How an idiosyncratic (zero-beta) risk can greatly increase the firm’s cost of capital

Abstract

The celebrated capital asset pricing model (‘CAPM’) brought numerous appealing insights and spawned a new theory of capital budgeting. One key intuition is that there is ‘no penalty for diversifiable risk’ – that is, any risky payoff that has zero-correlation with the wider economy, and hence zero-beta, is treated as ‘risk-free’. Does that mean that managers can bet the firm on a spin of the roulette wheel without attracting a higher CAPM discount rate? Our re-interpretation of CAPM reveals that potential financial losses which are conventionally regarded as firm-specific ‘unpriced’ risks can bring a large increase in the firm’s beta and CAPM cost of capital, despite having zero-beta and making only negligible difference at the aggregate market level. This mathematical result clashes with textbook expositions but is easily demonstrated and can be traced to authoritative but overlooked parts of the theoretical CAPM literature.

JEL Classification: G11, G12

Keywords

CAPM cost of capital priced diversifiable risk

1. Introduction

Firm-specific ‘idiosyncratic’ risks are defined as any influence that affects the firm’s cash payoff in a random ‘noise’ way that is uncorrelated with the economy and market. Suppose that a firm listed on the stock market learns that its highly unique and profitable product is causing injury to users. A class action against the firm has been lodged and has potential to bankrupt the company. What does this news, reported to shareholders and financial markets, do to the firm’s cost of capital?

The usual capital asset pricing model (CAPM) understanding is that such risks are uncorrelated with the market’s return and are therefore ‘zero-beta’ and thus ‘unpriced’.

That statement ‘sounds right’, but what does it mean when we say that a risk is ‘unpriced’, and can it truly be that such a worrying albeit clearly firm-specific risk goes ‘unpriced’?

In the following analysis, we use simple CAPM equations to show that the potential financial payout from the class action, viewed as a zero-beta random component of the firm’s future cash flow, must cause the firm’s cost of capital to increase, possibly very greatly.

The remainder of this article is devoted to explaining this result formally along with its interesting implications. The theoretical foundations of our argument are found to exist already, having been clearly stated in widely overlooked works by Fama (1977) and Ingersoll (1987).

Recent corporate disasters have revealed that some idiosyncratic risks taken on by firms amount to ‘betting the firm’, or close to that. Volkswagen took on a paradigmatic idiosyncratic risk when it gambled that its ‘switch’ software that tricked diesel emissions testing equipment would not be revealed.¹ Similar zero-beta risks are taken by mining companies. The collapse of Vale-BHP Billiton Vale Mariana dam led the Brazilian government to fine the joint venture company US$4.8 billion along with other class action payouts. Commonly unrecognized idiosyncratic risks include, for example, the infamous cost overrun of the Boston ‘big dig’ megaproject, which was attributed largely to geotechnical and other engineering design mistakes² unrelated to market conditions. The risk of such technical mistakes made by the firm’s engineers or consultants is a distinctly firm-specific risk.

Such large zero-beta risks threaten firm survival, which of itself suggests that real-life managers will not discount such risks at the risk-free rate, contrary to CAPM theory. However, for the purposes of our analysis, we imagine strict adherence to CAPM logic, meaning that all of the zero-beta elements of the firm’s payoff are priced by discounting them at the risk-free cost of capital regardless of their downside potential.

1.1. CAPM narrative

The CAPM is widely accepted and taught in MBA programmes and is said to be influential in the corporate finance practice of capital budgeting. Surveys including Fernández (2009), Graham and Harvey (2001) and Jacobs and Shivdasani (2012) have noted how deeply the CAPM is entrenched in finance theory and corporate finance practice. There is no comparably simple but elegant asset pricing method, nor is there any theoretically coherent yet practicable way of capital budgeting. Fifty years after its invention and first appearance in MBA classes, CAPM remains as widely taught and respected as ever:³

Given this background, here we adopt the Sharpe-Lintner CAPM as the default model for expositional purposes and for illustrating the issues involved in assessing a firm’s cost of capital. (Jagannathan et al., 2017: 264)

One of the celebrated concepts that came with CAPM is the distinction between diversifiable and undiversifiable risk. A risk is called diversifiable if it has zero-correlation with the stock market’s average return or payoff. Clear examples quoted in textbooks include firm-specific events such as industrial accidents, the CEO being hit by lightning and, in a most salient example, the firm’s new satellite launching successfully and then being demolished by space junk (Brealey et al., 2020).

Under headings like ‘Don’t be fooled by diversifiable risk’, CAPM expositions suggest that there is no penalty when firms take ‘diversifiable risk’, known also as ‘firm-specific’ or ‘idiosyncratic’ risk. Instead, the firm’s cost of capital is said to depend on just its ‘systematic’ or undiversifiable risk, captured by its CAPM ‘beta’:

. . . we have defined risk as the asset beta for a firm, industry, or project. But in everyday usage, ‘risk’ simply means ‘bad outcome’. People think of the risks of a project as a list of things that can go wrong. For example: . . . a plant manager worries that the new technology will fail to work . . ., a telecom CEO worries about the risk that a satellite will be damaged by space debris . . . Notice that these risks are all diversifiable. For example, the Iridium-Cosmos collision was definitely a zero-beta event. These hazards do not affect asset betas and should not affect the discount rates for the projects. . . .Sometimes financial managers increase discount rates in an attempt to offset these risks. This makes no sense. Diversifiable risks should not affect the cost of capital. (Brealey et al., 2020: 229)

This statement represents the common and widely accepted view that the firm’s cost of capital is not increased by its diversifiable risk, no matter how large or probable the potential loss. Practitioners, however, suggest otherwise.

Finding that firms discount projects at rates higher than the firm’s cost of capital, Jagannathan et al. (2016) hypothesize that firms are averse to firm-specific risk:

Almost two-thirds of respondents report that idiosyncratic risk is important in determining their firm’s discount rate (65.4%), about the same number as for market risk (63.4%). (p. 458)

Firms’ aversion to ‘idiosyncratic risk’⁴ is attributed to their capital providers not being fully diversified (i.e. not behaving rationally as assumed under CAPM) or to the firm’s own managers being concerned with personal reputations and hence conservatively avoiding risks of firm-specific (i.e. manager-specific) project failure.

In this article, we provide a CAPM-based explanation for why indeed managers should account for firm-specific risk in their chosen discount rates.

Their allowance for firm-specific zero-beta risks is shown to be rational and correct under CAPM, contrary to conventional CAPM teaching.

