Where:

R_p          = Expected Return on Portfolio

R^f         = Risk Free Rate of Return

O_p        = Standard Deviation, or total risk, of Portfolio Returns

O_m       = Standard Deviation, or total risk, of Market Returns

R_m       = Expected Return on Market

The capital market line is a critical concept of modern portfolio theory because it forms the basis of many pricing models in modern finance and capital budgeting, including the capital asset pricing model (which will be proven below). There are several important concepts about the capital market line, however, that must be understood.

1. First, the “market portfolio,” represented by the tangent point in this model, does not refer to the S&P 500 (although the S&P 500 is often used as a proxy), but to a very specific market portfolio that is comprised of all risky assets in the economy (i.e. bonds, public stock, private stock, art, commodities, real estate, labor capital, etc.) held in proportion to their actual market weights. This market portfolio, by definition, is highly diversified, and, because of its diversification, has eliminated all idiosyncratic, or company specific risk. Therefore, the total risk of the market portfolio, represented by the term (O_m), only reflects market, or systematic risk. No company specific risk is priced.
2. Second, the individual portfolios (i.e. the portfolios on the line), do not refer to any portfolio, such as a portfolio of bonds, but to a very specific portfolio combination comprised of the risk free asset and the market portfolio. At one extreme, represented by the intersection at the vertical axis, is a portfolio 100% allocated to the risk-free asset. At the other extreme, represented by the tangent point on the Markowitz frontier, is a portfolio 100% allocated to the market portfolio as previously defined. Investors can also leverage the market portfolio by borrowing money at the risk-free rate and investing the proceeds into the market, thereby extending the line beyond the market portfolio.
3. Third, since the individual portfolios in the CML simply represent combinations of the risk-free asset and the market portfolio, these individual portfolio combinations, by definition, are perfectly positively correlated with the market return (i.e. have a correlation coefficient equal to 1.0). Therefore, the “beta” of these portfolios is simply represented by the relative standard deviation of the portfolio and the market (i.e. O_p/O_m = Beta). Furthermore, these individual portfolios must, by definition, be fully diversified portfolios. Therefore, the total risk of the individual portfolios, represented by the term O_p, only reflects market, or systematic risk. No company specific risk is priced.

This last point is particularly important because it allows us to reconcile the capital market line with the single index model (a pricing model in finance), which forms the basis of the capital asset pricing model. To demonstrate this concept (this is the most complicated math in this article), we begin by rearranging the terms from the equation above and generalizing the subscript from p (a portfolio) to i (a security), thereby rewriting the equation of the CML for any security in terms of realized returns as follows:

R_i – R_f = O_i/O_m*(R_m-R_f) + e_i

Where:

R_i         = Return of Security i

R_f         = Risk Free Rate

O_i         = Standard deviation of security i

O_m       = Standard deviation of market portfolio

R_m       = Return of Market portfolio

e_i          = Error term of returns (i.e. idiosyncratic or company specific risk)

By definition, the variance, or risk, of this equation can be expressed as follows (note that R_f drops out because the risk-free asset is constant and has no risk):

O_i^2 =(O_i/O_m*p_i,m)^2*(O_m^2) + O_e^2 + 2Cov(R_m,e_i)

Further notice that the non-systematic risk component, represented by e_i, of this portfolio is, by definition, zero because the portfolios in the CML are fully diversified. Accordingly, the 2Cov(R_m,e_i) term and the O_e^2 term drop-out from this equation, and the standard deviation of the security simplifies to the following equation:

O_i = (O_i/O_m*p_i,_m)*O_m

Substituting the above equation into capital market line, we discover the following:

R_i = R_f + ((O_i/O_m*p_i,_m)*O_m)/O_m)*(R_m-R_f)

First, notice that O_i/O_m*p_i,m is simply the definition of Beta. Therefore, the capital market line can be alternatively expressed as follows:

R_i = R_f + B_i*O_m/O_m*(R_m-R_f)

Finally, the O_m terms cancel and the final equation simplifies to the capital asset pricing model:

R_i = R_f + B_s * ( R_m – R_f)

This demonstrates that the capital market line is fully consistent with capital asset pricing model. However, further, notice that the correlation coefficient between the security (which plots on the CML) and the market is, by definition, 1.0 (see point 3 from above). Therefore, if we replace B_s, with O_i/O_m*p_i,_m (i.e. the alternative formulation for beta), and substitute the correlation coefficient with 1.0, we obtain the capital market line:

R_i = R_f + O_i/O_m*(R_m-R_f)

Again, this demonstrates that the capital market line is fully consistent with the capital asset pricing models. More importantly, however, we can equate these two relationships for fully diversified portfolios that plot on the CML, and derive the following mathematical relationship:

R_f + B_s*(R_m-R_f) = R_f + O_s/O_m*(R_m-R_f)

Solving for B_s, we discover that

B_s = O_s/O_m

Therefore, for fully diversified portfolios that plot on the CML, the Beta is equal to the standard deviation of the portfolio divided by the standard deviation of the market. This is Butler-Pinkerton’s Total Beta. This shows that Total Beta is consistent with CML when the correlation coefficient is 1.0 (the thrust of Larry’s argument). However, this concept does not apply to individual securities. To understand this concept, let us review a related concept in modern finance: the security market line.

Security Market Line (SML)

A related concept to the CML is the security market line, also referred to as the capital asset pricing model. The SML is a pricing equation for individual securities that is based upon the mathematical relationships derived from the CML (see above). In particular, the equation for the SML is expressed as follows:

E(R)_s = R_f + B_s * ( R_m – R_f)

Where:

E(R)_s   = Expected Return on the Stock

R_f         = Risk Free Rate of Returns

B_s        = Beta of the Stock

R_m       = Expected Return on the Market

The equation tells us that the expected return on an individual stock is simply equal to the risk free rate (first term of equation) plus the equity risk premium multiplied by the stock’s beta (second term of equation), or systematic risk factor. This line is effectively the index model derived from the CML. Again, recall that the beta of a stock can be expressed as follows:

B_s = O_s/O_m * p_s,m

Where:

B_s        = Beta of Stock

O_s        = Standard deviation, or total risk, of stock returns

O_m       = Standard deviation, or total risk, of market returns

p_s,m      = Correlation of stock and market returns

Therefore, substituting this equation into the original equation, we obtain the following:

 E(R)_s = R_f + O_s/O_m * p_{s,m *} ( R_m – R_f)

Where:

E(R)_s   = Expected Return on the Stock

R_f         = Risk Free Rate of Returns

O_s        = Standard deviation, or total risk, of stock returns

O_m       = Standard deviation, or total risk, of market returns

p_s,m      = Correlation of stock and market returns

Notice that this equation is essentially the capital market line, except that the correlation coefficient does not drop from the equation. The correlation coefficient does not drop from the equation because unlike a diversified portfolio on the CML, an individual security does not necessarily have perfect positive correlation with the market. More importantly, since investors in the Markowitz framework are presumed to be mean-variance efficient and hold the market portfolio (or some combination of the market portfolio and the risk free asset as predicted and described by the capital market line), these investors, by definition, only care about how an individual security’s total risk contributes to the total risk of their portfolio on the CML. Therefore, they price investments using economic relationships derived or related to the CML (namely the index model/capital asset pricing model). This model, as derived from the CML, includes the correlation coefficient. Investors require it because they only care about their systematic risk.

Understanding Larry Kasper’s Argument

If one accepts the premises of Modern Portfolio theory, then Larry Kasper’s arguments regarding Total Beta should become very clear. In particular, modern portfolio mandates that all investors are mean-variance efficient. Therefore, these investors diversify and develop portfolios that plot on the Markowitz Efficient Frontier. More importantly, since a risk-free asset exists, these investors further optimize by holding only one risky portfolio (i.e. the market portfolio), and allocate between this portfolio and the risk-free asset based upon their risk preferences (as demonstrated in the capital market line). Since investors plot on the capital market line, are mean-variance efficient, and well diversified, they should price individual risky assets on the basis of the economic relationships that exist on the capital market line. The capital market line tells us that the relevant economic relationship is the single-index model, or capital asset pricing model, which is defined as follows:

E(R)_s = R_f + O_s/O_m * p_{s,m *} ( R_m – R_f)

