Post on 27-Feb-2018
transcript
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What does this course offer?
Scientific tools and techniques forvaluing financial assets
We focus on• Valuation of publicly traded firms
• Value of equity• Value of total company (debt+equity)
• Valuation of investment projects
Valuation Alex Stomper
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Relevance of valuation
Valuation and portfolio management• Central role in fundamental analysis• Useful input for technical analysis
Valuation and corporate finance• Corporate objective: value maximization• Capital budgeting• Mergers and aquisitions• Other corporate restructurings
Valuation
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Valuation and market efficiency(1)
In an „efficient“ market, the market price is thebest estimate of the true value of an asset.
Deviations of market price from true value arerandom.• Weak form: the current price reflects all information in
past prices (i.e., past prices do not help to identifyunder or over value stocks)
• Semi-strong form: the current price reflects all publicinformation
• Strong form: currect price reflects all information,public as well as private.
Valuation
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Valuation and market efficiency(2)
Strong-form market efficiency is theorecticallyimpossible: Grossman and Stiglitz (1980).
However, the market is usually smarter thanyou think.
Therefore, compare your valuation results withthe market price whenever possible• Do you know something that the market does not
know?• Or do you make a mistake?
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Approaches to valuation
Discounted cashflow valuation, relates thevalue of an asset to the present value ofexpected future cash flows on that asset.
Relative valuation, estimates the value of anasset by looking at the pricing of 'comparable'assets relative to a common variable likeearnings, cash flows, book value or sales.
Real options approach to valuation,quantifies the value of managerial flexibilityusing option pricing models.
Valuation
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Generic DCF valuation formula
The value of an asset is determined by the presentvalue of expected future cash flows generated by theasset.
where CFt is the cash flow in period t, r is theappropriate discount rate.
Underlying principle: valuation is additive!
∑= +
=N
ttt
rCFEV
1 )1()(
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Key components in DCFvaluation Relevant cash flows
• All cash flows from and to investors (inflows andoutflows)
• Difference between cash flow and profit/loss Appropriate discount rate
• Account for the time value of cash flows (earlier cashflows are more valuable)
• Account for the uncertainty (risk) of cash flows Matching principle: different valuation models
use different combinations of cash flows anddiscount rates.
Valuation
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DCF valuation models
Free cash flow valuation modelFree cash flow valuation model Capital cash flow valuation model Adjusted present value model Divident discount model Economic-profit-based valuation model
Valuation
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Key steps in FCF valuation
Estimating current free cash flows Estimating growth rate Estimating cost of capital Estimating residual value (company value at
the end of the explicit forecast period)
n
n
tt
t
WACCRV
WACCFCFEV
)1()1()(
10 +
++
=∑=
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I. From earnings to free cashflows EBIT (earnings before interest and taxes)
- Corporate tax rate* EBIT= NOPLAT (net operating profit less adjusted taxes)
+ accounting deductions that did not involve a cash outflow (depreciation,amortization)
- accounting income that did not involve a cash inflow= Gross Cash Flow
- Gross investments (net investment + depreciation)
= Free Cash Flow
- (1-Corporate tax rate)* Interest expense
- Repayment of principal
= Free Cash Flow to Shareholders
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Firm value vs equity value
Value of firm:
Value of equity
∑= +
=N
tt
t
WACCFCFEV
1 )1()(
∑= +
=N
tt
e
t
kFCFEEE
1 )1()(
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Taxes in FCF valuation FCF are calculated as if firms are all-equity
financed Marginal corporate tax rate is applied to EBIT,
without taking into account that interestpayments are tax deductiable
Debt tax shields are recognized through thediscount rate
Alternatively, one can include the debt taxshields in cash flows, then the discount rateshould be the pretax discount rate (Capitalcash flow valuation model, see later).
