NPV (All Equity)

Company Value

( ) ( ) ( )( ) ( ) ( ) ( )

[ ] [ ] [ ]( )[ ]2

212,12,121

2,1212

22

22

12

12

212122

212

1

2211

2211

12

,2

σ

ρσσρσσσσσ

=

≡==+

++=

⋅++=

+=+=

RV

rhowwwwww

RRCOVwwRVwRVwRV

REwREwRERwRwR

P

P

P

P

Annuity

t

t

rg

grC

grCPV

)1()1(

)()( ++

−−

−=

Equivalent to perpetuity at time 1 –

perpetuity at time t

Portfolio TermsRisk = covariance / correlation

μ = mean return =

VAR =

StdDev

= SQRT(VAR)

∑=

T

ttrT 1

1

( )∑=

−T

ttrT 1

21 μ

Project Valuation

Perpetuity

grCPV−

=Cashflow

Growth rateDiscount rate

Interest Rates

( )( ) ( )ir +=++ 111 πnominalreal inflation

ir ≈+π

T

TAPREARr ⎟

⎠⎞

⎜⎝⎛ +== 1

effective annual rate

NPV

( )∑ +=

t

trCPV

1 1

IRR

r such thatNPV = 0

Payback

AccountingRate of Return

RoI

= Book IncomeBook Assets

EAC

equivalent annual cost

also known asbreak-even rental

torAnnuityFacCOSTSPV )(

=

( )[ ]NWCCAPXDEPRECNOPATFCF

TEBITNOPATΔ−−+=

−= 1

Bonds & Fixed Income

Treasury Securities1yr <= T-Bills

10yr <= T-Notes10yr > T-Bonds

BondsPrice = P(C,T)

coupon%

terminalperiod

Pricing

( ) ( )

( ) ( )∑

∑

∑

=

=

++

+=

++

+=

+=

T

tTrtr

C

SEMI

T

tT

Tt

t

Tt

FP

rF

rCP

FPVCPVP

2

12

22

2

1

211

11

)()(coupon face

Zero-Coupon BondsUse ZCBs

to get r-values for eachyear (spot values) when

calculating bond prices if there isnon-flat term structure.

(could use annuity only when term-structure is flat)

B(0,t) is equivalent to$1 ZCB for t years

Discount Factor

Yield toMaturity

What value of r givesthe market price P

equal to the discountedcash flows for the bond?

t

ryield curve

Interest Rate Term StructureGraph of YTM for

ZCBs

over time

ForwardRates

Expected interest rates in the future

t

ZCB

$1

2r

3r( )331 r+

( )221 r+ ( )321 r+×

( ) ( )( ) 3

22

2

33

32 111

DFDF

rrr =

++

=+

DurationThe weighted average of thetime taken to get payments

Interest Rate SensitivityInterest rates are more sensitive:

-

when maturity is longer-

when the coupon is lower

ZCB DurationThe duration of a ZCB is the same

as its time to maturity

( )

rrD

VV

rC

Pw

wtD

tt

tt

T

tt

+Δ

⋅−=Δ

+⋅=

⋅=∑=

1

11

1

ratio of change in value

ratio of change in interest rate

cashflow at time t

( )ttrtB

+=

11),0(

RD

= YTM on the debt of the company

( ) ( )∑= +

++

=T

tTt YTM

FYTMCP

1 11

Market price

Potential problem:multiple IRRs

( )FMAFA RRRR −+= βDivisional Leverage

Company

1

1

111

11

,,,

VDL

ED

ADE

=

βββ

2

2

222

22

,,,

VDL

ED

ADE

=

βββ

%1w %2w

2211

,,,,

AAAC

C

CDCECCC

wwVDLED

βββ

ββ

+=

=

( )( )

( )gWACCgFCF

WACCT

T −+⋅

×+

11

1

Firm ValueComprises of value of all its projects -

The present discounted value of all its cashflows.

CorporateValuation

LeverageGearing = Leverage = D/V

Asset Beta = unleveredEquity Beta = levered

Tax Rate

Taxes PaidEBT

TaxThe tax rate to be used

may not be the corporate tax rate. Strictly speaking it

should be the effective tax rate

Earningsbefore tax

( )VER

VDTRWACC ED +−= 1

Leverage

Debt

DebtIn principle the market value of the debt, but in

practice this hard to find. Book value is a valid proxy unless

company is in distress.