The question of how to account for a zero-beta risk in the discount rate is answered directly by the CAPM’s own logic. Specifically, when we apply the payoffs or ‘certainty equivalent’ form of CAPM (e.g. Grinblatt and Titman, 1998), the correct CAPM discount rate comes out in the analysis as a mathematical implication of the calculated asset price. This is a strong methodological argument for preferring the payoffs form of CAPM over the usual alternative of simply assuming some ‘appropriate’ discount rate.

The misconception that the CAPM discount rate is not affected by zero-beta risks arises, as we explain later, via an oversimplified interpretation based on what is implied superficially by the returns form of CAPM. By looking closely at the payoffs form of CAPM, we get a clearer and more correct interpretation of the conventional CAPM returns expression.

In short, any practitioner who applies the payoffs form of CAPM will not make the mistake of ignoring zero-beta risks. Using the payoffs form (see our examples), practitioners will correctly allow for zero-beta risks without knowing it. That is what we show in our analysis.

1.2. Correct CAPM findings

Our first conclusion, which is uncontentious in theory, is that a firm that takes on favourable (i.e. positive expected cash flow) ventures that are uncorrelated with the market or ‘zero-beta’ will reduce its overall cost of capital, because the new ventures are discounted at the risk-free rate (other ventures have higher costs of capital, so the average is reduced).

Our much less obvious CAPM conclusion is that a firm that encounters some new diversifiable risk of loss, such as becoming the target of a hefty legal claim by consumers or regulators, or even its own shareholders, must then be subject to a higher CAPM cost of capital than an identical company not exposed to that ‘diversifiable risk’.

To dispel any doubts, our analysis begins with a simple numerical counterexample, relying on nothing but elementary CAPM calculations. After working through that example, we provide more general algebraic interpretation of what it means under CAPM when a risk is called ‘diversifiable’. The conclusion of this analysis is that all diversifiable zero-beta risks affect the firm’s cost of capital, either up or down.

Put succinctly, a zero-beta risk increases (reduces) the firm’s cost of capital when the expected cash flow in question has negative (positive) mean. This is a very simple result mathematically and is easily proven using CAPM equations, but seems to be largely unrecognized in CAPM discussions.

Going back to our original example, in a legal action, where the firm is being sued and can only lose, the expected (i.e. mean) cash flow is negative. The higher the probability of losing, the larger negative the mean.

2. Proof by example

The CAPM fact that a zero-correlation (zero-beta) payoff should be discounted at the risk-free rate is clearly correct – under CAPM. But that does not imply that the firm’s overall cost of capital will not suffer when the firm encounters a new idiosyncratic or firm-specific risk. In this regard, much accepted CAPM doctrine is misleading or wrong. An easy way to quickly prove that is by example.

Consider a firm that pays stochastic cash payoff or value $V$ at period end. The firm starts the period (the single period assumed in CAPM) with an expected terminal payoff of $E [V] = 100$ and a payoff variance of $var (V) = 1600$ .⁵ Suppose also that its payoff correlation with the market return $r_{M}$ is $ρ = 0.75$ , implying therefore that its payoff covariance with the market is

cov (V, r_{M}) = ρ \sqrt{var (V) var (r_{M})}

Let $var (r_{M})$ take the realistic amount 0.026 (based on the last 50 years of S&P 500 data, equivalent to a standard deviation of 16.1% p.a.)

\begin{array}{l} cov (V, r_{M}) = ρ \sqrt{var (V) var (r_{M})} \\ = 0.75 \sqrt{1600 \times 0.026} \\ = 4.84 \end{array}

By the usual CAPM pricing formula, as shown in Lintner (1965), Brealey et al. (2020), Grinblatt and Titman (1998), Fama and Miller (1972) and Luenberger (1998), the firm’s CAPM price at period start is

P = \frac{E [V] - k cov (V, r_{M})}{R_{f}}, R_{f} \equiv (1 + r_{f})

(1)

where $k$ is an exogenous market constant driven by the market’s aggregate risk aversion and $R_{f}$ is the return factor on the risk-free asset.

To make our calculations as realistic as possible, we use constants $k = 1.85$ and $R_{f} = 1.067$ . Although not necessary to our theory, these are calibrated like $var (r_{M})$ against the last 50 years of US financial market history.

The constant $R_{f} = 1.067$ is just the 50-year average 10-year annual bond return. To find a realistic value for the market constant k, we need some further CAPM analysis. Letting $V_{M}$ represent the aggregate market payoff (i.e. the sum of the payoffs $\underset{j}{} V_{j}$ of all stocks $j = 1, 2, \dots$ on the market), we can use equation (1) to write an equation for the price, $P_{M}$ , of the aggregate market

P_{M} = \frac{E [V_{M}] - k cov (V_{M}, r_{M})}{R_{f}}

(2)

Since

\begin{array}{l} cov (V_{M}, r_{M}) = P_{M} cov (r_{M}, r_{M}) \\ = P_{M} var (r_{M}) \end{array}

and $E [R_{M}] \equiv E [V_{M}] / P_{M}$ , equation (2) can be rewritten as

\begin{array}{l} k = \frac{E [R_{M}] - R_{f}}{var (r_{M})} \\ = \frac{E [r_{M}] - r_{f}}{var (r_{M})} \end{array}

Plugging in the 50-year empirical estimates of these three market parameters, the market $k$ is

k = \frac{0.1145 - 0.0666}{0.026} = 1.85

The firm’s asset price is therefore

\begin{array}{l} P = \frac{E [V] - k cov (V, r_{M})}{R_{f}} \\ = \frac{100 - 1.85 (4.84)}{1.067} \\ = 85.4 \end{array}

and the firm’s CAPM market-implied cost of capital is thus

E (r) = \frac{E [V]}{P} - 1 = \frac{100}{85.4} - 1 = 17.1 %

(3)

At this time, still at period start, the firm is made aware of a new risk. It receives the unwelcome news that it is subject to a legal claim by parties affected by its historical, albeit long ceased, activities (the James Hardie asbestos case in Australia is a realistic example). No other firm is affected, making this example a prototypical case of a ‘diversifiable risk’.

The class action has two possible outcomes. If the firm loses, which independent legal opinion rates as a 50% probability, it will lose all of its expected payoff this period (from damages paid and legal costs). If it wins, its payoff will be unaffected, since its legal costs will be paid by the litigants.

The court case and its effect on the firm has zero-correlation with the market aggregate, leaving the firm’s payoff covariance with the market unchanged. The relevant statistical formulae define the firm’s covariance with the market as the sum of the covariances of each of its separate activities’ payoffs with the market, and this particular activity (a court case) has zero covariance with the market, hence adding nothing to the total covariance which remains unchanged (we have assumed that the covariances of the firm’s other activities with the market are independent of the outcome of its court appearance).

