)
Price (k$)
20 40 60 80 100
5
10
15
20
IRR vs Volatility
Avg Dividend Volatility (%)
IRR (%)
'JHVSF  'JMUFSFE FTUJNBUFT PG UIF *33 MFGU QBOFM BOE USBOTBDUJPO QSJDFT SJHIU QBOFM XJUI POF TUBOEBSE EFWJBUJPO CPVOET
0 50 100 150 200
8
10
12
14
16
Evolution of IRR
Deals
IRR (%)
BC IRR
Filtered IRR
1−std bounds
0 50 100 150 200
6
7
8
9
10
11
12
Evolution of Prices
Deals
Price (k$)
Exact Mean
Observations
Filtered Mean
1−std Bounds
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5 10 15 20
8
9
10
11
12
Price after 3 Dividend Payments
Deals
Price (k$)
Ex−ante Expectation
1−std Bounds
Observed
Filtered
1−std Bounds
0 5 10 15 20 25 30
0
5
10
15
20
Term Structure after 3 Dividend Payments
Period
µt+τ(%)
Ex−ante
1−std Bounds
After 3 Pyaments
1−std Bounds
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 $PODMVTJPOT
 "O &%)&$3JTL *OTUJUVUF 1VCMJDBUJPO

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 $PODMVTJPOT
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EFQFOE PO ɮ
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QSJPS EJTUSJCVUJPO PG ɮBSF /BOE O XIJDI XF
LOPX UP CF PCTFSWBCMF
1SJPS BOE QPTUFSJPS WBMVFT PG UIF
NFUBQBSBNFUFST úBOE ġ
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WBSJBODF ʤɮ PG USBOTJUJPO QSPCBCJMJUZ ɮ XF
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5IF 7BMVBUJPO PG 1SJWBUFMZ)FME *OGSBTUSVDUVSF &RVJUZ *OWFTUNFOUT +BOVBSZ 
 5FDIOJDBM "QQFOEJY
5BCMF  1PTUFSJPS NFBO BOE WBSJBODF PG FYBNQMF USBOTJUJPO QSPCBCJMJUJFT BU UJNF U JO FBDI PCTFSWBUJPO SPVOE
3PVOE Ȇɮ ʤ
ɮ Ȇɮ ʤ
ɮ
 QSJPS    
    
    
    
    
    
    
    
    
    
    
    
    
USVF WBMVFT  
5BCMF  1PTUFSJPS NFBO BOE WBSJBODF PG FYBNQMF &4$3 EJTUSJCVUJPO QBSBNFUFST MPHTDBMF BU UJNF U JO FBDI PCTFSWBUJPO SPVOE
3PVOE ȆNʤ
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     
     
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     
     
     
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     
     
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USVF WBMVFT   
 "O &%)&$3JTL *OTUJUVUF 1VCMJDBUJPO

