The impact of the Covid-19 pandemic on the hotel Industry’s economic performance: Evidence from Portugal

This paper estimates the impact of the Covid-19 pandemic on the economic and financial performance of the Portuguese mainland hotel industry. For that purpose, we implement a novel empirical approach to gauge the impact of the pandemic during the 2020–2021 period in terms of the industry's aggregated operating revenues, net total assets, net total debt, generated cash flow, and financial slack. To that end, we derive and estimate a sustainable growth model to project the 2020 and 2021 ‘Covid-free’ aggregated financial statements of a representative Portuguese mainland hotel industry sample. The impact of the Covid pandemic is measured by the difference between the ‘Covid-free’ financial statements and the historical data drawn from the Orbis and Sabi databases. An MC simulation with bootstrapping indicates that the deviations of the deterministic from the stochastic estimates for major indicators vary between 0.5 and 5.5%. The deterministic operating cash flow estimate lies within plus or minus two standard deviations from the mean interval of the operating cash flow distribution. Based on this distribution, we estimate the downside risk, measured by cash flow at risk, at 1294 million euros. Overall findings shed some light on the economic and financial repercussions of extreme events such as the Covid-19 pandemic, providing us with a better understanding of how to design public policies and business strategies to recover from such an impact.


Introduction
The outbreak of the Covid-19 pandemic (hereafter, referred to as the 'pandemic') triggered unprecedented global disruptions in ☆ Authors gratefully acknowledge the useful and valuable comments and suggestions from Celeste Eusébio, Fernando Tavares, Paulo Alves, and Rui Neves on an earlier draft, and research seminar participants at CICEE -Research Center in Economics & Business Sciences, Portugal (October 2022). We are thankful for helpful comments from an anonymous editor and two anonymous reviewers. We also thank GOVCOPP at the University of Aveiro for providing access to AMADEUS/ORBIS and SABI databases, and INE (Statistics Portugal) for providing access to the "Sistema de Contas Integradas das Empresas" (SCIE) database. The authors are solely responsible for any inexactitudes, omissions, and errors. This paper is part of the research program "PTDC/EGE-ECO/4993/2021 -THE PORTUGUESE HOTEL INDUSTRY: PREPARING FUTURE AHEAD OF COVID-19 PANDEMIC" submitted to Fundação para a Ciência and Tecnologia (FCT Portugal), which did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. tourism and hospitality ecosystems and these sectors plunged into a severe economic and financial crisis. Unsurprisingly, such economic, financial and social impacts have a greater negative impact on economies like that of Portugal, which are much more dependent on the performance of the tourism sector than other countries. 1 At the global hotel industry level, most have been forced to downsize their operations, resulting in significant economic imbalances. Ultimately, some of them may have been driven into financial distress or even insolvency. For example, the 11.1% operating revenue average annual growth rate of the Portuguese mainland hotel industry during the 2014-2019 period was abruptly interrupted by pandemic lockdown measures, travel bans, and other restrictions, triggering unprecedented disruptions and plunging the sector into economic turmoil, with grievous social-economic externalities [1]. Notably, it led to a generalized scale-down of hotel operations, plummeted revenues, fueled workforce layoff, and exposed the ecosystem to economic and financial disarray [5,6]. Although prospects for the recovery of the tourism sector remained relatively unsteady, governmental policymakers and hotel ecosystem participants alike strove to map out alternative feasible recovery paths (and pace) for designing post-pandemic resilient and sustainable recovery strategies for the industry (e.g., Ref. [7]).
The most recent research on the topic examines either public policy aimed at mitigating the impact of COVID-19 on tourism (e.g., Refs. [8][9][10]), or explores particular performance dimensions of specific segments of the hotel industry, such as listed hotels (e.g., Refs. [11][12][13][14]). 2 Yet, the development of blueprints to support the design of prospective strategic scenarios requires the availability of a comprehensive and quantitative assessment of the magnitude of the economic, financial, and social impacts of the Covid-19 shock. To our knowledge, no such assessment has been carried out, motivating us to perform this study.
The main research objective of this paper is to quantitatively assess the potential economic and financial shocks on the performance of the Portuguese hotel industry. Data are based on approximately 1000 hotels in mainland Portugal in the period 2020-2021.
Besides providing evidence of the aggregated pandemic's impact on the economic and financial condition of the industry, this work also contributes to the literature by applying a novel methodological approach. First, it derivates and estimates a deterministic business model for the sector, anchored on the maximum sustainable growth rate (SGM) conceptual framework to gauge the impact of the pandemic during the 2020-2021 period in terms of operating revenues, net total assets, net total debt, operating cash flow, and financial slack. 3 Next, this approach combines with a robustness check that incorporates stochastic variability into the deterministic base case by applying Monte Carlo (MC) methods with bootstrapping to measure the downside risk of our hotel industry sample using the cash-flow-at-risk (CFaR) conceptual framework.