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Investment expenditures Gross investment
= capital expenditures + change in working capital Working capital = current assets (inventory, cash and
account receivable) - current liabilities (account payable,short-term debt)
Capital expenditures minus depreciation is called netcapital expenditures
Gross investment minus depreciation is called netinvestment
If depreciation is the only noncash expense/income thenFCF = NOPLAT - net investment
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Estimating current FCF:example
EBIT 1500Tax rate 40%NOPLAT 900Depreciation 300Gross cash flow 1200Capital expenditure 500Change in working capital 100FCF 600Interest expense 120repayment of principal 150FCF to shareholder 378
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II. Estimating growth rate
Look at historical growth rate Look at forecasts by analysts Look at fundamental drivers of growth rate
• Reinvestment rate:IR = Net investment / NOPLAT
• Return on invested capital (ROIC) ROIC = NOPLAT / Invested capital⇒IR*ROIC = Net investment / Invested capital
= capital growth rate
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Invested capital
Invested capital = total assets – excess cash –marketable securities – noninterest bearingshort term liabilities
Excess cash and marketable securities areexcluded because it is easier to value themseperately
noninterest bearing short term liabilities areexcluded because they are financed bysuppliers and their costs may have alreadybeen reflected in NOPLAT
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ROIC: example
NOPLAT in year t 900total asset at the end of year t-1 3000excess cash (t-1) 20marketable security (t-1) 100noninterest-bearing liabilities (t-1) 150Invested capital at the end of year
t-1 2730ROIC in year t 0.3
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Determinants of ROIC
CapitalInvestedvenues
venuesEBITtROIC Re
Re)1( ××−=
Profit margin Asset turnover
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Fundamental growth rate
The case of constant ROIC gNOPLAT = IRt * ROIC
Return on new invested capital (RONIC)≠ ROIC
gNOPLAT = IRt * RONIC The case of changing ROIC
gNOPLAT = IRt * ROICt+1+(ROICt+1 -ROICt)/ROICt
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Forecasting FCFs: example
year 2004 2005 2006 averageinvested capital(beginning) 400.00 450.00 530.00
NOPLAT 150.00 200.00 205.00
ROIC 0.38 0.44 0.39 0.40
Net investment 50.00 80.00 70.00
IR 0.33 0.40 0.34 0.36
FCF 100.00 120.00 135.00
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Forecasting FCFs: example
Forecast FCFs in the next three years,assuming alternatively that
(1) IR and ROIC are the same as in 2006(2) IR and ROIC are the same as in 2006,
RONIC equals to 0.3(3) IR and ROIC equal to the average levels in
2004-2006
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III. Estimating cost of capital
The expected FCFs to the firm are discounted usingthe weighted average cost of capital (WACC) toget the firm value.
The expected FCFs to equityholders are discountedusing cost of equity to get the equity value.
The weight of each financing form is defined as theratio between the market value of that financing formto the total market value of the firm.
Noninteresting-bearing liabilities are not consideredwhen computing WACC.
The tax advantage of debt financing is reflected inWACC.
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WACC
where kd = pre-tax cost of debt kp = cost of preferred stock ke = cost of equity t = corporate tax rate D/V = target debt ratio using market values P/V = target preferred stock ratio using market values E/V = target equity ratio using market values V = market value of the firm (D+P+S)
VEk
VPk
VDtkWACC epd ++−= )1(
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Weights in WACC The target weights instead of the current weights should
be used. Weights should be calculated using market values. The market value may not exist, especially for the debt.
Possible estimation procedure:• Identify all payment obligations to debt holders• Estimate the credit risk of debt-type financing instruments• Find market-traded instruments that have similar credit risk
and time to maturity• Use the market returns to discount the outstanding
payments to debt holders
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Cost of debt/preferred stocks Cost of debt = expected return on debt *(1-t)
• Expected return on debt ≠ coupon rate• Expected return on debt ≠ promised yield
Investment-grade debt (debt rated at BBB or better): useyield to maturity of the company‘s long-term, option-freebonds
If the bond rarely trades, use the average yield to maturityon a portfolio of long term bonds with the same creditrating
Below-investment-grade debt: use CAPM to estimate theexpected return
Adjust for interest tax shields Preferred stocks: preferred dividend devided by the
market price of preferred stocks
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Cost of equity: CAPM
whererf = risk-free rate
ß = the sensitivity of the stock return to market return E(rm) = expect return of the market portfolio E(rm)-rf = market risk premium
])([ fmfe rrErk −+= β
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CAPM: implementation (1)
Risk-free rate: use long-term goverment bond Market risk premium: historical data
• Use the longest period possible: short-term estimatesare very noisy (annual standard deviation of stockreturns 20%).
