beta

1. Total Beta or Beta?
Peter J. Butler, CFA, ASA
While it does not necessarily have to be one or the other, this was the primary question facing
panelists and the audience at the ASA Advanced BV Conference in Boston on October 20, 2009.
In the interest of full disclosure, I am a strong proponent of total beta as the Butler Pinkerton
Calculator (BPC) relies upon it for its underlying theory and calculations (Note: I will add my
own commentary in italics when commentin uo o e pnlt pi s
g pn t r aes ’ o t.
h is n )
With that out of the way, the following is a synopsis of the very unique debate (in order of
panelist appearance) held in Boston in front of approximately 350 appraisers.
First as a reminder, total beta is equivalently: B tr r s/σ where r is the correlation coefficient
e /o σ m
a
between a stock and the market and σrepresents the standard deviation of either the stock or the
market.
D . hiT fls“ oaB t: smaig e S mme i l ”
rC r oa i T tl e E t t B t y
s l: a i n a tc l
r ay
After taking the audience through some statistical concepts, Professor Tofallis concluded that
od a bt cnonscr li ( wt t r av vlit (s/σ ) in the following
ri r e ofud or ao r i h e t e o ti σ m
ny a e t n ) h e li aly
equation:
B t=rs/σ
ea σ m
In other words, if r is close to zero, a very volatile stock (with a high total beta and high total
risk) will have a beta (and systematic risk) close to zero.
Poesr oai t npi e ot e ’ i t it adce r er t tni t t t
rf o T f l h o t u bt s n a ly n id e a h h i c e h a
s ls e nd a s bi t s c a d ad a
s c’ cr li ( wt t m re i(ee l )h m sus b s tt wt nbeta. He
t ks or ao r i h a ts gnr l t ot nt l tii i i
o e tn ) h e k ay e a e asc h
then mentioned that since total beta does not contain the correlation (r), it should be more stable
over time. (This is one big advantage of total beta over beta. The BPC calculates both total
beta and beta. In an attempt to deal with the volatility of beta, I recommend running the BPC
(or doing the calculations yourself) for all five-trading days of the week for the most appropriate
look-back period This will allow you to (generally) see the volatility in beta and the relative
stability in total beta).
Professor Tofallis concluded his presentation with the comment that conventional beta is not a
robust statistic – conclusion which all panelists agreed with. He also indicated that he wanted
a
to perform additional research on the relative stability between beta and total beta. (I
subsequently invited Professor Tofallis to perform this research using the BPC, if he so desires).
1

2. Mr. P t B t r“ h B t r ikr nC l l o: h C o e
e r ul : T e ul Pn et a ua rT e h i ”
e e e o c t c
I commented upon traditional factor models and their lack of empirical data for company-
specific risk (CSR). I also cited various court cases which indicate that judges, for lack of a
btr oda te o gess sd y pr srt “r ghif results into line with their
eew r,r i d f us ue b apa e o bi t ri
t er e i s n e nal
cet oj t e, hn t r a ao i u f lo oh tc”
ln ’ b cvsw e o evl t nn t a t d t r k.
i s ei h ui ps i ei
I then keyed in on the total cost of equity (TCOE) equation developed by Professor Damodaran:
TCOE = risk-free rate + Total Beta*Equity Risk Premium
(Note: Mr. Larry Kasper has a major problem with total beta in this equation). When referring
to this equation, I cited five other PhDs (as well as numerous appraisers and a portion of a
Harvard Business School (HBS) case study) who have no problems using total beta in this
equation for the analysis of private companies. (o : a oce to netet ak’ sg
N t I l id w i s n bnsuae
e s t v m
o t a bt w e cl l i f e ncute’eu yr kpe i s(ERP). Mr. Roger
f o l e hn a u t g o i on i qi i rmu
t a c an r g rs t s m
Grabowski took issue with this claim. However, I also attended the AICPA BV Conference in
San Francisco in November and heard Professor Damodaran refer to Goldman Sachs’ of use
t abt w e dt mn g f e n on y E Pin one of his slides).
o le hn e r i n ao i cut ’ R
t a e i rg rs
I cited sources for peer-review (AICPA textbook, PPC textbook, CCH textbook, NACVA and its
Current Update in Valuations, the HBS case study, numerous independent
endorsements/testimonials, including by some holding doctorates in finance, the Sharpe Ratio
(Refuted by Mr. Grabowski, but used by Professor Meulbroek, Covrig and McConaughy in their
derivation of total beta. In fairness to Mr. Grabowski, he cites Dr. Sharpe himself to refute the
Sharpe ratio –I will touch upon this below), investment banks (Refuted by Mr. Grabowski.
However, there seems to be conflicting information about this as noted above)).
I concluded my presentation by referring to its sub-tl “ h C o e.T e hi ibten
ie T e hi ” h co e s e e
t, c c w
empirical data for CSR and TCOE (using total beta) and no data for CSR (using completely
subjective factor models). I cautioned the audience to make the choice wisely.
L ry K se:“ n ma u Fn ig f m teB t rPn etn Mo e
ar apr A o l s idn s r o o h ul ikr e o dlfor Company
Specific Risk Premiums –TCOE, Total Beta, Diversification Discount”
(Mr. Kasper had 74 slides for a 30 minute presentation. Therefore, I will key in on major
points).
Mr. Kasper started his presentation with the following comments: If total beta is wrong, then:
 TCOE is wrong
 Butler Pinkerton Model is wrong
 Diversification discount is wrong
2