Maximum sustainable growth framework
Tourism and hospitality industries are exposed to a wide spectrum of risks. Therefore, enterprise risk management (ERM) -risk identification, measurement, and managementis instrumental in helping firms to manage their value creation objectives, particularly in terms of mitigating financial distress and optimizing risk portfolio (e.g., Refs. [25][26][27][28][29]). 4 At the hotel firm level, the measurement of the expected impact of downside risk factors on value creation should be a primary managerial concern (e.g., Refs. [27,[33][34][35]. Further, there is abundant and compelling evidence that the volatility in corporate accounting aggregates, such as net income and operating cash flows, is related to value creation (e.g., Smithson & Simkins, 2005 [33]). Under well-diversified firm ownership, risk management can be expected to be positively related to a firm's value, which could limit the expected costs of financial distress, manage financial slack, reduce tax liability, and mitigate suboptimal resource allocation (e.g., Ref. [34]).
To estimate the economic and financial impact of the Covid-19 pandemic on the Portuguese mainland hotel industry in 2020 and 2021, we develop a novel empirical, methodological approach based on the SGM framework. SGM builds on the percentage-of-sales method's standard assumptions that the stocks of the balance sheet accounts are optimized in relation to the current level of sales and vary in proportion to sales; and that depreciation and amortization are not an available source of funds because it is assumed that the same amount is applied in restoring fixed assets operational functionality (e.g., Refs. [36][37][38]). 5 Under the maximum annual percentage increase in operating revenuega firm can sustain, keeping constant at the pre-1 According to Instituto Nacional de Estatística [1], the direct and indirect contribution of the tourism industry to GDP dropped from 11.8% in 2019 to 6.6% in 2020 and rose to 8.0% in 2021 (see also [2][3][4]). 2 For further details on the pandemic's impacts on the tourism industry, see, e.g., Refs. [15][16][17][18][19][20][21][22][23][24]. 3 [30], the two concepts have distinct meanings: risk relates to objective probabilities, whereas uncertainty relates to subjective probabilities (see also [31,32] pandemic levels (2019): (i) fixed assets utilization, proxied by the net fixed assets-to-operating revenue ratio; (ii) after-tax operating revenue profitability, measured by the net income-to-operating revenue ratio; (iii) capital structure gauged by the debt-to-equity ratio; and (iv) the retention rate of earnings, measured by the complement of the dividend payout ratio, and without resorting to incremental external funding. Financial slack (FS) is a readily available liquidity cushion in the form of excess cash holdings and debt capacity, which provides financing flexibility by mitigating the impact of adverse liquidity shocks, and financial distress and by moderating suboptimal allocative behavior, namely in the form of underinvestment (e.g., Ref. [12]). 6 Therefore, for precautionary reasons, firms tend to accumulate liquid assets, such as cash and equivalents, as an 'insurance' against liquidity shortfalls arising in adverse states of cash flow generation and to avoid asset fire sales, raising externally costly unanticipated funding, or incurring inefficient underinvestment (e.g., Refs. [39][40][41][42]).
Under this framework, firms with higher asset systematic riskiness and costlier access to external capital markets tend to carry larger cash holdings on their balance sheets. We measure excess cash holdings as the difference between "Cash & Equivalents" and the "Liquidity Buffer" balances (e.g., Refs. [43][44][45][46]). 7 Conceptually, debt capacity is the incremental borrowing required to sustain the capital market's perception of a firm's current aggregate asset systematic riskiness. Or restated, the maximum amount that could optimally be borrowed at the current risk-adjusted marginal cost of debt (e.g., Refs. [39,47,48]).

