Cost of Capital Slides

Estimating Cost of Capital


Anders Vilhelmsson

Department of Business Administration, Lund University

September 2009

Cost of Capital


Aim of the 2 lectures

I Cover chapter 10 in the book

I Cover relevant research, particularly from 2004 (when thebook was updated) until 2009

I Slides can be found after the lecture at www.nek.lu.se/nekavi

Cost of Capital


Valuation

I Easy in theory, the total value of a company is the presentvalue of all future cash �ows

I V = CF11+k +

CF2(1+k )2

+ CF3(1+k )3

+ CF4(1+k )4

+ CF5(1+k )5

...

I However, k is unknown and may not be constant over time

Cost of Capital


WACC

I WACC = DV kd (1� Tm) +

EV ke

D = Value of debtV = Enterprise valuekd = Current borrowing rate (tax deductible)Tm = Corporate (marginal) tax rate (e.g. 26.3% in Sweden)E = Value of equityke = Cost of equity

I Example 1 on the board

Cost of Capital


WACC

I Why do we need 2 full lectures to do the above calculations?I kd and especially ke are unobservableI We need theory (models) to estimate kd and keI In practice it may also be non-trivial to calculate (target) DVand E

V

I We will put most e¤ort in estimating ke correctly since theuncertainty is largest in this number.

Cost of Capital


Cost of debt

I Primary problem, non-�at term structure of interest ratesI In principle 1 year CF should be matched with 1 year debtrate, 2 year CF with 2 year rate and so on

I In practice match with the duration on the company�s CFsI Growth stocks, high duration, value stocks low durationI The book recommends about 10 years for all companies

Cost of Capital


Duration

I Macaulay�s Duration:

D =n∑t=1

�t � PV (CFt )V

�=

n∑t=1

�t � CFt/(1+r )

t

V

�I V = Enterprise Value

I Do loan example on the boardI What happens with the sum is in�nite (e.g. CF from a stock)?

I D =n∑t=1

�t � PV (CFt )V

�+ (n+Dcv )

PV (CVn)V

I Dcv = 1r�g Duration of continuing value, derive on the board.

I Do stock example on the board

Cost of Capital

Duration

Figure: Source: Own calculations

Term structure of interest rates


Estimate cost of debt

I Use Yield to maturity (YTM) on long term bondI YTM > kd but small di¤erence for BBB companies and betterI P = C

1+ytm +C

(1+ytm)2+ C(1+ytm)3

... C+P(1+ytm)n

I Solve for YTM but this is an n:th order equation (numericalsolution)

I Calculate YTM in Excel example

Cost of Capital

Cost of debt vs YTM

00.05 0.10

0.150.20

0.250.30

0.350.40 0.45

0.50

0 .01

0.25

0.13

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Recovery rate

YTM as a function of Recovery rate and defualt prob. cost of debt is 6%

Default probability

YTM

Figure: Source: Own calculations


Estimate cost of debt

I What to do with companies that only have untraded or shortdebt?

I Find credit ratingI Compare to traded long bonds with the same credit rating

I What to do with companies with <BBB rating?I Use BBB cost of debt and add 0.5% units (motivated by 0.1higher CAPM beta)

Cost of Capital


Estimate cost of equity

I Estimating the cost of equity is the same thing as explainingthe cross section of stock returns

I Why do companies have di¤erent expected returns?I Theory: Because of di¤erent exposure to systematic riskfactor(s)

I CAPM, FF3, momentum, liquidity, risk aversion (APT)

Cost of Capital


The CAPM

I Security market line (SML) E [Ri ] = rf + βi (E [Rm ]� rf )E [Ri ] = Expected return on asset irf = risk free rateE [Rm ] = Expected return on the market portfolioβi =

cov (Ri ,Rm )σ2m

systematic risk in asset i

I Problem: E [Ri ],E [Rm ] and βi are all unobservable and rfvaries with maturity

Cost of Capital


Estimating the risk free rate

I Same thing as with the cost of debt (Match each cash �ow)I Make sure cash �ows and cost of capital uses the samecurrency

Cost of Capital


Estimating the market risk premium

I Interesting working paper athttp://papers.ssrn.com/sol3/papers.cfm?abstract_id=1473225

I Compares the recommended market risk premium from 150di¤erent textbooks

I CAPM actually gives the market risk premium asE (Rm)� rf = γσ2m assuming CRRA utility