3. Mr. Kasper then used most of his time trying to prove that total beta is wrong. (I encourage all
to take a look at his slides. Below, I will show how he failed in his mission).
(The issue which is readily apparent, however, in his writings as well as in this debate is Mr.
Kasper has somehow believed that I have called for total beta (total risk) and the BPC to be used
for the investment analysis of publicly-traded stocks. We all know that CSR is at least partially
diversified away for public stocks, so it makes no sense that we would promote total beta (total
risk) in this manner – this is what Mr. Kasper would have you believe. Mr. Kasper actually
yet
thought that (as two CFAs) we were claiming that a stock will have a different rate of return
depending on if a diversified investor or an undiversified investor owned the stock.
We have only promoted total beta as a proxy for private companies where total risk is either
priced or, at a minimum, some level of CSR is priced. In other words, Mr. Kasper believes that
total beta violates financial theory. It does –BUT only for publicly-traded stock where stocks
are priced as part of portfolios; Total beta does not violate financial theory, however, for private
companies where CSR is priced on some level).
P oesr a d rn “ iversification, Cost of Equi a dV le
rf o D mo aa : D
s t n a ”
y u
Poesr D m dr ,nh ol d et o m n r o MrK se s r eti , ae h
rf o’ a oa n i i n i c cm et y n . apr pe n t n m d t
s s a s y r a ’ s ao e
following statement related to the comment that the Butler Pinkerton Calculator and/or total beta
could not withstand a serious Daubert Challenge (at a 95% confidence level):
T iivl t n I o’ee hv 9% cn dne ead g h cag i m
h s a a o. dntvn ae 5 of ec rgri t hnen y
s ui i n e
pocket…o w rsohtf c
r od tt e et
a f .
(Out of curiosity, does anyone think that the subjective factor models provide 95% confidence?)
Professor Damodaran then described the mean-variance framework (the background for the
CAPM) and mentioned that the risk of an investment is not the risk of it standing alone, but the
risk that it adds to your overall investment portfolio (as calculated by beta). Thus, in this CAPM
world, the expected return of an individual investment is equal to:
Expected return = Rf + beta*ERP
B t“ ee ” x t h v leh C P w r .These rebels, instead of choosing to invest in
u R bl ei w o i a t A M ol
, s s ot e d
the market portfolio (a completely diversified portfolio) choose to invest all of their wealth (or at
least a sizable portion of their net worth) in a private business. If the completely undiversified
investor wants a return comparable to what he or she would make as a diversified investor, then
the expected return is equal to:
Expected return = Rf + total beta*ERP
Professor Damodaran brought out important implications of this equation:
3

