SOCIETY OF ACTUARIES Exam FETE Financial Economic Theory and Engineering Exam (Finance/ERM/Investment) Exam FETE MORNING SESSION

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1 OCIEY OF ACUARIE Exam FEE Facal Ecoomc heory ad Egeerg Exam (Face/ERM/Ivesme Exam FEE MORNING EION Dae: hursday, Ocober 9, 9 me: 8:3 a.m. :45 a.m. INRUCION O CANDIDAE Geeral Isrucos. hs examao has a oal of pos. I cosss of a morg sesso (worh 6 pos ad a aferoo sesso (worh 6 pos. a he morg sesso cosss of 9 quesos umbered hrough 9. b he aferoo sesso cosss of quesos umbered hrough 9. he pos for each queso are dcaed a he begg of he queso.. Falure o sop wrg afer me s called wll resul he dsqualfcao of your aswers or furher dscplary aco. 3. Whle every aemp s made o avod defecve quesos, somemes hey do occur. If you beleve a queso s defecve, he supervsor or procor cao gve you ay gudace beyod he srucos o he exam boole. Wre-Aswer Isrucos. Wre your caddae umber a he op of each shee. Your ame mus o appear.. Wre o oly oe sde of a shee. ar each queso o a fresh shee. O each shee, wre he umber of he queso ha you are aswerg. Do o aswer more ha oe queso o a sgle shee. 3. he aswer should be cofed o he queso as se. 4. Whe you are ased o calculae, show all your wor cludg ay applcable formulas. 5. Whe you fsh, ser all your wre-aswer shees o he Essay Aswer Evelope. Be sure o had all your aswer shees sce hey cao be acceped laer. eal he evelope ad wre your caddae umber he space provded o he ousde of he evelope. Chec he approprae box o dcae morg or aferoo sesso for Exam FEE. 6. Be sure your wre-aswer evelope s sged because f s o, your examao wll o be graded. ourez le caher d exame pour la verso fraçase. 9 by he ocey of Acuares Pred he U..A. 475 N. Margale Road Exam FEE-Fro Cover chaumburg, IL 673-6

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4 Facal Ecoomc heory ad Egeerg Formulae hee May 8 Morg ad aferoo exam booles wll clude a formula pacage decal o he oe aached o hs sudy oe. he exam commee fel ha by provdg may ey formulas, caddaes would be able o focus more of her exam preparao me o he applcao of he formulas ad coceps o demosrae her udersadg of he syllabus maeral ad less me o he memorzao of he formulas. he formula pacage was developed sequeally by revewg he syllabus maeral for each major syllabus opc. Caddaes should be able o follow he flow of he formula pacage easly. We recommed ha caddaes use he formula pacage cocurrely wh he syllabus maeral. No every formula he syllabus s he formula pacage. Caddaes are resposble for all formulas o he syllabus, cludg hose o o he formula shee. Caddaes should carefully observe he somemes-suble dffereces formulas ad her applcao o slghly dffere suaos. For example, here are several versos of he Blac-choles-Mero opo prcg formula o dffereae bewee srumes payg dvdeds, ed o a dex, ec. Caddaes wll be expeced o recogze he correc formula o apply a specfc suao of a exam queso. Caddaes wll oe ha he formula pacage does o dcae where he formula occurs he syllabus, or does provde ames or defos of he formula or symbols used he formula. Wh he wde varey of refereces ad auhors of he syllabus, caddaes should recogze ha he leer coveos ad use of symbols may vary from oe par of he syllabus o aoher ad hus from oe formula o aoher. We rus ha you wll fd he cluso of he formula pacage o be a valuable sudy ade ha wll allow for more of your preparao me o be spe o maserg he learg objecves ad learg oucomes. Facal Ecoomc heory ad Egeerg Formulae hee May, 8

5 Copelad, Weso, hasr, Facal heory ad Corporae Polcy Dv ( + s Re v + m Dv + ( W & + I Dv Re v ( W & I Re v ( W & I ( + s NI Re v ( W & dep A I dep Re v ( W & dep ( I dep ( + s NI A s ( + NPV NPV N FCF I ( + N FCF I ( + IRR B WACC weghed averagecos of capal b( τ c + s B + B+ FCFfor cap. budgeg ( Re v VC FCC τ c ( Re v VC FCC dep I ( Re v VC FCC( τ c + τ c( dep I ( Re v VC FCC dep( τ c + dep I EBI ( τ + dep I earg before eres ad axes c γ γ g a N ( γ p geomerc reurs + ( + γ p N arhmec reurs ER ( j Rf + ER ( m R f β j R E( R + β δ + ε j j j m j R R ( R R β + ε j f m f j j R γ + γβ + ε where γ Rm R f R p excess reur o porfolo p ( Rp Rf ' p p p Facal Ecoomc heory ad Egeerg Formulae hee May, 8

6 R α + β R + ε m [ ] ER ( ER ( + ER ( ER ( β z m z α ER ( ( β z ER ( Rf b ER ( m R f + semb ( + hehml ( λ ER ( ER ( m z R p ( + rp D AR ( A( + AGP ( P R ˆ γ + ˆ γ β + ε R γ + γ l( sze + γ ( Boo + ε Mare [ ] ER ( ER ( + ER ( ER ( β z m z ER ( ER ( + ER ( ER ( β z, I I z, I, I R E( R + b F bf + ε where R radom rae of reur o he h asse ER ( he expeced rae of reur o he h asse b he sesvy of he h asse s reurs o he h facor F he mea zero h facor commo o he reurs of all asses ε a radom zero mea ose erm for he h asse w R wr we( R + wb F wb F + wε p w chose o be a large umber wb for each facor R we( R + wb F wb F p Facal Ecoomc heory ad Egeerg Formulae hee 3 May, 8

7 R R we( R p we( R p ER ( λ + λb λ b E( R R λ b λ b f ER ( R δ R b δ R b b f f f Cov( R, δ Var( δ where Cov( R, δ he covarace bewee he h asse s reurs ad he lear rasformao of he h facor Var( δ he varace of he lear rasformao of he h facor C (,, Nd ( Nd ( where A B A B d + l( A V B V d d V s d d V V ρ V V + V A AB A B B [, ] Q Q MAX Q capacy V( η q( m MAX p( e m U( a, e V( η m a e f ( p,... p η f( p,... p η m m V( η V( η far game ε p( r c + ( p( dr c p( r c + ( p( r c d r( d + c c p rrd r d rd ( c c d > ad r( < c c p p E( p η p pj, + E( pj, + η p j he acual prce of secury j ex perod j, + j j, + j j, + pj pj where p j, + E( pj, + η he predced ed-of-perod prce of secury j gve he curre formao srucure η ε j, + he dfferece bewee acual ad predced reurs Facal Ecoomc heory ad Egeerg Formulae hee 4 May, 8

8 E( ε j, + E rj, + E( rj, + η submargale E( p η p j, + j p j Er ( η > j, + margale E( p η p j, + j p j Er ( η j, + radom wal f( r, +,..., r, + f( r, +,..., r, + η r E( r r, r,..., r ε j, + j, + j, + j j, j, E( r E( r ( r E( r Cov( r, r r E( r r E( r f( r dr j, + j, + j j j, + j r j j j, + j, + j j j ER ( ˆ ( ˆ ˆ j β j Rf + ERm βm R f β j E( ε j where ε ( ˆ j Rj E Rj β j ER ( ˆ β he expeced rae of reur o he jh asse durg hs me perod, gve a predco j j R f of s sysemac rs, ˆ β j he rs-free rae of reur durg hs me perod ER ( ˆ β he expeced mare rae of reur, gve a predco of s sysemac rs, ˆm β m m ˆm β he esmaed sysemac rs of he jh secury based o las me perod s formao srucure η Rj aj + bjrm + ε j R α + β ( R R + β ( RLE RE + β ( HBM LBM + ε j j j m f j 3j j he chage eargs per share for he jh frm NI ˆ ˆ j a + bj m + ε j where m he chage he average EP for all frms (oher ha frm j he mare NJ aˆ + bˆ m j, + j + where ab ˆ, ˆ coeffces esmaed from me-seres fs of R α + β ( R R + β ( RLE RE + β ( HBM LBM + ε j j j m f j 3j j m + he acual chage mare average EP durg he (+h me perod N abormal performace dex API ( + ε j N j where N he umber of compaes a porfolo,,..., Facal Ecoomc heory ad Egeerg Formulae hee 5 May, 8