Cash-flow-at-risk
Another valuable tool to assess non-financial firms' downside risk is cash flow at risk (CFaR). CFaR is a composite measure of the maximum decrease in expected cash flows associated with the uncertainty of risk factors, given a pre-defined confidence level, for a given period, which Stein et al. [49] define "as the probability distribution of a company's operating cashflows over some horizon in the future, based on information available today". Taking a prespecified timeframe and statistical confidence level, the CFaR approach to downside risk measurement estimates the maximum shortfall of cash a firm is willing to accept and, therefore, its overall liquidity risk over a given period (e.g., Refs. [27,[49][50][51]. 8 , Moreover, since all risk exposures can be aggregated into a single metric, CFaR provides quantitative information, at least accurate on average, helping to guide managerial decision-making (e.g., Refs. [34,53,54]). Indeed, 'it is the "lower tail" of the cash flow distribution that can have costly consequences, such as insufficient funds to carry out the company's investment program or even bankruptcy' [54]; CFaR provides a measure of such lower tail effects, which we estimate by bootstrapping the Portuguese mainland hotel industry model for robustness-checking purposes, following Alexander [55].
It should be emphasized that the data panel used in the deterministic methodological approach features an inherent statistical significance limitation, which inhibits inferential testing. The estimation of CFaR with Monte Carlo (MC) simulation has been addressed in the literature, namely, through examples that suggest that this numerical tool is effective for solving problems in finance that involve closed-form analytical solutions that are too complex or impossible to determine (e.g., Refs. [56,57]). Besides the benefit of efficiently dealing with complexity, another advantage of MC is its inherent randomness, which is essential for simulating real-life random systems [58]. This method is, therefore, an obvious choice for tackling the CFar estimation as a robust check in our study's approach. Hence, we use MC bootstrapping computational methods to perform the number of trials necessary in each simulation experiment to generate a numerical approximation to the true distribution of the output variable at the standard 95% confidence level. 9

Research design
This paper examines the impact of the Covid-19 pandemic period during the 2020 and 2021 sampling periods on the Portuguese mainland hotel industry. Specifically, we estimate the pandemic effect in terms of total net assets, total net debt, operating cash flow, and financial flexibility, for a representative sample of Portugal's mainland hotels (hereafter, referred to as the 'sample').
The empirical implementation strategy is designed in five steps. First, we estimate aggregate balance sheets, income, and operating cash flow statements for the 2014-2021 sampling period, drawing data from the Orbis/Sabi databases. Second, we derive a steadystate version of SGM and estimate the operating revenue sustainable growth rate to forecast the sample's financial statements for 2020 and 2021. 10 Regarding the third step, we project 2020 and 2021 aggregate balance sheets, income statements and cash flow statements, which are unconditioned by the occurrence of the Covid-19 pandemic. Fourth, we measure the (deterministic) impact of 6 Henceforth, we use 'financial flexibility' and 'financial slack' interchangeably. 7 See Panel D of Appendix II. It should be noted that cash holding balances are firm-specific and determined by firm characteristics, such as size, cash flow generation profile, growth opportunities set, and firm positioning in its life cycle. 8 Conceptually, CFaR is a metric similar to value-at-risk (VaR), which "summarizes the worst loss over a target horizon that will not be exceeded with a given level of confidence" [52]. Specifically, VaR describes the quantile of the projected distribution of gains and losses over a target horizon. Specifying c as the selected confidence level, VaR corresponds to the 1-c lower tail level. Unlike CFaR, VaR does not include the time value of money because the short time horizon does not require periodic cashflows to be discounted [26]. 9 See Ref. [59] for a comprehensive description of the use of the MC approach in risk analysis. 10 See, Appendix I for the derivation of the sustainable growth rate model.
the Covid-19 pandemic as the difference between the projected and observed 2020's and 2021's aggregated operating revenues, net total assets, total net debt, operating cash flow, and financial slack. Fifth, we run a Monte Carlo simulation experiment to check for the robustness of the deterministic 2019 base case in terms of the output variables required to compute the CFaR.