• Use geometric average instead of arithmetic average• Adjust for survivorship bias.• Nomally used numbers: 4.5-5.5%
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CAPM: implementation (2)
Beta: estimated from the market model
• Normally five-year monthly data are used• Adjustment for low trading frequency (Dimson(1979))
Market portfolio• In theory, all assets must be included• In practice, well-diversified stock indexes are used as a
proxy: S&P 500, MSCI world index, MSCI Europe index,etc
itmtit rr εβα ++=
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Beyond CAPM: APT model
where Fk = the k-th systematic factor that drives security return, = risk premium of the k-th factor
Difficulty in implememtion: not clear• What are the factors?• How to measure them?
∑
∑
=
=
+=
++=
n
kkikfi
n
kkiki
rrE
Fr
1
1
)( λβ
εβα
kλ
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Beyond CAPM: Fama-French model
where are exposures to the market portfolio, size portfolio andbook-to-market portfolio repectively, are returns on themarket portfolio, small stock portfolio, large stock portfolio, highbook-to-market stock portfolio, low book-to-market stock portfoliorespectively.
An empirical model designed to capture the sizeand book-to-market effects in stock return
Theoretical fundation still not clear
321 ,, βββ
)]()([)]()([])([)( 321 LHBSfmfi rErErErErrErrE −+−+−+= βββ
LHBSm rrrrr ,,,,
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Capital structure and cost ofcapital Modigliani and Miller theorem: In a perfect market without tax,
capital structure has no impact on either the firm value or the costof capital.
An easy way to understand this fundamental result in corporatefinance: In a perfect market without tax, capital structure has noimpact on the expected cash flows to the firm.
In the MM world, when the more expensive equity is substituted bythe less expensive debt, the cost of equity increases according,leaving the weighted cost of capital unchanged.
In a world with tax, debt increases the cash flows to the firm byreducing taxes. This is not reflected in FCFs, therefore it must bereflect in WACC.
In the real world (with both tax and market imperfections), anoptimal capital structure is determined by the trade-off between thetax advantage of debt and the cost of high leverage.
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Leverage and equity beta
Industry beta is often used to improve theestimation of company beta.
However, firms in the same industry may havedifferent leverage ratio
Procedure for inferring beta from comparablefirms• Estimate beta for each comparable firm• Back out the unlevered beta• Calculate the relevered beta using the target leverage
ratio
Valuation
Two alternative leverage policies
The relation between levered- and unlevered-betadepends on the assumed leverage policy.
MM assumption(Modigliani and Miller 1963): constantdebt value
ME assumption (Miles-Ezzell 1980): constant debt ratio Note that ME assumption is different from MM
assumption even if expected growth rate is zero. Failing to recognize this difference has led to confusion
even among experts (Fernandez 2004, Cooper andNyborg 2006)
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Unlevered beta: constant debt level(1)
If the debt value is constant, then interest tax shieldsshould be discounted by cost of debt, therefore
)1()(
/)1(1/)1(
/)1(11
)1()1(
)1(
)1(
tED
EDtEDt
EDt
DtEDt
DtEE
VDt
VE
VtD
VV
VD
VE
tDVVVDEV
DUUE
DE
DE
UD
UEU
LD
L
UU
LD
LEC
UTSUL
−−+=⇒
−+−+
−+=
−+−+
−+=
−+=⇒
+=+=⇒
+=+=+=
ββββ
ββ
ββ
βββ
βββββ
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Unlevered beta: constant debt level(2)
If , then we have• Hamada (1972) formula
• Relevered beta
0=Dβ
EDtEU /)1(11−+
= ββ
])1(1[EDtUE −+= ββ
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Unlevered (levered) cost ofequity
Unlevered cost of equity can be derivedeither using unlevered beta or directlyfrom the following formula
)1(
)1()(
/)1(1/)1(
/)1(11
DEtDkWACC
tEDkkkk
EDtEDtk
EDtkk
uL
duue
deu
+−==>
−−+=⇒
−+−+
−+=
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Unlevered beta: example
A privately-held company has a leverage ratio(D/E) of 60%.