9 ε j abormal performace measured by devaos from he mare model V ( α [ ( ] ( r μα + λ where r he rs-free rae of reur μ( α he valuao schedule used by he mare o fer he expeced ed-of-perod value from he sgal α λ he mare s adjusme for he rs of he projec max mze E U ( W subjec o W X + βv + Y ( α V( α M where W he erepreeur s al wealh V M he value of he mare porfolo β he fraco of he mare porfolo owed by he erepreeur Y he amou vesed he rs-free asse α he fraco of he projec he erepreeur reas W ˆ he ucera ed of perod wealh of he erepreeur Wˆ ˆ ˆ αμ ( + ε + βm + ( + r Y α [ μ+ ˆ ε μ( α + λ] + β Mˆ ( r V + M + ( + r( W X + μ( α λ where ˆM he gross reur of he mare porfolo EUW ( ( EU ( W ( μ+ ε μ( α + λ+ ( α μα α EUW ( ( EU ( W ( M ( + r V β EU ( W ( ε λ ( αμ + α EU ( W M where μα μ α E ( D X D ( ( p ( ( ( ( ( V D + τ D + X r D f X dx + + β X D f X dx D X + D V( D + μτ pdβ ( X D f( X dx + r X D ED ( VD ( + τ pdβ + r D V ( D τ p + β ( D V D ( τ pd ( β r Facal Ecoomc heory ad Egeerg Formulae hee 6 May, 8

10 D ( A V D ( ( τ p + βa D ( τ p + r τ p + r β (+ r A β r + β + r + + τ p ( + r I + D C+ Np C+ P e e Q M max mze ( τ p D + pem + X D Q+ N P+ τ pdl max mze L τ pd + X D P+ τ pd+ I C τ p P L+ I C ( τ p + X D ( P + τ D + I C p DX ( max( I C+ L,lX τ [ ] p P D( x C+ X I τ D( x [ ] P P D( x + τ pd( x L τ p D I C+ L old V + a p old P V ( E+ + a+ b P + E where P mare prce of old share P ( E+ + a+ b + a P + E P E ( E+ b ( + a P + E P + E ( E+ b E ( + a P P + a + E( B ssue ad ves E b ( E+ b < ( + a ora> P ( + P E maager s wage before soc spl W ( z αp( z + βp ( m, P Facal Ecoomc heory ad Egeerg Formulae hee 7 May, 8

11 maager s wage wh soc spl aouceme s W (, z αpˆ (, z + βp (, p W ˆ α P P γ maxmze wage ˆ(, ˆ γ Pz P α PP [ ] Pz ˆ(, cz ( γ + where γ α ( γ mare value of he frm afer he spl aouceme γ γ [ ] [ ] Mz (, Pz ˆ(, P (, ( + cz ( + cz ( γ γ γ γ M ( z, ( γ γ γ γ M(, z ( μ K M( z M( z M( z M( z where M ( z pre-spl value of he frm, ( l u l K M z ( l M( z γ + γ K ( γ V ( X EX X ˆ X s + Yˆ ( s s + s m m ( ( V ( Xˆ ( ( C m X where ( he expeced oal broerage commsso E X( C he cos of execug he spl V( X ˆ( C X ˆ( EX ( N ( F F Xs+ Ys ˆ + YF V Y EX X Yˆ YC m m s (, (, m, s + s m + F s Facal Ecoomc heory ad Egeerg Formulae hee 8 May, 8

12 ˆ Xs + Ys m m Xs + Ys m m+ ( Fs s + sm EV ( Ym C s + s + F m s Xs+ Ys EY Y E( X( X, Yˆ, Y Y m m m m m s + sm s + s + F ˆ ( m s m m Y ( s + s F m s Ym + + K sm sm dα B ( [ V( B(, D ] α( D( dd D d d B B α α ( [ V( B(, D ] ( ( α( D( < dd dd D D [ ] ( D ( V( B(, D ( V( I α ( ( ( d α V B dα B α B dd d B dv B α + α dd dd D D d dd d D ˆ( εα, D ( a D + D α α D X αε ( γ ( α ( εγ c( s, p, a [ ] max mze U s c( s, p f ( s, p a dsdp subjec o [ ] c( s, p, a V c(, s p f (, s p a dsdp G( a V [ ] λ [ ] U [ s c(, s p ] [ (, ] λ [ (, ] V [ c(, s p ] max mze U s c(, s p f (, s p a dsdp + V c(, s p f (, s p a dsdp G( a V U s csp + V csp or λ [ ] max mzeu ( + λ V c( s, p f ( s, p a dsdp G( a V a c( s, a [ ] subjec o [ ] max mze U s c( s f ( s a ds V c( s f( s a dsg( a V Facal Ecoomc heory ad Egeerg Formulae hee 9 May, 8

13 [ ] V c( s f ( s a ds G ( a a c( s, a [ ] + λ [ ] V[ c( s ] fa ( s a ds G ( a [ ] + λ [ ] + μ [ ] max mze U s c( s f ( s a ds V c( s f ( s a ds G( a V + μ [ ] [ ] U s c( s fa ( sa λ+ μ V c( s f( s a U s c( s f( s a V c( s f( s a V c( s f ( s a a fa ( sa γ ( ( δ + δc λ+ μ δ f ( sa δ ( f ( a sa γ γ ( + ( δ ( λ+ μ δ f( s a cs c( s, p, a [ ] λ [ ] [ (, ] (, ( Maxmze U s c(, s p f (, s p a dsdp + V c(, s p f (, s p a dsdp G( a V + μ V c s p f s p a dsdp G a [ ] [ ] U s c(, s p fa (, s p a λ+ μ V c(, s p f(, s p a [ ( ] [ Ρ ] a U sc Ρ fa ( p a f ( ( a p a f a.. pa λ + μ + μ + + μ V c( f( p a f( p a f( p a m s b a + ε j j j s m p qa + ε for,.., j j j j c( p,..., p β β p + c( s, p, m, a( m, m( m [ ] max mze Espm,, U s c( s, p, m a( m subjec o (for all m spm [ ] [ ] am ( a ha maxmzes sym [ ] mm ha maxmze [ ] mm ( he ˆ ( E V c( s, p, m G a( m a( m V, E V c(, s p, m a G( a, for each m E V c(, s p, mˆ a G( a sym, for each m V( X P( X C( X X X X Facal Ecoomc heory ad Egeerg Formulae hee May, 8

14 ( β Es ( α λ( β CovsR (, M + r f ( β Es ( α λ( β CovsR (, M max mze αβ, + r f a E( W + α + βe( s b Var( W + βcov( W, s + β Var( s V subjec o [ ] ( β Es ( α λ( β CovsR (, M max mze + αβ, + r f [ ] μ( a E( W + α + βe( s bvar( W + βcov( W, s + β Var( s V μa + r μ f Es ( λcovsr (, m ( ( (, + ( + r [ ae s b Cov W s βvar s ] [ α β ] β a + E( s b Var( s V λacov(, s RM Cov( W, s β bvar( s Var( s f r ( + D V + db Bew ( D P(,,, rf, V ( + r D+ dd f Bew B D P( V + db,,, rf, V P( V,,, rf, V + D( PX + PY D dd + D B V PV (, D,, rf, V V [ ] V q( s V( s I ds s a [ ] V q( s V( s I D ds E sa sb [ ] V q ( s V ( s I ds + q ( s Dds D sa sb Facal Ecoomc heory ad Egeerg Formulae hee May, 8

15 [ ] V q( s V( s I ds s b V EFCF ( U ρ where V U he prese value of ulevered frm(.e. all equy EFCF ( he perpeual free cash flow afer axes ρ he dscou rae of a all-equy frm of equvale rs V U EFCF ( EEBI ( ( τ c or VU ρ ρ NI + D (Re vvcfcc dep( τ + Dτ V d c d c E( EBI ( τ Dτ + ρ L c d c D d B b b V V + τ B L U c V L ( τ c E( EBI B + τ c I ρ I I VL + + B + B I I I I I VL B B I + B V L + B + + I I I I I V L I > ( τ c E( EBI B > ρ( τc I I B weghed average cos of capal WACC ρ( τc I Facal Ecoomc heory ad Egeerg Formulae hee May, 8

16 B WACC ρ( τc V NI D d τ c ( D d EBI + ( τ c I I I I V I L NI ( D + ( τ d I c I B + τ c ρ I I I I VL + B + B NI B ( ( c b ρ τ ρ B ρ+ ( τ ( ρ s c b B WACC ( τ c b + s B + B+ B WACC ρ( τc B + G V V τ B L U c V U EEBI ( ( τ ( τ ρ c ps payme o shareholders ( EBI D( τ ( τ d c ps payme o bodholders afer persoal axes D( τ oal cash paymes o supples of capal EBI ( τ ( τ D( τ ( τ + D( τ d c ps d c ps d pb pb V L E( EBI ( τ ( D d ( τ pb ( τc( τ ps c τ ps + V ρ D d ( τ pb where B ( τc( τ ps G B ( τ pb ( τ ( τ ( τ pb c ps b b U ( τc( τ ps + + B ( τ pb Facal Ecoomc heory ad Egeerg Formulae hee 3 May, 8