Data
Economic, financial, and operating data for this research were drawn from INE (Statistics Portugal), Sabi, and Orbis, covering the 2010-2021 sampling period. However, to minimize the spillovers of the financial assistance program signed between Portugal and the International Monetary Fund, the European Union, and the European Central Bank, encompassing the application of a three-year economic adjustment program (2011-mid-2014), we restricted the sampling period to 2014-2019.
Results of summary statistics and parametric tests for equality of means document that the variables used in the deterministic model do not exhibit, at the standard confidence levels, statistically significant differences in means between the sampling subperiods of 2010-2019 and 2014-2019. These results support the consistency of using the 2014-2019 subsampling period for this empirical analysis and segmenting hotels by star category. 11 To be included in the sample, a hotel must comply with the following criteria: (i) be included in the National Register of Tourism Enterprises (RNET) database, with an assigned fiscal number; (ii) be established and operating on Portugal's mainland (iii) be a star classified hotel or hotel-apartment; (iv) be active for the entire sampling period. 12 The population of hotels and hotel-apartments was drawn from the INE database for the 2019-2021 period (Table 1).
A search in the RNET database yielded a sample of 1282 hotel units with assigned fiscal numbers and star classifications that met the above-mentioned criteria. Finally, we identified 972 entities in Orbis/Sabi databases that matched that set and reported complete data for the 2019-2021 period, which corresponded to 1057 hotel and hotel-apartment units. For this sample, we extracted economic, financial, and operating data at the hotel business firm level (Table 1). 13 Data on income statement items, such as labor costs, other operating costs, financial revenues, and financial expenses, were drawn from the Sabi database. In addition, data on the interest coverage ratios and the debt spreads associated with the synthetic credit ratings were collected from Prof. Aswath Damodaran's website 14 ; 10-year government bond yields for Portugal and triple-A rating countries were collected from the European Central Bank's -Statistical Data Warehouse.

SGM modeling and estimation
Aggregate balance sheets, income, and cash flow statements were estimated using the variable specified in Appendix II. Financial slack is modeled as the sum of excess cash holdings (ECH) and debt capacity (DC). ECH is measured as the difference between the book value of cash and equivalents, and the minimum cash balance requirements (e.g., Ref. [44]). 15 We use the defensive interval ratio (DIR) to estimate the short-term liquidity needs in terms of the number of days a hotel could operate resorting only to its current assets (e.g., Ref. [60]):

DIR = Currednt Assests t Daily Operating Expenses t
(1) where current assets include cash, marketable securities, and net receivables; and daily operational expenses are measured by the sum of the cost of sales, operating costs, and net interest rate divided by 360. We define DC as the maximum amount of borrowing lenders are willing to extend to an investment-grade rated firm based on its interest coverage ratio and the debt spread associated with its rating notation (see, e.g., Ref. [61]). Hotel's DC is specified as: where r F denotes the rate of return on a riskless asset; and spread, the debt's default risk premium. We estimate hotel synthetic rating notations, interest coverage ratios, and the debt default spreads associated with them, using the model by Damodaran [62], parametrized according to Aswath Damodaran's website (see Table 2). The model uses the operating income (EBIT) and the net interest expense as inputs to estimate the interest coverage ratio, which is extensively used by Standard and 11 Test results are available from the corresponding author upon request. 12 The National Register of Tourism Enterprises (RNET) provides a database on tourism firms with a valid license, including data on, e.g., firm denomination, type of tourism unit, capacity, location, and age: http://business.turismodeportugal.pt/pt/Planear_Iniciar/Licenciamento_Registo_ da_Atividade/Empreendimentos_Turisticos/Paginas/rnet.aspx. 13 Data on the sampled firms for the year 2021 were drawn from the Orbis/Sabi databases on September 30, 2022, when only 915 of the 972 sample firms had disclosed their data to the database. Therefore, we estimate 2021 financial statements as a direct proportion, using 915/972 as a constant of proportionality. 14 Data available at: http://pages.stern.nyu.edu/~adamodar/New_Home_Page/dataarchived.html. 15 The "Liquidity Buffer" (the minimum cash balance requirement) is specified as the product of the 2019 defensive interval ratio by the daily operational expenses.
Poor's and Moody's, two leading international rating agencies.