I σ2m can, at least historically, be observed but not γ (therelative risk aversion)

Cost of Capital


Figure: Source: Fernández (2009,WP)



Figure: Source: Fernández (2009,WP)

Cost of Capital


Estimating Beta

I Lets look at the recommendations given by the bookI Use at least 60 data pointsI Use monthly dataI Use SP500 or MSCI world index as market portfolio

Cost of Capital


At least 60 data points

I Number of data points is a trade o¤ betweenI Precision and possible time variationI If you think that beta is constant over time use all data youhave

I 60 data points in not a magic number, happens to be 5 yearsof monthly data

Cost of Capital


Use monthly data

I Use monthly frequency - good idea if stock is very illiquid(traded infrequently)

I For e.g. the 30 stocks in Dow Jones you can use daily of evenintra-daily (15-30 minute data)

I Andersen et al. (2006) (Dow Jones 30 between 15 minutesand 1 day),

I Lewellen and Nagel (2006, JFE) daily and weekly on all NYSEstocks

I You can also adjust (Dimson 1979, JFE) for infrequent tradingI Bid ask Bounce can be �xed by calculating returns usingmidquotes instead of transaction prices(bid price + ask price)/2

I Currently there is a shift towards use of higher frequency inbeta estimation

Cost of Capital


Use a broad market portfolio

I Use a broad value weighted stock index to calculate betas,otherwise their is no theoretical foundation.

I Never use a local market index, in e.g. Finland you wouldbasically measure a stock�s sensitivity towards Nokia

Cost of Capital


Industry betas

I Idea: Improve precision in beta by using the mean beta of theindustry (adjusted for leverage)

I Assumes that companies in the same industry has the samesystematic operational risk

I Di¤erent betas within an industry is only due to di¤erentleverage

I Master thesis topic: How well does this assumption holdempirically?

Cost of Capital


How to calculate an industry beta

I First compute betas for all companies in the industry withregression analysis

I Unleverage beta withVu

Vu+Vtxaβu +

VtxaVu+Vtxa

βtxa =D

D+E βd +E

D+E βe) βe = βu +

DE (βu � βd ) +

VtxaE (βtxa � βu)

assume βd = 0 and βu = βtxa) βe = βu(1+

DE )

I Do calculations on the board

Cost of Capital


How to calculate an industry beta

I Average βu over all companiesI Relever to each companies target D

E ratioI Example on the board

Cost of Capital


Other factor models

I Eugene Fama and Kenneth French 3 factor model FF3, Famaand French (1992, JoF)

E [Ri ] = rf + βi ,m (E [Rm ]� rf ) + βi ,smbE [SMB ] + βi ,HMLE [HML]SMB is the return on a small stock portfolio minus a big stockportfolio (small minus big)HML is the return on a high book to market minus a low book tomarket portfolio (high minus low)

I Is SMB and HML capturing risk exposure or misspricing?I Still open research question, enough papers to be the topic fora separate course

I Momentum, Jegadeesh and Titman (1993,RFS) and Liquidity,Amihud (2002) are other prominent factors

Cost of Capital

Sample period is from March 1990 to April 2004

Panel A: 25 portfolios sorted on Book-to-market and size

λMKT λSMB λHML λMOM λMIMRA R2

CAPM -0.626 0.27

[-0.87]

CAPM+MIMRA -0.935 -0.020 0.64

[-1.51] [-2.07]

FF3 -1.602 0.141 0.353 0.60

[-2.92] [0.46] [1.24]

FF3+MIMRA -0.988 0.174 0.289 -0.023 0.66

[-1.41] [0.57] [1.03] [-2.61]

FF3+MOM+MIMRA -0.666 0.178 0.325 1.895 -0.025 0.68

[-0.83] [0.59] [1.16] [2.11] [-2.80]

Source: Nyberg and Wilhelmsson (forthcoming, The �nancialreview)

Sample period is from March 1990 to April 2004

Panel B: 25 portfolios sorted on Book-to-market and size and 30 industry portfolios

λMKT λSMB λHML λMOM λMIMRA R2

CAPM -0.031 0.00

[-0.06]

CAPM+MIMRA -0.202 -0.012 0.15

[-0.41] [-1.23]

FF3 -0.144 0.110 0.033 0.03

[-0.29] [0.36] [0.11]