The term O_s/O_m*_p,m is also referred to as Beta.  In the case of a diversified portfolio on the CML, the correlation coefficient of the portfolio and the market is, by definition, 1.0, and, therefore, the Beta of a diversified portfolio on the CML simplifies to O_s/O_m, or Butler-Pinkerton’s Total Beta. However, in the case of an individual risky asset, such as a privately owned business, the correlation coefficient does not always drop-out from the equation because the correlation coefficient for a private business is not always 1.0. Therefore, in the case of an individual risky asset, the pricing equation from the CML cannot simplify to Total Beta, but remains beta, unless the correlation coefficient of the security and the market is 1.0. Accordingly, Beta is the relevant risk metric within the context of modern portfolio theory because it quantifies the amount of risk that will actually contribute to the risk of the portfolio that investors hold on the CML. If investors priced individual risky assets using any metric other than Beta (i.e. such as Total Beta), then these securities, by definition, would be incorrectly priced unless their correlation coefficient with the market portfolio was equal to 1.0. This is the thrust of Larry Kasper’s argument, and is very compelling if we assume modern portfolio theory applies to all markets.

Understanding Butler-Pinkerton’s Argument

Peter Butler primarily argues against Larry Kasper’s arguments by suggesting that individual investors in the marketplace for privately owned businesses are not fully diversified, and, therefore, require compensation for all risk (both systematic and idiosyncratic). In particular, Peter Butler states that individual investors in the private market are undiversified price setters (a direct violation of the primary assumptions of CAPM and modern portfolio theory) and, therefore, require compensation for their total risk due to their inability (or conscious decision not to) diversify. Peter Butler then concludes that due to their lack of diversification that these investors require compensation equal to Total Beta, which is defined as follows:

TB = O_s/O_m

Where:

TB       = Total Beta

O_s        = Standard deviation of security

O_m       = Standard deviation of market

Therefore, in the private market, the relevant pricing equation is no longer traditional CAPM, but an “improved” version of CAPM using Total Beta, which is expressed as follows:

R_i = R_f + O_s/O_m*(R_m-R_f)

Notice that this equation is effectively the equation for the capital market line. Therefore, in effect, Peter Butler is suggesting that non-diversified investors in the private market (due to their lack of diversification) require compensation above traditional CAPM, In particular, these investors will require their excess return (i.e. return in excess of the risk-free rate) per unit of total risk (i.e. standard deviation), also known as the Sharpe Ratio, to equal that of the market, irrespective of the securities correlation with the market. In modern portfolio theory, investors only require an excess return per unit of Beta (i.e. systematic risk) equal to that of the market. Consequently, Total Beta results in a higher required rate of return than conventional beta, thereby lowering the price of these investments and causing them to plot above the security market line as contemplated in modern portfolio theory (which should not happen if the assumptions of modern portfolio theory hold). Appraisers should understand that Total Beta, by definition, makes the following assumptions regarding the private marketplace;

1. Investors in the private marketplace do not diversify, even though empirical research demonstrates that diversification is highly advantageous, and, therefore, price investments outside of the framework of modern portfolio theory (i.e. since modern portfolio theory presumes that all investors are diversified).
2. Investors who are diversified cannot or do not enter the market for private businesses and, therefore, compete away the idiosyncratic risk that is, according to Peter Butler, being priced by this market.
3. Investors in this marketplace, because of their lack of diversification, therefore, price investments such that their excess return per unit of total risk equals that of the market portfolio as contemplated in the capital market line, which presumes that all investors are fully diversified and hold the market portfolio.

These assumptions may or may not be unreasonable (discussed later). However, some interesting questions are raised if these assumptions are accurate. First, why do investors in the marketplace elect not to diversify when there are clear economic benefits from doing so? Secondarily, if individual investors in the private market do not diversify, what is preventing outside investors from entering this market and extracting the “free-lunch” that exists due to the pricing of idiosyncratic risk. In fact, if idiosyncratic risk is fully priced, there would seem to be a wonderful investment opportunity for large individual investors: namely, enter the market for private businesses, purchase, say, 30 private businesses with low correlation that are being priced for “full” risk, create a diversified portfolio, eliminate the non-systematic risk component, and earn an excess return per unit of total risk that far exceeds that achievable in the marketplace. Clearly, something is preventing this from happening, or Total Beta is incorrect.

More importantly, if assumptions 1 and 2 are accurate, then we still need to demonstrate empirically whether assumption 3 is valid; that is, do undiversified investors really demand a return premium per unit of total risk equal to that of the market portfolio? If this is not the case, then we cannot really use Total Beta confidently. This problem is further compounded because we suffer a big weakness when we infer Total Beta from the market; that is, unlike the coefficient on regular beta, the coefficient term on Total Beta cannot be tested for statistical significance..