The firm’s new mean payoff is now, however, $E [V] = 50$ because it now has only a 50% chance of collecting its previously assumed mean of 100 and a 50% of earning zero payoff after costs.

Upon being informed by the firm of its new risk, the market revises the firm’s price to

\begin{array}{l} P = \frac{E [V] - k cov (V, r_{M})}{R_{f}} \\ = \frac{50 - 1.85 (4.84)}{1.067} \\ = 38.5 \end{array}

and effectively imposes a new cost of capital of

E (r) = \frac{E [V]}{P} - 1 = \frac{50}{38.5} - 1 = 29.9 %

Thus, contrary to accepted but oversimplified CAPM doctrine, the firm pays for its new zero-beta risk in two ways. Its mean payoff drops from 100 to 50 and its cost of capital increases from 17% to 29.9%, the combined effect of which is a price decrease from 85.4 to 38.5.

It is hard to overstate the importance of this simple but strict CAPM analysis. It shows that the firm is indeed penalized for its ‘diversifiable’ court case risk, potentially very heavily. Moreover, it appears that the effect of the lower payoff mean on the firm’s value can swamp any effect on the payoff covariance (Best and Grauer, 1991; Chopra and Ziemba, 1993), in which case the penalty for a reduced mean can be the dominant driver of the cost of capital.

3. General explanation

The preceding example calculations show that a reduction in mean payoff with no change in payoff covariance brings a higher CAPM cost of capital.

The general CAPM law is that a lower payoff mean $E [V]$ , of itself, brings a higher CAPM cost of capital.

We can explain this result as follows. The firm’s CAPM cost of capital is embedded in its equilibrium price. Substituting for $P$ in equation (3) gives

E (R) = \frac{E [V]}{P} = R_{f} [\frac{E [V]}{E [V] - k cov (V, r_{M})}] = R_{f} {[1 - k \frac{cov (V, r_{M})}{E [V]}]}^{- 1}

(4)

This equation reveals what drives the firm’s cost of capital under CAPM equilibrium. Specifically, the only characteristic of the firm’s cash flow that affects its cost of capital is its ratio of risk to mean

\frac{cov (V, r_{M})}{E [V]}

(5)

and $E [R]$ is strictly increasing in that ratio. That is why a lower mean payoff, with no change in risk or covariance, prompts an increase in the firm’s equilibrium cost of capital. The general conclusion is that the effect of any risk depends on how it alters the firm’s ratio of payoff covariance to mean or covariance per unit of mean. That last phrase, ‘per unit of mean’, is crucial.

3.1. What about beta?

The ratio (equation (5)) is the best known Fama’s ratio because its importance in CAPM was clearly identified by Fama (1977):

. . . differences in the values of $β_{i}$ [firm beta] are the source of differences in the values of $E [{\tilde{R}}_{i, t}]$ for different firms. . . . only the ratio of $cov ({\tilde{V}}_{i, t}, {\tilde{V}}_{M, t}) / E [{\tilde{V}}_{i, t}]$ changes from one firm to another, and this ratio of covariance to expected value is the source of differences in the values of $E [{\tilde{R}}_{i, t}]$ for different firms. It follows that the relative risk measure $β_{i}$ is directly related to $cov ({\tilde{V}}_{i, t}, {\tilde{V}}_{M, t}) / E [{\tilde{V}}_{i, t}]$ , the ratio in which the firm produces contributions to the aggregate risk and to the expected value of invested market wealth.⁶ (p. 7)

The relationship between Fama’s ratio (5) and beta is easily uncovered. We first write the CAPM is its ‘returns’ form CAPM

E (r) = r_{f} + β [E [r_{M} - r_{f}]]

(6)

in which $r \equiv V / P - 1$ is the stock return and the firm’s beta is

β = \frac{cov (r, r_{M})}{var (r_{M})}

(7)

The CAPM expression (6) is merely the alternative way to write equation (1).

In what follows, we first show how the firm’s beta is driven by its ratio (5) of risk to mean, and then plug in the assumed values in our numerical example to show how a (so-called) ‘diversifiable risk’ can greatly increase beta.

Beta defined as usual in equation (7) can be rewritten using again just simple CAPM statistical algebra as

\begin{array}{l} β = \frac{cov (\frac{V}{P} - 1, r_{M})}{var (r_{M})} \\ = \frac{1}{P} \frac{cov (V, r_{M})}{var (r_{M})} \\ = \frac{R_{f}}{E [V] - k cov (V, r_{M})} \frac{cov (V, r_{M})}{var (r_{M})} \\ = \frac{R_{f}}{var (r_{M})} {[\frac{E [V]}{cov (V, r_{M})} - k]}^{- 1} \end{array}

(8)

Hence, it appears that beta is driven by a firm characteristic that is more fundamental than itself, namely, the same ratio of risk to mean

\frac{cov (V, r_{M})}{E [V]}

that was revealed above.

To complete our demonstration, let’s find the firm’s beta before and after the news of the legal action. Plugging in the relevant values, the pre-news beta is

\begin{array}{l} β = \frac{R_{f}}{var (r_{M})} {[\frac{E [V]}{cov (V, r_{M})} - k]}^{- 1} \\ = \frac{1.067}{0.026} {[\frac{100}{4.84} - 1.85]}^{- 1} \\ = 2.18 \end{array}

Similarly, after the news, when the only change is a drop in expected payoff to 50, the post-news beta is

\begin{array}{l} β = \frac{R_{f}}{var (r_{M})} {[\frac{E [V]}{cov (V, r_{M})} - k]}^{- 1} \\ = \frac{1.067}{0.026} {[\frac{50}{4.84} - 1.85]}^{- 1} \\ = 4.84 \end{array}

Plugging these calculations of beta into the usual CAPM returns equation (6), the pre- and post-news costs of capital are

\begin{array}{l} E [r] = r_{f} + β [E [r_{M}] - r_{f}] \\ = 0.067 + 2.18 (0.115 - 0.067) \\ = 17.1 %, \end{array}

and

\begin{array}{l} E [r] = r_{f} + β [E [r_{M}] - r_{f}] \\ = 0.067 + 4.84 (0.115 - 0.067) \\ = 29.9 % \end{array}

respectively, exactly as obtained already. It is fair to say we suggest that no CAPM exposition is complete without the new expression for beta shown in equation (8), which to our knowledge first appeared in Johnstone (2017).

The end result is that we have demonstrated coherently that a ‘diversifiable risk’ can add markedly to the firm’s beta and cost of capital, contrary to conventional wisdom and accepted finance doctrine.

