Probabilistic Modeling of Small Business Recovery after a Hurricane: A Case Study of 2017 Hurricane Harvey

Hurricane Harvey was a Category 4 hurricane that made landfall on August 25, 2017, between Port Aransas and Port O’Connor with sustained winds over 130 mph (GLO 2020). It caused catastrophic flooding and 68 direct deaths. The economic loss is estimated at $120 billion (2017 USD), primarily from flooding in the Houston metropolitan area and southeast Texas, where many areas received more than 40 inches (1,000 mm) of rain over a four-day period. Fig. 1 shows 39 counties in Texas eligible to receive individual assistance (IA) and public assistance (PA) as the part of this presidentially declared disaster (4332-DR).

The storm displaced more than 30,000 people, prompted more than 17,000 rescues, and shut down ports, trade, tourism, agricultural production, and general businesses across most of the Texas coast. The effects on oil and gas production rippled across the nation, with a gas price spike averaging about $0.33 a gallon (Energy Information Administration 2018). FEMA’s National Flood Insurance Program (NFIP) received over 92,000 claims and disbursed more than $8.92 billion to claimants (FEMA 2019). The Small Business Administration (SBA) has disbursed over $2.7 billion in home loans and almost $291 million in business loans to assist disaster recovery efforts. The cost of Harvey’s destruction was offset by an increase in business activities related to emergency repair, reconstruction, and restoration efforts, combined with an influx of funding from federal aid and insurance payments. In its aftermath, cars and trucks damaged or destroyed by flooding must be repaired or replaced, lifting sales in the auto industry. Many businesses, in addition to repairing damaged facilities, replaced some or all of their equipment and inventory. Meanwhile, thousands of evacuees stayed in hotels or rental units before returning to their flood-damaged homes, and purchasing furniture, household goods, electronics, and clothes. Memberships (to clubs, sports centers, parks, theaters, and museums), telecommunication services and entertainment were among the industries hit the hardest, whereas health services, food and beverages, rental housing, motor vehicles, furniture, and clothing sectors fared better.

Furthermore, 71% of manufacturing firms reported revenue and/or production losses from Harvey and an average closure of 5.5 days. In addition, 27% expected more difficulties in finding and hiring workers over the subsequent six months (Federal Reserve Bank of Dallas 2017). Overall, the net effect on the state’s economy may be much less severe than many have expected (Texas Comptroller of Public Accounts 2018).

The aim of the survey is to collect firm-level data on operations, storm impact, resilience, and recovery of small businesses impacted by Hurricane Harvey. The survey instrument was submitted to and approved by the Human Research Protection Program of Texas Tech University (IRB2019-463). The survey was administered through Qualtrics Panels, a 3rd-party survey vendor (Qualtrics) over two waves.

Respondents were recruited directly through Qualtrics based on the following inclusion criterion:

This criterion ensures that respondents had intimate and detailed knowledge of the key operational and financial characteristics of businesses. Moreover, by requiring at least 12 months of tenure prior to and 6 months poststorm, we ensure that respondents are qualified to report on decisions related to both preparation and recovery. Prior to launching the survey, Qualtrics ran a prelaunch analysis to evaluate and possibly improve aspects of the survey to optimize the quality of the responses. Based on their evaluation, Qualtrics estimated a predicted duration of 13.5 min for the survey. The fact that an actual average response time was 12 min indicates that respondents in our survey were spending the amount of time consistent with the expectation.

Wave 1 data collection was conducted in November 2019, generating 260 unique and random survey responses. The subsequent analysis allowed us to refine certain questions and identify coverage gaps. As the result, Wave 2 data were collected between October 2020 and November 2020, and consisted of 360 additional unique survey responses. Overall, across both waves the total sample is 620 respondents. Based on self-reported zip code, about 200 different zip codes from within the 39 impacted counties are represented across the sample (see Fig. 1). As seen in Fig. 1, the geographic area comprising all survey respondents spans a large area of the 39 impacted counties. Overall, we view this sample a providing reasonable representation of the geographic area most impacted by Hurricane Harvey.

After consenting and meeting the inclusion criterion, respondents were first asked some general demographic questions about themselves including how long they had been working at the company prior to Hurricane Harvey, if they were still working at the same company (if not, how long after Harvey they worked there), how many people worked under their management, and how many people were at a more senior level. Next, they provided general characteristics about the business itself including industry, annual gross revenues, number of employees, and number of locations. We then asked them a series of questions about their business operations and financing, followed by questions about the financial impact of Hurricane Harvey and questions about the company’s level of preparedness and investments in resilience. The last set of questions related to the built environment and infrastructure. The mean time spent on the survey were roughly 12 min, indicating that respondents did spend sufficient time to provide thoughtful responses.