17 ( τ c G ( B ( τ pb ER ( j Rf + ER ( m R f β j where ER ( j he expeced rae of reur o asse j R f he rs-free rae ER ( m he expeced rae of reur o he mare porfolo Cov( R j, Rm β j Var( R B βl + ( τc βu ER ( bj Rf + ER ( m R f βbj m s ( EBI R bj B( τ c L E R Cov R ( s f + λ ( s, m ( τc ( τc B Cov( s, Rm Cov( EBI, Rm Cov( Rbj, Rm L L L R f + λ ( τc Cov( EBI, Rm λ ( τc B Cov( Rbj, Rm ( EBI ( τ c E( Rbj B( τ c U R V + λ ( τ Cov( EBI, R E( EBI ( τ f c m c V V + τ B L U c V ( B P + Er ( rf + Er ( m r f β where Er ( he saaeous expeced rae of reur o asse Cov( r, rm β he saaeous sysemac rs of he h asse Var( rm Er ( m he expeced saaeous rae of reur o he mare porfolo r he o-sochasc saaeous aualzed rae of reur o he rs-free asse f d dv + d + V d d V V d dv dv V lm V V V Facal Ecoomc heory ad Egeerg Formulae hee 4 May, 8

18 r V r V V Cov( r, rm Cov( rv, rm β, βv Var( r Var( r m VCov( rv, rm V β βv V Varr ( V m m r f VN( d e DN( d where he mare value of equy V he mare value of he frm s asse r he rs-free rae he me o maury D he face value of deb (boo value f L( V + rf d D + d d V β Nd ( βv β VN( d β VN( d De N( d ( e V r V f r f Nd D βv ( N( d V R + ( R R N( d β s f m f V V ( ( s Rf + N d Rv Rf B V βb β V VB B N ( d N ( d V R + ( R R β b f m f B V + ( ρ ( b Rf Rf N d B B b s Rf ( Rf N( d V + ρ B V V + + B V V + ( ( ρ Rf N d Rf V B+ R f( + ( ρ Rf [ N( d + N( d ] Rf + ( ρrf [ N( d + N( d ] ρ V Facal Ecoomc heory ad Egeerg Formulae hee 5 May, 8

19 s B ρ+ ( ρ b dv V μ( V, d + dw VFVV V+ rvfv V rfv+ FV + C (, (, (, (, V FVV V + rvfv V rf V + C ( ( ( ( r FV ( A+ AV+ AV ( r BV ( A + AV+ AV r α V B ( p C B pb[ ( α VB] B r BV ( C + ( C ( V r r V + where p B ( V V B r DC( V αvb ( V V B r r B( V A + AV + A V B( V c( C c( C ( V r r V B r V ( V V ( V + B( V DC( V V ( V + B p B αv p L U c U c B c B B M ( + r γ V + γv f V D M ( + r γ V + γ( V c f V < D where γ, γ posve weghs r he oe-perod eres rae V, V he curre ad fuure value of he frm D he face value of he deb c a pealy pad f barupcy occurs f V < D where a M a γ( + r V + γva f D < D Va + r (ell he ruh V b M a γ( + r + γv a f D< D + r (le D he maxmum amou of deb ha a usuccessful frm ca carry whou gog barup V a M a γ( + r + γ( V b c f D < D V a (le + r V b M a γ( + r + γv b f D< D (ell he ruh + r γ ( V V < γ c a b Facal Ecoomc heory ad Egeerg Formulae hee 6 May, 8

20 dv( + + P( + P( u ( + P ( where u ( + he mare requred rae of reur durg he me perod dv ( + dvdeds per share pad a he ed of me perod P+ ( prce per share a he ed of me perod P ( prce per share a he begg of me perod Dv( + + ( P( + V ( + ( + u where Dv ( + oal dollar dvded payme ( dv( + V ( he mare value of he frm ( P( EBI ( + + m ( + P ( + I ( + + Dv ( + R ( + Dv ( + + ( P ( + R ( + Dv ( + + ( + P ( + m( + P ( + R ( + EBI ( + I ( + + V ( + V ( EBI( + I ( + + V ( + + ( + u Y d ( EBI rdc ( τ c rdp ( τ p where Y d he ucera come o he h dvdual f corporae come s receved as dvdeds EBI he ucera cash flows from operaos provded by he frm r he borrowg rae, whch s assumed o be equal for dvduals ad frm D he corporae deb D persoal deb held by he h dvdual c τ c he corporae ax rae p τ p he persoal come ax rae of he h dvdual Y g ( EBIrDc( τ c( τg rdp( τ p where Y g he ucera come o he h dvdual f corporae come s receved as capal gas he capal gas rae for he h dvdual τ g Y g ( EBI rdc( τ c rdp ( τg + rdp( τ p τg Facal Ecoomc heory ad Egeerg Formulae hee 7 May, 8

21 dv Y g Y > d Y g corporae deb r( τ c( τ g Dc Y g persoal deb r( τ p D p dv j Rj Rf δ + δβ j + δ ( R P f + ε j j where δ a cosa δ fluece of sysemac rs o R j he sysemac rs of he jh secury δ fluece of dvded payme o j j R j β j P he dvded yeld of he jh secury ε j a radom error erm R f he rs-free rae cos of eral fuds r ( τ ( τ A c d where r A he pre-ax reur o vesmes real asses persoal dvded come ax rae of he h dvdual τ c corporae effecve margal ax rae EBI + mp + B I + Dv EEBI ( V Dv+ + τ d E( EBI V B mp Dv + B mp + E( EBI EBI I+ + [ ] E f( I f ( I E ( E( EBI E( I + f( I I+ + + E ( EBI f( I + E ( ε ε EBI I f I I ( + ε + f( I + ε I + f( I + γε + γ γ E( ε + [ EBI E( EBI ] Dv a + c ( Dv Dv, + U where Dv he chage dvdeds c he speed of adjusme o he dfferece bewee a arge dvded payme ad las year s payou j Dv he arge dvded payou au a cosa ad ormally dsrbued radom error erm Facal Ecoomc heory ad Egeerg Formulae hee 8 May, 8

22 P ( P P P ( P P + dv( B g B C A g A C PB P dv A g arbrage prof π P + dv ( dv + P + P P B A B A π ( ( P P + dv A B DY a+ aβ + a3age + a4inc + a5dr + ε where DY dvded yeld for he h dvdual s porfolo β he sysemac rs of he h dvdual s porfolo AGE he age of he dvdual INC he gross famly come averaged over he las hree years DR he dfferece bewee he come ad capal ga ax raes for he h dvdual ε a ormally dsrbued radom error erm Dv β Dv + β NI + β NI + Z 3 where Dv he chage dvdeds perod Dv he prevous perod s dvdeds NI hs perod s eargs NI las perod s eargs Z uacpaed dvded chages (he error erm R α + β R + ε where he oal reur (dvdeds ad capal gas o he commo soc of he jh frm j j m j R j β j a cosa erm R m sysemac rs ε j he abormal performace of he jh secury P a+ bdv + cre + ε where P he prce per share RE reaed eargs ( NI P a + b + ε ( NI P ε Dv aggregae dvdeds pad ou he error erm where ( NI P he eargs / prce rao for he frm ( NI P a me dex he average eargs/prce rao of he dusry ER ( j Rf + ER ( m R f β j ε he error erm DY DY R γ + R γ β + γ + ε j m j m j DY j m where R he rae of reur o he jh porfolo j γ a ercep erm ha should be equal o he rs-free rae Facal Ecoomc heory ad Egeerg Formulae hee 9 May, 8 R f accordg o CAPM

23 R m he rae of reur o he mare porfolo β j he sysemac rs of he jh porfolo γ he dvded mpac coeffce DY he dvded yeld o he jh porfolo, measured as he sum of dvdeds pad durg he j prevous year dvded by he ed-of-year prce DY he dvded yeld o he mare porfolo, measured over he perod of mohs j m ε he error erm ER ( j Rf a+ aβ j + a3( DYj Rf where ER ( j he expeced before ax reur o he jh secury he before-ax reur o he rs-free asse R f β j he sysemac rs of he jh secury a he cosa erm a he margal effec of sysemac rs a 3 he margal effecve ax dfferece bewee ordary come ad capal gas raes DY he dvded yeld (.e. dvded dvded by prce for he jh secury j [ ] [ ] [ ] R λ + β MF + λ + β MB + λ + β HML + λ + λ d + ε p F F 3F 3 4 p, p where MK he excess reurs o he CRP value-weghed porfolo MB he dfferece bewee average reurs o small mus bg equy capalzao porfolo HML he dfferece bewee average reur o hgh mus low boo equy o mare equy porfolo d he equally weghed yeld of socs porfolo p mus he mare dvded yeld p, λ he rs premum correspodg o he h rs facor λ he coeffce o he dvded yeld measure 4 PN E E PN P( N NE + W where P E he pos exprao share prce N E he umber of shares ousadg afer repurchase P he pre-aouceme share prce N he pre-aouceme umber of shares ousadg P he eder prce W he shareholder wealh effec arbuable o he eder offer F P NE fraco of shares repurchased N W ( F ( P P F P + P NP P P E P P Chew, he New Corporae Face, Noe Hardy, Ivesme Guaraees + w + w LN( μ, log ( μ, w w N w w Facal Ecoomc heory ad Egeerg Formulae hee May, 8