A comparable publicly-traded company with aleverage ratio of 40% has an equity beta of 1.2.
The comparable firm is assumed to maintainthe current debt level (in value) in the future.
The corporate tax rate is 35%. What is the beta of equity for the private
company?
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Unlevered beta: constant debt ratio(1)
If the debt ratio is constant, then interest taxshields should be discounted usingunlevered cost of equity, therefore
WACCpretaxED
DkED
Ekk
EDD
EDE
VV
VV
VD
VE
VVDEV
DEu
DEU
UL
TSU
L
UU
LD
LEC
TSUL
=+
++
=⇒
++
+=⇒
=+=+=⇒
+=+=
βββ
ββββββ
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Unlevered beta: constant debt ratio(2)
Relevered beta
Levered cost of equity
WACC of the levered firm
)( duue ED ββββ −+=
)( duue kkEDkk −+=
DEDtkkWACC d
uL +−=
Valuation
Constant leverage ratio:example
Redo the previous exercise assumingthat the comparable firm is going tomaintain the current debt ratio in thefuture
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IV. Estimating residual value Method 1
Method 2
Sincethese two methods are equivalent.
Since the reinvestment rate in the residual period may be differentfrom that in the explicit forecast period, FCFT+1 may not equalFCFT*(1+g)
Method 1 automatically takes this into account.
gWACCRONICgNOPLATRV T
T −−= + )/1(1
gWACCFCFRV T
T −= +1
)/1(*)1(* 111 RONICgNOPLATIRNOPLATFCF TTT −=−= +++
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Residual value: example
FCFT = 100, NOPLATT = 200 From year T+1 on, g = 5%, RONIC=
12% WACC = 10%
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Residual value: other methods
If RONIC = WACC, then method 1 reduces to theconvergence formula
This method assumes that new investment in theresidual period does not creat any value
Liquidation value: only if liquidation is very likely Replacement cost: no good economic reason
WACCNOPLATRV T /1+=
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Other DCF valuation model
Capital cash flow valuation model Adjusted present value model Economic-profit-based valuation model Discount dividend model
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Capital cash flow valuationmodel
CCF = FCF + interest tax shield Firm value is derived by discounting the CCF using
unlevered cost of equity Given that interest tax shield are discounted using
unlevered cost of equity, it follows that unleveredcost of equity equals pretax WACC.
Advantage: no need to adjust discount rate forchanges in financial structure.
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CCF valuation model: example
Example: Current FCF of a firm is 200,interest expense is 50, tax rate is 40%.The firm has a pretax WACC of 10% andan expected growth rate of 5%. Valuethis firm using CCF valuation model.
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Adjusted present value model Valuation by components approach:
where VU is the expected FCF discounted using unleveredcost of equity, VTS is the present value of interest tax shields
Possible discount rates for interest tax shields:• Pretax cost of debt: varying or risky debt• Unlevered cost of equity: costant debt ratio (in this case, APV
model is equivalent to CCF valuation model) Advantage:
• No need to adjust discount rate for changes in capital structure• Can easily be combined with a variety of valuation models
TSUL VVV +=
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APV model: example
year1 year2 year3 year4
Unlevered cash flows 100 100 1000 1000
debt 0 0
2000 (at
8%)
2000 (at
8%)
unlevered cost of equity 0.14
tax rate 0.34
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Economic-profit-based valuationmodel
Economic profit (or Economic Value Added) = Invested capital * (ROIC-WACC) = NOPLAT – invested capital * WACC
Firm value and economic profit
Important message: a project creates value forshareholders iff its return is higher than its cost of capital
∑∞
= ++=
100 )1(
)(t
tt
WACCprofitEconomicEcapitalInvestedV
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Discount dividend model General model
where DPS is dividend per share, ke = cost ofequity
Gordon growth model: for stocks with a stablegrowth rate
where E(DPS1) is expected dividend nextperiod, g is growth rate in dividends forever
∑∞
= +=
10 )1(
)(t
te
t
kDPSEP
gkDPSEPe −
= )( 10
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Gordon growth model
Works best for companies• in stable growth• in stable leverage• pays out dividend regularly
Limitations• Extremely sensitive to the input for growth
rate• Extremely simple growth pattern
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Two-stage growth mdoel
Assumption: high growth rate (g) in the first nperiods and normal growth rate (gn) in therest periods
)()1()(
])11(1)[((
)1()1()(
11
10
nen
e
n
e
n
e
ne
nn
tt
e
t
gkkDPSE
gkkgDPSE
kP
kDPSEP
−++
−++−
=
++
+=
+
=∑
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How does it work?
