3.1 Overview of the Valuation Framework Used in the Sample
The Flow to Equity method is used for the DCF valuation in all reports analyzed: the levered Free Cash Flows or Flows to Equity (FtE) are discounted by the levered cost of equity to the value of equity. The cash flow forecast consists of a detailed forecast, on average, for the next 5 years followed by a perpetuity setting (steady-state) covering all years beyond the detailed forecast, except for about 10% of the cases where the detailed forecast is followed by a convergence period. For both alternatives, the present value of the cash flows beyond the end of the detailed forecast or the end of the convergence period is calculated as the present value of a growing perpetuity (terminal value).
With the exception of a few older cases, the
Capital Asset Pricing Model (CAPM) is applied to derive the RADR. For most valuations, the risk-free rate is determined according to the procedure suggested by the
Institute of auditors in Germany (
Institut der Wirtschaftsprüfer in Deutschland, IDW) that refers to the yield curve to obtain a single risk-free rate by averaging and rounding as shown in Becker et al. (
2018, pp. 138–139). This procedure has been subject to criticism (see, for example, Bassemir et al.
2012; Drukarczyk and Schüler
2021, pp. 254–255), but will not be replaced by a more precise method in our paper. The market risk premium is regularly chosen from the bandwidth recommended by the IDW at the valuation date, and will also not be addressed. The transition from the unlevered cost of equity to the levered cost of equity depends upon the transition from the unlevered beta value to the levered beta value. Exemptions are the valuation of unlevered companies and a few older valuations that do not employ the CAPM. This beta transformation is not shown in detail in the reports. When we reconstructed the valuations manually, we saw that the predominant way to link unlevered and levered beta values was to follow Harris and Pringle (
1985) in recent years in that they link unlevered and levered cost of equity, since Harris and Pringle do not apply the CAPM.
The information about both expected interest payments and expected levels of debt employed, as well as the link between unlevered and levered cost of equity is incomplete. This prevents us from modeling the impact of a change in terminal value of equity on the levered cost of equity of preceding years. When the modified terminal value of equity exceeds the reported terminal value of equity, the leverage ratio (debt/equity) would decrease, leading to lower levered cost of equity and a higher value of equity at the valuation date. In order to avoid mixed results, we refrain from correcting only the cases for which we can adjust the discount rates and not correcting all other cases.
Since we are interested in the inflation rate used in the valuations by German auditors, we will focus on the terminal value, i.e., the present value of all future expected cash flows after the cash flow forecast has ended, as there is no information about the inflation rates used by the valuators for the years within the detailed forecast horizon or the convergence period. For the growing perpetuity representing all years after the end of the detailed cash flow forecast or the end of the convergence period, the valuators have to determine a growth rate. Growth is driven either by capital expenditures—ignoring an impossible infinite growth due to cost decreases (real growth)—or price increases (inflation). The Fisher-equation is usually applied to rates of return free of the risk of default (i).
^{1} Applying it to growth rates, skipping the expectation operator for the sake of a simpler presentation, and assuming a covariance between real growth and inflation of zero the relationship between expected real growth (g
_{R}), expected inflation (π) and expected nominal growth (g
_{N}) is as follows:
$$g_{N}=\left(1+g_{R}\right)\left(1+\pi \right)-1$$
(1)
It can also be used for linking nominal (index N) and real (index R) RADR (r) with the expected inflation, assuming a covariance between r and π of zero:
$$r_{N}=\left(1+r_{R}\right)\left(1+\pi \right)-1$$
(2)
Purchasing power equivalence can be illustrated for the terminal value (of equity), TV, with E
_{R} denoting the value of equity in real terms and E
_{N} the value of equity in nominal terms for a simplified cash flow-based model and by ignoring any other challenges caused, for example, by the tax regime:
$$TV=\underset{E_{R}}{\underbrace{\frac{FtE_{R}}{r_{LR}-g_{R}}} }=\underset{E_{N}}{\underbrace{\frac{\overset{FtE_{N}}{\overbrace{FtE_{R}\left(1+\pi \right)} }}{\underset{r_{LN}}{\underbrace{\left(1+r_{LR}\right)\left(1+\pi \right)-1} }-\underset{g_{N}}{\underbrace{\left[\left(1+g_{R}\right)\left(1+\pi \right)-1\right]} }}} }$$
(3)
Since the year 2005, German auditors do not use a economy-wide nominal growth rate for filling out Eq.
3, but include the expansion-induced increase of retained earnings in the numerator in an equivalent manner for the vast majority of the cases, by assuming that the NPV on this additional investment is zero (see Becker et al.
