Total Beta Proponents Response to Eric Nath - BVMarketData

Letter to Editor
This leter to the editor is in response to Eric W. Nath’s leter to the editor which was published on page
51 in the Spring 2011 Business Valuation Review.
In that letter, Mr. Nath shows a simple chart with two stocks going in uniformly opposite directions. He
correctly calculates the same standard deviation for both stocks and, therefore, the same total beta
equal to 1.2. He then states that,
“This is just a general example of how there are, in fact, an infinite number of ways in
which the standard deviation of returns for Company A and Company B could be
identical, meaning identical Total Betas, but with returns moving in completely opposite
directions. Total Beta pays no attention to whether the prospects of a given company
are positive or negative. Total Beta pays no attention to anything other than historical
volatility, and is blind as to whether volatility in a particular case might be a good thing
or a bad thing. How can anyone maintain with a straight face that the total risk of
Company A and Company B are the same It looks to me like Total Beta cannot be true,
but is simply a statistical non sequitur.”
Mr. Nath’s aleged criticisms realy point to the CAPM (which almost al appraisers use in al
assignments) as Total Beta is merely an extension of CAPM theory for undiversified investors. For
example, calculations of Beta are also based on historical stock prices. Beta calculations are also
dependent upon the same relative standard deviations as Total Beta. Moreover, Betas can equal each
other, yet follow divergent paths.Is Beta, therefore, a statistical “non sequitur” Of course not.
Ironically, Mr. Nath seems to criticize the use of history yet likes history to distinguish between alleged
“good” and “bad” volatility to determine the prospects of a given company. What Mr. Nath has failed to
realize, however, is that according to modern portfolio theory (MPT) - ALL volatility is bad. Again, total
beta applies MPT to undiversified investors. It is really as simple as that. Naysayers should not hold
Total Beta to a higher standard than CAPM Beta. For a more technical response to Mr. Nath’s incorrect
criticisms, please see below.
Let’s look at the folowing Chart of aleged “good” and “bad” volatility:
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In the Chart above, which appears very similar to the chart that Mr. Nath created, Company A or the
aleged “good” volatility company has a historical beta of negative 0.31 (rounded). Visualy, we think
you can see that it is negatively correlated with the S&P 500 or the market. For the aleged “bad”
volatility company– Company B, as you might gues, the “bad” volatility company has a positive 0.31
beta, the mirror-image of Company A.
Before we addres Mr. Nath’specific statements above let’s develop an equation for the future price of
a non-dividend paying stock. The equation below is the most common equation for stock price and is
used in the Black-Scholes option pricing equation…
S t = S 0 exp(( t + W t )
Stock price today is S 0 and we want to forecast S t , which is stock price at some future time t. The Greek
symbol, mu (represents the rate of return that we expect the stock to earn. In our case, this would
be the discount rate that we used to price the stock at time zero (t = 0). The Greek symbol, sigma
(represents expected volatility. In most cases, we use historical stock prices to measure sigma. The
symbol W t is a Brownian motion with mean zero and variance t, and is the random variable that is the
source of all risk.
With our stock pricing equation in hand let’s isolateMr. Nath’s statements above…
“This is just a general example of how there are, in fact, an infinite number of ways in which the standard
deviation of returns for Company A and Company B could be identical, meaning identical Total Betas, but
with returns moving in completely opposite directions.”
The statement above is correct. The expected value of W t is zero, which means that we expect the stock
to earn the discount rate (over the time interval [0,t]. But note that the variance of W t is non-zero,
which means that in all cases (noting that in continuous time W t will never equal zero) the stock will
actually earn greater than the discount rate or less than the discount rate. At time zero looking forward
we do not know the path that the stock will follow but from the stock price equation above we do know
the possible paths and their attendant probabilities.
“Total Beta pays no attention to whether the prospects of a given company are positive or negative.”
The statement above is incorrect. If W t were not random then S t would be known with certainty and mu
(would be the risk-free rate. The fact is that from the perspective of time zero looking forward we do
not know the value of W t (i.e. risk) which is why the discount rate is greater than the risk-free rate. The
mere fact that the discount rate is greater than the risk-free rate says that we don’t know whether the
actual prospects for the company are positive or negative and this uncertainty is the reason why the
equity risk premium (r m –r f ) is greater than zero. If the equity risk premium is a function of volatility,
and volatility gives us the distribution of possible results (good and bad), then by definition the discount
rate obtained via the CAPM (or Total Beta) takes into account the probabilities of both positive and
negative prospects for the company being valued.
