Corporate Finance Math Sheet

8/10/2019 Corporate Finance Math Sheet

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Corporate Finance Professor Shepherd

Math Work Sheet

Miscellaneous (Background Equations Alphabetied!

Balance Sheet

Assets = Liabilities + Equity

Book "alue= Value of Co.s Assets Total Liabilities taken from Balance Seet

Capitaliation (#Cap$! %ate

= annual net o!eratin" income#cost $or %alue&Cash %atio

= Cast + Sort'term liabilities#Current liabilitiesCurrent %atio

= Current Assets#Current Liabilities&ebt'Assets %atio$te i"er te ratio( te i"er te bur)en on te com!any of interest !ayments an) re!ayin" )ebt.

= Total Liabilities#Total Assets&ebt'Equit %atio

= Total Liabilities#Total Stockol)ers Equity

&i)idend *ield %atio= *i%i)en) !er Sare#stock !rice+ross Profit

= Sales Cost of oo)s Sol)+ross Profit Margin

= ross ,rofit#SalesEarnings per Share

*i%i)e net earnin"s by - of saresBasic - of earnin"s can be eiter a%era"e - of sares issue) an) outstan)in" in a "i%en year*ilute) - of sares an) - of sares tat /oul) a%e been outstan)in" if all stock options ,ere e-ercisedan) all

con%ertible securities /ere con%erte) to stocks

Market.to.Book %atio= Stock ,rice#Book Value !er Sare/et Asset "alue of a Stock

SeeBook Value/et 0perating 1nco2e (/01!

= 0!eratin" 1ncome 0!eratin" E2!ensesPoison Pill Flip.0)er Clause

B#3 or S#3 is allo/e) to obtain eac - of sares of te acquirin" com!any( equalin" te current !urcase !rice

by$4& 5ulti!lyin" te ten current !urcase !rice times te - of sares for /ic te ri"t /as e2ercisable6 an)$7& *i%i)in" tat !ro)uct by 89: of te ten current !er sare market !rice of te C#S.

Price.Earnings (P'E! %atio= 5arket ,rice !er Sare#Earnin"s !er SareProfit Margin

= ;et 1ncome#Sales%eturn on Equit

= ;et 1ncome#a%era"e S3s equityShareholder3s Equit

= Assets ' LiabilitiesSurplus

Ca!ital Sur!lus= Assets Liabilities State) Ca!ital $Ca!ital !rotecte) %ia ,ar&

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Earne) Sur!lus= Accumulate) ,rofits Accumulate) Losses *i%i)en)s !ai) out of Earne) Sur!lus

4otal Assets

= Current Assets + Lon" Term Assets $stuff not to be con%erte) into case /#in 4 yr&Working Capital

= Current Assets Current Liabilities

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"aluing Fir2 0utput (Chapter 5!

1. &iscounting and Present "alueA. Co2pounding

4. ormula *etermine tefuture%alue you /ill a%e at te en) of te year for eac)ollar in%este)( an) subtract one from tat amount

i. Effecti%e annual rate =$4 + Annual ,ercenta"e Fate&m 4

;ii. m = number of !erio)s in eac year6 ; = te total number of !erio)s.

C. Present "alue4. Basic *iscountin" to ,resent Value

a. For2ula 1n calculatin" ,V /e mo%e back/ar)s trou" timeGfrom a future )ateback to to)ayGtrou" te !rocess of discounting9

i. P" of a :u2p Su2

o ,V = Sum $4 + r&n

o ,V = !resent %alue( S = sum( n = te number of years( an) r =interest rate#)iscount rate

*. Annuities4. P" of an annuit

o ,V = , HHHH,HHHHH r r$4 +r&n

E. "aluing a Perpetuit4. >ormulaa. ,#r $,ayment#)iscount rate&

b. Te calculation assumes te !er!etuity /ill continue in)efinitely.c. As te )iscount rate e2cee)s 49:( te incremental %alue of eac ne/ years !ayment

is only a fraction of 4:7. Valuin" a !er!etuity inten)e) to be sol)

a. &i)idend )alue ; )alue of the salei. S $current )i%i)en) %alue$4 + r& + S $sale + r