2018, pp. 170–178). As a consequence, FtE does not only consist of dividends paid, but also of this portion of the earnings retention assuming this increase in retained earnings equals its value contribution. The growth rate left in the denominator must be predominantly induced by inflation. However, whether any of this (remaining) growth is caused by real growth can remain unclear, because the reports can be quite unspecific in this regard. We refrain from speculating about this problem, which cannot be solved quantitatively with the available data, and assume below that the remaining growth rate is inflation-induced. Nevertheless, we solve Eq.
1 for the implied real growth with using the general expected inflation rate, if the company specific growth rate were a nominal growth rate. This rate is company specific as there is no single rate that captures all the expected changes in prices for input and output factors for each company. We use the variable π
_{C} for the company specific inflation rate in the following. Since 2010, a number of reports show an additional increase in retained earnings to the amount of π
_{C} times book value of equity (E
_{BV}), and since 2011 a tax payment on the inflation-induced increase in terminal value. The latter is defined by π
_{C} times half of the personal income tax (in German:
Abgeltungsteuer) times the terminal value. Although the reports can be quite vague in motivating the former effect, we treat it as being inflation-induced in our analysis, because its definition (π
_{C} E
_{BV}) is based on this interpretation.
3.3 Estimation of Expected Inflation Rates
The nominal FtE, the nominal RADR, and π_{C} are given in the reports. The RADR, in nominal terms, contains the economy-wide rate of inflation expected by the market (π). A consumer price index is used as the general inflation rate on a regular basis. This is justified since the expected nominal return to investors (RADR) defines their consumption potential in nominal terms. We need to estimate expected inflation rates at the valuation date for two reasons. First, they can serve as a point of reference for the company specific rates. Second, we will need them to transform the RADR in nominal terms into the RADR in real terms.
In contrast to realized price changes, inflation expectations are not directly observable and have to be extracted from surveys or prices of inflation-linked securities or derivatives (Deutsche Bundesbank
2015). While surveys offer the opportunity to learn about the inflation expectations of market participants, inflation expectations drawn from security prices can be affected by risk premia (Deutsche Bundesbank
2014, p. 75;
2015, pp. 46–47). However, due to the underlying market transactions, financial products offer an inflation estimate free of strategic under- or overestimation. Based upon the existing literature on the topic, as for example European Central Bank (
2021), Koester et al. (
2021), Speck (
2016) or Schüler and Lampenius (
2007), we use several estimators of expected inflation:
1.
Survey of Professional Forecasters (SPF): The SPF is a quarterly survey of the
European Central Bank (ECB) among professional forecasters from the financial sector and research institutions located in the European Union. It provides an established data set for (long-term) inflation expectations in the Eurozone (Möhrle
2020). We use the average long-term (five years ahead) Euro area annual
Harmonised Index of Consumer Prices (HICP) inflation point estimate.
^{2} The series is available beginning with the third quarter of 2002. Since the data is only available on a quarterly basis, we use the expected rate of a quarter for all valuation dates falling within this quarter.
2.
Implied inflation rates: Expected rates of inflation can be inferred from linking expected real interest rates and nominal interest rates on German government bonds by solving Eq.
2 for the expected inflation rate. We use the monthly data for expected real interest rates of German government bonds with a remaining maturity of ten years published by the Deutsche Bundesbank.
^{3} The series is available for all years within our sample period. Since the data is only available on a monthly basis, we use the expected rate of a month for all valuation dates within this month. It should be noted that the Deutsche Bundesbank itself uses survey data on the expected inflation rates according to the forecasts sold by Consensus Economics Incorporated (Deutsche Bundesbank
2022) for calculating the real interest rates. This data set is not available to us. Thus, we essentially estimate these survey-based expectations in a retrograde manner. Other than the Eurozone-based SPF, which serves to forecast the inflation in the Eurozone, the comparison of nominal with real interest rates on German governmental bonds serves to estimate the German inflation rates. One could argue that the latter is in line with the domicile principle set forth in IDW standards. Since one could also argue that the Eurozone is a larger market served by many of the companies analyzed, our use of both indicators covers both arguments.
3.