Only considerInterest bearing

debt

D =

LT Debt

+

ST Debt(if ST debt is not related to workng

captial)

- Cash(if cash is not

used for working capital

it could be used to pay off debt holders)

EquityThe market value of the equity of the company

E = share price * number of shares outstanding

VE

VD

EDA βββ +=

Can be assumed to be 0 if debt is risk-free

CAPM

Equity Beta

Given by the covariance of the stock with a give index

(usually via regression analysis)

Risk Free Rate

Given by short term treasury bills (up to 1 year maturity) in the US, or gilts

in the UK.

Market Risk Premium

The difference between the return expected from

investing in shares and the risk free rate.

Typically ~ 5%

Modigliani-Miller• MMI –

Capital Structure Irrelevant• Perfect Capital Markets:

•

Individuals can borrow at the same rate as corporations.• No bankruptcy costs / distress costs• No agency problems• Symmetric information•NO TAXES

This allows us to make RA

= WACC

Thus RA

does not change due to capital structure –

but this does not hold in the real world of taxes etc.

Cos

t of c

apita

l

wacc

Optimum Leverage

RA

D/VRisk of bankruptcy due to debt

payments makes debt more risk

BetaA measure of how much a stock

contributes to portfolio risk, i.e. how much the stock moves when the market moves.

1 = same as market0 = unrelated to the market-1 = inverse to the market

( )FMEFE RRRR −+= β

Dividend Growth Model

( )( )gR

gDVE −+⋅

=10

Dividend Growth ModelAKA the Gordon Growth Model or constant growth model. Assumes constant growth, ok for mature or

stable industries

Terminal Value

Based on the PV of a constant FCF (as in last period) with constant growth

Note: Perpetuities bring back 1 period (T = last period of FCF model)

V / EBITDA

Market Value of EquityBook Value of EquityRatios

Use a ratio for a given industry or similar firms

and apply.

Ratios / PerpetuitiesRatios can be equivalent to a

perpetuity resulting inRatio = r –

g=> beware same assumptions as

DGM

Portfolio TheoryActual Returns:

Expected Returns:

Variance:

Standard Deviation:

Portfolio PerformancePortfolio Risk and Diversity

A portfolio of about 20 stocks can diversify almost all specific risk

Risk

# Stocks20

MarketRisk

Idiosyncratic RiskSpecific Risk

(Diversifiable Risk)

Market Risk(Systematic Risk)

w1

:w2

(Mix)Risk

(StdDev)

RP

0% 100%

RP rho

= -1 (max benefits from diversification)

-1 < rho

< 1 (some benefits from diversification)

rho

= 1 (no benefits from diversification)

Correlation

Risk of share

Market risk of share

Specific risk of share

βI x Rm

= +

Major Part

Risk of portfoli

o

Market risk of share

Specific risk of share

βp x Rm

= +

Negligible

Accounting StatementRevenues

- Costs=

EBITDA (operating income)-

Depreciation=

EBIT (operating profit)-

Interest Expense=

Pretax

Income-

Taxes=

Net Income-

Dividends (& share buy-backs)=

Addition to Retained Earnings

( ) ( )bRoER

bgR

bEP

EE ⋅−−

=−−

=11

Plowback Ratiob = plowback ratioDividends = E(1-b)

RoE

> r : positive growth opportunitiesRoE

< r : value being destroyed

Free Cash Flows

Equity FCF(EFCF)

Operating FCF(FCF)

( ) ( ) ( )TT

WACCFCF

WACCFCF

WACCFCFFCFV

+++

++

++=

111 221

0 L

( ) uesNetDebtIssTInterestFCFEFCFNetDebtNWCCAPXonDepreciatiNetIncomeEFCF

+−−=+Δ−−+=

1

Note: VE

= Free Cash Flows available to equity holdersFunds available in company after:-Building up NWC-New investments-Paying off old debt / Issuing new debtFunds available in form of dividends

or share repurchases

EDA VVV =−

ADE VVV =+

Note: VA

= Free Cash Flows of the companyFunds that would be available to equity holders if D = 0:Funds from company free cash flows= NOPAT+ Depreciation-

Change in NWC ( = current assets –

current liabilities)+ New investments

Calculating Share ValueTo value shares, divide dividends and repurchases by number of shares outstanding

Taxes and NOPATIf actual taxes are known then we can use EBIT –

Tax ExpenseInstead of NOPAT

Abnormal ReturnsAbnormal returns: αi

But if capital markets are efficient then

( )[ ]FMiFii RRRR −+−= βα

Efficient MarketHypothesis

( )( ) 0=⇒

−+=

i

FMiFi

ARERRRR β

Abnormal Returns

( )DAAE ED ββββ −+=

( )DAAE RREDRR −+=

( ) ( ) ( )[ ]BPVAPVABPVGain +−=

Mergers & Acquisitions

Overall / Economic Gain

Cost of CASH Acquisition / Premium (A buying B)

( )( ) ( ) CashAPVABPVCostGainBPVCashCost

−−=−−=

Market ValueMarket Value (MV) could be a

combination of PV and expectation (P) of takeover premium (C):

MV = PV + P*C

Stock AcquisitionsNote: that the cost to firm A cannot be calculated from the stock price

ratio.

