A Discounted Cash Flow variant to detect the optimal amount of additional burdens in Public-Private Partnership transactions

Graphical abstract

subsequent enhancements, DCF is now long since a prominent method within the so-called income approach to property appraisal. Moreover, it may be devoted not only to identify the value that can be ascribed to a property, since it is suitable also to support decisions about the feasibility and viability of investing in real estate because its structure reflects the economic fundamentals on which the same investment decisions rely [20].
The International Valuation Standards, published by the namesake Council, provide a concise definition of the DCF; it is as follows: "A financial modelling technique based on explicit assumptions regarding the prospective income of a business or property" [21,p. 300]. The development of a DCF analysis entails the need to outline a scheme (Fig. 1) wherein investment costs, operating revenues, and operating costs are projected over the time span under investigation, and processed jointly with other related parameters (Table 1). This leads to identifying the cash flows. Sharing the principle that deferred values are not immediately comparable, the cash flows, in turn, should be discounted back to the present by means of a conversion factor represented by the discount rate. Among the other essentials of DCF, the so-called judgment criteria play a prominent role. The Net Present Value (NPV) expresses the sum of all the ingoing and outgoing cash flows, once they have been discounted at a given discount rate. Therefore, it could be said that the NPV provides a measure of the additional monetary resources generated by implementing the investment. Besides, the Internal Rate of Return (IRR) is a specific discount rate, which has the property to make null the NPV. It represents a relative measure of the profitability of the self-same investment. The equation for the NPV calculation is as follows: where CF is the amount of the cash balances, which in turn overall constitute the cash flow to be discounted at the rate r, while i represents each time unit of analysis during the time span n.
The discount rate estimation is a highly debated issue. The literature points out the prevalence of two schools of thought within the branch of property investment valuation. The first one uses the yields of long-term Treasury bonds as a source [22], relying on the empirical finding that the construction and real estate sectors are generally less risky than other markets. The second approach bases itself on the principle that the discount rate should reflect the combined cost of all the funding sources employed to ensure the covering of investment costs, looking at equity and debt as the main ones; this perspective singles out the concept of Weighted Average Cost of Capital (WACC) [23].

Method remodeling within the scope of public-Private partnership
Before discussing the DCF variant we propose, it appears appropriate to make an introductory digression about the scope of application. Investment projects concerning both private properties and collective infrastructures and facilities are ever more frequently carried out under the umbrella of Public-Private Partnerships (PPPs) [24]. This phenomenon is induced by several circumstances, such as the paucity of public funds intended to provide public goods and services at the urban-wide scale, as well as the will of public bodies to benefit from the entrepreneurial ability of private entities during the production process of the aforementioned goods and services [25]. It has been argued that PPPs lend themselves to drive the decision makers toward investment transactions characterized by a higher degree of social and environmental sustainability [26]. It means that, within the PPP transactions, the private counterparts may be asked to bear additional burdens, in order to bankroll the provision of benefits in behalf of the community. In the matter of the social dimension of sustainability, the literature focuses on the concerns related to the equitable access to the resources [27], as well as to the strengthening of the so-called social capital [28], even through the provision and maintenance of infrastructures and facilities [29]. Besides, the environmental sustainability concerns may relate to the adoption of mitigation measures at the urban-wide scale, as well as to the implementation of solutions aimed to reduce the greenhouse gas emissions at the building scale. A specific branch of action, which ties together the social and environmental aspects, pertains to the provision of energy-efficient affordable dwellings [30], as a way to counteract the phenomena of housing poverty and fuel poverty.
The negotiation process that commonly precedes the signing of a PPP agreement entails the need to be supported by evaluation evidence. The DCF analysis may provide a suitable aid, particularly in order to outline the feasibility degree of a project, as well as to rank the design alternatives according to their economic viability. Nevertheless, to pursue the aforementioned goals, the traditional DCF framework requires being adapted. This article aims to outline a way the DCF may be remodeled, allowing us to organize more equitable PPP transactions.
The DCF variant discussed herein takes advantage of a specific perspective, which can be adopted in order to estimate the discount rate. It strongly differs from the two schools of thought discussed in the previous paragraph, although it still relies on the fundamental concepts of cost of capital and risk. In his 1990's essay, Kincheloe [31] argues the supremacy of the WACC as a method to estimate the discount rate in the real estate appraisals, so he too belongs to the second of the aforementioned schools of thought. Among the reasons for endorsing this conclusion, the ability of the WACC in representing the capital structure implied in an asset or in a project plays a remarkable role. Especially, the author criticizes the adoption of a discount rate relying on the investor's expected return on equity. He adversely judges the fact that this alternative method, although correctly applied to a different kind of cash flow, usually leads to underestimating the market value of a property. The previously mentioned study by Ling [23] argued that this last criticism is, at least partly, misplaced. Therefore, we can choose to apply the WACC, as the discount rate, to a certain kind of cash flow, namely the beforedebt cash flow from operations. Otherwise, we can opt for the developer's expected return as the discount rate to be applied to a different kind of cash flow, specifically after the debt reimbursement. In the both cases, we are able to get the same result if the costs of funding sources, which contribute to define the discount rates, are specified correctly over the whole time period of analysis.
Estimating the discount rate by referring to the return expected by the developer on its invested equity carries a remarkable benefit. According to the fundamentals of economic theory, if the financial model (see the previous Fig. 1) explicitly includes all the pertinent costs, namely the remuneration owed to all the employed factors of production, then the NPV represents a kind of extra profit. Indeed, such factors include also the entrepreneurship and the involved properties À which can be fairly remunerated by dedicated cost items À as well as the financial capital À whose expected remuneration is expressed precisely by the discount rate. This awareness opens the doors to pursue our original goal, that is to say, limiting the extra profit by imposing additional burdens on the private entities involved in PPP transactions, so they may become more equitable. Consistently with the above-described conceptual framework, within the customization we propose here, the discount rate estimation relies on the Capital Asset Pricing Model (CAPM). The rationale of the CAPM is summarized in its three basic terms: a market risk premium rate, additional to a base rate, allowing to express the return achievable from somehow risk-free investments, and a beta parameter, which specifically takes account of the not diversifiable risk [32]. The standard equation is as follows: where k e is the discount rate estimated as the expected return on equity, rf stands for the rate of return of the most risk-free investments, rp is the market risk premium rate and b is a parameter varying around unity, which represents the degree of exposure to the average market risk. The literature long since has discussed a broad range of data sources for gathering information in order to estimate the terms constituting the CAPM. Moreover, several studies provide their own estimates too. As an example, you can see the appendices in Damodaran [33]. The structure of the discount rate, according to Eq. (2), and some exemplifying estimating sources are further shown in Fig. 2.
Once known the values of NPV and IRR for the project under evaluation, the DCF variant entails essentially an optimization problem, which can be solved through simple one-shot equations, or else by iterative calculations if additional constraints must be considered. Since the costs usually precede the revenues, the NPV turns out to be a decreasing function of the discount rate. Let us assume to face an economically viable project; therefore, for a given discount rate, we achieve a positive NPV and an IRR higher than the same discount rate (Fig. 3, on the left side). Additional burdens, which lower the NPV and the IRR, may be charged to the project developer, until the self-same NPV comes close to zero (Fig. 3, on the right side). An IRR exactly equal to the expected return on equity, namely to the discount rate k e , represents the limit condition to be satisfied in order to judge the project still economically  viable. If such additional burdens are foreseen in the form of a lump-sum payment a, which will occur at time t, as is the case of public works built by the developer in behalf of the community, their calculation is as follows: which represents just a mere deferral of the NPV amount. Otherwise, if the additional burdens are foreseen as a periodic payment, as is the case of a concession fee c to be paid on a yearly basis from time t to time n, the calculation is as follows:

Method validation
The DCF variant outlined here has been tested and validated through a case study application. The case consists of the rehabilitation of a sports complex located in the suburbs of Bologna (Figs. 4 and 5), a medium-sized city in Northern Italy. The case perfectly fits the circumstances for which the DCF variant has been proposed: it is an urban renewal project that implies the need to incur property investments, to be carried out under the umbrella of a Public-Private Partnership. Here is the case history.
During the early nineties, a new indoor sports arena mainly meant to host basketball matches was built on the southwestern outskirts of Bologna. Its occasional and desultory usage led to a sudden physical and functional decay. In 2008, a private company came forward to renew the arena and the surrounding area. It submitted a project to the Municipality, in order to be awarded the right to carry out the works, according to a long-term ground lease, which is referred to as surface right in the national legal system. The ground lease was foreseen to last up to ninety-nine years. The project contemplated the rehabilitation of the building, in order to make it suitable to host other events beyond the sports competitions, such as concerts, exhibitions, and corporate conventions. Moreover, the feasibility of the renewal project was pursued by adding a new shopping mall.
The subsequent negotiation process among the involved public bodies, namely Municipal and Provincial authorities, and the private developer was intended to point out the transaction characteristics: the site management conditions, according to the entrepreneurial ability of the private entity, on the one hand; the community benefits to be charged to the same private entity, in the form of a yearly concession fee, on the other hand. Therefore, during the second half of 2010, the negotiation process was supported by the previously defined DCF variant.
An appraisal of the fundamental financial and macroeconomic parameters was carried out prior to developing the cash flows, according to the scheme drafted in the previous Fig. 2. Information and data were gathered from the sources there mentioned, as they are summarized in Table 2. The risk-free rate was estimated looking to the average yield offered by the long-term Treasury bonds, with a maturity of 30 years (the longest available in Italy), which were placed to the investors through auctions during the time span from 2005 to 2010. Both the nominal rate (rf n ) and the real rate (rf r ) were taken into account, the latter by referring to the inflation-indexed Treasury bonds. Hence, the inflation rate (i) was extrapolated according to the following calculation: The choice of the beta parameter (b) was suggested by the data disseminated by Damodaran (see http://pages.stern.nyu.edu/$adamodar/). Accordingly, on the whole, the project was considered more      21 19 prone to risk than the average. Finally, the risk premium rate (rp) was inferred from the data included in several periodic reports, released by survey institutions and focused on the companies listed on the Milan stock exchange. The listed companies taken into account were those operating in fields akin to the project, namely construction sector, real estate, utilities, leisure and entertainment industry. Instead than applying the inflation rate to the operating costs and revenues, we opted for the use of a real (deflated) discount rate. The outcome was a 6.77% expected return on equity, net of the inflation. The Tables 3-5 show the cash flow for the first couple of decades in the time span under analysis, while for the complete cash flow please refer to the supplementary materials here enclosed. The results were fairly good since the NPV turned out to be positive (about 238 thousand Euros). According to the previous Eq. (4), this allowed to estimate the maximum yearly amount of the concession fee (about 26 thousand Euros), to be paid from the beginning of the management after the completion of the refurbishment works (year 8), until the expiration date of the ground lease (year 99). As may be observed in the complete cash flow, and as further shown in Fig. 6, the concession fee represents exactly the amount of resources which lowers the NPV until it becomes null. 5.5% 6.0% 6.5% 7.0% 7.5% 8.0% 8.5% 9.0% Net Present Value (thousand Euros) Discount rate (%) NPV (withou t the concession fee) NPV (with the concession fee) Fig. 6. Comparison between the NPVs with and without the concession fee.