5IF 7BMVBUJPO PG 1SJWBUFMZ)FME *OGSBTUSVDUVSF &RVJUZ *OWFTUNFOUT  +BOVBSZ 
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XIJDI GPMMPXT GSPN UIF EFàOJUJPO PG UIF
CFUB EFOTJUZ XJUI NFBO ú
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PCTFSWFE UIF QPTUFSJPS WBMVFT PG úBOE ġ
DBO CF DPNQVUFE BDDPSEJOH UP FRVBUJPO 
BCPWF 5IF QPTUFSJPS WBMVFT PG UIF NFBO BOE
WBSJBODF PG ɮBSF UIFO DPNQVUFE VTJOH UIF
OFX WBMVFT PG úBOE ġ
5IF SFTVMUJOH WBMVFT GPS UIF FYBNQMF
EFTDSJCFE JO DIBQUFS  BSF HJWFO JO UBCMF 
GPS  JUFSBUJPOT VTJOH  OFX PCTFSWBUJPOT
JO FBDI SPVOE
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PG UIF DPOEJUJPOBM EJTUSJCVUJPO PG &4$3U
JO UIF QBZNFOU TUBUF
*G PCTFSWFE &4$3 EBUB DBMM JU Z GPMMPX B
MPHOPSNBM QSPDFTT PG NFBO NBOE QSFDJTJPO
Q UIFO JUT MJLFMJIPPE GVODUJPO JT HJWFO CZ
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QSPQPSUJPOBM UP UIF QSPEVDU PG B (BNNB
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XJUI B /PSNBM EJTUSJCVUJPO XJUI NFBO Ȇ
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NFBO &4$3 &&4$3U BOE BSJUINFUJD
WBSJBODF PG &4$3 BU UJNF Uʤ
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5IF 7BMVBUJPO PG 1SJWBUFMZ)FME *OGSBTUSVDUVSF &RVJUZ *OWFTUNFOUT +BOVBSZ 
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BOE TJHNB TDBMF PG UIF MPH EBUB UIVT
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NGPMMPXT B (BVTTJBO OPSNBM EJTUSJCVUJPO
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UIF QSJPS WBMVF PG NBOE UIF QSJPS WBMVF
PG QSFDJTJPO ŋJT TFU UP B TNBMM OVNCFS
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5IF WBMVF PG QSFDJTJPO QBSBNFUFS QGPMMPXT
B (BNNB EJTUSJCVUJPO GPS XIJDI XF OFFE
UP EFSJWF UIF TIBQF B BOE SBUF C NFUB
QBSBNFUFST 8F àSTU HJWF QB QSJPS FYQFDUFE
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QSFDJTJPO FH UFO UJNFT Q
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QSJPS LOPXMFEHF BOE UIF OFX JOGPSNBUJPO
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EFTDSJCFE JO DIBQUFS  BSF HJWFO JO UBCMF 
GPS  JUFSBUJPOT VTJOH  OFX PCTFSWBUJPOT
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3PCJDIFL BOE .ZFST  IJHIMJHIUT B
DPODFQUVBM QSPCMFN XJUI VTJOH B TJOHMF
EJTDPVOU SBUF UP BEKVTU CPUI GPS UJNF WBMVF
PG NPOFZ BOE SJTLJOFTT PG DBTI áPXT -FU
UIF DFSUBJOUZ FRVJWBMFOU PG UIF FYQFDUFE
DBTI áPX BU UJNF J&U[$J] JT $∗
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UJNF 5IVT UIF DPSSFDU QSFTFOU WBMVF PG UIF
VODFSUBJO DBTI áPX $JJT úJ&U[$J]
(+SG)J 5IJT FYFSDJTF
DBO CF SFQFBUFE GPS FBDI DBTI áPX PG BO
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 "O &%)&$3JTL *OTUJUVUF 1VCMJDBUJPO

5IF 7BMVBUJPO PG 1SJWBUFMZ)FME *OGSBTUSVDUVSF &RVJUZ *OWFTUNFOUT  +BOVBSZ 
 5FDIOJDBM "QQFOEJY
PCUBJO JUT WBMVF
7U=
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WBMVBUJPO GPSNVMB UIBU EPFT OPU EJTUJOHVJTI
CFUXFFO UJNF BOE SJTL QSFGFSFODFT BOE
JOTUFBE VTFT B TJOHMF DPOTUBOU EJTDPVOU SBUF
UP WBMVF UIFTF DBTI áPXT BT
7U=
5
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(+L)J 
XIFSF LDBO CF JOUFSQSFUFE BT UIF *33 PG UIF
JOWFTUNFOU BT JU JT UIF DPOTUBOU SBUF UIBU
BQQMJFT UP BMM GVUVSF DBTI áPXT
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FRVBUJPO  BOE FRVBUJPO  UP CF FRVBM
XF NVTU IBWF
úJ&U[$J]
(+SG)J=&U[$J]
(+L)J
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5IJT JNQMJFT
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B DPOTUBOU SBUF PG  −(+SG)
(+L) 5IJT JNQMJFT
UIBU UIF JOWFTUPS XPVME EFNBOE B MPXFS
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TUBSU XJUI B DPOTUBOU ƣGPS BMM QFSJPET BOE
UIFO DPNQVUF UIF .-& FTUJNBUFT PG UIFTF
QBSBNFUFST CZ àMUFSJOH PWFS QSJDFT "T DBO
CF TFFO GSPN UIF UBCMF JUFSBUJOH PWFS 
 QSJDFT BMNPTU DPNQMFUFMZ FMJNJOBUFT UIF
FGGFDUT PG JOJUJBM DIPJDF PO UIF àMUFSFE QSJDFT
5IBU JT NPEFM GPSFDBTUFE QSJDFT VTJOH .-&
FTUJNBUFT BGUFS JUFSBUJOH PWFS  QSJDFT
BSF BMNPTU FOUJSFMZ JOEFQFOEFOU GSPN JOJUJBM
DIPJDF PG QBSBNFUFST 4JNJMBS SFTVMUT BSF
PCUBJOFE GPS EJGGFSFOU DIPJDFT PG ɫWBMVFT
"O &%)&$3JTL *OTUJUVUF 1VCMJDBUJPO 