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b. I4.99#4+r = I4.99#4 + .4 = 9or P year of "ro/t( )i%i)en) !ayment = )i%i)en) from year before O$4+ "ro/t rate&. Ten fin) ,V of tat )i%i)en) $multi!ly )i%i)en) O interestrate from Table ?'N usin" year P&

iii. $?& Sum all of tese ,V %alues for initial "ro/t yearsi%. $N& 0nce it sto!s "ro/in"( use $*i%i)en)#interest rate& O ,V from Table ?'N

$usin" te number of "ro/t years + 4 as te year&%. $8& A)) %alues )urin" "ro/t !erio) $Ste! ?& + !er!etuity after initial "ro/t

$Ste! N&b. E2.i. Assum!tions

o 1nitial Earnin"s = I4.99( "ro/in" at N: for 8 years.o Stable Earnin"s tereafter( )iscounte) Q 49:

ii. Calculation

o Calculation

S $*i%i)en) Value&

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$4 + r&n

E2& R4 I4.99#$4 + .4&4= .J9J4

R7 I4.9N#$4 + .4&7= .K8J8

R8 I4.4#$4.4&8= .7@N

Ten you a)) tem all u! = N.98J

Ten calculate te !er!etuity of 4.4 once initial "ro/tceases $,#r'"& andten multi!ly by te discount factor

' Standard Perpetuit ,ith constant gro,th>

I4.4#.49 = 44.9

' &iscount Factor

?#$4 + .4&@= .8@N8

' %esult I44.9 2 .8@N8 = @.@9N@

Lastly( a)) te t/o -s to"eter for a com!lete %aluation of anin%estment /it )ifferin" "ro/t !erio)s = @?=9=

3. /et Present "aluea. ,ositi%e ;et ,resent Value


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B. Measur2ent of %isk60f E-pected "alues and Standard &e)iations4. >irm A $


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c. "ariance is te e2!ecte) square) )e%iation of te e2!ecte) returns.N. Standard &e)iation is te square root of our %ariance. @K: of te results of any )istribution

of outcomes normally fall /itin one stan)ar) )e%iation of te mean an) J8: fall /itin t/ostan)ar) )e%iations.

a. Because /e a) to square our )e%iations to eliminate ne"ati%e numbers( /e a%earri%e) at results tat are out of scale /it te ori"inal results. To eliminate tis( /ecalculate te square root of eac result

i. X899 = 77.?@ii. X?(999 = 8N.

b. Tus( >irm As ST* is 77.?@: of its e2!ecte) %alue an) >irm Bs ST* is 8N.: ofits e2!ecte) %alue.

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N. Appling the Standard &e)iation> 1;* BETA

K


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' 47: = 8: + $B 2 E,&' 47: = 8: + $B 2 49 8&' 47 = 8 + B8' = B8' #8 = B

?9 G B

Capital Structure (Chapter !

1. Modigliani H Miller3s 1rrele)ance IpothesisA. Perspecti)es

4. /et 1nco2e Perspecti)ea. E-a2ple

i. Set'U!o >irm 4 An all equity firm /it I499(999 in%este) tat as an

e2!ecte) return to sareol)ers of I47(999.o >irm 7 )oubles >4s by borro/in" an a))itional I499k Q K:

interest an) /oul) a%e an e2!ecte) return to sareol)ers ofI4@(999.