4. 1) When selling a private business or asset, the best potential buyer, other things being
equal, will be a diversified investor or an entity with diversified investors (a publicly
traded firm).
2) Private business owners who are fully invested in their own businesses are holding onto
these businesses at a discount, especially if going public or selling to a publicly-traded
company is an option (as diversified investors will pay a higher price for the same asset
as an undiversified investor since the diversified investor can shed the inherent
unsystematic risk).
3) Total beta should provide little explanatory power for expected returns of publicly-traded
firms, especially those that are widely held by institutions and have large market caps. (I
have always mentioned this in my presentations and in writing,ot r t MrK se’
cn ayo . apr
r s
assertions).
4) Total beta is not the appropriate measure of risk if an asset is being valued for a potential
buyer who is mostly diversified. (Note: When you placed a CSRP on a private company
in the past, how did you analyze the relative diversification, or not, of the likely pool of
buyers? While total beta does capture maximum TCOEs and CSRPs, these are data
points we have never had before. Total beta allows one to better think through these
issues).
Professor Damodaran then noted the problematic issues related to the build-up model (BUM) –
the typical approach used by many, if not most, appraisers.
T e rf srae t B M a r i fr i s rfrh fl wn r sn:
h poe o cld h U
s l e “ c eo d at ”o t o o i e os
ep s e e l g a
1) Dependence on historical data (Note: recently the professor added a feature to his
website, www.damodaran.com, which calculates the forward-looking ERP (updated
monthly)).
2) Double counting or triple counting risk and/or internal consistencies (For example, there
may be a liquidity premium in some (or all) industry risk premiums (IRPs) if small,
lightly/inefficiently traded companies are included; it is likely there is a liquidity
pe i i t “i pe i ”f s a e dce a these are the companies that are
rmu n h s e rmu o m lr eis s
m e z m r l l
lightly and/or inefficiently traded; Since both size premiums and IRPs calculate expected
returns from beta and compare it to actual returns, these two metrics may be double-
counting risk in some fashion; it is next to impossible to separate size risk from CSR;
Potential for questionable guidelines in the IRP; and the fact that the BUM does not
account for leverage in the IRP such as how betas and total betas can be adjusted for
4

5. leverage. Of course, after you get through the gauntlet above, you still have to
completely guess at CSR).
Professor Damodaran ended his presentation with the comment that the diversification discount
is separate from the illiquidity discount. In other words, it is perfectly logical to use a higher
discount rate to capture the absence of diversification and also apply an illiquidity discount to
value. The same cannot be said of build-up models. (Note: the BPC allows appraisers to see the
average daily reported trading volume for each week so appraisers can make the (subjective)
determination if there is a liquidity premium in the pricing. If the stock does trade in an
inefficient market (i.e., may have a liquidity premium), then the appraiser may want to exclude
the guideline from further consideration).
Professor Tofallis essentially passed on his turn for rebuttal (o :Po s rT f l’
N t rf s o ls
e eo ai
presentation was not controversial for any of the panelists).
Rebuttal from P t B t r “ h B t rPn etn C l l o:Si (yfr teB s
e r ul : T e ul ikr
e e e o a ua r tl b a) h et
c t l
Choice”
A t ha n MrK se s r eti , kydi o a e pi s blv w r taken out of
f r er g . apr pe n t n I ee n n f o tI eee e
e i ’ s ao w n i e
cn x MrK sece D . hre sm nl aef m16 ad u e t fl wn qo :
ot t . aprid rS a ’ e i pprr 94 n pld h o o i ut
e. t ps a o l e l g e
Thus far nothing has been said about such a relationship for individual assets.
Tp a yh E ,Rvl s s c t wts g astwll above the capital
yi l t R σ a e as i e i i l s s i ie
cl e u oad h ne e l
market line, reflecting the inefficiency of undiversified holdings. Moreover, such
points may be scattered throughout the feasible region, with no consistent
rl i si btente epc drtr a dt a r k(R. However, there
e t nhp e e hi xet e n n o li σ )
ao w r e u t s
will be a consistent relationship between their expected returns and what might be
best called systematic risk as we now show.
Mr. Kasper implied that the bolded statement above was a critical blow to total beta since total
beta captures total risk, or σ and develops a consistent relationship between expected return and
,
risk.
However, of course, there is no relationship between expected return of an individual stock and
its total risk – a portfolio. In the quote above, Dr. Sharpe was setting the stage for beta and the
in
C P (l s set l t et c) hr as c’ cn i t no rkt aw l
A M p ae e h a sn ne w e
e e s e e t ks otb i fi o e-diversified
o r uo s l
portfolio is its beta (systematic risk) and its expected stock return is based on the CAPM. Private
companies, on the other hand, have either completely priced CSR and, therefore, total risk, or at
least partially priced CSR. Therefore, total beta reference points are useful in making
appropriate determinations of the cost of capital for private companies.
I also commented upon another quote taken out of context by Mr. Kasper from the 4th edition of
the Brealey and Myers textbook:
5