In relative valuation, you try to figure out thevalue of the firms being analyzed by looking atthe market values of similar or comparablefirms.
Steps in relative valuation• Identify comparable firms• Calculate the „multiples“• Compare the multiples and control for factors that
might affect the multiples Implicit assumption: market is on average right
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Most popular multiples Earnings multiples
• Price/earnings ratio and variants• Value/EBITDA• Value/FCF
Book value multiples• Price/book value (PBV, or market-to-book equity)• Value/book value• Value/replacement cost (Tobin‘s Q)
Revenues multiples• Price/sales• Value/sales
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Price / Earnings ratio
PE = market price per share / Earnings pershare
Price can be• Current price (most of the time)• Average price for the year
Earnings per share (EPS) can be• EPS in most recent financial year• EPS in trailing 12 months (trailing PE)• Forecast EPS next year (forward PE)
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PE ratio across countries: July 2000
Developed markets
source: Damodaran 2002
Emerging markets
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Determinants of PE ratio
Other things equal, PE ratio is higher forfirms with• High growth potential• High payout ratio• Low cost of equity (low equity risk, low risk free rate)
gkgratioPayoutEPS
gkgDPS
EPSPPE
ee −+=
−+== )1()()1(
00
0
0
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PE ratio: regression analysis Advantage of regression analysis: the informatíon in the entire
cross-section instead of a few comparable firms can be used Problem: the coefficients may be unstable Example: regression results for Compustat sample
(Damodaran 2002)
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PEG ratio PEG = PE / Expected growth rate in earnings A simple way to control for the influence of growth
rate on PE ratio But not completely neutralize it since PE is not a
linear function of expected growth rate
No standard time frame for measuring expectedgrowth rate
)()1(
gkggratioPayoutPEG
e −+=
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Value multiples V / EBITDA = (E + D) / EBITDA V / FCF = (E + D) / FCF FCF = EBIT (1-t) – (CAP EX – D&A) - ∆ working capital
= (EBITDA – D&A)(1-t) - (CAP EX – D&A) - ∆workingcapital
= EBITDA(1-t) + t (D&A) – CAP EX - ∆working capital
Advantages• Less firms with negative EBITDA than firms with
negative earnings• not influenced by difference in depreciation schemes• Not influenced by differences in capital structure
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Determinants of V/FCF ratio
Stable growth case
Two stage growth case
gWACCg
gWACCFCFgFCF
FCFV
−+=
−+= 1
)()1(
0
0
0
0
)()1()1()1(
))1(
)1(1)(1(
0
0
nn
nnn
n
gWACCWACCgg
gWACCWACCgg
FCFV
−++++
−+
+−+=
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Value multiples: Example
Consider a firm with the followingcharacteristics• Tax rate = 33%• Capital Expenditure/EBITDA=30%• Depreciation&Amortization/EBITA=20%• Cost of capital=10%• No requirement for working capital• Stable growth rate=5%
Calculate V/EBITDA & V/FCF
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Price-to-book ratio
Price-to-book ratio (market-to-book ratio)=market value of equity / book value of equity
For a stable growth firm
Since g=(1-Payout ratio)*ROE, we can further derive
)(*
)(* 1
0
1
0
0
gkratioPayoutROE
gkBVratioPayoutEarnings
BVPPBV
ee −=
−==
gkgROEPBV
e −−=
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Value-to-book ratio
Definition
For stable growth firm
debtofvaluebookequityofvaluebookdebtofvaluemarketequityofvaluemarket
valueBookValue
++=
gWACCgROIC
gWACCBVROICgtEBIT
gWACCBVFCF
BVV
−−=
−−−=
−=
)()/1)(1(
)( 0
1
0
1
0
0
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Value-to-book ratio: example
Example: Consider a stable growth firmwith the following characteristics:ROIC=12%, WACC=10%, g=5%.Estimate its Value-to-book ratio.