>irm -4 ;et 1ncome @?8===

>irm -7 0!eratin" 1ncome Less interest I7N(999 K(999 = @?===

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Q K: = ;et 1ncome

Te tra)itional a!!roac looks only at net income( an) %alue) earnin"s !er sare. Un)er tis assum!tion( if >irm0ne an) >irm T/o are i)entical( te e2!resse) earnin"s soul) be ca!italie) at 47:

ii. Better to borro/ ri"t Set ca! rate $!re')etermine) !otential rate of returnon an in%estment& at 47:

o Equity of >4 47(999#.47 = I499ko Equity of >7 4@(999#.47 = I4??k

iii. Tus( >7 as I4??k /orti%. ,roblem /it tis is tat it )oes not account for variance

o E2!ecte) earnin"s of associate) /it >7 are more %olatile tan >4b#c of fi2e) )eman)s of cre)itors re"ar)less of /eter te firm asa "oo) year or not.

%. Effect of Le%era"e on ,rofits

o >irm 4' I499(999 Equity

Scenerio A $Ba)& B $;ormal& C $"oo)&

Assume) 0!eratin" Earnin"s I7(999 I47(999 I77(999

Earnin"s to S#3 I7(99 I47(999 I77(999

Fate of Feturn 7: 47: 77:

o >irm 7' 89: Equity 89: *ebt $sli"ly mo)ifie) from abo%ee2am!le&. To fit tese -s /it te abo%e e2am!le Must )ouble te I.

A B C

Assume) 0!eratin" Earnin"s I7(999 47(999 77(999

1nterest $IN(999& $N(999& $N(999&

Earnin"s to S#3 $I7(999& K(999 4K(999

Fate of Feturn 'N: 4@: ?@:$4K#89kloan& =?@ FoF

o Le%era"e %astly increases te %ariance on returns( so in%estors /oul)be e2!ecte) to )eman) a i"er return#ca! rate 4N:.

7. 4he /et 0perating 1nco2e Approach (Modigliani and Miller Io2e.Made :e)erageApproach!>

a. Point> Cor!orations /ill say teyre earnin" more because teyre le%ere)( but tis iscra! because you can )o tis yourself an) "et te same result.

11. Capital Structure in the %eal World Fela2in" te Assum!tions in te 5Z5 3y!otesis49


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A. &ebt and 4a-es4. 0,ning all the debt and equit in a le)eraged fir2 increases total after ta- ,ealth of

SIs because interest paid on loans reduces corporate ta-es

7. 1nterest 4a- Shield E%ery )ollar !ai) as interest re)uces cor!orate ta2 by ?8: $te cor!. ta2rate&. So( interest tax shield = .35 x (interest payents)

a. Te fun)s sa%e) from bi" broter are ten a%ailable to te S3 or for rein%estment inte firm.

b. Ca!italiin" te *e)uctions from Cor!orate Ta2

,V Ta2 Siel) = calculate) interest ta2 siel)#interest rate on )ebt

?. 4he 12pact of E-pected Bankruptc Costsa. ;ot on a linear rate( but on an u!/ar) slo!in" rate

B. Weighted A)erage Cost of Capital4. Calculations

a. Basic Com!onents

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i. Fate of return on )ebt = face %alue of )ebt#%alue of te firm 2 te interestrate

ii. Fate of return on equity = %alue of equity#%alue of firm 2 ca!italiation rateb. Basic WACC

c. Ad)anced WACC A)Mustin" for te Ta2 Siel)

i. Ste! 4 Calculate After ta2 cost of )ebt

After ta2 cost of )ebt = $interest rate&$4 cor!. ta2 rate&

ii. Ste! 7 Calculate A)%ance)


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111. Capital Structure in the Courts1V. Wages and E-cessi)e &ebt

Co22on Stock (Chapter !1. :i2ited :iabilit of Stockholders11. 4he Proble2 of &ilution

A. Si2ple &ilution>


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o Creates free ri)er !roblem 0l) in%estor as no incenti%e to in%estb#c e /ont "et anytin" out of it( Must a))e) losses.

ii. Sell 4??.?? ne/ units at I(899 eac6 at te en) of te )ay( 7??.?? units /illsare a %alue of I4(89(999 or I(899 !er unit.