Note that the inclusion criterion required these respondents to report being in a managerial role and having input and influence over key decision-making. In Table 1, one sees that the vast majority of respondents (78%) had at least four other people under their management, with 37% reporting having over 10 people under their management. Moreover, a majority of respondents (63%) also reported having at most three people more senior than them, with 25% reporting having no one else more senior to them (see Table 2). We view this as corroborating evidence that indeed these individuals assumed managerial roles in their respective companies. Whereas they were required to work at the company for a year leading up to Hurricane Harvey and at least six months after, the average tenure at the time of Harvey was 7.5 years, and most respondents (90%) reported still working for the same company at the time they were surveyed. It is also worth noting that 48% of them were self-employed.

Tables 3–5 present some key characteristics of small businesses represented in our study. As shown in Table 3, more than half of the firms (57%) had annual revenues of less than $1M. The mean (median) number of employees in these firms was 84 (36). As for the industry category, our sample is a relatively diverse mix with the sample being roughly uniformly split over the reported categories. The survey form initially classified small businesses into 21 industry sectors as defined in the North American Industry Classification System (NAICS). They include: Agriculture, Forestry, and Fishing and Hunting; Construction; Manufacturing; Transportation and Warehousing; Wholesale Trade; Retail Trade; Finance and Insurance; Mining, Quarrying, and Oil and Gas Extraction; Information; Accommodation and Food Services; Other Services (except Public Administration); Professional, Scientific, and Technical Services; Real Estate and Rental and Leasing; Health Care and Social Assistance; Administrative, Support, and Waste Management; Arts, Entertainment, and Recreation; Educational Services; and Utilities. Following convention in the economics literature, the categories were consolidated into eight broader industry sectors (as displayed in Table 4) which has the effect of increasing statistical power (Ewing and Wunnava 2004; Bureau of Economic Analysis 2017). In addition, we regrouped across the two waves of the survey to account for small differences in the categorical industry choices. In terms of how many locations their business had, a majority of those surveyed (54%) reported their company had just one location (see Table 5).

Tables 6–9 report key characteristics elicited on financial operations of the surveyed small businesses. A majority of the surveyed firms reported having access to some kind of bank funding (65%), with about half also reporting access to owner funding (48%), and about a quarter (28%) having access to investor funding (as seen in Table 6). We also asked about lines of credit, and 71% reported having access to a line of credit, and 74% of those reported using it (as shown in Tables 7 and 8). Last, we measured business exposure to the Houston market by asking respondents to report the fraction of sales generated from the Houston area, and the fraction of suppliers located in the Houston area. On average, about 65% of sales were from the Houston area, and 64% of their suppliers were located in the Houston area, as indicated in Table 9.

In terms of recovery, most of the firms (87%) reported that the business fully recovered after Hurricane Harvey. Of those firms that did recover, one sees in Table 10 that there is substantial variation in the time reported to recover. Here, we provided categorical choices, as a common practice, to shorten the response time and reduce recall bias (e.g., Asgary et al. 2013). Many firms recover quickly (37% reported recovering within three months), whereas others take much longer (22% reported taking 10 months or longer). In Collier et al. (2020), about 50% reported that they had not recovered within a year. Note that the question on business recovery was open to respondent’s own definition/interpretation, which would vary greatly. Some might base their answer on return of revenue or profit in comparison to prestorm levels and others on restoration of key production capacities (e.g., equipment, employees, and utilities). It is also expected that comparison could be made to one’s peers affected by the same storm. Such variability in how recovery is defined and how it relates to business characteristics was further examined in a follow-up study through interviews of small construction contractors in Houston, TX (see preliminary result in Sahu et al. 2022). In terms of preparedness which reflects a business owner/manager’s own assessment of their coping capability and outlook of possible consequence, Table 11 illustrates substantial variation in whether the surveyed individual felt the company was prepared for Harvey; this ranged from about 26% reported A great deal of preparedness, to 18% reporting A little preparedness, down to 9% reporting Not at all prepared. In a more objective manner, the most popular mitigation measures adopted were backing up of computer files, use of emergency generators, installing flood barriers, and performing a review of their property.

About half of the respondents reported that the company proactively made investments in resilience leading up to Harvey. To ensure consistency in their understanding of resilience, we included a robust checklist of 14 common mitigation measures ranging from install storm shutters, to backup computer files, to procure additional insurance coverage. These measures represent three broad categories of actions: physical, operational, and financial., in terms of building damage, 63% reported that the business sustained some building damage because of Hurricane Harvey. In addition, utility outages were very commonly reported (as seen in Table 12): 83% reported power outages, 68% reported water outages, 82% reported internet outages, and 90% reported transportation disruptions.

