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Lecture 2. Valuation and the Cost of Capital. Valuation.

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Lecture 2

Lecture 2

Valuation and the

Cost of Capital


  • Last time we argued that information about state-contingent cash flows and how to price them was crucial for valuation, and therefore, both for management and market opinions about what the firm should do and how to value what the firm is doing.

  • Now we’ll be more precise about how information is used in valuation.

Weighted average cost of capital wacc model of valuation
Weighted Average Cost of Capital (WACC) Model of Valuation

  • Framework defines “free cash flows” as focal point of valuation

  • Discounting forecasted free cash flows requires one to take account of asset risk (unlevered beta) as well as effect of debt

  • This framework allows us to measure the firm’s cost of capital, and to evaluate projects.

  • Also allows estimation of divisional cost of capital.


  • Equity holders require rate of return RL

  • They contribute Equity/V = 1 – L proportion of the firm (or division or project)

  • Debt holders require rate of return RD

  • They contribute D/V = L proportion of the value of the firm (or division or project)


  • WACC is used by the market to discount expected Free Cash Flows in order to arrive at value of the firm (or division or project). Because it measures the cost of capital it is also used to evaluate prospective projects.


  • Debt is tax-favored relative to equity, so that in order to provide holders of debt a return of RD the firm must earn RD(1-t)

  • Formula for WACC:

    WACC = RD(1-t)L + (1-L)RL

Deriving wacc
Deriving WACC

By definition:

Vt = {E[FCFt+1] + E[Vt+1]}/(1+WACC)

By using iteration, we can express value in terms of WACC and expected free cash flows.

Alternatively, we can write:

1+ WACC = {E[FCFt+1] + E[Vt+1]} / Vt

Defining free cash flows
Defining Free Cash Flows

FCF = Gross after-tax cash flow – Gross investment

Gross after-tax cash flow = Net operating profit less adjusted taxes attributable to EBITA (which is approximately equal to EBITA(1-t)) + Dep.

Gross after-tax cash flow = EBITA(1-t) + Dep.

Defining fcf cont d
Defining FCF (Cont’d)

NOPLAT = Net operating profit less adjusted taxes

NOPLAT = Gross after-tax cash flow less Depreciation


Gross investment = NINV + Depreciat.


where NINV is net investment


  • Note that EBITA(1-t) does not take interest on debt tax shield into account. That is taken into account later.

  • EBITA(1-t) is an approximation, since tax consequences of income is a complicated concept in practice. In particular, NOPLAT does not include non-operating income or subtract taxes on non-operating income.

Thinking about free cash flow
Thinking about Free Cash Flow

  • Assume no non-operating assets and no goodwill amortization

    Where does FCF go?

    FCF = Net income – NINV – principal payment + after-tax interest + principal payment

    FCF = {NI – NINV – PP} + {ATI + PP}

    The first part is the part of free cash flow the firm keeps or pays as dividends

    The second part is the part of free cash flow that goes to debtholders

Taking new financing into account
Taking New Financing into Account

FCF = {NI – NINV – PP} + {ATI + PP}

V = Value of old equity + Value of new equity + Value of old debt + Value of new debt

FCF + V = {NI – NINV – PP + New debt + New Equity Issues +

Value of old equity} +

{ATI + PP + value of old debt after payment of PP}


This is the same as saying that FCF is the only leakage of value from the firm (other than taxes) and that is received by those that have equity or debt claims on the firm.


RD = Expected market return on debt

RD ={E[interest]+ E[PP]+ E[capital gain]}/D

RL = Expected market return on equity

RL = {E[Dividend] + E[capital gain]}/(V-D)

Recall that

1 + WACC = {E[FCFt+1] + E[Vt+1]} / Vt

Almost there
Almost there…

Assume no expected new debt or equity

Retained earnings add to equity value

E [FCF + V] = {E[Dividend] +

E[Future value of current equity]} +

{E[(1-t)Interest] + E[PP] +

E[Value of old debt]}

E[FCF+V] = (1+RL)(V-D) + RD(1-t)D+D


E [FCFt+1+Vt+1]/Vt = RLt [(V-D)/V]t +

[(V-D)/V]t + RDt(1-t)[D/V]t + [D/V]t

Thus, if you are in a steady state,

WACC = RD(1-t)L + (1-L)RL


  • This says that market returns and market values today reflect expectations of FCFs in the future (note that left hand expression can be solved iteratively based on future expected cash flows)

  • Taxes are total effective marginal corporate rate (federal, state, local)

  • Steady state assumes constancy of expected returns on equity and debt and constancy of leverage. Is this reasonable?