The economic Gain is required to calculate true cost, since share price

may change with merger.

True cost can also be calculated from computing gain to shareholders

of company B.

Cost of STOCK Acquisition / Premium (A buying B)

( ) ( )( ) ( ) ( )ABPVxAPVABPVCostGain

BPVABPVxCost⋅−−=−

−⋅=

Does the Market Value of the firm reflect takeover premium?

( )[ ] ( ) ( )[ ]BPVBMVBMVCashCost −+−=Premium paid

over market valueDifference between

MV and value as separate entity (PV)

x = fraction of combined firm stock going to shareholders of B

Dubious Motives•Agency –

empire building, larger companies, prestige, perks, compen.•Diversification –

declining cash rich industry. Funds should be returned to shareholders.•Increase EPS –

number of shares traded may not be equal.

Valid MotivesShareholders are better off•Value Creation

•Operating synergies –

horiz: market power and vert: market foreclosure.•Complimentary resource synergies•Cheaper external financing•Correct management failure

•Wealth Transfers•Bondholders to shareholders•Employees to shareholders (wage concessions)•Customers to shareholders (market power)•Taxes to shareholders (unused taxshield)

Notes on FCF

Essentially cash generated before payment is made to debt holders

•Sunk costs –

ignore, but market value if sold is relevant. •Opportunity costs –

(incremental) should be included (not allocated). Excess capacity is not free.•Inflation –

cashflows

need to be matched to rates of return (nominal usually)•Financing costs –

taken into account in WACC (e.g. dividends and interest)

•Depreciation is not

inflated in nominal/real calculations.

RoE

= Net IncomeStockholders Equity

Forward RatesNomenclature:

( )3,2,032 fr ≡

Interest Rates•S$£

: $/£

spot exchange rate ($x:£1) •F$£

:$/£

forward exchange rate•r£

: nominal interest rate £•r$

: nominal interest rate $•i£

: UK inflation rate•i$

: the US inflation rate.[S$£

x S£$

= 1]

Covered Interest Parity

( ) ( )

( )$

$££$£

£$£$£

11$

111111$

r

FrS

rSS

+→

+→+→→

To UK +1 year Back to US

( )( ) $£

$£

£

$

11

SF

rr

=++

No riskless

arbitrage

( ) ( ) ( )( ) $£

£

$$£$£$£ 1

1S

rr

SESEF++

≈′⇒′=

Expected future

spot rateEstimating

future forward rates

( )( )

( )$£

$£

£

$

£

$

$£$

$££

11

SSE

iEiE

PPSPSP

′≈

++

≈⇒≈Purchasing Power Parity

RoI(rate of return)

1

1

0

1 −++

=

−=

CCCRoI

estmentInitialInvCashOutRoI

tL

( )mkt

stki

stkmktcorrσ

σβ ⋅=

,

( )ttt r

DF+

=1

1

discountfactor

( )gWACCFCFV

−=

V2.2 -

Licence / CopyrightCreative Commons

Attribution-ShareAlike

3.0 http://creativecommons.org/licenses/by

-sa/3.0/

Created ©

Matt McNeill 2007mmcneill.semba2008@london.edu

Contributions: Mike Rizzo, Mark Carroll

Correlationbetween two

stocks

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

VD

RRTRWACC

TS

DA 1

ERDRDRTVR EDDUA +=⋅+

VER

VDRR EDA +=

( )TSVV LU PV−=

( )( )TRREDRR DAAE −−+= 1

Real additionalcashflows

from TSWhen T=0

Unlevered

DTSA RRR ≤≤Tax Shields

Thus annual tax savings are:

RTS

approximates to RD

when risk of NOT using tax shields is minimal

( )