We first verify the independence of observed recovery time from a variety of class variables as defined or derived from a firm’s characteristics, action, and impact. Using the Pearson’s chi-squared test, the null hypothesis that the frequency distribution of recovery time has no association with a given class would be rejected when the test statistic, ${\chi }^2$, exceeds the critical value at a significant level with a certain degree of freedom. In such cases, this class variable could be a candidate to stratify the sample and differentiate recovery functions. We ask the respondents to select from a set of categories (i.e., first column in Table 13 below). Use of intervals in survey allow for quicker response, fewer data entry error, and/or increased response rate. This is generally accepted best-practice for survey implementation of large-scale panels. For example, in the 2020 FEMA National Household Survey (FEMA 2020), rather than entering an exact amount, the respondent was asked to give a ballpark figure for the amount they have set aside for emergency by selecting from the following ranges: $1 to $99; $100–$399; $400–$699; $700–$999; $1,000–$1,499; $1,500–$2,999; $3,000–$5,000; more than $5,000. In studying the impact of COVID19, Pierel et al. (2021) asked small businesses to indicate their recovery time in less than 1 week, 1–4 weeks, 2–3 months, 4–6 months, 7 months–1 year, 1–2 years, or longer than 2 years. As with any survey design, some ex-ante subjective judgment is required when formulating questions. However, these specific bins were selected with the aim of balancing the appropriate level of granularity while also spanning a reasonable recovery timeframe for most small businesses (e.g., Lee 2019). High concentration of data points in certain bins may call for setting closer thresholds and vice versa. However, we note that our examination of the data reveals we did not have such an issue with too much/little concentration in any single bin (the frequency ranged from 37% in the largest bin of 1–3 months to 10% in the smallest bin of 10–12 months). Overall, we view this as ex-post support that our threshold choices for recovery time were reasonable. For statistical analysis, we convert the categorical recovery times as reported to the numerical values as midpoints for parameter estimation (i.e., model fitting). When the recovery time is greater than 12 months, we approximate it to 18 months (the midpoint between 12 months and 26 months when the first wave survey was initialized).

The class variables we examined include industry sector, investment in resilience, levels of preparedness, line-of-credit, power outage, and building damage ratings.

The observed count of firms corresponding to the recovery time and its industry sector classification are tabulated and analyzed. For example, 10 out of 92 (or 10.9%) construction firms report in our survey that they recovered within 7–9 months after the storm. For the same timeframe, only three out of 77 (or 3.9%) small businesses in hospitality, food, and entertainment services recovered. Results from the Pearson’s chi-squared [${\chi }^2(35)=41.12$, $p=0.22$] fail to reject the null hypothesis at 95% significant level. Thus, the recovery time of firms appears independent from the industry sectors to which they belong. We are able to draw a similar conclusion when comparing individual sectors with the entire sample. The closest exception, though, is hospitality, food and entertainment services sector [${\chi }^2(5)=10.58$, $p=0.06$] in which a large number of firms are on the two extreme of recovery spectrum: 45.5% recovered within the first 3 months, whereas 16.9% had not recovered at the time of survey. Our findings echo a similar conclusion in Webb et al. (2002) about the absence of significant effect of economic sectors on long-term business recovery. Therefore, we do not stratify the sample based on industry sectors.

Investment in resilience, levels of preparedness, line-of-credit, power outage, and building damage have significant association with reported outcome of recovery time as shown in Table 14. Among them, building damages appears to be the strongest driver: 55.8% of firms with none or minor damage recovered within 3 months, and 76.6% did so within 6 months. In contrast, only 8.9% and 28.6% of those with major building damages recovered within 3 months and 6 months, respectively. The correlation between building damage and business recovery was also observed in the Northridge Earthquake (Dahlhamer and Tierney 1998), the Loma Prieta Earthquake, and Hurricane Andrew (Webb et al. 2002). However, content damage, not building damage, was found to be a strong predictor in Hurricane Matthew (Watson et al. 2020). Note that the question on level of preparedness was purposely designed to be vague, allowing for open interpretation by respondents based on the small business’ unique traits and contexts. Because of the diversification of small businesses and industries represented in our sample, we opted for such a general question with an intent to capture the mixed effect of perceptions of threats, protective actions, and stakeholders that ultimately drive decision making in disasters (Lindell and Perry 2012). It is similar to a question in FEMA National Household Survey (FEMA 2020): How confident are you that you can take the steps to prepare for a disaster in your area, not at all confident, slightly confident, somewhat confident, moderately confident, or extremely confident?