3.2. ‘But diversifiable risks have zero-beta’

A risk or risky payoff that has zero-correlation with the market has zero-beta, so how can the firm’s beta increase, and its cost of capital increase accordingly, when it takes on such a risk? The answer again takes only simple CAPM algebra.

Treat the firm as the sum of its old self and a new zero-beta business activity called ‘court case’. Its new beta is therefore the price-weighted average of the old business’ beta of 2.18 and the court case’s beta of zero.

The old business has price 85.4. The court case has a negative price given by its expected cash flow of −50 discounted at the risk-free rate (since it has zero-beta).⁷ Thus

P_{court case} = \frac{- 50}{1.067} = - 46.9

which is of course the firm’s lost value, 85.4 − 38.5. So its new beta is the price-weighted average

\frac{85.4 (2.18) - 46.9 (0)}{38.5} = 4.84

as already found.

This analysis has an important general implication. Specifically, any firm-specific activity or risk that has negative expected payoff, and which is uncorrelated with the market, will increase the firm’s cost of capital.

Such risks come as part and parcel of many businesses. For example, the highly firm-specific events of the star CEO having a heart attack, or the company’s product causing injury, or an employee being harmed by the firm’s work environment are negative expected payoff zero-beta⁸ events. The higher the probability of such firm-specific loss, the higher the firm’s cost of capital.⁹

We claimed in our introduction that firms can be identical except for their different diversifiable risks and still have different costs of capital. Moreover, the firm with more diversifiable risk can have the higher cost of capital. That is evident when we think of our example firm before and after it learns of its new litigation risk. The only difference between the ‘two firms’ lies in their respective diversifiable risks, and yet the firm facing an additional diversifiable (zero-beta) risk has the much higher cost of capital.

4. To whom are firm-specific risks costless?

There must surely be a sense in which so-called diversifiable risk is truly diversifiable, since by the central limit theorem the outcomes of the many idiosyncratic activities of the many firms in the market must tend to cancel each other out.

Our interpretation is that ‘diversifiable risk’ is essentially costless at the market level, but not at all at the firm level. That is not a contradiction because the firm is understood as negligibly small relative to the market aggregate, so its cost of capital can increase greatly with no discernible effect on the market cost of capital.¹⁰

Consider once more the risk of just one firm facing a major legal challenge that may cause it great loss, albeit not having any effect on other firms in the market. That potential firm loss does three things at the market level. First, it adds nothing to the variance of the market payoff $var (V_{M})$ because it is uncorrelated (has zero-correlation) with the market payoff. Second, it has a very small negative effect on the mean of the market payoff, but that effect is negligible in a large market. Finally, it has negligible effect on the total price of the market $P_{M}$ because the firm is so small relative to the market.

The net effect is that the ratio (5) applicable at the market level

\frac{cov (V_{M}, r_{M})}{E [V_{M}]} = \frac{1}{P_{M}} \frac{var (V_{M})}{E [V_{M}]}

(9)

is virtually unchanged, which means that the market’s expected return (i.e. its cost of capital) is unaffected by the firm’s legal risk. So, in a practical sense, the firm’s legal risk is ‘diversifiable’ and inconsequential in the eyes of investors who hold the market portfolio. The firm’s firm-specific risk makes no practical difference to the market cost of capital, despite greatly adding to its own cost of capital.

4.1. Many zero-beta risks at market level

The argument that a firm’s legal exposure, and other firm-specific zero-beta payoff risks taken in the market, is costless needs clarification. If all firms have such risks of large loss, the market-level cost of capital is clearly affected despite the fact that, according to the law of large numbers, the variance of the sum of payoffs from these risks has variance approaching zero.

Let’s consider how this can occur in a hypothetical context where all firms take on zero-beta risks of large loss. Perhaps the best example of a zero-beta risk is a bet on the roulette wheel. Apart from the fact that the casino’s probability of paying out on winning bet might be very slightly dependent on the economy, this bet is as close as imaginable to a paradigmatic diversifiable risk.

Suppose that every firm on the stock market announces that it has plans for a long night at the local casino and intends to make repeated $$ X$ bets on Red.

Each $$ X$ bet is on a different independent spin and has negative expected money value. The variance of the sum of all firms’ payoffs approaches zero, by the law of large numbers; however, the sum of expected payoffs is heavily negative.

If the market is informed of this forthcoming mass outing to the casino, the market’s forward-looking Fama ratio (equation (9)) is unchanged in its risk parameter $var (V_{M})$ , but has greatly reduced mean $E [V_{M}]$ . The market cost of capital is thus higher, despite these new risks being as purely idiosyncratic zero-beta risks as might ever exist.

5. Fama (1977)

There is much research on empirical estimates of beta and beta’s statistical association with other characteristics of the firm; however, there has been little consideration of what CAPM itself says directly about the fundamental parameters or characteristics of the firm that determine its beta.

As far as CAPM is concerned, the fact evident in equation (8) is that beta is driven fundamentally by the ratio of the firm’s payoff covariance to mean

\frac{cov (V, r_{M})}{E [V]} \equiv \frac{1}{P_{M}} \frac{cov (V, V_{M})}{E [V]}

(10)

Fama’s CAPM logic was clarified in a contentious yet hugely cited paper by Lambert et al. (2007). That paper sets out straightforward CAPM payoffs-based algebra revealing the importance of the Fama ratio (10) as the minimal sufficient determinant of the firm’s cost of capital.

There is, however, little recognition in common CAPM expositions and applications of the role of the payoff mean $E [V]$ . An everyday example occurs in capital budgeting where the expected future cash payoff $E [V]$ is discounted at some given rate based on an off-the-shelf plug-in beta (e.g. an industry beta) regardless of its amount $E [V]$ . Any re-assessment of the amount of the expected payoff is taken as irrelevant to beta and the discount rate, thus explicitly violating Fama’s CAPM logic. Put another way, the practice is to calculate the asset price as the discounted present value

P = \frac{E [V]}{E [R]}

(11)

with some arbitrary plug-in discount factor E[R] chosen independently of the amount $E [V]$ ; for example, $E [R]$ might be the documented industry average discount factor. Fama’s CAPM exposition shows that this practice is not theoretically correct under CAPM because the numerator $E [V]$ affects the cash flow beta and hence affects the theoretically appropriate discount factor $E [R]$ .¹¹

Finance discussions tend to dwell on empirical determinants (or mere ‘correlates’) of the firm’s cost of capital (i.e. the market-imposed discount rate), rather than extracting information about those determinants from the CAPM itself.