Inflation swaps: Inflation buyers and inflation sellers exchange fixed and variable rates on the same notional amount. The floating rate is determined by the inflation development during the contractual period. The annualized fixed rate (swap rate) can be interpreted as the daily available estimator for the expected inflation for a given future date (Deutsche Bundesbank
2015). Additionally, swap rates for different horizons enable us to calculate implied forward inflation rates. A prominent swap rate, used, for example, in Deutsche Bundesbank (
2014,
2015) and Speck (
2016), is the 5Y5Y forward inflation rate. It is the annualized inflation rate over a five-year period beginning in 5 years’ time. The rate can be derived from inflation-linked swap rates (ILS) of contracts with a maturity of 5 and 10 years. Generally, forward rates between year m (with m > 0) and year
n (with
n > m) can be calculated using the inflation-linked swap rates as follows:
$$\begin{array}{l} \left(1+\pi _{0\rightarrow n}\right)^{n}=\left(1+\pi _{0\rightarrow m}\right)^{m}\left(1+\pi _{m\rightarrow n}\right)^{n-m}\\
\left(1+\pi _{m\rightarrow n}\right)^{n-m}=\frac{\left(1+\pi _{0\rightarrow n}\right)^{n}}{\left(1+\pi _{0\rightarrow m}\right)^{m}}\\
\pi _{m\rightarrow n}=\left[\frac{\left(1+\pi _{0\rightarrow n}\right)^{n}}{\left(1+\pi _{0\rightarrow m}\right)^{m}}\right]^{1/n-m}-1 \end{array}$$
(4)
It should be noted that the calculation used in Speck (
2016, p. 6) is based upon logarithmic rates. This is also true for the 5Y5Y rate provided by Bloomberg.
^{4} It can be shown that Eqs.
4 and
1 in Speck (
2016) are equivalent. One should remember to re-transform logarithmic rates, like the 5Y5Y rate, into their discrete equivalent, because the usual valuation framework is based upon discrete planning and discounting. Otherwise, a valuation error will occur.
Bloomberg offers quotes on inflation swaps for 1 through 10 years, as well as for 12, 15, 18, 20, 25, 30, 35, 40, 45 and 50 years. Since we focus on inflation effects in the terminal value, we calculate the forward inflation rate starting at the end of the detailed forecast horizon or the end of the convergence period for each valuation until year 30. With reference to Eq.
4, we set m equal to the last year of the respective forecast horizon; this is in line with our assumption to refrain from analyzing inflationary effects prior to the steady-state. Then, we have to decide how to use the data available for all periods after m. Since the risk-free rate used in most of the reports is derived from the German yield curve, which covers maturities up to 30 years, we pragmatically use the swap rate for year 30 (
n = 30). Finally, we apply Eq.
4 and calculate the annualized forward rate that is assumed to be constant for all periods after period m. For a detailed forecast for the next 5 years, for example, we apply the rate 5Y25Y. We acknowledge that other approaches might be possible. We do not analyze the matter any further here. Since the data needed is available beginning in June 2004 on a daily basis, we can apply the specific rates for each valuation date after this date only.
There is also the possibility to derive inflation expectations from the difference of the yields of (equivalent) nominal and inflation-linked government bonds (Deutsche Bundesbank
2015). We refrain from using this so-called
Break-Even Inflation Rate (BEIR) because the number of inflation-linked government bonds is limited; currently, only 5 German inflation-linked bonds have been issued, and data is available only from 2009 onward.
We refrain from picking one of the three estimators presented here, and instead report most results for all of them and test the differences in the results regarding their statistical significance. Nevertheless, we think the inflation-linked swaps provide considerable potential: they cover a set of different periods that also allow for the calculation of forward swap rates. Therefore, they are not only quarterly point-forecasts like the SPF rates, and are not solely survey-based like, again, the SPF rates and also the implied inflation rates, but derived from market transactions.
3.4 Research Approach
We use the following measures to answer our research questions I through III:
Critical period t′: This is the point in time in which the nominal cash flow starts to decrease, if the company is not able to pass on inflation completely, i.e., d < 1 (see Appendix). We use the variable g
_{C} to measure the combined rate of growth that depends upon the level of d:
$$g_{C{,}t}=\left[\left(1+d\pi _{C}\right)\frac{CI_{t-1}}{FtE_{t-1}}-\left(1+\pi _{C}\right)\frac{CO_{t-1}}{FtE_{t-1}}\right]-1$$
(5)
$$\begin{array}{l} t'=1+\ln \frac{d\pi _{C}CI_{0}}{\pi _{C}CO_{0}}\cdot \left(\ln \frac{1+\pi _{C}}{1+d\pi _{C}}\right)^{-1}\;\forall t'\geq 1\\ t'=1+\left[\ln \left(dCI_{0}\right)-\ln \left(CO_{0}\right)\right]\cdot \left(\ln \frac{1+\pi _{C}}{1+d\pi _{C}}\right)^{-1} \end{array}$$
(6)
CI stands for cash inflow, and CO for cash outflow. For our empirical analysis, we use revenues for CI and the difference between revenues and EBIT for CO, each for the first year of the steady-state, in order to get results that are not influenced by corporate taxes, interest earned on liquid funds or cash flow effects of debt financing. Besides, we need to use this simplified cash flow measure, because the change in net working capital is not shown in the reports. It should be noted that in the case of an incomplete passing on of inflationary effects, the company specific growth rate cannot be constant over time.