[ ]DRTEBITT

DRInterestInterestTEBITT

InterestEBITTTax

D

D

⋅⋅−⋅=⋅=⋅−⋅=

−⋅=

TS

D

D

RDRTTSPV

DRTTS⋅⋅

=

⋅⋅=

)(

DTSA RRR ≤≤

Bankruptcy Costs (BC)• FCF insufficient to meet RD

.D (interest)•

Direct Costs

–

Legal, Accounting, Trustee, Management fees etc•Indirect Costs

–

Production inefficiencies (e.g. supplier terms), lost investment opportunities, talent loss etc•Typically 1-20%, μ=3-4%, young firms

Financial Distress Costs (FD)•Reduced financing capacity (D & E)•Higher cost of capital•Loss of customers / suppliers / talent

)PV()PV()PV( ACBCFDTSVV UL −−+=Agency Costs (AC)

• Debt Overhang•New E raised goes to D shortfall if project successful. E insures D.•Soln: Issue more D, use E to buy back D, convertibles

• Overinvestment•FCF which should go to D is risked by E on risky project. •Downside goes to D, upside to E (ltd liability)

• Risk Shifting•2 +ve

NPV projects with different risk•D fear funds will be allocated to hi-risk (ltd liability of E means D loses)•D thus require higher RD

and no projects now have +ve

NPV•Soln: Debt covenants etc

)PV()PV()PV()PV(

TSACTSBCFD

<<<<

Optimum Leverage

VU

D/E

V PV(TS)

PV(TS)-PV(BCFD)-PV(AC)

( )BCR

stExpectedCoBC P)PV( =

Adjusted Present Value (APV)Value project

as if all equity financed –

use Operating FCF, discount at RA

= RE

all equity firm. Add PV(TS)

generated by new project

Note: APV assumes D constant over time, iterative WACC assumes D/V ratio constant, so values may be slightly different.

)()( TSPVAllEquityNPVAPV +=

( )( )

[ ]EA

EE

RRWACC

gRTEBITV

==

−−

=

:firmequity allfor

1

When RTS

= RA

When RTS

= RD

2 Period Binomial ModelFor European Call Option

Note:

that Δ,B in each period willChange depending on outcome of previous period (Dynamic Replication)

C

K

St

$

Long Call

C

K St

$Short Call

P

K

St

$Long Put

0

X

TerminologyLong

= Buy the right to…Short

= Sell the right to…

Call

= …

buy at given pricePut

= …

sell at given price

Strike

= Exercise = KPremium = Cost = P, CStock Price = S

Option typesEuropeanCan exercise only on given maturity date

AmericanCan exercise any time up to (and on) the given maturity date.

maturity = expiration

Options

Combining options requires going long / short on options with different strike prices.

Long Call = +45˚

(L R)Short Call = -45˚

(L R)Long Put = +45˚

(L R)Short Put = -45˚

(L R)

C(X)+

P(X)X

St

$ e.g. Straddle

PayoffProfit

= Long C(X) + 2 x Short C(Y) + Long C(Z)

Y St

$ e.g. Butterfly

X Z

V2.2 -

Licence / CopyrightCreative Commons

Attribution-ShareAlike

3.0 http://creativecommons.org/licenses/by

-sa/3.0/

Created ©

Matt McNeill 2007mmcneill.semba2008@london.edu

Contributions: Mike Rizzo

P

-X

St

$Short Put

K

{ }K-S0,Max=C

Call OptionS>K = In the moneyS=K = at the moneyS<K

= out of the money

Put OptionS<K = In the moneyS=K = at the moneyS>K

= out of the money

{ }S-K0,Max=P

Arbitrage PrincipleIf two combinations of assets have same cashflows

in every period and every outcome, then they must have the same price.

This can be used to calculate replicating portfolio for use in pricing options

Option Pricing

Replicating PortfolioUse arbitrage principle.Δ

= proportion of stock (aka

hedge ratio / option delta)B = value of risk free BondsCall: Δ

> 0, B < 0 (long S, short B)Put: Δ

< 0, B > 0 (short S, long B)

Binomial Model

P not known, thus cannot use weighted average for C0

P%

1-P%

{ } ( )BRSKSC FUUU ++Δ⋅=−= 1,0max 111

{ } ( )BRSKSC FDDD ++Δ⋅=−= 1,0max 111

Using replicating portfolio, solve simultaneously for Δ, B BSC +Δ⋅= 00

( ) BRSC

BSt

FUU

U

++Δ⋅=

′+Δ′⋅=

111

1

( )BS

BRSC

D

tFDD

′′+Δ′′⋅=++Δ⋅=

1

11 1

BSC +Δ⋅= 00

{ }( ) BRS

KSCt

FU

UU

′++Δ′⋅=

−=

1

,0max

2

22

( ){ }

( ) BRS

KSCBRS

tFM

MM

tFM

′′++Δ′′⋅=

−=

′++Δ′⋅=

1

,0max1

2

21

2

( ){ }KSC

BRS

DD

tFD

−=

′′++Δ′′⋅=

22

2

,0max1BSV +Δ⋅=value of

option

European Call option

Black-Scholes

Black-Scholes

Assumptions•Stock variance σ2

is constant•Rf

is constant•No dividends•Frictionless trading (no transaction costs)Note: σ

is not an observed quantity in market, and BS is often used to find implied volatility.