o Value after te sale I4.85( or I(899#unito Loss on ori"inal unit I7(899o Loss on ne/ unit I9o Eli2inates the free rider proble2but creates indifference$te

in%estor is no/ in)ifferent e can in%est anoter I49k an) o/n7.??? units /ot a total of I4(899 an) lose I7(899. 1f e fails toin%est( is ori"inal unit is /ort I(899 an) is loss is te same.

iii. Penalt &ilution Sell 799 ne/ units at I8(999 eaco Value after sale I4.85( or I8(K??#unito Loss on ori"inal unit ' IN(4@o ,rofit on 7 ne/ units I4(@@o 1n%estor as a loss of I7(899 but if !re%ious in%estor )i)nt

contribute a))itional money( as loss of IN(4@i%. E-tre2e Penalt &ilution> Sell N99 ne/ units at I7(899 eac

o Te 1n%estor /o buys N ne/ units as no/ in%este) I79(999 forI4(899 of macines( losin" a total of I7(899. Te free'ri)er /o

)oes not in%est as a unit tat is no/ /ort I?(899( so e as lostI@(899.o 5V Te IN9 e2ercise !rice /oul) !rucase 72 of c#s tat te s#/oul) be able to !urcase for IN9 on te market6 in oter/or)s( "et )ouble tat /ort in c#s( or IK9 /ot of c#s.

Corporate &ebt (Chapter !

ond !uotations"

#dv Current Riel) = Annual *ollar 1nterest ,ai) O 4995arket ,rice

O coul) also be cas inflo/s( )e!en)in" on /ic si)e of te t2 you are /orkin" from.

So( if you !urcase) a bon) /it a !ar %alue of I499 for IJ8.J7 an) it !ai) a cou!on rate of 8:( tis is o/ you])calculate its current yiel)

CR = .98 2 I499 O 499: = 8.74: IJ8.J7

SimpleCurrent Riel) = yearly !ayment ,rice

1. Contract 1nterpretation11. Contract 4er2s111. 4he 1ndenture 4rustee and the 4rust 1ndenture Act1V. Capital :easingV. Asset.Backed Financing

Chapter 7 (Preferred Stock!

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1. 1ntroductioni. .

B. 4he Dses of Preferred Stock4. For !e"elopent#$tage Copanies7. For Pu%lic Copanies

a. >or !ublic com!anies( ,#S is ta- fa)oredi. 9: of )i%i)en)s recei%e) on A;R stock are ta2 )e)uctible

ii. 1f reci!ient o/ns 79: or more of te stock( K9: of te )i%i)en)s are

)e)uctible.iii. 1f reci!ient o/ns K9: or more of te stock( 499: is )e)uctible.?. For the &ssuer

11. &i)idends111. Altering the Preferred Contract1V. Board &uties

0ptions and Con)ertible Securities (Chapter !

1. 1ntroductionA. :ong and Short Positions and Position &iagra2s

4. :ong Position is /ere an in%estor buys an o!tion e2!ectin" tat its %alue /ill increase.a. Lon" ,ositions in Call 0!tions

i. Benefits o!!ose) to buyin" stocko Buyer of stock bears te risk tat its !rice /ill )ecline rater tan

increase. ,uts an entire !urcase !rice of I499 at risk.o 0!tion buyer only risks te amount s!ent to !urcase te call

o!tions.ii. 4i2e."alue te )ifference bet/een intrinsic %alue an) market %alue. Te

lon"er a security as time to increase in %alue te "reater its time %alueo 0%er time tis %alue "ra)ually )ecreases to ero as te !ossibility of

stock increases )iminises.o An American o!tions time %alue is "reater tan tat of a Euro!ean

o!tion because it can be e2ercise) at any time rater tan at a sin"lemoment.

b.