FF3+MIMRA 0.019 0.154 -0.008 -0.027 0.31

[0.04] [0.50] [-0.03] [-3.12]

FF3+MOM+MIMRA 0.176 0.141 0.013 1.211 -0.025 0.33

[0.31] [0.46] [0.04] [1.34] [-2.93]

Source: Nyberg and Wilhelmsson (forthcoming, The �nancialreview)


In defence of beta

I Builds on solid theoryI Assumes multivariate normal distribution or investors withpreferences for only mean and variance (both assumptions arewrong)

I FF3 purely empirical evidence, no theory, size premiumvanishing

I rejecting FF3 does not really support CAPM we have morethan 2 competitors (evolution/creationism)

I CAPM may hold conditionally (beta should be forwardlooking)

I E¤ect is too small to save CAPM according to Lewellen andNagel (2006, JFE)

Cost of Capital


Importance of model selection

I How important is the selection of factor model? Lets try to�nd out!

I Calculate cost of equity for J&J using CAPM and FF3 inExcel.

I Is the J&J results typical or not? Possible master thesis topic.

Cost of Capital


Is the cost of equity time varying / are stock returnspredictable ?

I E [ri ]� rf = βi [E (Rm)� rf ]I E (Rm)� rf = γσ2m

I Three possible sources of time variationI Time varying betasI Time varying risk aversionI Time varying volatility

Cost of Capital


Time varying betas

Figure: Source: Andersen et al. (2004) AA is Alcoa, ALD is Alliedcapital corporation, DD is DuPont, and DIS is WaltDisney.The samplecovers theperiod from 1962:3 through 1999:3.

Cost of Capital


Time varying betas

Figure: Source: Andersen et al. (2004)

Cost of Capital


Time varying betas

Figure: Source: Andersen et al. (2004) The sample covers the periodfrom 1993:2 through 1999:3. We calculate the realized quarterly betasfrom daily returns.

Cost of Capital


Time varying betas

Figure: Source: Andersen et al. (2004) The sample covers the periodfrom 1993:2 through 1999:3. We calculate the realized quarterly betasfrom 15 minute returns.

I Conclusion - No consensus but tentative yesCost of Capital


How important is the increased precision from 15 minutereturns?

I Daily sampling gives uncertainty (95% CI) of about 1, 15minute of about 0.2

I Simple illustration of e¤ect on valuation, sayE (Rm)� rf = 3%, rf = 2% Company with constant growthin dividends of 2%, last dividend 1$. Point estimate of beta1.5.

I Daily sampling gives beta between 1.0 and 2.0, 15 minutesampling gives beta between 1.4 and 1.6,

I How much will this e¤ect the equity value of a company witha constant growth of dividends of 2%, last dividend 1$.value = 1/(k � g)

Cost of Capital


How important is the increased precision from 15 minutereturns?

I Daily beta Capital cost between 5% and 8%. Value between1/(k � g) = 1/(0.050� 0.02) : 33. 33$ and1/(0.080� 0.02) = 16. 67$

I 15 minute beta Capital cost between 6.2% and 6.8%. Valuebetween 1/(k � g) = 1/(0.062� 0.02) = 23. 81$ and1/(0.068� 0.02) = 20. 83$

Cost of Capital


Time varying risk aversion

Figure: Source: Bollerslev et al. (2009, JEc)

Cost of Capital


Time varying risk aversion

Figure: Source: Bollerslev et al. (2009, JEc)

I Conclusion - yes (not everyone agrees)

Cost of Capital


Time varying variance

Figure: Variance from 1960-2000

I Conclusion - Yes clear consensus

Cost of Capital


Stock return predictability

I Emerging consensus that stock returns are predictable (achange since the book was written)

I Taken as evidence of time varying risk premium, not asevidence against EMH

I Remember EMH says risk adjusted returns areunpredictable, not regular returns

I Conclusion: The cost of equity is time-varying but it isextremely hard to estimate over short periods of time so wemay be better of using a constant cost of equity

Cost of Capital


Hybrid �nancing

I Mix of debt and equity such as convertible bonds, options,CDS instruments etc.

I Can be broken down to basic parts using replicating portfoliosI E.g. Convertible bond = Corporate bond + Call option, Calloption = Risk free bond + company stock

I More on this on the real option lectures

Cost of Capital

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