( ) ( ) ( )21 NPVN dKdSC ⋅−⋅=European Call option

( )

( ) ( )TfRKK

Tdd

TT

KS

d

+=

−=

+⎟⎠⎞⎜

⎝⎛

=

1PV

2PVln

12

1

σ

σσ

μ

d2

d1

Normalcumulative

densitydistribution

( )( ) ( ) ( ) ( )21 NPVNPV dKdDSC ⋅−⋅−=

Incorporating dividends into BS

Put-Call Parity

( ) ( ) ( )KKSK PVCP +=+

K

S

$

K S

$K

S

$K

S

$K

S

$+ += =

( )( )

( ) ( )KKR

KK TF

CP1

),0B(PV

−=+

=

( ) ( ) ( )21 NPVN dKdSP −⋅+−⋅−=European Put option

( )( ) ( ) ( ) ( )21 NPVNPV dKdDSP −⋅+−⋅−−=

Incorporating dividends into BS

Derived from binomial with infinitely small periods (and assumptions)

American Options•Can be exercised before maturity date•Call options

–-

No dividends = European Call-

Price: calc euro call options maturing at all dividend paying & expiry datesand choose largest

•Put options –-

no dividends: may still exercise early-

value larger than euro put

WarrantsEquiv. To Call option except:•

Issued by company so company gets purchasing price•

On exercise company issues new shares

and gets exercise price

•

Delayed equity issue (mitigates signalling problem)• Exec stock options are warrants•

Equity rights issues are “special warrants”.

Treat same as Options for pricing but need to adjust for share increases and purchase price

n = number of existing sharesm = number of warrantsr = number of shares per warrant

(conversion ratio)K = exercise priceλ

= dilution factor (% fraction of E that goes to new stockholders)

( )

[ ]( )

mrnmr

RTnKEWnKEW

mrKEVmEV

f

TT

TT

+=

⋅=−⋅=

+=⋅+=

λ

σλλ

,,,,C0,max

warrantsof price

0BS0

00

Black-Scholes

Shorthand

S = Stock price at time 0K = Strike price / exercise priceT = Time to maturity (years)Rf

= Risk free rateσ

= volatility

( )σ,,,,CBS fRTKSC =

Value of all equity firm

Value of all warrants

Convertible BondsEquivalent to a package of a:

straight bond + warrant

•Mitigate agency problems of debt•Mitigate signalling problems•Can obtain debt at lower current

cost (coupon discount related to value of warrant)

n = number of existing sharesm = number of bondsr = conversion ratio: number of shares per bondFB = face value of each bondFB /r = conversion priceKB = conversion value: market price of bond

divided by r (strike price of each share).Usually calculations worked out with complete company values:

FD = Face value of debt (#bonds * FB

)KD = Exercise (strike) price of debtE0 = Value of initial equityE0

*

= Adjusted value of initial equity ET * = Adjusted value of equity at maturity

Value of firm (ET

*)

Valu

e of

Con

verti

ble

(CB)

FD FD

/λ

default holdsbonds

convertsto shares

FD

λ

λFK

FE

D

T

=⇒

>⋅ ∗Conversion exercised if:

convertible = warrant + straightbond bond

( )TCWCB ,B+=

See bond pricing on other side

( )σλ ,,,,C *0BS fD RTKEW ⋅=

( )( )

( ) ( )B0

T

FmSnE

EE

EE

⋅+⋅=

−⎟⎟⎠

⎞⎜⎜⎝

⎛−=

=

0

0*

0

**0

dividendsPVmaturity

before couponsPV

PV

existing equity

monies raised by convertible

issue

Need to solve recursively

[where F’

= CB???] [ ]

⎥⎦⎤

⎢⎣⎡ −⋅=

−⋅=

0,max

0,max

*

*

λλ

λ

DT

DT

FE

FECB

Equity Issues•IPO = Initial Public Offering, a company’s first

offering of shares to the general public.