;o u!!er boun) on !otential !rofits( but tere is a limit to o/ muc you can lose

Te !rice of te o!tion7. Short Positionis /ere an in%estor antici!ates tat a stocks %alue /ill )ecrease o%er time

an) terefore borro/s I to buy stock at its !resent %alue an) ten !ay back len)er /it teamount recei%e) from te sale /en te stock )ro!s an) terefore !ocketin" te )ifference.

a. Calculation

$alue of Short =%roceeds of short sale & 'eplacement Cost

b. Unlimite) )o/nsi)e Z limite) u!si)e11. Io, 0ptions 0perate

A. Call 0ptions$See abo%e&

B. Put 0ptions ,ut o!tions "i%e te ol)er te ri"t to sell $!ut& te un)erlyin" asset to te /riter of teo!tion at a !re%iously s!ecifie) !rice.4. Calcluation

$alue of %ut (ption ) *xercise %rice & +arket %rice of Stock

;ote te %alue of te !ut o!tion can ne%er be less tan ero. 1f !rice of stock "oes u!beyon) te o!tion !rice( te ol)er sim!ly )oes not sell.

111. "aluation of 0ptionsA. Factors 4hat Enter into Pricing 0ptions

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4. *xercise %rice Te i"er te e2ercise !rice in realtion to te market !rice of teun)erlyin" stock( te lo/er te %alue of a call o!tion

a. E2.i. E2ercise !rice of I799 for a current stock !rice of I499 is likely to be

/ortlessii. E2ercise !rice of I499 for a I499 %alue) stock !rice = a 89#89 cance te

stock !rice /ill rise abo%e te e2ercise !rice )urin" te life of te o!tion.7. Stock %rice == Te i"er te stock !rice relati%e to te e2ercise !rice( te more %aluable te

call o!tion /ill be.a. 0nce te stock !rices rise abo%e te e2ercise !rice( te e2cess of te stock !rice o%er

te e2ercise !rice re!resents te miniumum %alue of an o!tion.?. *xercise %rice Te lon"er te life of te o!tion( te "reater its %alue( all oter tin"s bein"

equal.N. $ariance of the ,nderlying Stock-s $alue 4his is the 2ost i2portant factor in

deter2ining option )alues. Te "reater te %ariance in te %alue of te un)erlyin" asset( temore %aluable te call o!tion /ill be.

8. nterest 'ate Te %alue of te o!tion is !artly relate) to o!!ortunity cost o/ mucinterest can you earn on your money in%estin" it some/ere else )uirn" te life of teo!tion

a. Tus( /en interest rates are i"( not !urcasin" te stock an) lea%in" te o!tion in

e2istence increases te %alue of te o!tion as you "et your I trou" in%estin" it in"o%t treasuries.B. Biono2ial 0ption "aluation Method $Sim!le ''ot on exa&

4. +ain is t,ice as 2uch ,hen buing ,ith borro,ed @97. E2. Current stock !rice = I499( 4 yr Euro!ean call o!tion Q I499( Stock !rice as a 89#89

cance bein" eiter IK9 or I479 in 4 yr /# a 49: i'rate.a. lt. 1 Call 0ption9 Buy 4 o!tion /ic in 4 year /ill be /ort eiter I9 $lo/& or

I79 $i"&.b. lt. 2 :e)eraged Stock Purchase9 Borro/ )iscounte) !resent %alue of lo/ bon)

$IK9& /# 49: i'rate = K9#4.4 = I7.?. Ten buy one one sare of stock at its current!rice I499( bot for one year.

i. 1n%estment in o!tion = 499 7.? = I7.7ii.

Possible 0utco2eso Lo/ $IK9& stock %alue loan re!ayment $IK9& = I9o 3i" $I479& stock %alue loan re!ayment $IK9& = IN9

?. :a, of 0ne Price ,ayoff on le%ere"e) in%estment is 7 times te !ayoff of 4 call o!tiona. Value of 7 calls = current %alue of sare $I499& bank loan $I7.?& = I7.7

b. Value of 4 call = 7.7#7 = I4?.@?N. 0ption &elta'Iedge %atio

a. Te number of sares nee)e) to re!licate one call o!tion.b. E2. 1n te !recee)in" e2am!le( 7 calls are re!licate) by a le%ere) !osition on 4 sare.

