Measuring short-term risk of initial public offering of equity securities: a hybrid Bayesian and Data-Envelopment-Analysis-based approach

Abstract

This paper offers a methodology to estimate an unconditional probability density function (PDF) for the stock price of an initial public offering (IPO), at a short-term post-IPO horizon. The resultant PDF is unique to the IPO of interest (IPOI) and serves to model the short-term post-market uncertainty associated with its price. Such a methodology is unprecedented in the IPO risk literature since the ex ante quantification of the short-term uncertainty associated with the stock price of a newly public firm was viewed as burdened by the lack of sufficient accounting and market history at the IPO stage. This gap is addressed here through recognizing that common in most IPO cases are the scarcity of hard data and abundance of soft data (strong prior belief), and that one can combine Bayesian inference and Data Envelopment Analysis (DEA) to develop a unique risk quantification setting that befits and serves these two characteristics of IPOs. In this setting, DEA serves to quantify the prior belief, to be subsequently updated in the Bayesian phase. This paper remains the first of its kind which unravels the IPO risk analysis from such perspective. It develops an iterative process that uses DEA to design a multi-dimensional similarity metric to find the ‘comparables’ to IPOI, and thereof the closest comparable to it, whereupon Bayesian inference is employed to utilize the information available from these comparables to sequentially update and revise the IPOI’s prior PDF. The validity of the proposed risk methodology was examined by backtesting analyses.

Appendix: Robustness testing of phase I

As indicated in Sect. 4, the three tests presented here pursue to demonstrate the robustness of Phase I and can be considered as additional checks to ensure accuracy. Phase I incorporates certain steps to address specific characteristics of input data such as negative data and non-discretionary factors. To this end, we have benefited from important works published by Emrouznejad et al. (2010), Silva Portela et al. (2004) and Sharp et al. (2007). In addition, we needed to take into account the cases where the IPO of Interest (IPOI) is an efficient DMU itself. Recall that the set of comparables for an ordinary (inefficient) IPO consists of its efficient peers as well as other inefficient peers that share the same efficient peers. The notion of slacks-based measure of ‘super-efficiency’ in DEA was utilized to tackle the cases of efficient IPOI (Tone 2002; Cooper et al. 2007). A detailed description of these steps can be found in Paradi et al. (2018). The robustness testing, which only concerns Phase I, focuses on examining the contribution of these additional steps. Undoubtedly, a DEA model capable of handling efficient IPOIs or negative data would be a more inclusive model, capable of covering a larger sample of IPOs. It is, therefore, interesting to study and visualize how the addition of such steps impacts the results, and whether the expected benefits are yielded through the added complexity.

The first test to this end consists of three different “Runs.” Figure 1 visualizes the results. In Run 1, we exclude the two steps of outlier detection and efficient IPOI treatment, where the latter deprives the model of its capacity to handle efficient IPOIs. In Run 2, the model is enriched by adding the efficient IPOI treatment capability. In Run 3, outliers are detected as well, using the “Jackstrap” approach proposed by De Sousa and Stosic (2005). It is clear from the graph how the incorporation of the two steps of outlier detection and efficient IPOI treatment smoothes out the erratic changes in the number of comparables per IPOI, making the model more robust in selecting comparables. The outcome depicted on the lower pane seems to be more intuitive as these IPOs are from the same industry and it is more in line with expectations to have a similar count of comparables per IPO. Furthermore, under RUN 1, the number of comparables grows with the sample size. While this observation alone does not provide enough evidence to conclude instability, it raises the question of whether the model is sufficiently sensitive to the differences between the DMUs/firms. Furthermore, the rather increasing trend shown in the first pane of Fig. 1 can be interpreted as that a more recent IPOI is more likely to be linked with a larger group of comparables. This upward trend seems incompatible with practical intuition since given sufficient data per IPOI, one does not expect a randomly selected latter IPOI to be associated with more comparables, relative to a randomly selected former IPOI which took place a few years earlier.

Fig. 1

Number of comparables associated with individual IPOIs

A comprehensive theoretical discussion focusing on how the addition of the steps described above improves the model’s soundness is considered out of scope here; yet, we illustrate some of the underlying theoretical concepts, that add to the sophistication of the model but clearly increase robustness, through the exemplary graphs shown in Fig. 2.

Fig. 2

The data presented in Table 1.5 of Cooper et al. (2007) has been used to construct this graph. Note that the output ‘Inpatients’ has been eliminated which facilitates a 3D visualization. It is illustrated how the shape of the frontier changes by the exclusion of the efficient DMU G. The production possibility set, which is capped by the efficient frontier, spans a smaller space subsequent to the removal of the efficient DMU G. The segmentation of the frontier changes as well; the number of efficient hyperplanes decreases in this example

In Fig. 2, once the efficient DMU G is excluded, the production possibility set shrinks. Moreover, the inefficient DMUs which were previously associated with either of the two hyperplanes GLJ and DGJ, are now jointly enveloped by the new and larger hyperplane DLJ. It is this mathematical property that is primarily responsible for the differences between the first and second panes of Fig. 1. Under RUN 1, once an IPOI is identified as an efficient unit, it is excluded from the pool of candidates of any succeeding IPOI. Since the eliminated efficient IPO could potentially remain as an efficient unit if it were preserved in the pool of candidates, its removal could, therefore, impact the comparables associated with the subsequent IPOIs.

The second testing carried out to assess model robustness focuses on whether the size of the set of comparables of a randomly picked IPOI changes significantly if its pool of candidates grows in size. First, the algorithm identifies the set of comparables for a given IPO of interest. Recall that this IPO is no longer the IPO of interest (IPOI) in any of the next iterations but a candidate comparable in the pool of candidates of any succeeding IPOI. Yet, we can continue to find and record the number of comparables for it using the same definition used in the case of IPOIs. Once all the iterations have been executed, a ratio called mean-to-union is calculated for each IPO, which acts as another summary measure to gauge the robustness of the model. The numerator of this ratio is the average size of all the comparable sets identified for a particular IPO, across all the iterations. The denominator is the size of the union of all the comparable sets identified for it across all the consecutive iterations. The closer this ratio is to unity, the less is the variability in the size of the set of comparables selected for the IPO. The ratio would be equal to unity if the set of comparables remains intact for an IPO through all the executions. Ratios greater than 70% were observed for 72%, 66% and 65% of the IPOIs in RUN 1, RUN 2 and RUN 3, respectively.

In this connection, we note that it is not just the size of the comparables set that is expected to remain stable across the iterations but the composition of the set, as well. The detail of the testing carried out to assess this avenue can be found in Sorkhi (2015). Broadly, the outcome indicates that the composition of the set of comparables for a given IPO either remains intact across iterations or is replaced with newly-added IPOs, and not the existing ones. Further scrutiny of the data reveals that compared to the retained former comparables, the excluded former comparables tend to be positioned farther from the IPO under investigation.