Zachos and Chytis have turned accounting income into a slightly smaller euro figure. They identify Tables 7 and 8 as their key findings and give Tables 13 and 14 equal importance. Those four tables of p-values largely test whether the firms' average income exceeds zero. Making risk aversion an explicit choice within a performance metric is the interesting idea here. Yet the authors choose a single setting and leave it unchanged.

What does the adjustment measure?

Classic risk-adjusted return on capital (RAROC) divides expected return by economic capital, usually measured with VaR. The authors offer their version to accountants and analysts working in auditing, management accounting and investment. They describe it as "particularly useful in times of crises". Their objection to VaR is that it lacks coherence and can violate subadditivity under heavy tails.

They substitute a spectral risk measure in Acerbi's sense. This averages quantiles using weights, with the weighting function, or "risk spectrum", expressing the user's aversion to bad outcomes. Their choice is the power spectrum with exponent d = 1.2. From Zachos's earlier work they also take a variable Y, which introduces systemic risk into the spectrum. They fix it at Y = 1.

Revenue minus operating expenses minus asset impairments supplies the numerator. Bloomberg supplies both the income-statement figures and daily returns for the DAX 40 index and its 40 constituents, covering 2 January 2015 to 30 December 2024. The calculation has two variants.

At 99%, the historical index estimates are -5.61% for the power spectral risk measure (SRM), -4.68% for ES and -3.53% for VaR. The corresponding 95% estimates are -3.46%, -2.89% and -1.83%. The authors find that a more conservative measure produces a lower RAROC. Across firms, paired t-tests reject equality, with market-linked p-values around 0.00025.

The missing capital denominator

Equation 4.2 puts one minus the risk estimate beneath the accounting flow. Because that estimate is negative, the denominator becomes 1 plus a few percent. Adidas illustrates the effect at 99%. Its numerator is 12,682,000, with the tables expressed in thousands of euro. Applying the SRM gives 12,037,392; VaR gives 12,268,600 and ES gives 12,139,683.

The authors say this form is deliberate: it keeps the result in euro for direct comparison with financial statements. In doing so, it gives up the purpose of RAROC. Return divided by capital allows comparison between a capital-hungry desk and one requiring little capital. Reducing a euro figure by a firm-specific percentage tells you which firm earned more euro.

Even the reduction is small. At the 99% SRM, the haircut ranges from 0.6% for Vonovia in the market-linked variant to 10.5% for Commerzbank in the firm-specific variant. Across all calculations, the smallest is Vonovia's 0.2% under 95% VaR.

There is also a horizon mismatch. The -5.61% estimate at 99% comes from daily index returns and measures a one-day equity loss. The setup supplies no conversion from a one-day market-price loss to the capital required to support a full year of operating income.

Why the p-values were predictable

For the market-linked variant, each denominator is beta times a constant determined solely by the risk measure. The SRM-to-VaR ratio therefore comes to roughly 1.59 for every firm. Siemens Energy, with beta 1.606, has estimates of -9.01% and -5.67%; Vonovia has -0.61% and -0.39%. All three measures produce the same cross-sectional ranking.

Consider a paired t-test comparing x with x/(1+k) across 40 firms. Nearly all the differences share a sign, so rejection follows. Those differences are approximately proportional to each firm's income. The test thus largely asks whether the firms' average income is positive. Comparing the numerator with RAROC gives two-tailed p-values from 0.000245 to 0.000254, regardless of the measure. Comparisons between measures give 0.00025 to 0.00027. For the firm-specific variant, the respective ranges are 0.00061 to 0.00090 and 0.00067 to 0.00234.

These tests establish the significance of arithmetic.

The authors write that "the empirical results do not imply that one risk measure is universally superior." Their conclusion nevertheless suggests that VaR's "coherence issues" might explain the significant differences. The computation does not support that explanation. Subadditivity concerns the combination of positions; here, each number comes from a single return series and no positions are aggregated. SRM produces lower figures because it assigns greater weight to the tail than VaR. Table 2 had already settled that. For Adidas, the difference between SRM and VaR RAROC is under 2%.

Stricter risk measures flatter loss-makers

E.ON's firm-specific numerator is -17,879,000. At 99%, the SRM changes that to -16,610,376, compared with -17,085,010 under VaR. The same pattern appears for RWE (-4,043,500) and Porsche SE (-75,500). Any figure divided by a number above one moves toward zero. A loss-making firm therefore shows a smaller loss as its measured risk increases.

The authors acknowledge that the method "is not suitable, at least not without modifications" for loss-making entities. They leave this to a closing caveat. It is the central problem for a metric promoted for crises: loss-making firms are precisely where the adjustment changes direction. Three of the 40 firms already trigger that reversal in a market the authors regard as mature enough to turn off systemic risk.

The accounting numerators also require reconciliation. Allianz has 13,998,000 for 2024 and a ten-year average of 77,074,000. For Hannover Re, the figures are 3,252,700 and 17,719,691. Volkswagen's 2024 numerator is 272,135,000 thousand, or about €272bn. That scale is closer to revenue than operating profit.

A single choice of risk aversion: d = 1.2

The ability to tune the spectrum accounts for half the paper's case for spectral measures. But d = 1.2 is the sole value tested, as the authors acknowledge. Section 4 describes it as "mildly risk-averse". In Section 6 it becomes "mostly risk averse", reflecting "a rather conservative analyst".

We also could not find how the 95% or 99% level enters a spectrum described in the paper as using the entire distribution. When d exceeds one, the weight d·p^(d−1) increases with p. Equation (2.6) treats the variable as losses, placing the worst outcomes at high p. The tables instead show negative returns. The paper does not show which convention was used in the computation.

The authors turn off the systemic-risk channel because "there is no need to depict systemic risk" in a mature market. Yet the ten-year sample includes March 2020. The claim about crises has therefore not been tested on these data.

We could not replicate the study. It requires Bloomberg income-statement data for the 40 German firms, which we do not have. Using US stocks instead would test a different market.

A test ranking the 40 firms by return over SRM capital at an annual horizon would change my view. Those rankings would then need to produce allocation decisions different from the VaR rankings, out of sample. Explicit risk aversion deserves a place in performance measurement. What this paper demonstrates is that a larger denominator draws every figure toward zero.