Consider the question in Patton and Verardo (2012) concerning how betas react to firm-specific information. According to innate CAPM logic, the market will use information to reassess both the firm’s payoff mean and its covariance with future market conditions, and then reset beta accordingly, as per Lambert et al. (2007) and Fama (1977). Good and bad earnings surprises will tend to have opposite effects on the cost of capital according to their respective positive and negative effects on the assessed payoff mean. Ultimately, the firm’s new beta and cost of capital will hinge on the new value of (10).

Patton and Verardo (2012) set out a CAPM-based argument to warrant their empirical findings. In short, their high-frequency beta estimates, based on 25-minute observation intervals, indicate that firm betas (measured as short-term empirical returns correlations with the market return) increase during the short time interval of the firms’ earnings announcements. The explanation put is that firms announcing their earnings impart information from which the market learns about both the firm’s future prospects and the wider economy’s future prospects. Good (Bad) news about the firm is held to imply Good (Bad) news for the market as a whole, meaning that any firm’s earnings news, favourable or unfavourable, prompts an increase in the market’s assessment of the covariance between that firm’s profitability and the market’s future profitability, thereby increasing the firm’s beta. This ‘bellwether’ property of firm’s earnings, especially some firms,¹² is claimed to hold regardless of whether the news is Good or Bad, and to be stronger when the news is stronger.

This interpretation is clearly at odds with the CAPM’s internal logic. First, good and bad news about the firm will generally affect its expected future cash flow, and thus alter the denominator of (10), in one or other direction. Second, it is plausible that the firm’s earnings report will induce a lower assessed covariance (e.g. that would occur when a large favourable earnings surprise is attributed to the firm’s idiosyncratic profits, such as profits coming from a firm-specific technological breakthrough and occurring in an otherwise poor economy). Third, as emphasized by Lambert et al. (2007), the potential resolution of uncertainty provided by the earnings report can reduce the assessed covariance of the firm’s future earnings with the market (in theory, complete resolution would lead to certainty and zero-beta). Each of these three CAPM corollaries is essential to answering the Patton and Verardo (2012) question of how information arrival affects CAPM beta.

6. Ingersoll (1987)

Fama’s explanation of the drivers of CAPM beta seems widely unknown and also widely contradicted. A search of finance literature finds practically no mention of how the beta or systematic risk of a payoff is affected by its mean.

Reassuringly, however, this very point arises in the highly general decision theoretic CAPM exposited by Ingersoll (1987). Ingersoll’s CAPM admits any asset payoff distribution $f (V)$ and any utility function $u (\cdot)$ , and subsumes the conventional mean-variance CAPM as just one of the models admitted. In that context, Ingersoll (1987) gave the general definition of the ‘beta’ (p. 134) of asset $i$ as

b_{i} = \frac{cov (u^{'} (r_{M}), r_{i})}{cov (u^{'} (r_{M}), r_{M})}

where $r_{m}$ is the price-weighted average market return and $r_{i}$ is the individual asset return. An asset’s ‘Ingersoll beta’ is thus a standardized measure of the covariance of the asset’s return with another random variable, namely, the marginal utility of the market portfolio. Assuming to be typically positive, $cov (u^{'} (r_{M}), r_{M})$ , the necessary condition for a risk to have zero-beta, $b_{i} = 0$ , is

cov (u^{'} (r_{M}), r_{i}) = 0

(12)

and for that condition it is not sufficient that the risk is uncorrelated with the market return:

It is not necessarily true in this general context that any risk that is uncorrelated with returns on the market portfolio is non-systematic in nature. (Ingersoll, 1987: 208)

This is a highly important passage for how we understand ‘non-systematic’ risk in general and in the case of CAPM specifically.

To interpret his requirement (12) so as to understand conditions that imply $b_{i} = 0$ , Ingersoll (1987: 207–208) used standard statistical relationships and the concept of a conditionally fair game (Ingersoll, 1987: 15) to show that the covariance in equation (12) can be written as

\begin{array}{l} cov (u^{'} (r_{M}), r_{i}) = E [u^{'} (r_{M}) r_{i}] - E [u^{'} (r_{M})] E [r_{i}] \\ = E [E [u^{'} (r_{M}) r_{i} | r_{M}]] - E [u^{'} (r_{M})] E [r_{i}] \\ = E [u^{'} (r_{M}) E [r_{i} | r_{M}]] - E [u^{'} (r_{M})] E [r_{i}] \end{array}

(13)

For the purposes of our analysis, we can write Ingersoll’s condition (13) in terms of the firm’s payoff $V_{i}$ rather than return $r_{i}$

\begin{array}{l} cov (u^{'} (r_{M}), r_{i}) = E [u^{'} (r_{M}) E (\frac{V_{i}}{P_{i}} - 1 | r_{M})] - E [u^{'} (r_{M})] E [(\frac{V_{i}}{P_{i}} - 1)] \\ = P_{i}^{- 1} (E [u^{'} (r_{M}) E [V_{i} | r_{M}]] - E [u^{'} (r_{M})] E [V_{i}]) \end{array}

(14)

It follows from (14) that it is sufficient for $cov (u^{'} (r_{M}), r_{i}) = 0$ and hence $b_{i} = 0$ that:

(i) The expected payoff on the asset is independent of the market, implying $E [V_{i} | r_{M}] = E [V_{i}]$ for all possible $r_{M}$ ,

and

(ii) The expected payoff from the asset is zero, $E [V_{i}] = 0$ .

The first of these conditions, the payoff’s independence of the market, is part of the well-known understanding of an apparently non-systematic or diversifiable risk. The second condition is the far less obvious key to a full grasp of ‘diversifiable risk’, understood as risk that does not alter the firm’s cost of capital.

Condition (ii) brings Fama and Ingersoll together, and points to a deeper theoretical understanding of the risks that are not penalized and make no difference to the firm’s cost of capital. In terms of Fama’s ratio, conditions (i) and (ii) together imply that the risk being considered has zero effect on both the firm’s payoff mean and its payoff variance. Thus, the Fama ratio (10) stays unchanged, and the firm has unchanged beta and unchanged cost of capital.

Although (i) and (ii) are merely sufficient conditions for the firm’s cost of capital to remain unchanged, they are in practice necessary because any other conditions with the same outcome are pathological. Inspecting equation (13), we can see via the presence of the minus sign that a coincidence could leave $cov (u^{'} (r_{M}), r_{i}) = 0$ when (i) and (ii) are both untrue, but there is no general basis for that ever happening other than by some ‘lucky’ self-compensatory change in the variables in equation (13).

Returning to our examples of prototypical ‘diversifiable risks’ (e.g. the firm being sued by customers or the CEO being eaten by a shark), condition (ii) is not met (e.g. the threat of losing the court case has a negative mean payoff, not a zero-mean payoff).