Critical period t″: Point in time in which the nominal cash flow turns negative, if a company is not able to pass on inflation completely (d < 1):
$$t''=\frac{\ln \frac{CI_{0}}{CO_{0}}}{\ln \frac{1+\pi _{C}}{1+d\pi _{C}}}\;\forall t''\geq 1$$
(7)
Equation
7 follows from Eq.
5 by setting equal the CI-term and the CO-term on the RHS, both compounded over t″ years.
There are cases in which both t′ and t″ are negative, because the EBIT margin is very small. Thus, it is possible that even for d only slightly below 1 the change in EBIT is negative for the first forecast year. In our empirical analysis, we set t′ and t″ equal to 1 in these instances. Since the valuation reports do not apply different inflation rates to CI and CO, nor to components of CI and CO, we use one single company specific inflation rate for each case during our analysis.
The critical periods t′ and t″ illustrate the potential vulnerability of a company to difficulties in passing on increasing input prices. It is important for a valuator to check the assumption set to value the company accordingly.
Implied real growth rate (g
_{R}): As introduced above, we solve Eq.
1 for the implied real growth rate, if the company specific rate were a nominal growth rate. For doing so, we use the estimated general growth rate:
$$g_{R}=\frac{1+\pi _{C}}{1+\pi }-1$$
Additionally, in order to show the value effect of using a company specific inflation rate that differs from the general expected inflation rate, we also substitute the latter for the former for calculating a revised terminal value (TV*), and calculate the difference to the reported terminal value (∆TV = TV* − TV_{reported}). In order to show the impact at the valuation date, we show the difference between the reported value of equity (E_{reported}) and the modified value of equity (E*) at the valuation date (∆E). It could be assumed that the FtE at the beginning of the perpetuity is inflated by (1 + π_{C}) from the last period of the detailed forecast horizon or the convergence period, although this is not addressed in the reports directly. Therefore, we use two variants: for the first one, we do not adjust FtE (∆TVa and ∆Ea) regarding to this effect; for the second one, we inflate FtE by the general expected inflation rate and deflate it by the company specific rate (∆TVb and ∆Eb): (1 + π) / (1 + π_{C}). This is a simplification as our analysis to answer question (III) will show. We will report our results in % of TV_{reported} and E_{reported}. In line with our assumption mentioned above, we do not try to adjust the levered beta value and the levered cost of equity for discounting the terminal value to the valuation date. This is another simplification, because the ratio of debt to equity will change, if debt remains constant, and the value of equity will change with it.
The transition follows 5 steps: We omit the company specific inflation rate (1), solve Eq.
2 for the real cost of capital (2), add back the inflation-induced increase in retained earnings (π
_{C} times the book value of equity) (3a), estimate the nominal cash flow not affected by inflation-induced deviations between accrual- and cash-based effects (3b), deflate this estimated nominal cash flow to the real cash flow (4), and omit the inflation induced increase of the retained earnings and the taxation of the inflation-induced increase in company value (5).
Steps 1 and 2 adjust the denominator from r_{N} minus π_{C} as applied by the auditors to r_{R}. Steps 3a and 3b transform the reported (accrual-based) nominal cash flow to a nominal cash flow not distorted by inflation-induced deviations between accrual- and cash-based effects. Steps 4 and 5 transform this nominal cash flow to the real cash flow without inflation induced tax effects.
Steps 3a and 3b deserve a closer look. For illustration, we use the example used in Schwetzler (
2022b). The only driver of growth in the example is inflation (5%). This is comparable to our setting, because we do not address expansion-induced growth in our paper based upon the assumption that this part of growth is value neutral. The FCF with an inflation rate of zero can be interpreted as being the real FCF: 10.5. The nominal FCF is 9.25 in t = 1. The difference between earnings after tax (12.6) and nominal FCF (9.25) is 3.35. It is due to two effects: There is an increase in net working capital (NWC) in nominal terms, which does not occur in real terms. If the NWC consists of accounts receivable only, it increases because accounts receivable from previous sales that are paid by customers in t = 1 are smaller than new accounts receivables resulting from new, inflated revenues (in the example: 1.1, equal to the beginning balance of the NWC, 22, times the inflation rate, 5%). Secondly, replacement capex is higher (47.25) than the depreciation of previous investments (45) due to inflation. The inflation-induced increase in retained earnings (3.35 = 1.1 + 2.25) is equal to the inflation rate (5%) times the book value of equity (67). Thus, step 3a would add back 3.35 to 9.25, equaling the earnings after tax of 12.6. Step 3a reverses the retention of earnings defined by inflation rate times book value of equity. It is inflation-induced, and FCF would equal the earnings in a world without inflation (and without real growth).