MC approach
We use MC computational numerical methods with bootstrapping to produce forecasts of the 2020 and 2021 aggregated operating revenues, net total assets, net total debt, operating cash flow, financial slack, debt capacity, and excess cash holdings to check for the robustness of the deterministic model. A single independent variableoperating revenues sustainable growth rateis randomized, specified as a triangular distribution, and heuristically calibrated as follows: minimum = 0%; likeliest value = 12%; and maximum = 17%. 16 The MC simulation follows a bootstrap multiple-simulation approach, repeatedly simulating the model and then creating a distribution of the statistics from each simulation, as depicted in Fig. 1.

SGM estimation
Data in Table 1 document that during the Covid-19 shock, the hotel population diminished relatively to 2019. However, it only partially recovered in 2021 without achieving pre-pandemic levels. We estimate the 2019 'Covid-19-free' operating revenue sustainable growth rate, using the steady-state SGM model derived in Appendix I, at 12.0% (see Table 3). We used this growth rate to project pro forma aggregate balance sheets, income, and cash flow statements for the 2020 and 2021 periods (see Appendix II for the specification of the variables).
Panels A, B, and C of Table 4 present the estimation of the sample's aggregate financial statements for the 2020 and 2021 periods, based on the 2019 period's SGR estimate unconditioned by the Covid-19 outbreak. The table presents the 2019 base case (column [1]); the deterministic estimations for 2020 and 2021 aggregate pro forma balance sheet, income, and cash flow statements (columns [2,3]); the 2020 and 2021 real aggregate balance sheet, income, and cash flow statements (columns [4,5]); and the estimates of the impact of the Covid-19 pandemic on the output variables measured by the differences between the 2020-2020 and 2021-2021 real aggregate values and the deterministic estimates (columns [5,6]).
The deterministically estimated aggregate impacts inflicted on our Portuguese mainland hotel industry sample over the 2020-2021 pandemic period are presented in Table 5: The estimated economic impacts measured by the aggregated operating revenues, net income, and operating cash flow are: a 64.2% reduction (− 5317 million euros); a 160.1% decrease (− 1457 million euros); and a 56.0% drop (− 738 million euros), respectively. The financial repercussions, gauged by the variation in the non-current liabilities and the financial slack, are: a 29.9% increase (+2557 million euros); and an 87.1% decline (− 9316 million euros), respectively. The latter impact is the compound effect of the 53.1% fall in debt capacity (− 9444 million euros) and the 45.8% reduction in the excess cash holdings (− 129 million euros).
Despite the reported aggregate impacts of 2020 and 2021, we must recognize that, after a severe decrease in hotel demand in 2020, some recovery was seen in 2021. Accordingly, our model presents different impact variations between the estimated and real outcomes from 2020 to 2021, of which it is worth mentioning the following (i) a 50 million euro increase in operating revenues (1. To summarize, in 2021, net income and operating cash flow improved due to growing demand, whereas net total assets, net total debt, and financial slack worsened (Fig. 2). The launch of the vaccination campaign by the end of 2020 may have, to a certain extent, lessened tourist travel restrictions, creating the conditions necessary for some recovery in hotel occupancy rates in 2021 and, consequently, in operating revenues.  1]); the interest coverage ratios, synthetic credit ratings, and debt spreads are estimated under the assumption of Covid absence, and stability relating with the base case (see [2] and [4]); and the real aggregate values for the years 2020 r and 2021 r , respectively (see [3] and [5]). Damodaran  However, as shown in Fig. 2, the difference between 2020 and 2021 operating revenues is relatively marginal. We conjecture that this might have been the effect of the phasing out in 2021 of the governmental Covid-19 mitigating measures in place. Debt capacity and excess cash holdings are the two sources of financial flexibility required to mitigate potential liquidity shortfalls and suboptimal allocative behavior in adverse states of the world, such as during the Covid-19 outbreak (See Panel D in Appendix II for the specification of the variables). Results document that over the 2020-2021 period, the hotel industry suffered a major fall-off in financial flexibility due, first and foremost, to debt capacity contraction, which seriously curtails potential recovery funding needs, notably in the new context of central banking's monetary tightening policy. The breakdown of financial slack indicates that borrowing   [2]), the real aggregate values for the same period (see [3]), the impact of the Covid-19 pandemic on the output variables measured by the difference between the 2020-2021 real values and the aggregate deterministic estimates (see [4]), and under the following assumptions: the values of the variables used were taken directly from the databases, without intermediate estimations; Net interest expense t = (Net interest expense t-1 /Non-current liabilities t-1 * Funding needs t ) + Net interest expense t-1 ; Funding needs t = Total assets t -Equity t -Non-current liabilities t -Current liabilities t ; Paid out dividend t = Net income t -Δ (Equity t -Equity t-1 ) under the assumption that the issuance and repurchase of shares are equivalent to each other. capacity is its main determinant (see Table 5 and Fig. 2).
To sharpen the analysis of the results, we estimate the 2020 and 2021 impacts of the output variables at the hotel unit level (see Table 6). As expected, the impacts on accounting economic aggregates, such as operating revenues, net income, and operating cash flow, were more negative in the first year of the pandemic.  [2]); the deterministic estimates for the same period (see [3]); the impact of the Covid-19 pandemic on the output variables measured by the difference between the 2020-2021 the real values and aggregate deterministic estimates (see [4]); and the percent of estimates, calculated as [4]/ [3].    [2]); and the variation of those impacts in 10 3 euros (see [3]) and percentage (see [4]