Interpretation cont d
Interpretation (Cont’d)

  • For constant leverage assumption to be literally true firm would have to adjust debt and/or equity continuously to keep the market values of the two in fixed proportion. That is not realistic.

  • But, firm could be implicitly targeting both leverage and the return on debt if, for example, the firm were trying to maintain a given debt rating (BBB) and if firm had stable FCF risk.

Interpretation cont d1
Interpretation (Cont’d)

  • Note that RD is not promised return but expected return on debt (if debt is risky, promised return is higher)

  • RD is increasing function of leverage and FCF systematic risk. If leverage and FCF systematic risk are stable, then so is expected return on debt.

  • These are not good assumptions for young, growing firms or for LBOs, but not too bad otherwise.

Estimating r d
Estimating RD

  • If debt is not priced, look at comparable firms’ spreads or ratings on public debts (where comparable means comparable FCF risks and leverage); if highly rated don’t worry about ex ante vs. ex post

  • Similarly, could use scoring models (Zeta score) to estimate default risk premium, or KMV (Black-Scholes based) approach

  • Once you have yield, mark down one category to get expected return

Could you construct a beta of debt
Could you construct a beta of debt?

  • In theory, yes; in practice, no. Debt markets are not always trading enough, and price variation is small relative to transaction costs.

Estimating r l
Estimating RL

  • CAPM is still the best show in town, but the current state of the art is to use multiple factors (APM), allow coefficients to vary over time, and to “shrink” estimates to take account of biases that result from small sample size in estimating expected returns.

  • Recall that CAPM is not very useful in some cases (EM investments, where alternative approaches are better)

Arbitrage pricing model
Arbitrage Pricing Model

  • CAPM does not take account of missing markets for many stocks, or of human capital.

  • Additional factors (for which separate betas are estimated) include industrial production, short-term interest rate, yield curve, Baa-Aaa spread, which make a big difference to estimates of composite beta.

Apm details
APM Details

RL = Rf + [E(F1 – Rf )] beta1 +

[E(F2 – Rf )] beta2 +

[E(F3 – Rf )] beta3 + …

[E(Fk – Rf )] betak

where E(F) measures expected return of each factor and betas measure sensitivity of stock return of company to each factor.

Why use capm or apm at all
Why use CAPM or APM at all?

  • Does better at forecasting returns than simple average of historic returns.

  • Part of the reason for that is that average returns of all portfolios are measured with much more error than the covariance matrix of returns


  • Using a utility function and riskless rate estimate, construct two versions of efficient portfolios, one based on historical averages of mean returns, the other estimating expected return using APM.

  • Look at ex post performance of the two portfolios (M1, V1) vs. (M2, V2) evaluated using same utility function. (M2, V2) dominates

Practical tips for estimating r l
Practical Tips for Estimating RL

  • For U.S. publicly traded firms, estimates often differ (different methods, periods, market benchmarks).

  • Look for central tendencies

  • Do within-industry comparisons (since systematic operating risk should be similar for similar firms)

  • Use “shrinkage” to deal with sampling bias [(0.6 x estimate) + (0.4 x 1)]

Within industry comparisons
Within-Industry Comparisons

  • For non-public firms, or when checking robustness of estimates for publicly traded firms, it can be very useful to do within-industry comparisons

  • But these comparisons must take into account differences in leverage. How does one control for leverage? How does one “lever” or “unlever” an estimated beta?