24 f( x exp x πw ( log( x wμ w + w E e wμ+ w ( e + w wμ+ w w V e Y Y ( μ + a Y μ + ε ( μ + a Y μ + ε ( Y α + α μ + β π p p, + p,,, π π [ ] π ρ π ρ p( r Pr R( r Pr R( r + Pr R( r ( R R + ( R F ( x Pr( x Pr( x R r p ( r r log x μ ( r F ( ( x Φ p r r ( r f ( x p ( r log x μ ( r φ r ( rx ( r (from erraa shee { yδq μy } y( exp w ( + + y( where y( a y ( + z( μ [ y( ] e exp( ( [ exp( ( ] y w δ y Ε Ε Ε y q y y y u Mδ ( u exp uμ q q + ( q μ y y Ε [ y (] e Mq( wy exp μ y + ( ay Facal Ecoomc heory ad Egeerg Formulae hee May, 8

25 DM ( d δ ( + ( d DM ( d q d l( μ, ( y μ μ l( μ, + ( y μ 3 ˆ ( y ˆ μ where ˆ μ y l( μ, μ l( μ, ( Y μ μ l( μ, 3 ( Y 4 μ + l( μ, E μ l( μ, E μ l( μ, E ˆ ˆ a ( Y μ ( a l( μ,, a l( exp + π ( Y ( a μ ay l( exp π π l( + l( l a Y a ( Y ( a μ ay ( μ ( + where l N( μ,( h( a, ha (, ( a ( a Facal Ecoomc heory ad Egeerg Formulae hee May, 8

26 + l μ Z+ Z Z ε( a a [ ] F ( x Pr x Pr x R r p ( r r (,.., ( θ π( θ,.., f xx x f x x x dθ θ r Φ where f ( X θ s he desy of X gve he parameer θ ( θ,.., θ, θ +,.., θ ( r +, r ( r + ( r + ( r + ( r Θ L α m, L ( rr, + (, Θ ( ( (, + ( r ( ξ π ξ q θ ξ ( r r r ( r ( r ( θ Θ π ( θ q ξ θ ( r l x μ ( r p( r α + α ( Y μ + β ( ( F + F m F ( m F + ( + u F ( + u ( m u M F m m F s ( m c c erraa shee C p ( G F τ + x ( C p M + q G F oe: M should have d superscrp τ d d + x x ( C p F ( m m + q GF ( m erraa shee τ d x d x + τ ( + C q G F p M where < < d x r x r r τ ( ( + + d τ r r x r + r r x r r r x r C q G F p G F p M + e r u r P ( K d ( K d e p u d where e r u p u d + β AΦ Nα( α Facal Ecoomc heory ad Egeerg Formulae hee 3 May, 8

27 ξ log G ( μ + log( m Φ r Pr F + Vα e > G α V GF e α ( F ( α r Vα ( GF exp( zα + ( μ+ l( m e r ( β E X X > Vα + ( β α V CE α ( L α r r CE α( L E ( G F e F < ( GVαe α [ ] ( μ+ log( m + r e CE α( L e G Φ( zα α ( ξ CE α( X CE ξ( X ( α r ( ξ exp( ( μ l( ( E L e G F + m + Φ A ( y y y y y log( + ρ μ + φ log( + μ + ε y y y y ρ ρ ρ ρ where A (l GF ( μ + l( m + H B(, E Q F( ga65( H ga (, B(, d + 65 FEQ { ( ( ( ( ( } H F ga Φ d Φ d where 65 d ( log( ga65( + y ( y ad d( d( y PP : max P + α, G where P: sgle premum, α : parcpao rae, G: guaraeed payou, : value of he equy dex a me Aual Rache CAR: P + max α, Facal Ecoomc heory ad Egeerg Formulae hee 4 May, 8

28 Facal Ecoomc heory ad Egeerg Formulae hee 5 May, 8 AR: max, P α + CAR wh cap rae c: m max,, P c α + AR wh cap rae c: m max,, P c α + Hgh Waer Mar: max max, P G α + where ( max max,,.., H P G α + ( P G H P α α α ( ( { } d PP r P H e d K e d α Φ Φ where ( PP G K P α α ( ( ( ( d r H Pe d G P e d α α Φ Φ ( l P r d G P d α α + +, d d max, RP P α + ( r Q H E e RP max, r Q H PE e α + max, r r Q H P e E e α +

29 { } ( α Φ( Φ( αe e max, e d e d r d r Q where d r d +, d d { r d r α ( ( ( } H P e + e Φ d e Φ d r g EQ e + max α, e where g e : mmum accumulao facor { max ( (, g α } r E Q e + e ( α g r g e E Q e + ( e + α max, α ( α g gr e e + α BCK, α BC K : Blac-choles call-opo prce wh sre K, sarg soc prce. ad erm years (, d { ( ( ( ( r gr cr α Φ Φ 3 + α ( Φ( Φ ( 4 + Φ( + Φ ( 4 } P e d d e d d e d e d where where d d l r d g + + e ( α α l r d c + + e ( α α d d d d AR wh lfe-of-corac guaraee whou cap: d( PP r ( { ( ( ( ( } H α e Φ d + K e Φ d ( ( ( ( d + ( c e Φ d + Φ d P + α oole ad Herge, Isurace Idusry Mergers ad Acqusos r r + β ( r r f m f Facal Ecoomc heory ad Egeerg Formulae hee 6 May, 8

30 where r expeced rae of reur o he acquso r rs-free rae of reur r expeced rae of reur for he mare as a whole f m β measure of rs of a compay (boh deb ad equy relave o he mare as a whole D D E E r r + ( rf + β ( rm rf D+ E D+ E where r weghed average cos of capal WACC D E r requred reur o deb β bea of a compay s soc D mare value of a compay s deb E mare value of a compay s equy cos of capal requred capal (dscou rae afer ax eargs apprasal cos of capal NPV(cos of capal NPV(dsrbuable earg Excess capal +NPV(afer ax NPV(afer ax earg o he bu s ess + Excess NPV(crease RC rae eargs - Isurace capal +NPV(afer-ax earg o RC capal - NPV(afer ax earg o he bu s ess + Excess capal +NPV( RC (NPV( RC NPV( RC NPV(afer ax earg o he bu s ess + Excess capal + NPV( RC ((+dnpv( RC RC - NPV( RC NPV(afer ax earg o he bu s ess + Excess capal + RC -NPV( RC ( d value of Iforce ad Fuure Busess + adjused of boo value cos of requred capal where afer ax vesme eargs rae o capal d dscou rae RC requred capal oal reserve ( expeced loss where PLDF pad loss developme facor PLDF IBNR reserve ( expeced loss where RLDF repored loss developme facor RLDF rgeorgs, Real Opos NPV αec ( I ( + r...( + r [ ] Er ( r+ β Er ( r j j m [ ] Expaded( sraegc e prese value( NPV Drec( passve NPV + sraegc value + flexbly value dπa π A π A dα B + dk K α dk A A B A Hull, Opos, Fuures ad Oher Dervaves, Facal Ecoomc heory ad Egeerg Formulae hee 7 May, 8

31 z ε z ( z( ε N dx ad + bdz dx a( x, d + b( x, dz e μ ds μd + dz Δ μ + ε Δ φμ (, G G G G dg ( a + + b d + bdz x x x d μd + dz G G G G dg ( μ + + d + dz F e r ( df ( μ r Fd + Fdz dg ( μ d + dz l φl + μ, E b g e o μ μ varbg e e x l x φ μ, Facal Ecoomc heory ad Egeerg Formulae hee 8 May, 8

32 s s bu ug where u b g F HG u I KJ u l d μd + dz f f f f df ( μ + + d + dz f f f f f ( μ z Π f + f f Π f f + + r f f rf ( r f e Eˆ Ke ( Eˆ e r f Ke r r r ( ( c N d Ke N d ( ( r p Ke N d N d where d ( ( / + + / l K r ( ( / + / l K r d d r ( ( r c e N d e KN d ( ( ( r D Ke K r q c Ke + p+ e Facal Ecoomc heory ad Egeerg Formulae hee 9 May, 8

33 r ( ( c e N d Ke N d q q ( ( r p Ke N d e N d d ( ( / + + / l K r q ( ( / + / l K r q d d d ( r q d + dz e p ( rq Δ u d d ( ( r c e FN d KN d ( ( r p e KN d FN d r r c Ke + p+ Fe r f e pfμ + ( p f d f f F + F rf r HF e HA ( r q H F e HA F ( rrf H e H A x ' N ( x e π ΠΘ + Γ Θ+ r + Γ r [ N( d ] q e q r p e + c+ Ke a e ( ( f g Δ Facal Ecoomc heory ad Egeerg Formulae hee 3 May, 8

34 e p f ( g( Δ u d d f ( f + r q + f rf f f f f f f, j+ f, j f, j, j f, j+, j f f f,, j f j+ +, j ajf, j + bj f, j + cj f, j+ f+, j where, aj ( rq j j bj + j + r cj ( rq j j f f f +, j+ +, j f f + f f +, j+ +, j +, j f a f + b f + c f, j j +, j j +, j j +, j+ where aj ( r q j + j + r + r, b ( j j cj ( r q j + j + r where α f + β f + γ f f j, j j, j j, j+ +, j α j ( rq Z Z β Z j + + r γ j ( rq Z Z α f + β f + γ f f j +, j j +, j j +, j+, j Facal Ecoomc heory ad Egeerg Formulae hee 3 May, 8