•Primary Shares

–

new shares issued by company where money raised is invested in the firm•Secondary Shares

–

insiders selling stake in company where money raised goes to the previous owners

•SEO = Secondary/Seasoned Equity Offering, an equity issue by a firm that is already public.•Pecking Order: (1) Internal Funds,

(2) Debt, (3) Equity

Advantages of IPO•

Obtain cash –

bank finance, venture capital not enough•

Cheaper financing –

higher liquidity and lower info asymmetry (disclosure reqs.)• Other financing cheaper –

(as above)• Insiders can cash out• Easy future access to equity markets

Disadvantages of IPO• Costly –

admin fees (4%) & underwriters fees• Loss of control• Legal reqs. –

disclosure rules etc• Value of firm subject to external perception• Easier target for hostile takeovers

Underwriters•Investment banks which advise

the firm and provide independent

monitoring of quality of firm to the market.•Also handles:

•Roadshows

–

for signalling considerations and demand evaluation•Bookbuilding

–

bids during BB period (~2 weeks) can be revised and cancelled.•Price and Allocation

–

following BB period.

•Alternatives to BB might be allocations by auctions, but no discretion to underwriter on final price and allocations (Google)•UW formally buy shares from firm and sell to public at higher price. •Various sales models:

•Firm commitment

–

all shares (see Underwriter’s put)•Best Efforts

–

sale and return•All-or-none

•Price premium covers UW responsibility:•Market maker –

liquidity in first trading days•Research coverage after IPO

•Price premium covers UW risks:•Stabilising prices –

if they fall below offer price•Buying unallocated stock•Mitigated by green shoe

option (option to purchase additional ~15% shares from company at offer price if demand high within ~30 days of IPO)

•IPO Fees usually a fixed % of the issue (~7% in US, ~5% in UK, ~3.5% in Europe) •SEO fees about half IPO –

very varied•Rights fees usually ~2%•Underpricing

IPOs

(-12% UK, -15.8% US)•UW favour current / potential clients•Signalling/Reputation (future SEO)•Dispersion/Liquidity on lower prices•Attract less informed investors (mitigating winners curse problem)

Rights Issue• Rights = short warrants (~3 weeks) issued at zero price• UK ~60% equity issues, US <5%• Rights issued are proportional to shares owned•

Shares trade “cum rights”, but later can be split and traded separately.

• Exercise price drivers:•

low as possible

to ensure that always in money and all rights are exercised (income)•

not so low

as to signal the market that the managers think the share price will drop a lot in the next 3 weeks•

shareholders not bothered, since right value will always balance dilution of current share price.

Valuing RightsValue of exercising a right now

n = number of existing sharesi = issue ratio (rights issued per share)m = number of rights (m = i.n)r = number of shares per right

(r = 1)

(conversion ratio)

K = exercise priceλ

= dilution factor (% fraction of E that goes to new stockholders)

( )mrnES

nKEEnSE

TT

T

+=

+=⋅=

0

00

E0 = value of pre-rights company (all equity)ET = value of pos-rights company (all equity)S0 = value of each pre-rights stockST = value of each post-rights stock

FUW = underwriter’s fees

V0 = value of exercising a right immediatelyW0 = value of the option of exercising a right at maturity

KSV T −=0

iVSS T ⋅+= 00

Shareholders are not bothered

( )σλ ,,,,C10BS0 fUW RTnKFE

mW −⋅⋅=

Value of option of exercising a right at

maturity

Options increase value of the rights

00 WV <

Payo

ff

ST

Rights issues where K is closer to ST

have higher (W0

) option values since the downside is more limited(value of a call option decreases

as K is further from S)

K K*

( )σλ ,,,,P 0BS fUW RTnKFEUP −⋅=

UP

K S

C –

right ownerP –

firm

P –

underwriter

Preferred Stocks• Shares with fixed dividend (like debt)• Junior claim to debt, but senior to ordinary stocks• Dividend may not be paid, but only if ordinary stock dividend not paid•

Limited voting rights (usually become normal voting rights if dividend not paid)•

Often viewed as flexible leverage (classified as equity, limited control, cashflow flexibility if dividends not paid)•

Often bought by institutions, corporations or low income tax bracket (income tax < cap gains tax)

Forward & Future ContractsForward Contract

is commitment to deliver predetermined asset at a future time for a predetermined price.

Futures Contract

is a standardised forward contract traded on an organised exchange.

