Risk Models: Cost of Capital in Perspective, and the Limits of and Alternatives to CAPM.
Daniel Rivera Alves
Founding Partner, Elit Capital
Why there is no “free lunch”
There is a phrase that has crossed decades in financial markets and is often repeated without anyone knowing exactly what it means or where it comes from. In investing, diversification is often described as the only “free lunch” available[1]. The expression suggests something seemingly counterintuitive: that it is possible to reduce a portfolio’s risk without reducing its expected return in the same proportion, as long as the assets combined do not behave the same way.
This popular formulation, however, does not accurately describe what the mathematics shows. A portfolio’s expected return is exactly the weighted average of the expected returns of its assets; it suffers no proportional penalty at all. What is reduced, sub-additively, is the portfolio’s standard deviation, whenever the correlation between assets is below 1 (ρ < 1), with positive weights. Return aggregates linearly; risk does not. It is in this asymmetry, and not in any proportional trade-off, that the supposed free lunch lies.
This idea seems to offer a solution to every investor’s central dilemma: how to reduce exposure to risk without giving up return. This is precisely where the provocation of this section begins. If diversification were really as guaranteed a lunch as it seems, why do recurring financial crises show supposedly diversified assets falling together exactly when protection is most needed? The answer does not lie in any flaw of the algebra, which is impeccable, but in the assumption that the correlation (ρ) between assets remains stable over time, including in times of stress.
This is exactly why this discussion matters to those who value companies. Every discount rate used in M&A inherits the assumptions of a theory built for portfolios of liquid stocks. Understanding where these assumptions come from, and where they fail, is the problem this section sets out to clarify, before applying them to the cost of capital of a real corporate transaction.
CAPM: a first approximation
Before going deeper into the theoretical origin of the “free lunch”, it is worth placing the concept that derives directly from it and that every M&A professional knows well, present in practically every valuation report: CAPM, the Capital Asset Pricing Model[2].
In simple terms, CAPM seeks to identify the return an investor should require to accept the risk of investing in a given asset or company. The answer is given by an equation that adds to the risk-free rate — the return on an investment considered to carry the lowest risk in the market, such as government bonds — a risk premium. This premium is the product of the price of risk in the market (the equity risk premium, or ERP) and the asset’s amount of non-diversifiable risk (its beta). It is important to separate the two concepts from the outset. Beta does not measure the premium; it measures the asset’s sensitivity to market fluctuations. The ERP is the price of that risk. This distinction matters again later, when we discuss Aswath Damodaran’s implied ERP.
This required return corresponds to what, in valuation, we call the cost of equity (Ke), a central element in the calculation of the WACC, the Weighted Average Cost of Capital. The WACC combines the cost of equity with the after-tax cost of debt, and it is precisely this tax deductibility of interest that is the economic reason why a company’s capital structure affects its value:
WACC = [E/(D+E)] × Ke + [D/(D+E)] × Kd × (1 − t)
where E and D are, respectively, the market values of equity and debt, Kd is the pre-tax cost of debt and t is the marginal tax rate. Omitting the (1 − t) term is one of the most common mistakes in M&A valuations, and one of the most costly to a report’s credibility.
The WACC is used to discount a target company’s future cash flows to present value, and is fundamental in corporate valuations and in mergers and acquisitions. Behind practically every Discounted Cash Flow spreadsheet used in M&A, therefore, lies an application, often implicit, of CAPM.
CAPM did not emerge in isolation. It is the logical consequence of a broader theoretical framework developed more than a decade earlier: Modern Portfolio Theory.
Modern Portfolio Theory: the mathematical foundation
Modern Portfolio Theory, formulated by Harry Markowitz in 1952[3], is considered the founding milestone of modern quantitative finance. Its essential contribution was not merely to reinforce the common sense that diversifying is good, but to demonstrate mathematically that a portfolio’s risk depends on the covariance between its assets, and not just on the sum of each asset’s individual risk in isolation.
From this insight comes the very “free lunch” discussed at the opening of this section. By combining assets with correlation below 1, Markowitz showed it was possible to build an Efficient Frontier, a geometric curve gathering all combinations of assets that offer the highest possible return for each level of risk taken.
It was on this frontier that William Sharpe built, in 1964[4], the CAPM presented in the previous section. By introducing the risk-free asset and connecting it to Markowitz’s Efficient Frontier, Sharpe derived the Capital Market Line (CML), which relates expected return to the total standard deviation of efficient portfolios, and showed that, under certain assumptions (rational investors, homogeneous expectations, no transaction costs), every investor should hold the same portfolio of risky assets: the Market Portfolio. From the CML, Sharpe further derived the Security Market Line (SML), which prices each individual asset solely as a function of its beta. It is from the SML, not the CML, that the decomposition of an asset’s total risk into two parts arises:
- Systematic risk: market risk, non-diversifiable, associated with macroeconomic factors that affect all assets, and therefore rewarded[5].
- Idiosyncratic risk: risk specific to an asset or company, which can be eliminated through diversification, and therefore, according to the model, is not rewarded.