35 where α + r Z Z j ( rq + β j ( + r Z γ + r Z Z j ( rq + b m u u m g m αu m VL + u γ α b g λ + λ u m ( m u + m λ λ λ γv + αu + β L ω + αu + β 3 ω + βω + β ω + αu + αβu + αβ u + B 3 3 u exp πv v m L N M m m u F HG IO KJ Q P m l v ( u v K m w η where w m + m ( α β V + αu + β L ( ( V α u V + β V L L L Facal Ecoomc heory ad Egeerg Formulae hee 3 May, 8

36 ( ( E V + α + β V cov + L L m x y m ( λ cov λcov + x y cov ω + αx y + βcov r ˆ e E c (, ; / (, ; / ( e M a b Ke M a b e KN a q r r ( / + ( + / l r q a a a l( ( / K + r q + / b b b (, ; / (, ; / + ( Ke M a b e M a b e KN a r q r (, ; / (, ; / ( Ke M a b e M a b e KN a r q r (, ; / (, ; / + ( q r e r o M a b Ke M a b e KN a ( ( λ r λ ( ( ( ( ( ( ( max, max, q rq c p c+ e Ke H K : c e H / N y Ke H / N y λ r q+ / d q ( K l H / y + λ c c c do d ( ( ( ( ( ( q r q λ r λ H K: c N x e Ke N x e H/ N y + Ke H/ N y do c c c d do Facal Ecoomc heory ad Egeerg Formulae hee 33 May, 8

37 x ( H l + λ y ( l H + λ : ( ( q r q λ u ( / o ( ( λ ( / ( ( H > K c N x e Ke N x e H N y N y r + Ke H N y + N y + c c c uo u λ r λ ( ( ( ( H K : p e H / N y + Ke H / N y+ u q p p p uo u ( ( ( ( ( ( q r q λ r λ H K: p N x e + Ke N x + + e H/ N y Ke H/ N y + uo p p p u uo q r q λ ( ( ( ( ( r λ Ke ( H N ( y N ( y H < K : p N x e + Ke N x + + e H / N y N y d p p p do d q q r Y celb e N( a e N( a me N( a e N( a3 ( rq ( rq ( ( / m + + / l r q a a a a 3 ( ( / m / l r q ( rq l( m / / Y pelb e N b e N b e N b e N b ( r q + ( rq r Y q q max ( ( 3 o ( ( max / o / l r q b b b ( ( max / + / l r q b3 ( ( Facal Ecoomc heory ad Egeerg Formulae hee 34 May, 8

38 Y ( rq l( max / / ( + ( K max, τ τ M r rq r+ q+ 6 6 M e brqg r q b g ( ( r q + rq e e + ( r q ( r q ( ( ( + + rq r q + r q+ M l M qv o qu ( ( Ve N d Ue N d d ( ( ˆ o / o + U V + / l V U q q d d ˆ ˆ U + V U V ρ α d ( r q d + dz d ( rq λ d+ dz+ dp g φ( g v v v e Γ ( gv ( v l + r q + ω + θ g g ω l θ / v ( v v d ( r q d + ( dz Facal Ecoomc heory ad Egeerg Formulae hee 35 May, 8

39 d ( r q d Vdz + α dv a( V V d + ξv dz L d (( r q( d + (, dz [ K] V [ ] Cm + q ( Cm + K r ( q ( Cm K (, K C K dθ md sdz θ + f μ f + f z ( m f μ f + f z ( f f ( f f ( μ f f μ f f μr μ r df f μd + dz μ r λ μ r λ dθ dz f f d ( f g dz g g f f geg g dg rgd f fo g E g Facal Ecoomc heory ad Egeerg Formulae hee 36 May, 8

40 ˆ( γ f Ee f f P(, E( f N ( ( (, A P ( ( s EA s fo A EA A + + f ( ( [ ] c P(, E max( K, [ max(,] R c e E [ ] E max( K, E ( N( d KN( d V f UEU max, U f VN( d UN( d Europea call opo o a varable whose value s V [ ] c P(, F N( d KN( d F F l( + l( where d K K d d F value of F a me zero K sre prce of he opo P (, prce a me of a zero-coupo bod payg $ a me volaly of F F forward prce of V for a corac maurg me o maurg of he opo V value of V a me Facal Ecoomc heory ad Egeerg Formulae hee 37 May, 8 value of he correspodg pu opo p P(, [ KN( d F N( d ] B I forward bod prce FB P(, where B bod prce a me zero I prese value of coupos ha wll be pad durg he lfe of he opo volaly of he forward bod prce B Dy y where y volaly of he forward bod yeld y al value of y F y F forward yeld D modfed durao of he bod uderlyg he opo a opo maurg

41 [ ] c P(, E max( B K, where B bod prce a me E expeced value a world ha s forward rs eural wh respec o a zero-coupo bod maurg a me E ( B F B L( + Rδ max L, + Rδ L( + Rδ where value a me of a zero-coupo bod ha pays off L( + Rδ a me + + R δ G ( y E ( y y y G ( y y where E expecaos a world ha s forward rs eural wh respec o P (, forward yeld volaly y G ( R R τ E R R R R τ R ( R + G ( R + R where τ L prcpal R zero-coupo eres rae applcable o he perod bewee ad αv ρvwvw where V volaly of V W volaly of W ρ VW R forward eres rae for perod bewee ad correlao bewee V ad W R volaly of R W ( R m + m ( ρvrvrr ( E ( V E( Vexp R + m αv ρvwvw E ( V E ( V ( + ρ X Y V W ˆ r value a me of a eres rae dervave ha provdes a payoff of f a me ( E e f where r he average value of r he me erval bewee ad Ê expeced value he radoal rs-eural world R (, leˆ e r( dr m( r d + s( r dz dr μrd + rdz Facal Ecoomc heory ad Egeerg Formulae hee 38 May, 8

42 dr a( b r d + dz P (, Ae (, B (, r ( e B (, a a ( A (, exp a ( B (, + ( ab B (, R(, l A (, + Br (, ( dr θ ( d + dz 4a θ ( (, F + r(( P (, Ae (, P(, where l A (, l + ( F(, ( p(, θ ( dr [ θ ( ar] d+ dz a r d+ dz a θ a a ( F (, + af(, + ( e P (, Ae (, B (, r ( where e B (, a a ( P(, P(, 4a a a a l A (, l + B (, F(, ( e e ( e 3 [ θ ] dl r ( a( l( r d+ ( dz [ θ μ ] df ( r ( + af ( r d + dz du bud + dz prce a me zero of a call opo ha maures a me o a zero-coupo bod maurg a me s LP(, s p LP(, s N( h KP(, N( h p h l + P (, K p m [ α ] where p + Q, exp ( + j R m m j m jm α m l m jm Q e l P j R m, j m+ Facal Ecoomc heory ad Egeerg Formulae hee 39 May, 8

43 [ θ ] df ( r ( af ( r d + dz m pm+ Qm, j exp g( m + j x jm [ α ] ˆ P (, Ae ˆ(, B (, R where P(, B(, P(, + A e B B B+ P(, B(, + P(, 4a ˆ a l (, l l ( (, (, (, [ ] B (, Bˆ(, B(, + dp (, rp ( (, d+ v (,, Ω P (, dz ( where P (, prce a me of a zero-coupo bod wh prcpal $ maurg a me Ω vecor of pas ad prese values of eres raes ad bod prces a me ha are releva for deermg bod prce volales a ha me vω (,, volaly of p(, f (,, forward rae as see a me for he perod bewee me ad F (, saaeous forward rae as see a me for a corac maurg a me r ( shor-erm rs-free eres rae a me dz( Weer process drvg erm srucure movemes [ p ] [ p ] l (, l (, f(,, v (,, Ω v (,, Ω v (,, Ω df (,, d + dz( ( df (, v (,, Ω v (,, Ω dv (,, Ω dz ( m (,, Ω s (,, Ω s (, τ, Ω dτ where m (,, Ω saaeous drf of F (, sω (,, sadard devao of F (, m (,, Ω s(,, Ω s(, τ, Ω dτ df ( ξ ( F ( dz where F ( forward rae bewee me ad + as see a me m ( dex for he ex rese dae a me, smalles eger such ha ( ξ ( volaly of F ( a me γ ( volaly of he zero-coupo bod prce p(, a me m df( x ξ( vm( ( v ( + F( d + ξ( F( dz Facal Ecoomc heory ad Egeerg Formulae hee 4 May, 8