Emerging markets
Emerging Markets

  • Assumptions necessary for CAPM are not even close to true (distributional problems, institutional risks)

  • Best approach is to use sovereign spreads plus national stock volatility markup as starting point for equity premium, and to compute individual stock spreads over the sovereign spread based on a firm-level adjustment factor (relative volatility of that stock compared to others in the country is the one typically used).

  • There is little theoretical or empirical basis for any of this. Welcome to sausage making!

Levering or unlevering betas
Levering or Unlevering Betas

  • Two approaches are used, and they differ in their assumptions about how the capital structure of the firm evolves over time.

  • One approach assumes constant D; the other approach assumes constant L. Which one you use depends on which assumption is more realistic in the particular case (one can also use a combination of the two).

Constant d approach
Constant D Approach

  • The comparable levered and unlevered firms are related in the following way, under the assumption of constant D, and constant expected return on debt:

    VL = VU + tD

    where tax shield per period is worth tRDD, and is discounted as a perpetuity at the rate of RD

    Note Graham’s evidence of underuse of tax shield (firms could increase value of stock by 15.7% on average by raising debt to the “kink” he defines with virtually no risk of causing distress)

Constant d approach cont d
Constant D Approach (Cont’d)

Using WACC formula, one can show that, measured in terms of returns,

RL – RD = [1 +(1-t)D/(V-D)](RU – RD)

Measured in terms of beta

bL – bD = [1 +(1-t)D/(V-D)](bU – bD)

If the riskiness of debt is small, this becomes

bL = [1 +(1-t)D/(V-D)]bU

Constant l approach
Constant L Approach

  • This is not mathematically consistent, since WACC assumes constant L, not constant D. Constant L is better assumption anyway.

  • When you take this into account, the correct formulae become:

    RL – RD = [1 + D/(V-D)](RU – RD)

    bL – bD = [1 + D/(V-D)](bU – bD)

    bL = [1 + D/(V-D)]bU

    (See proof below)

Risk free rate measure
Risk Free Rate Measure

  • No ideal measure

  • Long-term bond without effect of inflation risk premium but with expected inflation included would be ideal, but this does not exist

  • A rough proxy is 10-year Treasury less 1%

Equity risk premium
Equity Risk Premium

  • This contributes more uncertainty to cost of capital than does estimate of beta!

  • Some economists see it as 2%, others use a number as high as 8%

  • Even after 150 years of stock prices, we really don’t know what the mean is, and to make matters worse, we have a sense that it should change over time (theory)

Equity premium estimates
Equity Premium Estimates

  • Short-sample problems with computing U.S. equity premium from average experienced returns (Fama and French 2002, Dimson et al. 2003). This is even more true for emerging market countries, and there are related concerns:

    • Distributions of returns not normal

    • Survivorship biases (“submerging” firms, markets)

  • Equity premium changes are unpredictable: At any point in time, your best guess of the future is the long-term retrospective mean (Goval and Welch 2004)

Equity premium estimates cont d
Equity Premium Estimates (Cont’d)

Dimson et al 2003, “Forward-Looking” Arith Mean, 1900-2002

AUS 7.6

ITA 5.6

CAN 5.5

DEN 2.7

FRA 5.6

GER 5.1

JAP 8.4

NETH 5.9

SAFR 6.8

UK 5.1

US 6.4

Equity premium estimates cont d1
Equity Premium Estimates (Cont’d)

  • Dealing with the conundrum of time:

    • Clearly, equity premium has fallen over time, and in theory should have (tech. change, reduced risk, rising P/E ratios), but when you carve up time you also get less accuracy

  • Sensible forward-looking procedure

    • Begin with bond yield risk premium (forward looking, and senior to equity in firms that issue them)

    • Use historical averages of equity premium, adjusted for current level of bond risk premiuim relative to its historical average

Divisional wacc
Divisional WACC

  • Find comparable stand-alone firms, in terms of FCF risk and compute their unlevered costs of equity, average them.

  • Determine appropriate leverage and effective marginal tax rate, and compute WACC from above formula (assuming constant L)

Does gaap matter
Does GAAP Matter?