The Real Debate & Empirical Support

We can debate the points of Total Beta and Beta all day, but the “real” debate really centers around whether the assumptions of modern portfolio theory or Total Beta are accurate. If we accept, for example, a strict interpretation of modern portfolio theory, then Larry Kasper’s arguments should govern this debate; that is, all investors are Markowitz Efficient investors, they form diversified portfolios, hold the market portfolio, plot on the CML and price individual securities using the capital asset pricing model that is derived from the CML. If this is the case, then investors cannot price investments using Total Beta because Total Beta, by definition, presumes that investors are undiversified, which is a violation of CAPM. Alternatively, if modern portfolio does not hold (i.e. real world investors do not hold market portfolio, plot on CML, etc., etc.) then perhaps an alternative pricing equation exists. Perhaps investors do price idiosyncratic risk. Perhaps investors even use a formulation of Total Beta. To evaluate theses questions, let us consider what empirical research has actually demonstrated. First, let’s start with the assumptions of CAPM beta and modern portfolio theory, and, then, we will proceed to Total Beta and idiosyncratic risk.

The Assumptions and Empirical Support of CAPM Beta

First, despite the ubiquitous use of CAPM Beta and modern portfolio theory, the empirical record in support of these models is very weak (see The Capital Asset Pricing Model: Theory and Evidence, Eugene F. Fama and Kenneth R. French). For example, early research of traditional CAPM beta statistically rejected the model by demonstrating consistent “alphas” in the market model, suggesting that CAPM was not capturing all risk factors. In addition, financial researches have widely documented that CAPM beta is too “flat” in terms of predicting returns; that is, CAPM beta overstates the expected return of high beta stocks and understates the return of low beta stocks (see the Ibbotson year book, for example). Moreover, later research, including the work conducted by Fama and French and others, has demonstrated that other factors including size, value, and momentum are all important factors (in addition to the traditional market beta) in terms of pricing securities.

More importantly, the models are subject to many unreasonable assumptions that do not hold up in practice. For example, CAPM Beta is based upon the following assumptions:

1. All investors aim to maximize economic utilities
2. All investors are rational and risk-adverse
3. All investors are broadly diversified across a range of investments
4. All investors are price takers, i.e. they cannot influence prices
5. All investors can lean and borrow unlimited amounts at the risk free rate
6. All investors trade without transaction or taxation costs
7. All investors deal with securities that are highly divisible into small parcels
8. All investors assume all information is available at the same time to all investors

In addition, modern portfolio theory prescribes the following additional assumptions:

1. All investors are only interested in maximizing mean for a given variance
2. Asset returns are jointly normally distributed random variables
3. Correlations between assets are fixed and constant forever
4. All investors aim to maximize economic utility
5. All investors believes about probability of returns match the true distribution of returns
6. There are no taxes or transactions costs

However, it is has been widely documented, for example, that individual investors do not own broadly diversified portfolios as contemplated by CAPM Beta. More importantly, investors rarely ever own the market portfolio has described by the capital market line. In addition, common sense dictates that individuals do not only care about mean and variance, but also care about other variables such as liquidity and taxes. Moreover, academic researches know that asset returns are not normally distributed about the mean; stocks, for example, exhibit the “fat-tails” problem, wherein large negative returns occur much more frequently than predicted by a normal distribution. Furthermore, investors do not have homogenous expectations about the future return distributions, and, often, focus on long-term returns instead of their single period return.

The point is that if (a) many of the underlying assumptions related to traditional CAPM beta and modern portfolio theory are violated in practice and if (b) the empirical record generally does not support CAPM beta, then is CAPM beta really the proper metric for pricing risk? More importantly, if these assumptions are violated in practice can we really make the claim that Total Beta is wrong using modern portfolio theory to prove that it is wrong (as Larry Kasper does)? Or, are investors in the marketplace using a different “tool” to price risk? Moreover, if investors are not using CAPM to price investments in the marketplace (i.e. given all the violations in the marketplace), then what metric are they using? Perhaps total risk is important to investors as the concept of Total Beta suggests. Let us evaluate this concept next.

The Assumptions and Empirical Support for Total Beta

Perhaps interestingly, and despite all the criticism of Total Beta, there is actually a wide body of financial research that provides both empirical and theoretical support for the notion that investors not only price systematic risk, but idiosyncratic risk as well, especially when they are non-diversified. For example, in a recent working paper published by the National Bureau of Economic Research, entitled “Entrepreneurial Finance and Non-Diversifiable Risk,” Hui Chen, Jianjun Mio, and Neng Wang demonstrate that “non-diversified entrepreneurs demand both systematic and idiosyncratic risk premium.” In the concluding remarks of the paper, these authors assert.