44 v ( v ( δ F( ξ ( F( + + δ df( δf( ξ( ξ( d + ξ ( dz F ( + δ F( m( Λδ where Λ he value of ξ ( whe here are such accrual perods ξ ( Λ s a sep fuco m( df ( ( δf Λm( Λm( d +Λ F ( + δ F( m( m( dz δ F( Λ Λ ( Λ m( m( m( dl F ( d+λm( dz m( + δf( δ F( Λ Λ Λ F where ε s a radom sample ε N(, j j j ( j+ F( jexp ( δ j +Λj ε δ j j+ + δf( j p ( δ (, (, ( p F ξ q ξ q q d + ( m( + δf( q df F ξ q, ( dz p p δ ( p F j λ j, qλ j, q λ q q j, q ( j+ F( jexp ( δ j + λj, qεq δ j j+ + δf( j q F q V ( p N τβ, q( G( γ( q + τ G( τ jgj( τ τ ( jgj j + j + + where γ ( N N N τ ( ( j jgj τ τ j jgj N N Vd ( p N τβ, q( G( γ( + τ G ( q d p N M τm, βmq,, ( Gm, ( γ( + τ G ( q m m, m, d Facal Ecoomc heory ad Egeerg Formulae hee 4 May, 8

45 λ jq, Λ δ α j q j, q p s qα q, q p α (... + ξ, q( ( q q df F dz F τ F + + Fτ G ( y yτ Fρ y, F, y y y, G ( y + F τ V + Vρ W, V, QL P (, s N ( d Rasmuse, Games ad Iformao, A Iroduco o Game heory bes respose: π ( s, s π ( s, s s s ' ' d ' domaed sraegy: π ( s, s < π ( s, s s doma sraegy: wealy domaed: Nash equlbrum: π ( s, s > π ( s, s s s s ' ', π π π π '' ' '' ' ( s, s ( s, s s ad ( s, s > ( s, s for some s, π ( s, s π ( s, s s ' ' pure sraegy: s : ω a mxed sraegy: s : ω m( a where m ma ( da compleely mxed: m > mmax sraeges: m mze max mze π ( s, s A s s maxm sraeges: max mze m mze π ( s, s s s Uewe (, ( U max mzev ( q( e w ( e e Facal Ecoomc heory ad Egeerg Formulae hee 4 May, 8

46 q w V ( q( e w ( e( e e q w e e U w ( e e U w U q U ( ( ( w e e Ue (, qe ( U max mzeu ( e, q( e e U U q + ( ( e q e U q U ( ( w e e max mze EV ((, q e θ w((, q e θ w(. subjec o e avgmax EU( e, w( q( e, θ EU ( e, w( q( e, θ U e Ce ( m mumewqeθ ((, w(. max mze EV ( q( e, θ C( e e U ( o vesgae U ( vesgae θ log( w + ( θlog( w ( θ log( w + ( θ log( w α w θ ( θlog( α w log( w ( θ log( w + ( θ log( w α w we α θ w we α θ ( α α θ θ ( θ we + ( θ we Facal Ecoomc heory ad Egeerg Formulae hee 43 May, 8

47 θ ( P E θ ( + ε θ P FE--7 FE--7 Noe ( ( F max, + max, F max, + max, ( FE-5-7 Noe FE-6-7 d μd + dz dr μ(, r rd + r dz Δr (, r (, (, Δ ( Δr (, (, Δ dr a( b r d + rdz dr a( b r d + dz, (a> dr a + b ( l r d + r dz dl ( a + b r + c l d + l dw dv M (, r d +Ω (, r dz M (, r V+ μ(, rv + (, r V Ω (, r (, rv r r rr dπ ( M (, r Δ M (, r d+ ( Ω (, r ΔΩ (, r dz dπ rπ d V + μ r λ r r V + r V rv ( (, (, (, r (, rr Facal Ecoomc heory ad Egeerg Formulae hee 44 May, 8

48 P ( + δ P ( P ( ( + δ δ e r ( P ( P ( P ( + P ( + + { + } P ( δ δ δ P ( + ( l + l( ( + + ( l r Noe: ypo ex r ( eher way wll receve full cred. ' dr ( f (, + d + dz l ( (... ( s r ( ( [ δ δ δ ] P ( P ( + P ( ( δδ.. δ..( δ + + ( + δ.. δ..( + δ δ [ ] ' ( (, ( ' ( ( ( (, ( dr f + + r f d + dz P ( + ( + δ.. δ ( + δ.. δ..( + δ +, j( P ( ( + δ.. δ..( + δδ ( + δ P ( + δ.. δ ( + δ.. δ..( + δ ( ( j δ δ ( + δ.. δ ( + δ.. δ..( + δ (cos φ( ' ' dr f ( + ( + r f ( d + ( dw (cos θ( [ ] ' ( dl r ( θ( l r d+ ( dw ( f(, β dr( ( α( βr( d + dw ( where α( + β f(, + ( e β [ θ( μ ] dr + ar d + dw du bud + dz p dp (, rp ( (, d+ (, P (, dz df (, (, (, d (, dz p p p dp rp d P dz (, ( (, + ( (, (, P (, (, L P (, + Facal Ecoomc heory ad Egeerg Formulae hee 45 May, 8

49 N L (, jδδ dl(, L(, Λ( jδ Λ( d +Λ( dz j + L (, jδ Δ L (, j Δ Λ j L (, j+ L (, jexp Λj Λj Δ+Λj ΔZ j+ + L (, j Δ j j j where j Λ caple C L P( [ F N( d R N( d ] where d δ + x F l + RX d d m L P R N d R N d L A R N d R N d m m where A P( m swapo ( [ F ( X ( ] [ F ( X ( ] m ( j P ( +, j P (, jexp r ( Δ+ ( j ΔZ( j ( ( + ( exp( ( + a b c d L (, j Δ Λ j L (, j+ L (, jexp Λj Λj Δ+Λj ΔZ j+ + L (, j Δ P ( + P ( + (, ; P h ( δ h ( where h ( + δ FE-8-7 V( E V( F V( D V( F D + P( V( F, D C( V( F, D m V ( F + D( + Facal Ecoomc heory ad Egeerg Formulae hee 46 May, 8 DF { } { } V ' ( E C+ V ( F D+ P V ( F, D C+ V D+ P R R R R R V ( E V ( F D+ P V ( F, D V D+ P ' N N N N N facevalue + prcpal forgve defaul pu assu m g revesme D ( P P NPV P ( D P B NPV + B value of regular deb + savg barupcy cos N N R R

50 FE-9-7 RBC C + C + C + C + C + C + C FE--7 Noe FE ( 3 a a 3b 4b Y x ω r MCaR ω ω DCaR j j j oal CaR CaR + CaR CaR ρ FE { } μ { } + { } mv { } NPV ( d V ( C ( + m V μ ( dv + { } V ( z ( znz ( ( + r + + ( z ( zn( z V{ } ( + r FE-5-8 Noe FE-38-7 c f ( w dw W VAR W α Δ ( ˆ se q c ( c ( f q FE-39-7 Noe FE-4-8 Noe FE-4-8 Noe FE-43-8 Facal Ecoomc heory ad Egeerg Formulae hee 47 May, 8

51 FE-44-8 XC Harcu sze of he losses XC+ LL where XC sum of excess capal LL remag lqud / surreder-able lables oal amou of avalable asses ( AA excess of % RBC avalable o mee ay remag lqudy demads seor deb + excess hybrd deb ad preferred soc leverage ECA + seor deb + hybrd deb + preferred soc Hybrd Rao U.. s a dard & pool ' s qualfyg hybrd U.. GAAP( cosoldaed capal + oal hybrd + oal seor deb Hybrd Rao Eurpoe Doubleleverage Doubleleverage U.. Eurpoe s a dard & pool ' s qualfyg hybrd Group Cosoldaed AC( excludg hybrd + oal hybrd + oal seor deb s a dard & pool ' s qualfyg hybrd + oal seor deb + oqualfyg hybrd U.. GAAP( cosoldaed capal + oal hybrd + oal seor deb s a dard & pool ' s qualfyg hybrd GroupCosoldaed AC( excludg hybrd + regulaory qualfyg hybrd capal FE-45-8 (equy +frachse oal shareholder reur crease e asses + crease frachse value + dvded crease frachse value + reaed prof + dvded frachse frachse growh rae + equy reur o equy reur o equy oal shareholder reur + frachse/equy (oal shareholder reur frachse growh rae E D D + E E COE E E E + F + ( + COE ( + COE ( + COE ( + COE F ROE COE E ( + COE { } { } ( + R A L + F A L + ( A ( R + m L ( R m + + F f f A A f L L where A balace shee asses a me L balace shee lables a me F frachse value R A acual asse reur R L acual lably reur R rs-free rae Facal Ecoomc heory ad Egeerg Formulae hee 48 May, 8 f A asse-relaed expeses as a proporo of A L lably-relaed expeses as a proporo of L ax pad as a proporo of pre-ax prof m A marg above LIBOR as asse swap m marg below LIBOR as lably swap L { } ( + R F ( A ( m + L ( m R ( A L + F f A A L L f