  • Earnings, per se, should not matter, since sophisticated investors, who should set prices, are forecasting cash flows.

  • But earnings is the signal that is released (and is correlated with FCF), so earnings news should, and does, matter (and firms care a lot about “managing earnings” because of its signalling.” But evidence suggests earnings matter to firms’ managers, per se (Graham et al. 2004).

  • GAAP rules, per se, should not matter; but there is lots of evidence that firms care a great deal about GAAP rules that can be unwound by anyone using firms’ accounts.

    • FAS 123 (Calomiris 2005)

    • FAS 133 (Pollock 2005)

Graham et al 2004
Graham et al. 2004

  • Survey of 401 Financial Executives: What drives decisions about earnings reporting and disclosure?

  • Earnings viewed as key for outsiders (even more important than cash flows), and managers are willing to sacrifice value to meet short-term earnings target (surprisingly, there is less concern about targets related to debt covenants, employee bonuses).

    • 55% of managers would avoid a very positive NPV project to meet this quarter’s consensus earnings forecast.

    • Smooth earnings are seen as crucial for market

    • Voluntary disclosure to reduce perceived risk is sometimes done, but also avoided to avoid setting precedents for future disclosure.

  • Not accounting fraud, but neither is this strategy long-term value maximizing.

Pollock 2005
Pollock 2005

  • FAS 133 creates differences in treatment depending on whether one qualifies for “hedge accounting” and this creates up front differences in earnings and the risk of accounting restatements related to future behavior, or retrospective judgments about failure to qualify

  • It shouldn’t matter at all, since people can do the math either way, but perception is that market is confused by either set of rules.

  • Restatements based on hedge accounting status changes have negative effects

Calomiris 2005
Calomiris 2005

  • ESO expensing should not matter much, since it can be reported (and added back) as a separate line (except insofar as earnings are triggers for contracts).

  • CEOs of Silicon Valley are up in arms, and see this as having big effects on their ability to maintain stock price and have access to funding.

  • Substantively, ESO “expense” doesn’t look like an expense (high ESO “expense” => higher P/E, controlling for other factors), and logically it is not an expense or an opportunity cost of the firm, although it should be viewed as a gross cost in managerial accounting.

Fcf vs eps conclusion
FCF vs. EPS Conclusion

  • We know how to value firms assuming that only FCF is what matters in numerator, but in practice, valuation seems not only to depend on FCF, or at least market participants that control firm decisions believe that it does not only depend on FCF.

Proof of constant l formulae
Proof of Constant L Formulae

Recall that FCF = Dividend + Retained earnings + PP + (1-t)DRD

Retained earnings are eventually paid out as dividends, so we can rewrite this as:

FCF + tDRD = Today’s dividend + future dividends resulting from retained earnings today + RDD

The value of the firm is the present value of these cash flows for all periods, which is a set of dividend and interest payments

Retained earnings are reinvested

We can divide the above into an FCF component paid as dividends and discounted at RU a tDRD component, discounted at RD (use of tax shield is roughly as risky as repayment of debt)

Proof cont d
Proof (Cont’d)

VLt = E[FCFt+1 ] / (1+RU) + tRDDt /(1+ RD) +

present value of expected VLt+1

Note that Dt = LVt , so

VLt = E[FCFt+1 ] / (1+RU)(1- tLRD)/(1+ RD) +

present value of expected VLt+1 divided by

(1- tLRD)/(1+ RD)

Successive substitution gives the result that present value is all future expected FCFs discounted by (1+RU)(1- tLRD)/(1+ RD)

We also know that present value is the value of expected future free cash flows discounted by WACC. Therefore,

1+WACC = (1+RU)(1- tLRD)/(1+ RD)

WACC = RU – tLRD(1+RU)/(1+RD)

Recall that WACC = RD(1-t)L + (1-L)RL

Thus, RU – tLRD(1+RU)/(1+RD) = RD(1-t)L + (1-L)RL

(1+RU)/(1+RD) is approx one, so this reduces to:

RU =LRD+ (1-L)RL