Robustness check: Monte Carlo simulation
As previously described, we followed a bootstrap MC multiple-simulation approach, repeatedly running the model. As such, ten simulation experiments were performed, each one with the number of trials required to generate a numerical approximation to the distribution of the output variables. 17 Results are summarized in Table 7. 18 In each experiment, cash flow at risk was estimated as the operating cash flow for which the accumulated probability of occurrence is 5%, or, in other words, the likelihood of exceeding CFaR is 95%. The Monte Carlo simulation results are very close to those obtained Table 8 | Robustness checks on the pandemic impacts (2020-2021). This table presents: the real aggregate impact for the 2020-21 period [1]; the deterministic estimations for output variables (see [2]); the Monte Carlo (MC) estimations for output variables (see [3]); the aggregate impact for the 2020-21 period considering MC estimates in 10 3 euros [4] and percent [5]; the differences between the pandemic deterministic and MC impacts as a percent of real [6]; deviation of the two estimates as a percent of the deterministic estimates [7]. Sources: ORBIS and SABI databases; authors' estimations.  17 The grand mean of the ten independent and identically distributed simulations is similar, under the linearity assumption, to generating ten samples from one simulation only. Nonetheless, the greater the number of simulation trials we run, the more the resulting means distribution will tend towards the true output variable distribution. 18 Detailed results are available from the authors upon reasonable request from registered Oracle Crystal Ball (64-bits) licensees.
with the deterministic approach described in section 4. Table 8 exhibits the robustness check estimates performed using Monte Carlo simulation methods with bootstrapping for the net total assets, net total debt, operating cash flow, and financial slack at risk, as well as the aggregate impact of the Covid-19 pandemic for the 2020-21 period. These suggest that, on average, the absolute deviations between deterministic and stochastic estimates at 95% confidence level are relatively minor, oscillating between 0.5 and 5.5%. In all iterations of the simulation experiment, the deterministic operating cash flow, as well as the grand mean, is within plus or minus two standard deviations from the mean interval of the operating cash flow distribution (see Fig. 3).