52 ( + R F A ( ( m + L ( ( m + ( s F ( R + s ( A L f A A L L f { } { } ( + R ( A L + F A + ( ( R m L + ( ( R m + + ( s F s ( A L f f A A f L L ( R + s g+ sg F A ( ( m + L ( ( m ( R + s ( A L f A A L L f FE-46-8 DL p( x( x A where p( x probably desy for losses ( x x> A DA q( y( L y where qy ( probably desy for losses ( y L> y D ( x A p( x dx L A L D ( ( A L yqydy D c c φ L Φ L dl c D A c c A da Aφ ( caφ( L ca A A where L he cv of losses A he cv of asses c A capal / asses rao Φ ( x he cumulave sadard ormal dsrbuo φ ( x he sadard ormal desy fuco d Φ( a ( + c Φ( a d L A Φ( ba Φ( b c A where a ( ( l( + c l( ( A c b + ( A A Φ ( x he cumulave ormal dsrbuo oe-perod expeced polcy-holder defc rao d zp( z dz where p( z he desy of c C he amou of capal a he ed of oe perod C c he amou of capal relave o he orgal expeced loss L [ ] c c( + p + + c( + p r + pcb g Facal Ecoomc heory ad Egeerg Formulae hee 49 May, 8

53 where r ad g are radom varables deog he aual reur o asses ad aual rae of chage value of he lables b curred loss rao C c + ρjcc j j oal capal μ μ D φ( μφ ( D c c d φ( cφ ( L c c dl φ( cφ ( D A ca c A da Aφ ( caφ( L ca A A F Φ( a Ee Φ( a l( + ( + where a E soc prce E exercse prce D LΦ( a ( + c LΦ( a L d Φ( a ( + c Φ( a L D AΦ( a LΦ( a L A L d A Φ( ba Φ( b c A where l( ( A c b + ( A A FE-47-8 Noe FE-48-8 Noe FE-49-8 Noe FE-5-8 Noe FE-5-8 defaul pu opo V( E F + V( A PV( L + O where V ( mare value PV ( he prese value E ower s equy A he asses L lables F he frachse value A agble asses O he defaul pu opo FE-5-8 Noe Facal Ecoomc heory ad Egeerg Formulae hee 5 May, 8

54 FE-53-8 Noe FE-54-8 Noe FE-55-8 ξ ρ FE-56-8 Noe FE-57-8 FE-58-8 wf ( w dw CE ( ρ Φ( ξ ρ E r + β ( E r D r f rm F b, g b, g b g b g b g L M N E s e φ e M ez s s b g b g b g L b g E s e b g b g zc s b gh N M + φ e ' r r + φ s + φ s s D rds s O QP E r ds s P M ez,, rds O P Q L M N O QP FE-59-8 dr r d rdw ( θ ( + λ + where λ w ( w( r( s ds + pb (, E B( B( B B p( B, E exp( rsds ( p( B, AB (, exp( rgb ( (, B γ exp ( b + γ AB (, ( γ + b(exp( γ( B + γ c (exp( γ ( B GB (, ( γ + b(exp( γ( B + γ where b + λ c θ γ b + γ ( 4c ϕ re C ( pb (, χ ( γ ( ϕ+ ψ + GB (,,, ( ϕ+ ψ + GB (, Facal Ecoomc heory ad Egeerg Formulae hee 5 May, 8

55 Xp + (, χ r ( ϕ ψ,, 4c ϕ re ( ϕ+ ψ γ ( ( φ( α( + ( φ( θ( α( λ( ( dr r d dw + b x ( x( τ E B ( B( τ B p( B, E exp( rsds ( G(, α( G(, F(, F(, G(, τ ( + φ( α( dτ G(, τ τ C ( PBNh (, ( XPNh (, ( p p PB (, where h ( + ( l p ( XP(, τ ( p [ G(, B G(, ] dτ G(, τ τ γ exp ( b + γ A(, ( γ + b(exp( γ + γ c (exp( γ G(, ( γ + b(exp( γ + γ where b + λ c θ γ b + M ( P max( P(, B, M (,, E P(, B B ( ( (exp( ( (, E M r s ds P B r r + ( θ ( + λ r ( + r ( ε N m PMAX ( P(, B, M (,, ( M ( exp r( ( P(, B N Facal Ecoomc heory ad Egeerg Formulae hee 5 May, 8

56 M ( PMAX ( P(, B, M (,, E P(, B B ( ( (exp( ( (, E M r s ds P B r r + ( φ( α( r ( + r( ε FE-6-8 Noe FE-6-8 Noe FE-6-8 Noe FE-63-8 share value of ag he projec share value of o ag he projec PV of asses place + PV of ewvesme umber of orgal shares + umber of ew shares PV of asses place umber of orgal shares valueof orgal asses + NPV of ew projec share value : facg projec wh rsless deb umber of shares FE-64-8 where WA Rs weghed amou Asses WA + cred equvale WCE rs capal weghed by asse caegores WCE weghed by cred equvales by ype of couer pary FE-65-8 ( E+ P( + r L E ( ( E+ P RI (, ( + Er ( ELa ( ( + hci (, a E ( ELa ( ( C I L + h a a I L a { E+ D}( + E( r E( L( a D( + r + hc( I, a E ( ELa ( ( C I L + h a a I L a Recommeded Approach for eg Regulaory Rs-Based Capal Requremes for Varable Aues ad mlar Producs, Facal Ecoomc heory ad Egeerg Formulae hee 53 May, 8

57 Noe mh, Ivesor & Maageme Expecaos of he Reur o Equy Measure vs. ome Basc ruhs of Facal Accoug E EV ( ROE IRR E ( + IRR x x x where E equy a me EV embedded value a me, usg dscou rae IRR IRR prcg eral rae of reur afer arge surplus ROE x reur o equy a me x eargs perod x E x ( x Bodoff, Capal Allocao by Percele Layer percele layer of capal ( α, α + j requred capal a percele ( α + j requred capal a percele ( α layer of capal ( aa, + b capal equal o amou ( a+ b capal equal o amou ( a j VaR( x oal requred capal [ x( α + j x( α ] α x( α loss amou a percele α x j seleced percele creme f( x dx where x loss amou y he capal ( F ( y x y y VaR(99% x y x y f( x dxdy F ( y y x y f( x dy ( F ( y y VaR(99% y f( x dy ( F ( y x y m( x, VaR(99% x x(% y f( x dydx ( F ( y Allocaed capal o loss eve x AC( x y x y f( x dy ( F( y y x AC( x f ( x dy ( F( y y y VaR(99% AC( x f ( x dy ( F( y y Facal Ecoomc heory ad Egeerg Formulae hee 54 May, 8

58 d dx AC x { ( } d y x dx f ( x dy ( F( y y d f( x y x y x d + y y dx dy dy ( F( y ( F( y dx { f ( x } y x f( x + dyf ( x ( F( x ( F( y y y x rf ( x dy r requred rae of reur o capal ( F( y y r y x y dy ( F ( y y x x + r dy ( F ( y y y x x + r( dy x ( F( y y premum e of expeses expeced loss + cos of capal P E [ L] + r ( allocaed capal corbued capal where P premum e of expeses E [ L ] expeced loss r requred rae of reur o capal [ ] r [ ] P E L + ( allocaed capal E L ( + r y x Px ( xf( x + r f( x dyxf( x + r ( F( y y y x Px ( f( x x r + dyx + r ( F( y y y x x + r dy x ( + r ( F( y y y x ( ( r ( Px xf x + dy ( + r x ( F( y y Facal Ecoomc heory ad Egeerg Formulae hee 55 May, 8

59 y x ( r ( dy x ( + r ( F( y y y x AC( x ( exp( x exp( x dy θ θ θ y AC( x exp( x θ d ( exp( x dx AC x θ θ { ( } + r( (exp( x x θ θ Hardy, Freelad ad ll, Valuao of Log-erm Equy Reur Models for Equy-Led Guaraees Y F μ + z where (,, z N ( Y α + α μ + β ( Y F Q w. pq. Q w. p. q where Q F μ + z ( ( α + α Y μ + α Y μ,,, Q F μ + α z, r, r ( ρ r, y μ ( ρ r y μ { P( >.5} { P( >.5} ( r I r + I r,, Facal Ecoomc heory ad Egeerg Formulae hee 56 May, 8

60 BEGINNING OF EXAMINAION Morg esso. (4 pos You are he Chef Iformao Offcer of he herwood Fores Isurace Compay. You have 4 sysem aalyss avalable o allocae o your projecs. Your curre approach s o allocae aalyss equally o he avalable projecs. Rch & Poor s Cosulg frm has approached you o offer her servces allocag he aalyss more effecvely. You are gve he followg: he probably of a aalys beg slled for he projec s 85%. Oucome Uly U( a, e Aalys slled for he projec, prof Aalys o slled for he projec, loss horage (o eough aalyss allocaed Excess (o wor for aalyss allocaed, loss Projec Aalyss requred A 5 B 5 he cos of Rch & Poor s Repor whch perfecly maches he supply ad demad of aalyss o each projec s 4,. he cos of Rch & Poor s Repor whch perfecly maches he supply ad demad of aalyss o each projec ad also perfecly maches he aalys sll o each projec s,. (a (b ( po Calculae he expeced prof from radomly assgg aalyss o each of projecs A ad B. (3 pos Evaluae he coss of he Rch & Poor s repors ad deerme whch you would purchase. EXAM FEE: Fall GO O NEX PAGE Facal Ecoomc heory & Egeerg Face/ERM/Ivesme Morg esso