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Tobin‘s Q ratio Definition
If Tobin‘s Q is smaller than 1, then a firm destroysvalue; if it is bigger than 1, then it creates value
Advantage: replacement costs provide a moreupdated measure of asset value than do bookvalues
Disadvantage: replacement costs are hard toestimate
placeinassetsoftplacementplaceinassetsofvalueMarketQsTobin
cosRe' =
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Revenue multiples Price-to-sales ratio
= market value of equity / total revenues• Internally inconsistent, since the market value of
equity is divided by the total revenues of the firm. => High leverage leads to low price-to-sales ratio Value-to-sales ratio
= market value of firm/ total revenues Advantages
• Available even for young or troubled firms• Not heavily influenced by acounting rules• Relatively stable
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Determinants of revenuemultiples
For a stable growth firm
gWACCIRinmoperatingtaxAfter
gWACCSalesIRtEBIT
SalesV
gkratioPayoutinmNet
gkSalesratioPayoutEarnings
SalesP
ee
−−=
−−−=
−=
−=
)1(*arg)()1(*)1(
*arg)(
*
10
10
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Choosing between multiples
There are many potentially useful multiples Which ones to use in valuation?
• Use a simply average of valuations obtained usingdifferent multiples
• Use a weighted average of valuations obtained usingdifferent multiples
• Rely entirely on one of the multiples• Most relevent one• Most accurately estimated one
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Choosing the comparison firms
Three possible choices• A few very similar firms• All firms in the same sector• All firms in the market
Regression analysis is necessary if youchoose the second or third approach
It is recommended to check whether the firm isover or under valued at both the sector andmarket level.
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Managerial flexibility (strategicoptions)
Managers react to changes in economicenvironment
DCF valuation and relative valuation do notexplicitly account for this.
Real options theory provides an usefulframework to quantify the value of flexibility.
This approach is particularly relevant for thevaluation of individual businesses and projects.
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Strategic options: examples
Option to postpone a project Option to abondon a project Option to temporarily shut down a
project Option to expand a project Option to downsize a project Option to change input or output factors
.....Valuation
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Certainty equivalent method Option pricing is based on the Certainty Equivalent Method
as opposed to the Risk-Adjusted Discount Rate Method. Instead of discounting the expected cash flowes using a
risk-adjusted discount rate, the certainty equivalentmethod discounts the certainty equivalent of futureuncertain cash flows at the risk-free rate.
The certainty equivalent of some uncertain payoff is definedas a sure amount of payoff that is considered to be asvaluable as the uncertain payoff.
This alternative method can be very useful even in theabsence of strategic options.
∑∞
= +=
1 )1()(
tt
f
t
rCFCEQV
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Obtaining certainty equivalents
How can we obtaint certaintyequivalents?• By looking at prices in the forward or futues
market (when forward or futures marketexists)
• Expected value minus dollar value of riskpremium (when risk premium and riskexposure are known)
• Expected value under the risk-neutralprobabilty (when markets are complete)
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Example: two-period gold mine
period1 2
output1000 1000
priceS1 S2
revenue1000S1 1000S2
Costs300 300
NCF 1000S-300 1000S2-300
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Example: two-period gold mine(2)
Suppose that risk free rate is 10%, thecurrent forwards prices are 320 for aone-year contract and 350 for a two-yearcontract. What is the value of this mine?