Such risks do not fit Ingersoll’s logically rigorous definition of diversifiable risks because they are not ‘fair games’, that is, specifically, they do not have expected (average) payoffs equal to zero. Instead, their mean payoffs are negative, meaning that despite having zero-correlation with the market, they increase the firm’s beta and cost of capital, as illustrated above. Thus, according to Ingersoll and Fama, the usual textbook version of an unpriced ‘diversifiable’ risk is best misleading.

Note that the ‘Fama condition’ is slightly more relaxed than Ingersoll’s conditions because Ingersoll’s (i) and (ii) require that the risk in question has zero impact on both of the firm’s payoff parameters $E [V]$ and $cov (V, r_{M})$ , whereas Fama’s ratio requires only that the ratio of those parameters, $cov (V, r_{M}) / E [V]$ , remains unchanged, which might occur by ‘coincidence’ when both parameters change.¹³

7. Improving CAPM intuitions

There is much that goes unknown or unstated about the CAPM concept of diversifiable risk and about what drives the CAPM cost of capital. It was shown above that the firm’s cost of capital is driven fundamentally by its Fama ratio of payoff covariance to payoff mean. The importance of that ratio was raised by Fama (1977) but let go, and, to our knowledge, was not elaborated upon in any detail until Lambert et al. (2007), which unfortunately did not cite Fama’s very closely related paper. The deeper version of Fama’s argument is found, as we have shown, in Ingersoll’s treatment of diversifiable risk.

The pleasing aspect of a better CAPM understanding that abides by Fama’s and Ingersoll’s expositions is that many CAPM implications become clear and neat. Some of these are as follows.

7.1. Idiosyncratic risk and the firm’s cost of capital

The message in this article can be verified by an intuitive CAPM argument. Take the firm’s payoff as the sum of the payoffs from its separate assets or sources of cash flow, each of which has its own CAPM price. The firm’s CAPM cost of capital is then the price-weighted average of the costs of capital of each of its assets. That is a well-known CAPM insight, based on the mathematical fact that CAPM asset prices are additive (cf. Luenberger, 1998).

Any payoff with zero market correlation is discounted correctly under CAPM at the risk-free rate. Also assume realistically that the firm’s overall cost of capital is higher than the risk-free rate. Now, if we add a new idiosyncratic payoff to the firm, specifically a payoff with zero market correlation and negative (positive) expected payoff, its CAPM price is negative (positive) and its effect on the firm’s price-weighted average cost of capital is unequivocally positive (negative).

The threat of an industrial accident, or damages payable if the firm is sued by customers or regulators, the threat of cybercrime attack and payouts, or one of the many other conventional ‘idiosyncratic risks’, usually by their nature have negative mean payoff and therefore cause an increase in the firm’s cost of capital.

If the firm becomes subject to a potentially ruinous idiosyncratic risk, its cost of capital can in CAPM principle increase massively. That conclusion is what marks our article apart from how the CAPM is commonly interpreted but is very easily confirmed.

7.2. Death of star CEO

One of the commonly cited firm-specific risks is that the CEO dies unexpectedly. There is empirical evidence of cases where such events have brought a downward shock to the firm’s stock price (e.g. Johnson et al., 1985; Larcker and Tayan, 2012; Salas, 2010).¹⁴

The unexpected demise or possibly resignation of a key employee is easily interpreted in a CAPM sense. There are two parameters that the market must revise when the bad news arrives. The obvious effect is on the firm’s expected or mean future payoff, which is naturally revised downwards when the news is bad. The less obvious effect is on the market’s forward-looking assessment of the firm’s payoff covariance with the market.

It is possible that the revised payoff covariance is rationally (Bayesianly) higher or lower.¹⁵ The CEO’s abilities, specialized knowledge or overall style might have contributed to a highly idiosyncratic income stream or to an income stream that is largely immune to economic downturns, either of which translates to a low payoff covariance with the market. If the CEO was sufficiently contrarian, the firm’s payoff covariance might have been not merely low, but negative. The loss of that CEO will lead the market to reassess the firm’s payoff market covariance upwards. Alternatively, if the departed CEO built real options like excess spare plant capacity that ‘come into the money’ in good times, a more conservative business model might produce future cash flows that have lower covariance with the economy and market. The revised covariance can clearly be higher or lower.

Ultimately the demise of the CEO and the prospect of a new CEO will cause immediate revisions in both the numerator (covariance) and denominator (mean) in the firm’s Fama ratio (10) and might lead to either an increase or decrease in the firm’s cost of capital. The fact that the Fama ratio of payoff covariance to mean might change up or down with such effects was explained in a related context by Lambert et al. (2007: 394).

7.3. Pure play

Consideration of Fama’s ratio brings clarity across much of everyday conversational finance. Let’s start with the widely used idea of a ‘pure play’. A pure play is loosely understood as a business activity which is internally very ‘undiversified’. An example might be a retail business that does nothing apart from buying shoes and selling shoes, all sales being made at the shop (an on-line seller is perhaps a different ‘pure play’ than a shopfront retailer).¹⁶

The assumption in finance is that a given pure play (Fuller and Kerr, 1981) has a given beta and inherent CAPM cost of capital, in the sense that two firms that both sell shoes from shops have ‘the same risk profile’ in a usually ill-defined sense.¹⁷ Fama’s logic questions the existence and even concept of a ‘pure play’. Two shoe shop businesses can have very different costs of capital because their differing fundamentals (e.g. operating leverage) imply very different Fama ratios.

It is well-known in finance that operating leverage affects beta. That effect can be identified most rigorously by considering how a change in operating leverage affects, respectively, the numerator and denominator in Fama’s ratio (10). The usual intuition is that a firm with higher fixed costs is higher risk and has higher beta and cost of capital.¹⁸ However, it is immediately evident from Fama’s ratio that an investment in fixed costs that succeeds by bringing a very large reduction in unit variable costs can increase the firm’s overall mean payoff by such an amount that its Fama ratio of payoff covariance to mean, and thus its beta, is reduced. That insight goes against conventional wisdom but is very easily obtained by thinking in terms of the firm’s Fama ratio.

In summary, the only way that two businesses can be said to have intrinsically the same CAPM cost of capital is for them to have the same Fama ratio. They can be in utterly different industries, and yet still have the same Fama ratio. Conversely, they can be in the very same line of business and yet have greatly different Fama ratios.

7.4. Twin securities

In the theory of real options, there is a mathematically well-defined definition of a ‘twin security’. To price an option on a real business activity, ‘real options’ theory considers the cost of capital on a ‘twin security’. The concept of a ‘twin security’ is more rigorous than the more mainstream concept of a matching ‘pure play’, but can be clarified further in terms of Fama’s ratio.