6. No one has shown that investors have paid a premium for diversified
firms…implying that there is no diversification discount.
Mr. Kasper confused diversified firms with diversified investors. He failed to provide the
following background in an attempt to show that various PhDs in finance, as well as Pinkerton
and myself, misunderstand financial theory. In my rebuttal, I quoted the entire quote from the 5th
edition:
What about diversification as an end in itself? It is obvious that diversification
rdcs i .I ’t t gi f mm ri ? T erul wt t s ru ets
eue r k s th a a r
s n a n o eg g h t b i h agm n i
n o e h i
that diversification is easier and cheaper for the stockholder than for the
corporation. No one has shown that investors have paid a premium for
diversified firms.
This quote relates to mergers of companies and their (inappropriate) quest to diversify (non-
strategically) since public stock investors can do it much easier. It has nothing to do with private
firms where often times the buyer pool is (relatively) undiversified. Public stock investors can
obtain diversification much easier than corporations –or for that matter, the private business
owner or likely pool of potential private business owners.
The diversified public stock investor will crowd out the undiversified public stock investor and
be willing to pay a higher price (since he or she is not exposed to as much of the individual
cm ays i ) n er a l e r unt na ud e ie pb ci et w u r u e
o pn’ rk ad a
s n o r e r h n ni rf d ul n s r ol e i .
w t a v si i v o d qr
Obviously, the seller, if economically rational, will accept the higher bid. In other words, if there
are undiversified investors in the public markets, they do not set the prices of stocks. Thus, Mr.
K se spc u a u etht ( w la nm ru P D )
apr sei s r m nt Ia e s u e s h s
’ o g a s l o :
have an unsupported and unfounded belief that private owners have higher costs
of capital than undiversified publii et s is without merit.
cn s r
v o…
Stated another way, undiversified public investors are price-takers and not price-setters and,
therefore, not important to the pricing of (efficiently traded) public stocks. Undiversified (or not
completely diversified) private buyers set the price of private firms, in contrast to diversified
public investors. T iesni l rft a o MrK se’i us i ttl e . h sh
hs s t l e e l f . apr s e wt oa bt T u, e
e ay u s l ss h a
has not proven total beta wrong! Therefore, contrary to his false assertion, there is a need
to rescale beta to total beta (e /o σ/m) to use as a proxy for private companies.
Btr r s σ
a
I then asked appraisers if they were on the left-s eo t “ gt s eo t fl wn
i r h r h i f h oo i
d e i ” d e l g
observations related to beta (left-hand side) and total beta (right-hand side):
 Correlated volatility (rs/σ ) v. Relative volatility (σ/σ )
σ m s m
 Covariance v. Standard deviation
 Well-diversified portfolio v. Stand-alone asset
6