This theoretical architecture brought undeniable benefits. It created a universal vocabulary — beta, volatility, the Sharpe ratio (1966, and therefore later than CAPM itself) — made it possible to compare managers and investment portfolios objectively, and provided the theoretical justification for the Market Portfolio on which, decades later, the passive fund and ETF industry would rest[6]. On the other hand, the theory also has structural weaknesses that become especially relevant when the same framework is used to value companies, and not only portfolios of liquid exchange-traded stocks. The following table summarizes these weaknesses and their direct consequences in M&A processes.
| Weakness | Description | Consequence in M&A |
|---|---|---|
| Parameter instability | Future returns and covariances are estimated from historical proxies. Small changes in the data produce extremely sensitive allocations and valuations: the classic garbage in, garbage out. | Wide value ranges; individual regression beta with a high standard error. |
| Neglected asymmetry | Variance treats gains and losses identically. | Ignores that the buyer fears loss, not volatility itself. |
| Dynamic, not constant, correlation | In systemic crises, the correlation between risky assets tends to rise and approach 1. | Diversification across a portfolio of holdings fails precisely at the worst moment. |
| Unrealistic liquidity assumption | The model assumes fractional, continuous trading with no price impact. | An untenable assumption for a private company; requires total beta and an illiquidity discount. |
These limitations are not merely academic. In The Black Swan, Nassim Taleb distinguishes the following two domains[7], Mediocristan and Extremistan, and in Chapter 17, “Locke’s Madmen, or Bell Curves in the Wrong Places”, directs this distinction specifically against the Markowitz and Sharpe framework:
- Mediocristan: a world where individual variations are limited and do not significantly change the total, for example, the height of a population. In this domain, the Gaussian curve usually works well.
- Extremistan: a world where a single extreme observation can dominate the entire result, for example, personal wealth, sales of a bestseller or market returns. In this domain, the Gaussian curve fails in potentially catastrophic ways.
For Taleb, financial markets — and, by extension, the value of a company exposed to business model disruption, regulatory change or competitive shocks — belong to Extremistan. Treating this environment with Mediocristan tools, such as variance, standard deviation and linear correlation, is not only imprecise but creates an illusion of control over risk that can lead managers to take excessive risks in the belief that uncertainty has been measured.
The risk of ruin — the possibility of total or irreversible loss in the face of an extreme event — is what this measuring instrument simply does not capture. It is this concept, more than any statistical refinement, that underpins the practical guideline presented at the end of this section.
Faced with this tension between the formal rigor of Markowitz and Sharpe and Taleb’s critique, Aswath Damodaran, the renowned NYU Stern professor, offers a more pragmatic view, and possibly the most useful one for those who need to estimate or assign a value range to a company. Damodaran acknowledges that CAPM’s assumptions are unrealistic, but argues that an imperfect, disciplined model is still more useful than no model at all. In fairness to Taleb, it should be noted that he does not intend to offer a pricing alternative; his recommendation concerns the structure of risk exposure, the Barbell Strategy discussed below, not the calculation of a discount rate. Damodaran’s relevant contribution lies in how to adjust CAPM without abandoning it.
Are there alternatives to CAPM?
The question is legitimate and deserves a direct, if brief, answer. Despite all the weaknesses listed, no alternative has established itself as a practical substitute for CAPM in pricing assets in M&A.
The sharpest critique comes not from outside the discipline, but from within it. Richard Roll showed as early as 1977 that CAPM is, strictly speaking, untestable: the true Market Portfolio, which includes every risky asset in existence, not just exchange-traded stocks, is unobservable, so any empirical test of the model is, in practice, a test of the efficiency of the index used as a proxy (typically the S&P 500 or the Ibovespa)[8]. It is a technically tougher argument than Taleb’s critique, precisely because it comes from within the statistical toolkit itself.
Along the same lines, Eugene Fama and Kenneth French showed empirically, from 1992 onwards, that beta alone poorly explains the variation in returns across stocks, proposing a three-factor and later five-factor model that adds size (small caps) and value (value versus growth) to market risk[9]. Outside academia, valuers of private companies often resort to the build-up model, which adds to the risk-free rate size and specific risk premiums estimated more discretionarily, and to the implied cost of capital itself, extracted from prices observed in comparable transactions.
None of these alternatives, however, offers the same operational simplicity as CAPM for arriving at a defensible discount rate in the day-to-day work of a valuation report. This is Damodaran’s parsimony argument: between a more complex, noisier model and a simple, consistently adjusted one, the latter tends to serve real decision-making better — which is why CAPM, despite all the criticism, remains the method most widely accepted by the market.
Notes:
[1]The attribution of the phrase to Harry Markowitz is common but not documented in a primary source. It circulates journalistically, without a precise reference. Worth noting are Warren Buffett’s counterpoint (“diversification is protection against ignorance”) and the origin of the broader economic proverb, “there’s no such thing as a free lunch”, popularized by Milton Friedman.
[2]Credit for CAPM is usually given only to William Sharpe (1964), but the model was developed almost simultaneously and independently by Jack Treynor (1961–62), John Lintner (1965) and Jan Mossin (1966).
[3]See full reference in Markowitz (1952).
[4]See full reference in Sharpe (1964).
[5]It is worth distinguishing systematic risk — non-diversifiable market risk, measured by beta — from systemic risk, a prudential regulation concept referring to the risk of a chain-reaction collapse of a financial system. The two terms are often confused in Portuguese.
[6]The index fund industry was made possible above all by the Efficient Market Hypothesis (Eugene Fama) and by John Bogle’s commercial execution, from 1976 onwards. MPT provided the vocabulary and the theoretical justification for the Market Portfolio, but not the industry itself.
[7]See full reference in Taleb (2008).
[8]See full reference in Roll (1977).
[9]See full reference in Fama and French (1992).
References
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