At the same time, however, the empirical research is still unclear as to whether total risk is priced. For example, in another National Bureau of Economic Research working paper, entitled “The Returns to Entrepreneurial Investment: A Private Equity Premium Puzzle?,” Tobias J. Moskowitz and Annette Vissing-Jorgensen, find that the returns to private equity are no higher than returns to public equity, even though entrepreneurial investment is extremely concentrated (i.e. poor diversification). Moreover, even if idiosyncratic risk is priced, none of these articles directly test whether this premium is directly proportional to the market price of risk (i.e. R_m-R_f/O_m) from the capital market line, as Total Beta presumes.

The point here is that we really cannot determine whether idiosyncratic risk is priced (as Butler-Pinkerton suggest) or whether systematic risk is only priced (as Larry Kasper suggest); that is, research provides support for pricing idiosyncratic risk, but other research does not. Moreover, none of the empirical research indicates that investors price idiosyncratic risk using Total Beta, or at least test whether Total Beta has any predictive power in terms of pricing idiosyncratic risk. Therefore, while Butler-Pinkerton may have uncovered a useful pricing tool, they have not exactly supported whether this pricing equation is used by investors.

Concluding Remarks & Areas for Further Research

This article has presented the underlying arguments for and against Total Beta. In particular, this article demonstrates that under a very strict interpretation of modern portfolio theory that Total Beta should not hold; that is, if investors truly optimize portfolios pursuant to Markowitz, hold the market portfolio, and plot of the CML, that these investors should rationally price investments using traditional CAPM Beta and not Total Beta. Furthermore, the only time in which Total Beta will price risk properly in the context of modern portfolio theory is when the correlation of the security and the market portfolio as defined by the capital market line is equal to 1.0. This rarely happens. However, if modern portfolio theory does not hold (i.e. if investors do not diversify), then Total Beta may be a useful concept. However, if this is the case, investors must assume that non-diversified investors require an excess return per unit of risk equal to that of the market portfolio. This article further suggests that the “real” debate concerning Total Beta is related to whether one accepts the premises of modern portfolio theory. If modern portfolio holds, then Total Beta does not hold. However, if modern portfolio theory does not hold, then Total Beta may hold. This article presented empirical research that challenges the assumptions of modern portfolio theory and presented other research in support of pricing models that consider idiosyncratic risk. These articles provide mixed results, but generally support the notion that idiosyncratic risk is priced, especially for non-diversified investors. These articles, however, do not verify whether these investors require compensation equal to Total Beta, as suggested by Peter Butler.

Appraisers should be aware of the factors discussed in this article before ardently supporting either model (i.e. CAPM Beta or Total Beta). The reality is that the financial community is still debating the merits of these models; that is the empirical record generally does not support traditional CAPM Beta under a strict interpretation of modern portfolio theory, and there is no empirical basis to support the assumption that Total Beta properly compensates investors for lack of diversification. Therefore, as appraisers, we really need to develop answers to many unresolved questions before using either these models blindly. These include the following questions:

1. Why do investors in the private marketplace elect not to diversify, especially when there are clear economic benefits from do so?
2. What is preventing outside investors from entering the private market an competing away idiosyncratic risk being priced as Butler-Pinkerton claim?
3. Why do investors in the public marketplace not follow modern portfolio theory or adhere to the assumption of the CAPM if these pricing models are accurate?
4. Do investors actually price idiosyncratic risk, and, if so, how do they price it?
5. Is Total Beta measuring idiosyncratic risk and is Total Beta able to reasonably predict market returns for undiversified investors?

Until these questions are answered, I believe that Total Beta and even traditional CAPM Beta should be used with relative caution by appraisers; that is, appraisers, in my opinion, should not automatically rely upon the company specific risk premiums suggested by Total Beta or automatically default to the cost of capital estimates obtained from traditional CAPM Beta because these equations are likely to provide false estimates of risk under the façade of mathematical accuracy. Unfortunately (or perhaps fortunately), our “reasoned judgment” is a better predictor of future required returns than the mathematical “precision” that these models provide. As Warren Buffet states in his essays.