Discussion and conclusions
According to Peter Drucker, "If you can't measure it, you can't manage it" [63]. Thus, the resilient and sustainable recovery from the Covid-19 pandemic requires the ex-ante substantiated assessment of the extent of the repercussions of the pandemic. The main research purpose of this paper is to contribute to that end. As such, we examine the economic and financial impacts on the performance of the Portuguese mainland hotel industry during the 2020-2021 Covid-19 pandemic period in terms of the industry's aggregate operating revenues, net total assets, net total debt, generated cash flow, and financial slack, using a deterministic approach and stochastic robustness checking. We found that, on average, over the 2020-2021 period, the Covid-19 pandemic inflicted an aggregate impact of: (i) a 64.2% reduction in operating revenues (− 5317 million euros); (ii) a 30.8% decrease in the industry's net total assets (− 7625 million euros); (iii) a 29.8% increase in the sector's indebtedness (+2557 million euros); (iv) a 160.1% decrease in net income (− 1457 million euros); (v) a 56.0% drop in the operating cash flow (− 738 million euros); (vi) an 81.1% decline in financial slack (− 9443 million euros). Overall, these (deterministic) findings, on the one hand, quantify the extent of the economic problem caused by the pandemic. On the other hand, they provide estimations of the economic thresholds to be overcome and the financial hurdles faced by the future sustainable recovery of the industry.
The robustness check, conducted through the MC simulation with bootstrapping, indicates that the deviations of the deterministic from the stochastic estimates are, at a 95% confidence interval: 1.1% for the operating revenues; 2.1% for the net total assets; 5.5% for the net total debt; 1.9% for the net income; 0.5% for the operating cash flow; and 0.5% for the financial slack.
Overall, not only the stochastic approach delivers comparable outputs for the variables at interest, but unlike the deterministic approach, only the output analysis of the simulation output can be used to understand what happens at the lower tail of the resulting cash flow distribution. In addition, the bootstrap MC approach allowed for an estimate of the uncertainty of the operating cash flow, resulting in an expected downside risk of the Portuguese mainland hotel industry over the 2020-2021 period of 1293 million euros. In other words, such is the maximum shortfall of our industry sample's generated net cash flow during the Covid-19 period. In this framework, the MC model provides more information than the deterministic model and is a valuable tool for assessing the effects of the Covid-19 pandemic on the Portuguese mainland hotel industry.
The 2020 and 2021 impacts on accounting economic aggregates, such as operating revenues, net income, and operating cash flow, were more negative in the first year of the pandemic, as expected. We conjecture that this is due to the initial rounds of government lockdown measures, travel bans, and restrictions. However, the massive vaccination deployment, and the governmental fiscal policy measures to provide emergency countercyclical support to households and firms, may have contributed to the downscaling of the impacts of those effects in 2021 (e.g., Ref. [64]). The financial impacts, measured, for example, by the stocks of net total debt and financial flexibility, reflect, among other factors, the funding needs associated with the negative cash flow generation over the 2020-2021 period, and the effect of the moratoriums on bank credit agreements enacted in March 2020 because of the Covid-19 health emergency. 19 We can conclude that the extreme uncertainty and volatility associated with events, such as the Covid-19 pandemic, can expose business activity to extremely adverse economic and financial consequences. Our findings are consistent with the claim that was the case in the Portuguese hotel industry.
A limitation of this study is that it focused only on hotels (ORBIS/Sabi databases do not cover very small accommodation units) and treated the sample as a whole without distinguishing hotel size or exploring possible recovery strategies. As such, future research on this topic should develop along two axes. The first is to use a difference-in-differences approach to study whether or not hotels grouped by star classification were impacted differently by the Covid-19 pandemic. The second, building on the conceptual framework that business strategies can be conceptualized as chains of real options, is to develop a randomized valuation framework to appraise the value creation potential of the post-pandemic recovery strategies of the hotel industry.

Author contribution statement
Mário Coutinho dos Santos, José Magano, Jorge Mota: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper

Appendix I. Sustainable growth model derivation
Following Van Horne (2002), Zantout (1990), and Higgins (1977), we derived a steady-state version of the sustainable growth rate model to estimate the 2020 and 2021 income statements, balance sheets, and cash flow statements for a sample of the Portuguese mainland hotel industry. The model was derived under the assumption that balances of balance sheet accounts are optimized in relation to the current level of sales and that depreciations were not an available source of funds because an application of the same amount in fixed assets is assumed to sustain their operational functionality. Debt Capacity t -Non-Current Liabilities t + Excess Cash Holdings t