S0 = current spot price (t=0)F = cost of T-period forward contract

Cost of carry:F – S0 > 0

Market is in ContangoF – S0 < 0

Market is in Backwardation

( )TrFS+

=10

Assuming both strategies of buying now and storing (at zero cost) and buying a forward contract on the asset are both 100% riskless

then:(risk of underlying asset already incorporated in S0

) ( )TrSF +⋅= 10

( ) ( )[ ] ( )TrSF +⋅+−= 1costs storagePVincomePV0

And now accounting for the costs of storage and income from the asset:

Futures• Extremely liquid due to use of exchange•

Firms can quickly rebalance risk management portfolios at low cost•

BUT: we do not know counterparty and default risk.•

THUS: exchanges require collateral and typically daily settlements (potentially requiring some cash now)

SwapsA swap is an agreement by which 2 parties exchange cash-flows of 2 securities (without changing their ownership)

Interest Rate Swap

is an exchange of interest payments on debt (most commonly the coupon swap: fixed rate with floating rate)

Currency Swap

is an exchange of payments in different currencies.

Interest Rate Swap Example•Assume that the floating coupon is 8% in first semester and increases 1% every period•The net payments from X to firm Y are

By entering the rate swap, Y borrows at floating rate (to which it has access) but eliminates interest rate risk.

Hedging•

Hedging

is obtaining insurance against some exogenous risk by taking an offsetting risk.• Risk Management

is defining an optimal set of hedges

We usually want to hedge:•Interest rate risk (inflation / real rate changes)•Currency risk•Fluctuation of commodity prices (inputs / complementary products)

Value of hedging usually depends on the need for a stable cashflow to take on other projects. •If the hedge will provide the capital for the stable project it is a good thing to do. Risk is bad.•If the hedge eliminates the chance of raising the capital for a project it is a bad thing to do. Risk is good.

Assume firm value V, depends on asset price S:

0S

US1

DS1

P%

1-P%

( ) ( ) ( )0

0101 %1%S

SSPSSPR DUS

−⋅−+−⋅=

Return on asset S for 1 period

0V

( )USV 1P%

1-P% ( )DSV 1

( ) ( ) ( )S

DU

RSVPSVPV

+⋅−+⋅

=1

%1% 110

Current value of the firm

( )TrSF +⋅= 10 r = risk free rate

0V

( ) ( ) ( )FVFSVSV UU =−− 11P%

1-P% ( ) ( ) ( )FVFSVSV DD =−− 11

Full Hedge

Underwriter’s PutWhen firm’s get a firm commitment from the underwriters, then it is equivalent to the firm putting a put option on the rights it is selling

UP is typically a very small part of the fee. Advice and monitoring is usually much more significant.

No Hedge

( )r

FVV+

=10

Optimal hedgingOptimal hedging

depends of structure of costs associated with low cash flow.•

Inability to service interest payments, increasing costs of financial distress and decreasing debt capacity (tax savings) of the firm.•Inability to take advantage of profitable investment opportunities•Inability to perform dividend smoothing.

•Note: These are only valid of there is a cost of obtaining outside financing.

Hedging InstrumentsPayoff structure should cancel underlying risk as much as possible

(*) Unless traded OTC, but then they are (much) less liquid.(**) However they do require a margin account.(***) Huge demand for swaps makes them extremely

liquid.

Payout PolicyThis is how a firm distributes cash to the shareholders in one of two ways:

Dividends

–

firm distributes cash (or stocks) to shareholders in proportion to number of shares held.

•DPS –

dividends per share•Dividend Yield: DPS / share price•Payout Ratio: DPS / EPS

Share repurchases

–

firm buys shares from shareholders (US = treasury shares / UK eliminated, unless reserved to balance stock options etc.)

•Open market repurchases•Fixed Price Tender Offer•Dutch Auction Tender Offer

Share repurchases should only be used to distribute extraordinary surplus cash-flow, but since mid 1980s US firms now redistribute 50%. Europe is 20%.

Payout Policy IrrelevanceIn perfect capital markets and dividends and cap gains tax are zero or the same, payout policy is irrelevant.

Dividends: shareholders could use dividends to buy more shares, or could sell shares for cash to simulate dividend.Share repurchase: shareholders total wealth unchanged whether they sell shares or not. Also equal to dividend distributions. If shares repurchased at premium there is no wealth transfer between shareholders unless some fail to participate in bid.

Payout Policy RelevanceWhen dividends and capital gains are taxed differently, payout policy is not

irrelevant.