7.  Portfolio managers v. Appraisers
 Partial risk v. Total risk
 No empirical data v. Empirical data
I have made my choice: I am on the right-side. (I also could have added: Unstable v. Stable
(relatively speaking)).
MrK se’R so s:
. apr ep ne
s
MrK se s t t t e i e t s t etrm D . hre ppr cua l (r od t
. aprte h h v w d h te n f
ad a e e am o rS a ’ ae acr e o w rso
ps ty
that effect). (For what it is worth, I encourage eeyn t ra D . hre pprWii F
vroeo ed rSap’ ae: la .
s lm
Sap (94, C p a ast r e: T er o m reeu i i udr od i s fi ,
hre 16)“ ai l s pi sA hoy f aktqib u ne cnio o r k
t e c lr m tn s”
Journal of Finance, 19 (3)).
Panel Discussion:
One of the major points discussed was the following:
 Total beta violates the CAPM (By definition, since total beta is used in the CAPM in
place of beta, it naturally violates the CAPM. However, the size premium and the CSR
also violate the CAPM. Most importantly, the mere existence of private companies
violates the CAPM. If a private business owner does not own the market portfolio (a
completely diversified portfolio) then he or she has violated the CAPM. Total beta
better models this violation).
MrR gr rb w k: T tl e vB t a dI us i te ul Pn etnMo e
. oe G a o si“ oa B t . e n s e wt h B t r ikr
a a s h e o dl”
Mr. Grabowski ended the debate by changing his role from moderator to an unchallenged
panelist –unchallenged in the fact that no other panelist saw his slides before he delivered them
to the audience. I will respond to his observations here, instead of having the opportunity in
Boston, which I would have appreciated.
 Investment banks do not use total beta to develop foreign country ERPs (I have already
commented upon this. I will leave it up to the reader to decide for himself/herself).
 BPM is based on the premise that most owners of private businesses are undiversified,
therefore the cost of capital of the private business should include that extra amount due
to the owner being undiversified. This leads to the unreasonable position that there are
at least two costs of capital for a business – cost of capital for investors that comprise
the
the pool of likely buyers and the current owner. (It has always been about the pool of
likely buyers for private companies. Sometimes, however, the current owner may be a
good proxy to gauge the diversification, or lack thereof, of the pool of likely buyers.
Along this line of thinking, I asked the Boston audience if anyone had ever asked to see
7

8. t ibs es w e’ pro os t etw e vl ng his/her private company. As
h r ui s o nr ot l t e n hn a i
e n s fi a m s u
far as I could see, no one raised their hand. I believe this diversification issue has
(generally) been given very little thought in our industry. My hunch is that when we as
an industry have placed a CSR on a company we have (generally) viewed it as a stand-
alone asset).
 No one uses TCOE = Rf + Total beta*ERP (We all know that CSR is
not completely priced for publicly traded stocks – otherwise there would be a
rl i si bten ad xet rt n T u, h w
e t nh e e σ n epc d e r. hsw y ould any academic/researcher
ao p w e u
ever consider using TCOE to analyze publicly-traded stock investments?)
 B t et a sui “ o-bc”m t d a sb c t et ao
e sm t s g l k ak e os r uj t o sm t n
a i e n o h e e i i
error. Thus, CSRP estimates derived from beta estimates are also subject to estimation
error. (True; I agree. What is not subject to estimation error in the world of
finance/valuation? The BPC, however, alerts the appraiser to this issue. Appraisers can
run the Calculator (or perform the calculations yourself) for all five days of the trading
week to gain a much better handle on this issue than solely relying upon printed sources
of capital which ignore the sensitivity. Moreover as pointed out above, total beta, being
generally more stable than beta, is relatively immune to this estimation error criticism).
 Poesr hres t : T eS a eR t de nt oe cssi
rf o S a
s p te “ h hr ao os o cvr ae n
as p i
w i oenet ete r ii o e” (Why would it? Individual assets –not part
h h n i s n r un sn l d.
c v m t vv
of a portfolio - violate the CAPM).
 Everyone would like a better method of estimating CSRPs –but is
BPM the method?
“ a ia ’s,o!
S y t i toJe
n ”
(By using the above quote, Mr. Grabowski believes that the BPM is not the method. I
believe, however, that Mr. Grabowski is mistaken if he believes that total beta and the
BPC are inferior to completely guessing at CSR. That, in a nutshell, is what the BPC is
up against – complete guess. Given this threshold, I am quite surprised at the intensity
a
of the misinformed criticism of total beta).
New Commentary by Peter Butler:
Since Shoeless Joe Jackson was somehow brought into the debate from the grave, I end this
paper with the following (adapted) qo f m “ Fe o D em ” baseball movie which
ut r A id f r s –a
e o l a
fr i ula o qo s sol s o:
ot t s l “ut ” he sJe
uo y s e e
8

9. “f o b i i te wl o ”
Iyu ul t hy i cme
d , l
While we have built the Calculator for ease of use and transparency, you do not have to use it. I
do, however, recommend that you use the theory behind total beta (while not throwing out all of
your other methods. It is a good idea, however, to carefully consider all of the issues related to
the BUM pointed out above) to build your own “ e ”–a field that eliminates some of the
fl
id
subjectivity in developing CSRPs and, therefore, TCOEs for private companies.
9