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I. Binomial model
S0
Su
Sd
p
1-p
C0
Cu= max(Su-X,0)
Cd= max(Sd-X,0)
X = exercise price, C = call option value, S = underlying value,
p = probability that underlying value goes up (irrelevant for valuation!)
p
1-p
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Binomial model (2)
The risk-neutral probability q is given by
The call option value is thus given by
du
df
duf
SSSrS
q
SqqSrS
−−+
=⇒
−+=+
)1()1()1(
0
0
f
du
rCqqCC
+−+=
1)1(
0
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II. Black-Scholes formula For European call and put, Black and Scholes (1973) derive the
following formula
S = underlying price, K = Exercise price, _ = annualized volatility of theunderlying, T = time to maturity, rf = continuously-compounded risk-free rate,N(.) = cumulative standard normal distribution
TddT
TrXS
d
dSNdNXeP
dNXedSNC
f
Tr
Tr
f
f
σσ
σ
−=
++=
−−−=
−=−
−
12
2
1
12
21
)21(ln
)()(
)()(
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III: Monte Carlo Simulation
Monte Carlo simulation can be used to valuemore complex options.• Step 1: simulate the distribution of underlying value
under the risk neutral probablity by generating alarge number of underlying price paths following
where is a normally distributed random variable.
Zr
titiitfeSRSS
~21
1,1,
2~ σσ +−
−− ==
Z~
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Monte Carlo Simulation (2)• Step 2: calculate the net cash flow in each period on
each sample path
where Xt is the production cost in period t.• Step 3: calculate the option value for each sample
path i
• Step 4: calculate the average option value over allsample paths
∑=
−=T
iit
tri NCFeV f
1
∑=
=N
tiVN
V1
1
)0,max( titit XSNCF −=
Valuation
96
Example: option to shut down
Value a gold mine with the followingcharacteristics• Produces gold in two periods• Temporary shut-down possible• Current gold price 300• Annual gold price volatilty 20%• Annually compounded risk free rate 10%• Annual production 1000• Annual Production cost 300
Valuation
98
Solution: Binomial model (2)
f
du
f
dduddd
f
uduuuu
f
rVqqVV
rVqqVSV
rVqqVSV
dudr
q
+−+=
+−++−=
+−++−=
−−+
=
1)1(
1)1()0,300max(
1)1()0,300max(
1
0
Valuation
99
Solution: Black-Scholes
tddt
trd
dNdNdNdN
tt
ft
σσ
σ
−=
++=
+
=
=+=
12
2
1
222*0.0953-
21
120.0953-
11
f
)5.0()300/300ln()](300e-)( 300[1000
)](300e-)( 1000[300 V0.095310%)ln(1 r
Valuation
100
Solution: Monte Carlo simulation
Function RAND() generates a random realization of a randomvariable uniformly distributed over the interval [0,1].
NORMSINV(U) generates a random realization of a randomvariable following a standard normal distribution.
()~)~(~
~
~
~21
1,
2
RANDUUNORMSINVZ
eR
RSS
Zr
tiit
f
=
=
=
=
+−
−
σσ
Valuation
101
Monte Carlo simulation: a samplepath
period t=1 t=2U 0.2679 0.7208Z -0.6193 0.5853R 0.9526 1.2121S 285.78 346.40Production cost 300 300NCF 0 46.40PV 0 38350.41V 38350.41
Valuation
102
Option to delay: example Panel A: Invest now
Panel B: wait one year and invest only in good state
-100
10 15 15 per year for ever
good
10 2.5 2.5 per year for ever
bad
0
-100 15 15 per year for ever
good
0 0 0 per year for ever
bad
Valuation
103
Option to delay (2)
Risk free rate = 5% per year. $1 invested in the market portfolio will be
worth either $1.3 (when the state is good)or $0.8 (when the state is bad) in oneyear.
Should we invest now or should we waituntil next year?
What is the value of the option to wait?
Valuation
105
Option to expand: example A project can generate the following CFs:
The firm has the option to double its capacity by investing another 140in year 1 if the economy looks good.
-140
good
bad
200
150
100
Valuation
106
Option to expand (2)
Risk free rate 5%. Risk neutral probabilities: q=0.6 in both
periods. What is the value of the project without
considering the option value? What is the value of the project after
considering the option value? What is the value of the option to expand?
Valuation