61 . (7 pos Palm ree Compay sars a ba ad borrows $M deposs, vesg hs moey a porfolo of rsy loas. Palm ree erally calculaes Ecoomc Capal o whsad a -- year cred loss eve o he loas over a -moh horzo. he ba has aual expeses of $.5M. All bes esmae assumpos below are realzed he frs year of operao for he ba. Palm ree s Hurdle Rae 5% Palm ree s Effecve ax Rae % Borrowg Rae o Deposs 5% Loa Porfolo Gross Eared Rae 7% Regulaory Capal $3M Eared Rae o Asses Bacg Capal 5% Expeced Aual Cred Losses o Loa Porfolo $M Percele aual cred losses Amou of loss $.8M $5.M $6.M $6.5M $8.M (a (b ( po Defe ad calculae he Reur o Capal, assumg he ba holds regulaory capal. (3 pos ( Defe ad calculae he Rs Adjused Reur o Capal (RAROC, assumg he ba allocaes Ecoomc Capal erally. ( Deerme wheher Palm ree s ba vesme added value o he frm durg he frs year. (c (3 pos ( Compare he advaages ad dsadvaages of IRR, Reur o Capal, ad RAROC as measures of performace. ( Recommed whch of hese measures he frm should use for performace measureme. EXAM FEE: Fall GO O NEX PAGE Facal Ecoomc heory & Egeerg Face/ERM/Ivesme Morg esso

62 3. (6 pos You ru a Moe Carlo smulao o prce a loobac pu opo o a 5-year reasury bod usg boh a Cox-Igersoll-Ross (CIR ad a Hull-Whe (HW model. o compare prces, he HW model ad he CIR model are fed such ha he al eres rae volaly s he same. I aes a hour o ge he resuls. he resuls are as follows: Ial Ieres Rae:.7 Log ru mea for Ieres Rae:.5 CIR opo prce:.78 HW opo prce:.83 (a (b (c ( pos Compare he wo eres rae models. ( pos Descrbe a effecve varace reduco echque for speedg up he compuaoal process. ( pos Expla why, uder hese codos, he HW model would geerae a hgher pu opo prce ha he CIR model. 4. (8 pos You repor o he acuary maagg a equy-led porfolo. he has ased you o esmae parameers for he Regme wchg Log ormal model wh wo regmes (RLN-, usg years of mohly soc prce daa for a dex used your porfolo. (a (b (c (d ( po Defe he RLN- model ad descrbe s ey feaures. (3 pos Expla how o use he daa o esmae he RLN parameers usg he maxmum lelhood esmao (MLE mehod. (3 pos Expla summary form how he Marov Cha Moe Carlo (MCMC mehod may be used o deerme he RLN parameers. ( po Descrbe he advaages ad dsadvaages of he MCMC approach, compared wh he MLE approach. EXAM FEE: Fall GO O NEX PAGE Facal Ecoomc heory & Egeerg Face/ERM/Ivesme Morg esso

63 5. (7 pos A frm wh o deb has he followg hree shor erm vesme projecs o cosder, each of whch has a al oulay of $9 wh cash flows over wo perods: Projec Projec Projec 3 Perod Cash Flow Year Year Year Year Year Year Afer ax Ne Icome Assume he weghed average cos of capal s %. (a ( po Descrbe he sreghs ad weaesses of he followg capal budgeg echques: ( ( ( (v Paybac Mehod Accoug Rae of Reur Ne Prese Value Ieral Rae of Reur (b (c (d (3 pos Evaluae he preferred projec, f ay, uder each of he capal budgeg echques assumg oly oe projec ca be seleced. ( po Recommed whch projecs o proceed wh, assumg he frm has he capacy o mae all vesmes ad desres o maxmze shareholder s wealh. ( pos Assumg a margal ax rae of %, he frm decdes ca crease s deb o 3% whou affecg s ably o borrow fuds; ( ( Calculae he compay s ew weghed average cos of capal. Descrbe how hs affecs he decso made (c above. EXAM FEE: Fall GO O NEX PAGE Facal Ecoomc heory & Egeerg Face/ERM/Ivesme Morg esso

64 6. (6 pos XYZ Isurace s cosderg a acquso of ABC Isurace, ad s coducg a acuaral apprasal of ABC. You are gve he followg: Rs free rae 3% Compay rs relave o mare 75% Mare expeced reur % Compay arge capal rao 5% Ieres o Capal, urplus, ad AVR 6% ABC adjused boo value a December 3, 9 $69.4 mllo Corporae Icome ax rae 5% 9 Mmum Requred Capal auory Prof o Exsg ad New Busess Uallocaed Expese Proxy DAC ax Asse he calculao of Corporae Icome axes aes o accou a Deferred Acquso Cos compoe. he auory Reserve ad he ax Reserve are equal. Calculae he Acuaral Apprasal Value of ABC Isurace a December 3 s, 9, usg CAPM, ad reflecg 3 years of fuure cash flows. EXAM FEE: Fall GO O NEX PAGE Facal Ecoomc heory & Egeerg Face/ERM/Ivesme Morg esso

65 HI PAGE INENIONALLY LEF BLANK EXAM FEE: Fall GO O NEX PAGE Facal Ecoomc heory & Egeerg Face/ERM/Ivesme Morg esso

66 7. ( pos (a ( pos Compare ad coras equlbrum ad arbrage-free eres rae models for boh realsc ad rs-eural scearos. (b (c ( po Assess wheher he Hull-Whe model s suable for valug bod opos. ( po Descrbe how a opo o a coupo-payg bod ca be decomposed o opos o a seres of zero-coupo bods he coex of a sgle-facor eres rae model. You eed o prce a.-year Europea call opo o a defaul-free bod ha wll maure 3 years usg he Hull-Whe oe-facor model. he sre prce s he cash prce ha wll be pad for he bod. Prcg parameers a.5.3 Curre erm srucure % sem-aual compoudg (fla curve Bod coupo rae 6% sem-aual Bod prcpal Opo sre uppose Δ.3 years ad R s he level couous rae he Δ perod. he bod prce he Hull-Whe model based o R s as follows: (, ˆ (, ( ˆ, P A e B R Where ˆ P, B, P, a l A(, l l ( e B(, B(, B(, +Δ P, B, P, 4a ( ( ( ( +Δ ( +Δ ( Ad ( Bˆ, B(, (, +Δ B Δ You are gve ha he crcal value of eres rae for whch he prce of he coupobearg bod equals he sre prce of he opo o he bod a opo maury s R 6.6% wh couous compoudg. (d ( pos Calculae he value a. R R. of he coupo due a me.5 gve EXAM FEE: Fall GO O NEX PAGE Facal Ecoomc heory & Egeerg Face/ERM/Ivesme Morg esso

67 7. (Coued (e (3 pos Calculae he value a. of he coupo ad prcpal due a me 3. gve R R. Gve he followg for Hull-Whe models: LP(, s l h + P, K p p ( as ( a p e ( e a a Where L s he prcpal of he bod, K s s sre prce for call opo ha maures a me o a zero-coupo bod maurg a me s. (f ( pos Deerme he value of he opo o he coupo payg bod a me. EXAM FEE: Fall GO O NEX PAGE Facal Ecoomc heory & Egeerg Face/ERM/Ivesme Morg esso

68 8. (7 pos You are he Chef Rs Offcer of Wdy Cy Isurace Compay he U, he surace arm of eraoal bag corporao IBG. Wdy Cy has radoally ssued producs coag oly eres rae rs bu s ow roducg hree ew, uque equy dex-led producs o he U mare durg a volale evrome for U eques. As a resul, Wdy Cy s curre pracce of evaluag rs based o regulaory capal has bee deemed adequae; IBG has bee usg ecoomc capal for several years may oher jursdcos. (a (b ( po Oule a repor o Wdy Cy s Board summarzg reasos o cosder usg a ecoomc capal approach. (3 pos ( ummarze he curre echques ad approaches ha Wdy Cy may use o calculae ecoomc capal for hese producs. ( Deerme he suably of each approach for he ew Wdy Cy producs. IBG has requesed ha ecoomc capal be calculaed o he bass of Excess Losses whch defes as ecoomc losses per u of coverage excess of he local jursdco s sauory reserves. Wdy Cy s modelg acuary has repored o you he followg depede excess loss dsrbuos for he hree producs: Produc A excess losses 3% of he scearos, excess losses of $5 per u 35% of he scearos ad $ per u 35% of he scearos. Produc B excess losses 94 ou of scearos ad ozero excess losses per u 6 of scearos of: $4, $, $7, $7, $8, $ Produc C Above a meda excess loss per u of $, excess losses per u were uformly dsrbued from $ o $5. (c (3 pos ( Calculae boh 9% E L L Dsrbuo for each produc. V ad he ( V > of he Excess Loss 9% ( Recommed ad jusfy whch measure should be used as he bass for allocao of ecoomc capal for Wdy Cy. EXAM FEE: Fall GO O NEX PAGE Facal Ecoomc heory & Egeerg Face/ERM/Ivesme Morg esso