With step 3b we aim at deriving the nominal cash flow without the inflation-induced deviations from a cash-based accounting and taxation setting that need to be addressed to prepare the transition to the cash flow in real terms. There are two effects in that regard, which sum up to 1.775 in the example: One is a tax effect of 0.675 (corporate tax rate, τ_{C}, 30% times the difference between replacement capex of 47.25 and the depreciation of 45). The second one is the inflation-induced change in NWC of 1.1. Because we already added back the inflation-induced increase in retained earnings of 3.35 in step 3a, we only need to subtract the difference between 3.35 and 1.775, i.e., 1.575. This is step 3b. As a result of 3a and 3b we have transformed the reported nominal cash flow (9.25) to the ‘undistorted’ nominal cash flow of 11.025. It is equal to the inflated real cash flow (10.5 times 1.05 = 11.025). Finally, we subtract the inflation-induced growth (10.5 · 0.05 = 0.525) in step 4. The resulting value is 10.5, the cash flow in real terms.
Due to a lack of data in the reports we need to set a pragmatic assumption for step 3b. We need to split up the inflation-induced increase in retained earnings into an estimation of the difference between capex and depreciation, which is assumed to cause the tax effect referred to above, and into an estimated increase in NWC. We try to solve this problem by using the relation of fixed assets to the sum of fixed assets and net working capital (variable a), using the sample of financial statements provided by the Deutsche Bundesbank.
^{5} We matched the industry for each company in our sample reports with the respective average ratio (2000–2019) for that industry in the sample of the Deutsche Bundesbank. Net working capital is defined as current assets without liquid funds minus current liabilities. We use the corporate tax rate for the steady-state, if available. If the rate is not available, we set it equal to 30%. Thus, for step 3b we subtract the following amount:
$$\pi _{C}E_{BV}a\left(1-\tau _{C}\right)$$
(10)
For the numerical example, this leads to (a = 45/67 = (47.25 − 45) / 3.35 = 0.672):
$$0.05\cdot 67\cdot 0.672\left(1-0.3\right)=1.575$$
Because all balance sheet items are supposed to grow by the inflation rate in that setting, variable a can be derived by either dividing the net investment of 2.25 by the total increase in retained earnings (3.35), or by dividing the beginning balance of the fixed assets (45) by the total assets (67). The latter can serve as a justification for our approach to estimate variable a empirically.
The reports provide incomplete information regarding the periodic level of interest-bearing debt and the change in debt for the steady state. Therefore, we set another pragmatic assumption by forgoing adjustments caused by debt financing.
We can expect a difference between the reported terminal value and the terminal value in a world without inflation, not only because the inflation rate implied by the nominal cost of capital differs from the company specific inflation rate regularly, but because of the other effects, too. As a simplification, assuming for steps 3a to 5 that those changes in the cash flow cause effective income taxes amounting to half of the personal income tax as in Schwetzler (
2022a), we can sum up as follows:
$$\begin{array}{l} TV_{\text{reported}}=\frac{FtE_{N}}{r_{N}-\pi _{C}}\\ \neq \frac{\left[FtE_{N}+\pi _{C}E_{BV}\left(1-0.5\tau _{I}\right)-\pi _{C}E_{BV}a\left(1-\tau _{C}\right)\left(1-0.5\tau _{I}\right)+\pi _{C}0.5\tau _{I}TV_{N}\left(1-0.5\tau _{I}\right)\right]\left(1+\pi _{C}\right)^{-1}}{r_{R}}\\ =TV_{R} \end{array}$$
(11)
It should be noted that the difference between π_{C} and π not only influences the implied real growth rate addressed in (II), but also the difference between TV_{R} and TV_{reported}. For company specific rates below the expected general inflation rate, the implied real growth rate will be negative and TV_{R} will be higher than TV_{reported}. Thus, (II) and (III) provide two different perspectives on the implications of company specific inflation rates different from π.