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“Total Beta pays no attention to anything other than historical volatility, and is blind as to whether
volatility in a particular case might be a good thing or a bad thing.”
The statement above is also incorrect. Volatility from the perspective of time zero looking forward is
always bad. The utility function for a risk-averse investor has a first derivative that is positive (more of a
good is always better than less) and a second derivative that is negative (marginal utility is decreasing).
Remember that from the perspective of time zero we deal in expectations because from our vantage
point we do not know whether W t will be positive (we will earn a rate of return greater than the
discount rate) or negative (we will earn a rate of return less than the discount rate). The utility function
for a risk-averse investor says that the deleterious effects on utility from a negative W t outweighs the
salubrious effects on utility from a positive W t , which again is why we have discount rates that are
greater than the risk-free rate. Both the CAPM and Total Beta equations reflect the utility function of a
risk-averse investor. The discount rate obtained via the CAPM (or Total Beta) by definition takes into
account the expected utility from rates of return that are a lot greater than expected, a lot worse than
expected, and everything in between.
Note that how one measures expected volatility is at the discretion of the appraiser. If the appraiser has
definite opinions as to expected future volatility then the appraiser may want to use that number. In
absence of any definite opinions or insights ex-post volatility (based on historical prices) is often used as
a proxy for ex-ante (future expected) volatility.
“How can anyone maintain with a straight face that the total risk of Company A and Company B are the
same”
Given that from the perspective of time zero looking forward we do not know the future value of W t ,
and from the risk-averse investor’s utility function that puts more weight on negative outcomesthan
positive ones, we can say with a straight face that the risk of Company A and Company B are the same.
From the vantage point of time zero we expect both companies to earn the discount rate. From that
vantage point we don’t know if actual volatility wil be a “good thing” or a “bad thing” but we do know
that since risk-averse investors put more weight on negative outcomes than positive ones ex-ante
volatility is always a bad thing. If this were not the case then all investors would be risk-neutral and the
expected return on all risky assets would be the risk-free rate.
From our work as appraisers, where we generally price total risk as opposed to just systematic risk, and
consistent with the classical definition of total risk which is the variance or uncertainty of returns:
Yes, we can confidently state with a straight face that we will calculate the same total risk premium for
both companies (to be used as proxies for our subject company) going into the future equal to:
Total cost of equity (TCOE) = risk-free rate + total beta*ERP
TCOE = 5% + 1.2*6% = 12.2%
Notice the rate is positive and greater than the risk-free rate. Whether or not we use 12.2% for our
subject company is another question and that depends on our assessment of its relative risk when
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compared to both companies. This is where Total Beta beats other costs of capital models hands down
–at least in our opinion. We now have–on a total risk perspective–both quantitative (think operating
results over time) and qualitative disclosure of risks (think reading Forms 10-K) on multiple specific
benchmarks to better support our determination of the appropriate discount rate. With these tools at
our disposal, most of the time, we feel we can explain the observed volatility in a stock. Moreover, Total
Betas are generally more stable across time than Betas, which is principally due to the volatility of the
correlation coefficient. This is another huge advantage that Total Beta has over merely using Beta to
develop a discount rate for private companies.
In any event, what rate of return would you use for your subject company, Mr. Nath, assuming you used
these two companies as proxies Please tell us how you would arrive at your conclusion
For what it is worth, we worked with Mr. Nath’s unlikely example because that is the hypothetical he
gave Total Beta proponents. We believe, however, it shows that using very extreme and unlikely
examples can provide aleged “evidence” for whatever one is trying to prove, especially when
disregarding the realities of the marketplace and MPT.
Finally, if Mr. Nath can tell us–with certainty based on historical stock prices - which one will provide a
better return from an investment standpoint, we would like to hear it and encourage him to enter the
investment management profession.
Respectfully submitted,
Peter J. Butler, CFA, ASA
Gary S. Schurman, CFA, CPA
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