There are several wordy definitions in the real options literature of a ‘twin security’. The most rigorous definition (e.g. Trigeorgis, 1996) is that two businesses are twin securities if their respective random cash payoffs are perfectly correlated (p. 17). Thus, if the random payoff from some business is $V$ , then its twin security is a business that pays amount $a + c V$ where $c$ is a positive constant. It is important to note, however, that the ‘perfectly correlated payoffs’ definition satisfies the inherent CAPM concept of a twin security if and only if $a = 0$ (and $c > 0$ ).

The Fama ratio of payoff $a + c V$ is by basic statistical laws

\frac{cov (a + c V, r_{M})}{E [a + c V]} = \frac{c cov (V, r_{M})}{a + c E [V]}

which equals $cov (V, r_{M}) / E [V]$ only when $a = 0$ . Thus, the payoffs must be directly proportional to one another.

Note that in the example calculations shown in Trigeorgis (1996: 153), the two businesses defined as twins have payoffs that are indeed proportional to one another, with a constant proportionality coefficient $c = 0.2$ , so they are indeed twins under CAPM.

Given that payoffs must be not merely perfectly positively correlated but must in fact be proportional with constant $c > 0$ , it is potentially misleading that the real options literature defines twin securities as simply ‘perfectly correlated’ (e.g. Copeland and Antikarov, 2001).

Confusion occurs, we suggest, because perfect correlation is not sufficient to ensure that assets have the same beta and cost of capital but is sufficient to allow the construction of a replica asset called a ‘twin portfolio’ (Trigeorgis, 1996). A twin portfolio is a portfolio of one asset combined with risk-free bonds that yields the same payoff, always, as the asset or business that it replicates. That portfolio, by construction, has the very same Fama ratio as the single asset replicated, but the two underlying single assets are not twins (because they don’t have the same Fama ratio).

This connection with Fama’s (1977: 7) explication of the ratio driving CAPM beta is not raised in the literature. Our conclusion once again is that it helps conceptually to call on Fama’s discussion. Two assets are mathematically ‘twin securities’ and have the same CAPM beta, if and only if they have the same Fama ratio – being perfectly correlated is not sufficient.

7.5. Firms in the ‘same risk class’

There are literally hundreds of statements in finance literature referring to firms having the ‘same risk profile’, being in the same ‘risk class’ or having ‘comparable risk’. Despite their lack of definition, these statements usually go unquestioned because they seem simple, right and obvious. For example,

Despite broad agreement that the cost of capital represents investors’ required expected rate of return on investment, which represents investors’ best available expected return on a comparable investment offered in the market, computing the cost of capital is not straightforward. (Jagannathan et al., 2017: 260)

Part of the difficulty of assessing a firm’s cost of capital is surely the problem of defining what is meant by a ‘comparable’ investment or one with the ‘same risk profile’ (which is the common catchphrase).

The CAPM fact is that any two uncertain future payoffs fall in the same risk class, or have the same risk profile, if and only if they have the same Fama ratio. We can say that they must have the same beta, but the Fama ratio is more informative than beta because it is expressed in terms of the underlying parameters that drive beta.

To see that this analysis does not fit easily with common CAPM teaching, consider this – has anyone ever seen an exam question in finance that asks the candidate to explain how two payoffs, one with very high covariance or correlation with the market and one with very low covariance with the market, can have the same beta? That question seems to be some kind of trick, or just unanswerable, but is immediately solved by reference to Fama’s ratio – the simple answer is that the first payoff has much higher mean.

7.6. Effect of firm’s operating leverage

The firm’s operating leverage, or ratio of fixed costs to variable costs, will obviously affect both the mean and market covariance of its payoff. Increases in operating leverage are usually understood as increasing the firm’s systematic risk and cost of capital (e.g. Brealey et al., 2020: 237–238). A more thorough analysis is easily obtained using Fama’s ratio. The short answer is that changes in the firm’s operations of any fundamental nature can drive its Fama ratio up or down. Suppose, for example, that higher fixed costs bring such productive efficiency and hence lower variable cost per unit that the firm’s mean payoff is greatly increased. If that effect overcomes any accompanying increase in the firm’s payoff covariance with the market, then the higher operating leverage may bring a lower Fama ratio and hence a lower CAPM cost of capital (Johnstone, 2020).

8. Pedagogical convenience

In a famous paper by Watts and Zimmerman (1979) on the ‘market for excuses’, the authors modelled theory as an economic good and set about identifying a positivist explanation of what drives the supply of, and demand for, ‘theories’.

One of the more unexpected ideas is that theory-providers and theory-supporters are deeply influenced by what is easy to teach and goes over well with the audience, for example, the MBA student market or the financial analyst profession (Watts and Zimmerman, 1979: 278–279).

This argument fits well with how we see CAPM pedagogy. The messages drawn from CAPM, at least in undergraduate teaching, are very seductive and also greatly oversimplified, with the effect that they are apparently very insightful but not difficult to teach or understand, albeit that they are wrong (see above).

The very notion of diversifiable versus undiversifiable risks is one of the salient attractions of CAPM. Students are enthralled with concepts and words like ‘unpriced risk’ and are usually very pleased when they can interpret the world through such a different hegemony or language. At a human behavioural level, they obtain a newfound self-assurance as they head towards the job market, often delighted with what they have learned.

This proven behavioural appeal of CAPM and its language was why Jagannathan et al. (2016: 458) intentionally omitted the words ‘idiosyncratic risk’ from their survey instrument – they wanted to prevent respondents from showing their preconditioning by offering pat answers straight from the CAPM narrative they absorbed in their MBAs.

In terms of pedagogical supply and demand, there is a good reason for finance instructors not to sacrifice classroom satisfaction by complicating the CAPM message in any way that makes it more rigorous and also harder to understand and teach. Instead, the CAPM is usually articulated at a level that combines apparent rigour with enough insight and ease of understanding to suit the intellectual tastes of a wide audience.

It is also important to remember that the CAPM was a revolution in its day (Bernstein, 1992), upsetting large parts of the academic establishment and finance industry, and was therefore hard to ‘market’ as a theory in the face of entrenched commercial interests, not to mention decades of accepted finance practice and lore. Modern finance based on CAPM, portfolio theory, options pricing and the other big ideas in finance led to new industries and displaced old ones. To win supporters and fend off criticism from disaffected practitioners, and some academic economists, it is natural that a theoretically watered down and popularized explanation of CAPM and its internal logic and corollaries took hold.