Te o!tion )elta is terefore [.i. 0ption &elta G

spread of possible option prices/spread of possible share prices ) 20 & 0/120 & 0 ) 1/2

C. Black.Scholes 0ption Pricing Model$A)%ance)#Com!licate)&a. &e2onstrates the da.to.da changes

1V. &isclosure 1ssues ,ith 0ptionsV. Con)ertible Securities &eal Protection and "enture Capital

A. &estruction of 0ption %ightsB. &ilution of 0ption %ights

i. For2ula$for )eterminin" a)Muste) con%ersion !rice&

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ACP G CP - CMP FM"'CMP

C5,Current 5arket ,rice Total Assets $!robably sometin" in te billions& multiplied by

4#total - of sares of C#S Z ,#S $!robably sometin" in te millions&AC,A)Muste) Con%ersion ,riceC,Con%ersion ,rice>5V>air 5arket Value

*i%i)en) %alue divided bytotal - of C#So i.e. )i%i)en) for IN995 /it 495 assets = N99#49 = IN9 !er

outstan)in" common sare.

b. E2am!lesSee p.304567. Dse of Con)ertible 0ptions in "enture Capital Financing Com!anies /# VCs can use t/o

)ifferent ty!es of Anti'*ilution Clauses to !rotectW its VCs $an) tereby encoura"e teirfun)in"&

a. >ull'Fatcet Clause 1f any sares are sol) cea!ly in a later roun)( ten tecon%ersion !rice of Series A stock is re)uce) to te cea!est !rice of any sares tatare sol) later on.

i. ,ersonifies te VCs bar"ainnin" !o/er in ne"otiatin" teir in%ol%ment.ii. E2.& 4 million common sares outstan)in"( !lus 789(999 con%ersion ri"ts at

I4#sareo Con%ersion /oul) yiel) con%ersion ol)ers 79: of com!anyo Su!!ose sale of 4 sare at 4 cento 789(999 con%ersion ol)ers coul) no/ !urcase 78 million sares(

out of 7@ million( or J@: if tere /as a full'ratcet clause. 789k#.94 = 78 5 A)) te e2tra 4 5 !re%ious sares to "et 7@

b. ormula

$;umber of Sares !re%iously 0utstan)in" 2 Con%ersion ,rice&+ $Consi)eration for a))itional sares& .

;umber of Sares 0utstan)in" after a))itional issue

iii. E2.& 0ne a))itional sare issue) for .94. ^ust factor in tis one a))itionalsare.

o - of sares !re%iously outstan)in" = 4(999(999o Con%ersion ,rice = 4.99o Consi)eration for a))itional sares = .94o - of sares outstan)in" after a))itional issue = 4(999(994.o =I.JJJJJJJJJ

i%. See o/ te effect of te 1. &istributions and 1n)estor WealthA. E-plaining &i)idend Practices

4. MHM 1rrele)ance Ipothesisa. >irst Seefi"ure J'4 on !a"e @7

i. Left an) si)e = /at eac sare is /ort before te )i%i)en)

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ii. Fi"t an) si)e = /at eac sare is /ort after te )i%i)en). Tis is 4#?less because of te lost !ro)ucti%e ca!acity of tat ca!ital.

b.


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ii. ;o/ )i%i)e tat by 4.?5 sares an) /e "et te )ilute) %alue !er sare )o/nfrom I499 !er sare to @.J7 !er sare.

iii. Tus te %alue of te )i%i)en) is @.J7 2 ?99(999 = I7?(9@(999.V. Stock %epurchasesV1. Spin.0ff 4ransactionsV11. 4racking Stock

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Corporate Finance Math Sheet