Then, we fit the data to various life distribution functions for both the entire sample and the split samples based on class variables. The unrecovered firms are right-censored. The Frechet and lognormal functions are best-fit in comparison. We adopt the lognormal models for its better-known qualities. Hence, the recovery cumulative distribution function (${\mathrm{F}}_{RT}$) and the recovery probability density function (${\mathrm{f}}_{RT}$) are specified aswhere $\mu$ and $\sigma >0$ = location and scale parameters of the distribution; $\mathrm{\Phi }$ and $\varphi$ are CDF and PDF of standardized normal distribution [i.e., $N(0,1)$]. The maximum likelihood estimatorswhere $r{t}_1,r{t}_2,\dots ,r{t}_n$ = sample’s recovery time as reported in the online survey.

We estimated $\mu$ and $\sigma$ for the whole sample ($\mathrm{N}=620$) and, for brevity, the split sample based on whether or not a firm made investment in resilience before Hurricane Harvey. Note that $e^{[\mu +({\sigma }^2/2)]}$ and $e^{\mu }$ are the mean and median recovery times, respectively.

We then set the time step as 1 month. Thus, the arrival rate function $\lambda (t)$ as shown in Eq. (6) is calculated and shown in Fig. 2.

When firms are considered as a single population, the conditional probability of recovery in the first month (i.e., $\mathrm{t}=1$) stands at 9%. In other words, about 1 out of 11 firms affected by the hurricane is expected to recover to the prestorm level. The probability reaches its peak value of 12% in month 3 before gradually trending down. If a firm still has not recovered after 3 years, the chance of recovery after that is as low as 5%. Meanwhile, the divergence between two groups of small businesses—those made investment in resilience (hereinafter referred to as resilient firms) and those that did not (nonresilient firms) is quite pronounced. For resilient firms, the conditional probability of recovery rises sharply during the immediate aftermath of the hurricane, peaking in month 5. After that, they are 50%–60% more likely to recover than the nonresilient ones.

The lognormal models described in the preceding section could be most useful when establishing certain benchmarks for specified business characteristics. The ability to compare itself with its peers under normal and crisis conditions is important to many business owners. In a related study, we interviewed 15 small construction contractors and saw a strong desire for tracking their own performance against that of their competitors after Hurricane Harvey (Keeler et al. 2022). Such an alternative (or supplementary) definition of business recovery could be useful only if peers or industry benchmarks are available. Underperformance, if detected, would call for a deeper analysis of its risk profile and possibly corrective actions. However, the averages (i.e., recovery time, recovery probability) given in these models only capture a part of the recovery process. Especially at the community level, a strong interest is to examine both the aggregated pattern of recovery and the uncertainty associated with it. To that end, we choose to create a synthetic community with a fixed number of small businesses each of which could be different in its financial, operational, physical property, and other attributes. When a hurricane hits, their recovery trajectories are likely to be unique but governed by a common nonhomogeneous Poisson process.

Fig. 3 below uses the arrival rate functions and plots the recovery process of 100 firms for the purpose of uncertainty quantification. These firms are randomly sampled from the lognormal distribution with parameters specified in Table 15. Each line in Figs. 3(a, c, and e) on the first column represents a single realization of the nonhomogeneous Poisson process in terms of the accumulative number of firms recovered from the hurricane in each month. Fig. 3(a) considers all firms having the same rates of recovery, whereas Figs. 3(c and e) are for resilient and nonresilient firms, respectively. Based on 100 realizations, Figs. 3(b, d, and f) show the percentages of firms that have recovered, accumulatively, in forms of mean, 5th-percentile, and 95th-percentile.

Three years after the hurricane, about 90% of all firms would have recovered. For resilient firms, the percentage is in the range of 93%–96%. In comparison, among nonresilient firms, that range from 87% to 89%. At the 12-month mark, 78% of resilient firms have recovered, whereas it is only 69% for nonresilient firms.

The model uncertainty is represented by 5th-percentile, and 95th-percentile bounds. The gap between the two bounds tends to expand as time elapses before contracting towards the end. Alternatively, we plot the coefficients of variance (CV) for the first 12 months. Nonresilient firms are shown to have the greatest variability in relation to their means. This could be attributed to larger uncertainties in model parameters for nonresilient firms. In practice, nonresilient firms may behave in a less predictive manner. Such high volatility, in a financial term, would correspond to higher risk to business performance or valuation.