Paradoxically, our explanation of how CAPM ‘prices’ diversifiable risks brings CAPM instruction closer to practical managers’ intuitions. Put very simply, when a firm is confronted with a zero-beta legal liability risk, its CAPM cost of capital can jump greatly, the more so the higher the expected cash outlay to the litigants. That theoretical result fits with even the most untrained intuition. If a firm is sued for a large part of its value and has a 50% chance of losing, its survival is effectively a coin toss that must intuitively leave it with a high cost of capital or risk premium. Our logical interpretation of CAPM agrees with that intuition.

8.1. How the misconception hides from view

When introducing the finance idea of diversifiable unpriced risk, we quoted the clear but misleading argument in Brealey et al. (2020) that warns against ‘being fooled by diversifiable risk’. To finish, we return to this book to show how it hides the correct CAPM interpretation. The explanation given by Brealey et al. (2020: 243–44) is that we can find the price of a risky venture by two equivalent methods. One method (detailed in a footnote) is to use the payoffs equation (1), where the numerator is described as a certainty equivalent. The other is that we can use the shortcut equation (11) by plugging in a discount factor $E [R]$ that includes both a risk premium and a premium for time or ‘waiting’.

Although these appear to be interchangeable alternatives, the problem is that they are not equivalent unless the discount factor $E [R]$ plugged into equation (11) is the very same discount rate as that implied by the price obtained when first finding P from equation (1) and thus extracting the CAPM price-implied discount factor $E [V] / P$ .

So, we cannot implement the shorthand method equation (11) without first obtaining the price-implied discount factor $E [R]$ from equation (1). Only after $E [R]$ is found from equation (1) and plugged into equation (11) can the two methods look as if they are the same. Any claim that they are the same is deceiving.

It is fair to say that the Brealey et al. (2020) explanation is a pedagogically convenient over-simplification that leaves everything seemingly holding together and produces some seductive conclusions around ignoring firm-specific risks, but ultimately misrepresents the economic substance of the CAPM.

All such misconceived interpretations of CAPM were explained and anticipated in the following quote by Lambert et al. (2007). The near-ubiquitous mistake of CAPM principle is to equate the firm’s return covariance and payoff covariance as if they are linear reflections of each other:

Because the CAPM is expressed solely in terms of covariances, this formulation might be interpreted as implying that other factors, for example, the expected cash flows, do not affect the firm’s cost of capital. It is important to keep in mind, however, that the covariance term in the CAPM is expressed in terms of returns, not in terms of cash flows. The two are related via the equation

cov ({\tilde{R}}_{j}, {\tilde{R}}_{M}) = cov (\frac{{\tilde{V}}_{j}}{P_{j}}, \frac{{\tilde{V}}_{M}}{P_{M}}) = \frac{1}{P_{j} P_{M}} cov ({\tilde{V}}_{j}, {\tilde{V}}_{M})

This expression implies that information can affect the expected return on a firm’s stock through its effect on inferences about the covariances of future cash flows, or through the current period stock price, or both. (Lambert et al., 2007: 390)

A remarkable aspect of the Lambert et al. (2007) paper is that it is cited more than 2750 times, and yet its fundamental message about what drives beta and the cost of capital is largely ignored – a recent exception to this oversight is Ellahie et al. (2021). The same point has been known and ignored, or just not known, since at least Fama (1977). It appeared also in Hull (1986: 446), in close to the same equations as Lambert et al. (2007).

9. Summary

The common CAPM narrative says that if the managers of a firm decide to bet a large amount on a longshot at the races, the firm-specific risk will be diversified by the market and the firm will not be penalized by being charged a higher market risk premium for its indulgence.

We argue, to the contrary, that the firm’s vow to bet large at the track will cause it a heavy penalty in its CAPM cost of capital while at the same time having negligible effect on the market’s overall risk premium.

The insights in this article contradict conventional wisdom but are easy to prove. They appear very simply, merely by interpreting CAPM via its payoffs or ‘certainty equivalent’ expression rather than by the far more usual returns format. These two alternative CAPM expressions are of course mathematically equivalent statements of the very same asset pricing model. Each can be derived from the other, but each reveals explicitly something new about what CAPM tells us.

When the firm faces a highly firm-specific risk with a potentially large payout (e.g. it pollutes groundwater and might be sued by regulators), the risk of a compensation payout is firm-specific and idiosyncratic (zero-beta). Viewed as an unknown future cash payout, it clearly has a negative mean, which means mathematically that its effect on the firm’s weighted average cost of capital must be upwards despite the fact that it is discounted at the risk-free rate (assuming of course that other parts of the firm have costs of capital higher than $r_{f}$ ).

A prototypical ‘diversifiable risk’ can cause a very large increase in the firm’s cost of capital yet make no discernible difference to the market cost of capital. That apparent contradiction occurs merely because the firm is so small relative to the market aggregate. While negligible at market level, the firm’s diversifiable risk is highly relevant to its own cost of capital.

A common mistake is to interpret a small firm’s negligible market level effect as implying a negligible penalty at the individual firm level. If the firm is a tiny weight in the market, a large increase in its cost of capital won’t be noticed at the aggregate market level.

A simple way to understand any threat of firm-specific loss is that it represents merely another source of random cash flow, specifically one with (1) negative CAPM price, because of its negative mean, and (2) a cost of capital equal to just $r_{f}$ (since its beta is zero). That new business, or newly recognized potential for loss, means that the firm’s overall cost of capital, given by the price-weighted average of the costs of capital of all its internal cash flow sources, must necessarily increase.

Any imaginable newly recognized loss with positive probability and zero-beta is known in textbooks as a ‘diversifiable risk’. For example, a drug company markets a new drug while worrying that it has negative side effects that may come to the fore when marketed widely. That new risk is nearly entirely ‘downside’ because its ‘payoff’ is highly unlikely to be positive. Like most similar risks, it has a negative expected payoff and zero-beta, and must under CAPM equations add to the firm’s cost of capital.

Finally, let us return to the original question of what it means to say that a risk is ‘unpriced’. That expression usually goes without any clear definition. One interpretation is that a risk is ‘unpriced’ if it makes no difference to the firm’s overall cost of capital. In this article, we say two things regarding why such interpretation is misleading. First, any risk that is truly uncorrelated with the market (i.e. zero-beta) must be discounted at the risk-free rate and might therefore be expected to lower the firm’s overall cost of capital. But second, if the expected payoff of the event in question is negative, then it has a negative CAPM price or weight, and the combination of being discounted at the risk-free rate and having a negative weight causes the firm’s cost of capital to increase.

Footnotes

Acknowledgements

We wish to thank the editor & anonymous referees for helpful comments and encouragement.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

ORCID iDs

Andrew Grant

David Johnstone

Notes

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