3FGFSFODFT
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5IF 7BMVBUJPO PG 1SJWBUFMZ)FME *OGSBTUSVDUVSF &RVJUZ *OWFTUNFOUT  +BOVBSZ 
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GPS MPBO JOTUSVNFOUT "MHP 3FTFBSDI 2VBSUFSMZ  m
●"OH " # $IFO 8 / (PFU[NBOO BOE - 1IBMJQQPV  /PWFNCFS &TUJNBUJOH
1SJWBUF &RVJUZ 3FUVSOT GSPN -JNJUFE 1BSUOFS $BTI 'MPXT
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&YQFDUFE 3FUVSOT 5IF +PVSOBM PG 'JOBODF  m
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JOWFTUNFOUT .FSJEJBN $BNQCFMM-VUZFOT BOE &%)&$ 3FTFBSDI $IBJS PO
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&%)&$3JTL *OTUJUVUF 1PTJUJPO 1BQFS +VMZ
●#MBOD#SVEF ' C -POHUFSN JOWFTUNFOU JO JOGSBTUSVDUVSF *O :BMF 40. BOE
&%)&$3JTL *OTUJUVUF "MUFSOBUJWF *OWFTUNFOU 4FNJOBST /FX )BWFO
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JO QSPKFDU àOBODF
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.D(SBX )JMM
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SFDFOU BEWBODFT JO TFRVFOUJBM .POUF $BSMP 1SPDFFEJOHT PG UIF *&&&  m
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NBSLFUT +PVSOBM PG 'JOBODJBM &DPOPNJDT  m
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"O &%)&$3JTL *OTUJUVUF 1VCMJDBUJPO 

5IF 7BMVBUJPO PG 1SJWBUFMZ)FME *OGSBTUSVDUVSF &RVJUZ *OWFTUNFOUT +BOVBSZ 
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BOE 3FUVSO PG /POUSBEFE "TTFUT GSPN $BTI 'MPXT 5IF $BTF PG 1SJWBUF &RVJUZ 'VOET
+PVSOBM PG 'JOBODJBM BOE 2VBOUJUBUJWF "OBMZTJT  m
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TFDPOE FE 0YGPSE 6OJWFSTJUZ 1SFTT
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0QUJNJ[BUJPO  m
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5IFPSZ BOE *OWFTUNFOU "OBMZTJT  FE 8FTU 4VTTFY +PIO 8JMFZ  4POT
●&TUZ # $  *NQSPWFE UFDIOJRVFT GPS WBMVJOH MBSHFTDBMF QSPKFDUT 5IF +PVSOBM PG
4USVDUVSFE 'JOBODF
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DBQJUBM TUSVDUVSF QPMJDZ GPS àOBODJBM JOTUJUVUJPOT "O JOUFHSBUFE BQQSPBDI +PVSOBM PG
'JOBODJBM &DPOPNJDT  m
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'JOBODJBM "OBMZTUT +PVSOBM m
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3BUF PG 1SPàU .BOBHFNFOU 4DJFODF  m
●(SBIBN + 3 BOE $ 3 )BSWFZ  5IF UIFPSZ BOE QSBDUJDF PG DPSQPSBUF àOBODF
FWJEFODF GSPN UIF àFME +PVSOBM PG 'JOBODJBM &DPOPNJDT  m
●)BMFZ $ 8  7BMVBUJPO BOE SJTLBEKVTUFE EJTDPVOU SBUFT +PVSOBM PG #VTJOFTT
'JOBODF \\^ "DDPVOUJOH  m
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JNQMJFE DPTU PG DBQJUBM *O 3FWJFX PG "DDPVOUJOH 4UVEJFT 7PMVNF  QQ m
 "O &%)&$3JTL *OTUJUVUF 1VCMJDBUJPO