Dividends: pay taxes on full cash distribution.Share repurchases: •Pay cap gains tax rate only (typically lower)•Only pay tax on part of distribution (the gain)•Only pay tax if chose to sell

Depends on investors tax bracket, e.g. pension funds are indifferent (but may like predictability of dividends)

Tax ClientelesInvestors with different dividend tax treatment will hold shares of firms with different dividend-payout ratios.

Different investors might pay different tax rates on dividend income and capital gains:•

In most countries individual investors pay higher tax rates on dividends than on capital gains.• Pension funds are tax exempt.•

Corporations typically pay lower tax rates on dividends than individuals.

This creates different tax clienteles for dividends: e.g. attracting institutional investors.

Signalling and Dividends• Investors react sharply to changes

in dividends–

Omission: -9.5%–

Reduction of more than 25%: -6.4%–

Reduction: -1.2%–

Increase: 0.7%–

Increase of more than 25%: 1.0%–

Initiation: 3.9%•

A superior firm has higher payouts to signal wealthy and confident. Less profitable firm cannot sustain large dividends over long run.•

A less profitable firm will eventually have to cut dividends, miss investments, issue equity or debt to finance dividends (inefficient!)

Implications –

select conservative ratios, and avoid raising dividends if risk of having to reverse it. Dividend change should be smaller than change implied by earnings (smoothing). Avoid dividend cuts if cost is small.

Signalling Methods•

Dividends: most effective since they set futurecommitment.•

Shares repurchases with auction: very effective, if theshares are bought at a premium and management precommits

not to tender

(i.e. not selling own stock at a premium).•

Open-market share repurchases: weakest signal.–

Shares are bought at their current market price.–

50% of announcements do not follow through, and10% repurchase less than 5% of the value announced.

Stock Splits•

Stock Splits: Increasing the number of the outstanding

shares by reducing its nominal value.•

Example: In a 2:1 split, investors receive two new shares

in exchange for each old one. The stock price drops by

50% (no money ever changes hands!).•

A rationale for a stock split is that it makes stocks cheaper

for small investors.•

However, this “liquidity effect”

is not well supported by

the existing empirical evidence: there is no significant

price response to a stock split.

Tax Cost of Excess Cash•

Excess cash is effectively generating a negative tax saving

for the investors.•

If firms have more cash than what they actually need to

finance their business and have financial flexibility, then

they should distribute it to shareholders.•

Important caveat: “cash required to finance the business”

and “cash required for financial flexibility”

are not exactly

well-defined quantities!

Forwards Futures Swaps OptionsCunstomised Yes No Yes No*Upfront Payment No No** No YesLiquidity Low High High*** HighDefault Risk High Low High Low

S1 S2 S3 S4Floating rate 8% 9% 10% 11%X’s payment $5 $5 $5 $5Y’s payment $4 $4.5 $5 $5.5Y pays to X $1.0 $0.5 $0.0 -$0.5

Real OptionsReal Options

These are options as applied to business decisions:

Follow-on investment

(Call option / BS)Timing options

(American calls / Binomial)Abandonment Options

(Put option / binomial)

( )σ,,,,CBS fRTKSC =Follow-On Investment

•Although a project may not look as if it will payoff at t=0, the volatility and upside risk profile may still make the option very valuable. •The risk downside is not relevant since option would not be exercised in that case.

S = NPV of FCFs

(at t=0)K = PV(expected investment) t=0T = Time of inventment

(years)Rf

= Risk free rateσ

= volatility (comparable stocks)

PV(FCF) PV(Inv)

upside of investment

t

InvT

FCFT+1 FCFT+2

…

Discount at RP

PV0

(FCF)

PV0

(inv)

Discount at RF

T0

Timing Options•Cashflows

equivalent to dividends from a stock. When a project’s forecasted cashflows

are sufficiently large the investment is made right away (option is called)

If C0 > S0-K then the option to defer the project and miss possible cash-flows is more valuable. i.e. Wait and see.

0S

US1

DS1

P%

1-P%

( ) ( ) ( )⎥⎦

⎤⎢⎣

⎡−

+−+⎥

⎦

⎤⎢⎣

⎡−

+= 1%11%

0

11

0

11

SDSP

SDSPR DDUU

P

Find expected return

S = Value of projectD = cashflows

from project during period 1Rf

= Risk free rateP% = chance of going high

To find P%,

set RP

= RF

(for example)

{ }( ) { }K-S0,max%1

K-S0,max%

1D

1U

⋅−+

⋅=

PPC

Abandonment Options•Exercised when value recovered from project’s assets si

greater than the PV of continuing the project for at least 1 more period.

Effect on CBS

if the given variable increases in value:

+-+++