5IF 7BMVBUJPO PG 1SJWBUFMZ)FME *OGSBTUSVDUVSF &RVJUZ *OWFTUNFOUT  +BOVBSZ 
3FGFSFODFT
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&RVJUZ 'VOET 443/ &MFDUSPOJD +PVSOBM
●,BQMBO 4 / BOE 3 4 3VCBDL  5IF WBMVBUJPO PG DBTI áPX GPSFDBTUT "O FNQJSJDBM
BOBMZTJT 5IF +PVSOBM PG 'JOBODF  m
●,BQMBO 4 / BOE " 4DIPBS  1SJWBUF &RVJUZ 1FSGPSNBODF 3FUVSOT 1FSTJTUFODF
BOE $BQJUBM 'MPXT 5IF +PVSOBM PG 'JOBODF  m
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*OGSFRVFOUMZ5SBEFE "TTFUT " #BZFTJBO 4FMFDUJPO .PEFM PG 7FOUVSF $BQJUBM
●-KVOHRWJTU " BOE . 3JDIBSETPO  5IF DBTI áPX SFUVSO BOE SJTL DIBSBDUFSJTUJDT
PG QSJWBUF FRVJUZ 8PSLJOH 1BQFS m
●1FUSJT ( 4 1FUSPOF BOE 1 $BNQBHOPMJ  %ZOBNJD -JOFBS .PEFMT XJUI 3
4QSJOHFS
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QSJWBUF FRVJUZ "WBJMBCMF BU 443/ 
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%JGàDVMUJFT 5IF +PVSOBM PG "MUFSOBUJWF *OWFTUNFOUT  m
●1IBMJQQPV - BOE 0 (PUUTDIBMH  5IF QFSGPSNBODF PG QSJWBUF FRVJUZ GVOET
3FWJFX PG 'JOBODJBM 4UVEJFT  m
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6OQVCMJTIFE XPSLJOH QBQFS
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%JTDPVOU 3BUFT 5IF +PVSOBM PG 'JOBODF  m
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*OEVTUSJBM .BOBHFNFOU 3FWJFX  m
●4IFBSFS " 5 BOE - 3 'PSFTU +S  *NQSPWJOH RVBOUJàDBUJPO PG SJTLBEKVTUFE
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●"NFOD / ' (PMU[ BOE - .BSUFMMJOJ 4NBSU #FUB  +VOF

●5JMM ) 8IP TBOL UIF CPBU +VOF
●6QQBM 3 'JOBODJBM SFHVMBUJPO "QSJM
●"NFOD / ' %VDPVMPNCJFS ' (PMU[ BOE - 5BOH 8IBU BSF UIF SJTLT PG &VSPQFBO &5'T
+BOVBSZ

●"NFOD / BOE 4 4FOEFS 3FTQPOTF UP &4." DPOTVMUBUJPO QBQFS UP JNQMFNFOUJOH
NFBTVSFT GPS UIF "*'.% 4FQUFNCFS
●6QQBM 3 " TIPSU OPUF PO UIF 5PCJO 5BY 5IF DPTUT BOE CFOFàUT PG B 5BY PO àOBODJBM
USBOTBDUJPOT +VMZ
●5JMM ) " SFWJFX PG UIF ( NFFUJOH PO BHSJDVMUVSF "EESFTTJOH QSJDF WPMBUJMJUZ JO UIF
GPPE NBSLFUT +VMZ
 "O &%)&$3JTL *OTUJUVUF 1VCMJDBUJPO

'PS NPSF JOGPSNBUJPO QMFBTF DPOUBDU
$BSPMZO &TTJE PO    
PS CZ FNBJM UP DBSPMZOFTTJE!FEIFDSJTLDPN
&%)&$3JTL *OTUJUVUF
 QSPNFOBEF EFT "OHMBJT #1 
 /JDF $FEFY   'SBODF
5FM      
&%)&$ 3JTL *OTUJUVUF&VSPQF
 'MFFU 1MBDF  -VEHBUF
-POEPO &$. 3#  6OJUFE ,JOHEPN
5FM    
&%)&$ 3JTL *OTUJUVUF"TJB
0OF (FPSHF 4USFFU  
4JOHBQPSF 
5FM   
&%)&$ 3JTL *OTUJUVUF/PSUI "NFSJDB
0OF #PTUPO 1MBDF  8BTIJOHUPO 4USFFU
4VJUF  #PTUPO ." 
6OJUFE 4UBUFT PG "NFSJDB
5FM    
&%)&$ 3JTL *OTUJUVUF'SBODF
 SVF EV  TFQUFNCSF
 1BSJT
'SBODF
5FM      
XXXFEIFDSJTLDPN