Analysis of Green Bonds

Fauß, Tobias Friedrich Article Analysis of Green Bonds Junior Management Science (JUMS) Provided in Cooperation with: Junior Management Science e. V. Suggested Citation: Fauß, Tobias Friedrich (2022) : Analysis of Green Bonds, Junior Management Science (JUMS), ISSN 2942-1861, Junior Management Science e. V., Planegg, Vol. 7, Iss. 3, pp. 668-689, https://doi.org/10.5282/jums/v7i3pp668-689 This Version is available at: https://hdl.handle.net/10419/294999 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Junior Management Science 7(3) (2022) 668-689 Junior Management Science journal homepage: www.jums.academy Analysis of Green Bonds Tobias Friedrich Fauß Eberhard Karls Universität Tübingen & University of Nottingham Abstract Issuing its first green federal security in 2020, Germany pioneered a unique twin bond concept to address potential liquidity risks compared to their conventional counterparts. A switch mechanism between green and conventional bonds was introduced that allows debt-neutral sale-and-purchase (switch) transactions by the issuing authority. The main goal of this dissertation is to provide a theoretical model that is capable to explain the effects of this twin bond concept on the pricing of green bonds. For this purpose, a stochastic liquidity premium following a Vasicek (1977) process, a constant green premium and a switch option, which is executed when the green bond price falls below the price of its conventional twin bond, are assumed. The model results confirm that this twin bond concept is a viable solution to mitigate illiquidity-induced costs for the green bonds. The main learning from the model is a potential positive value of the switch option before its execution. This implies that issuers adopting this concept could benefit from lower costs of capital compared to ordinary green bonds without a switch mechanism. For investors holding the green instruments, this implies a reduced exposure to liquidity risks. Keywords: Green bonds; German twin bonds; green premium; liquidity premium; switch transactions. 1. Introduction Most nations have acknowledged the risks of climate change and pledged to pursue mitigating measures. As of today, 193 Parties adopted the 2015 Paris Agreement on climate change with the commitment to limit global temperature increase to 1.5°C above pre-industrial levels, and all United Nations Member States committed to the 17 Sustainable Development Goals (United Nations,n.d.,2022). This transition to a more sustainable global economy requires a substantial amount of new investments. For example, the European Commission anticipates additional annual investment needs of approximately 2.3% of GDP (i.e., 336 bn. EUR) for necessary energy system investments (exclusive transport) in light of its 2030 Climate Plan and 1.6% of GDP thereafter, aiming to become climate neutral by 2050 (European Commission,2020). Similarly, the German Federal Government may need to increase its annual expenses from 200 bn. EUR to about 240 bn. EUR to become climate neutral until 2045, which is an additional percentage point of its 2019 GDP, cumulating to approximately 7% of GDP in total (Helmcke, Heuss, Hieronimus, & Engel,2021). In addition to funding from the public sector, private investments can play a crucial role to provide the required financial resources (European Commission,2020). One instrument to raise funding for this transition is the emission of green bonds. Such fixed income debt securities differ from conventional bonds as their proceeds are entirely dedicated to the financing of environmental or climate related projects (Ehlers & Packer,2017). Ehlers and Packer (2017) show that there is no single global definition for projects that fall into this category, but a range of different established standards and external verification procedures instead. For example, one widely accepted industry standard that is also adopted by the German Green Bond Framework are the Green Bond Principles issued by the International Capital Market Association (ICMA) (Finanzagentur GmbH, 2020). Other external validation concepts are second-party opinions by independent research institutes such as the Center for International Climate Research (CICERO), verification by auditors such as KPMG, certifications by organizations such as the Climate Bond Initiative (CBI) or green ratings, by rating agencies such as Standard & Poor’s and Moody’s (Dorfleitner, Utz, & Zhang,2021). Finally, there also exist regional standards such as the EU Green Bond Standard or China’s Green Bond Endorsed Project Catalogue (European Commission,2021;People’s Bank of China,2021). DOI: https://doi.org/10.5282/jums/v7i3pp668-689 T. F. Fauß /Junior Management Science 7(3) (2022) 668-689 669 The potential benefit of the above-outlined references is the reduction of asymmetric information and increase of transparency (Dorfleitner et al.,2021). The relevance of such additional information is supported by a CBI (2019) survey, which identified green credentials as one of the main drivers for green investment decisions. The global market for green bonds has strongly increased over the past years. Based on figures published by the CBI (2021), their issuance volume already surpassed the former annual maximum of 2020 (i.e., 294.4 billion (bn.) US Dollar (USD)) in Q3 of 2021, with 354.2 bn. USD Year to Date (YTD). Further, they forecast the annual volume to exceed one trillion USD by 2023. To put these numbers into perspective, SIFMA (2021) reports for 2020 a global long-term bond issuance volume of 27.3 trillion (tn.) USD, indicating a green bond market share of about 1% that year. While green bonds can help to finance sustainable investments, they may also incur additional expenses for their issuers compared to conventional bonds. These can be caused by internal costs due to screening, managing and reporting on their use of proceeds, as well as external costs for their certification and second-party opinion (Hachenberg & Schiereck,2018). However, these additional costs might be compensated if investors require a lower return (i.e., yield to maturity) for holding the green assets. In terms of prices, a lower yield means that a green bond can be issued at a higher price in comparison to a conventional twin bond with the same nominal value, which reduces funding costs. In fact, evidence for lower yields of green bonds is found by the majority of empirical studies (MacAskill, Roca, Liu, Stewart, & Sahin,2021). This implies that in spite of additional expenditures, issuers could even benefit from lower financing costs for their sustainable investments by issuing green bonds instead. One characteristic that potentially influences the yield of green bonds is their degree of liquidity (e.g., Zerbib (2019)). This is because investors may require a higher return for holding an illiquid asset (Kempf & Uhrig-Homburg,2000). The German Finance Agency (Finanzagentur) (FA), which is responsible for the issuance of German green sovereign bonds, argues that an excessive volume of green bonds may impede the liquidity of conventional bonds, while a deficient volume may impede their own liquidity (Finanzagentur GmbH, 2021a). In light of this trade-off, they pioneered a unique green bond concept to solve this issue in 2020, which in 2022 was adopted by Denmark as well (Dutch State Treasury Agency,2019;Finanzagentur GmbH,2020). In summary, this concept bases on the issuance of green bonds as twins to conventional bonds that coincide in almost all characteristics. This allows the introduction of a switch mechanism between both twins that has the function to secure a superior value of green bonds, which differ to conventional twins in its more restricted use of proceeds. Or in terms of yields, the yield of conventional bonds can serve as an upper limit for the yield of green bonds. For investors, this can imply additional certainty to sell a green bond for at least the price of its conventional counterpart. For issuers, this may imply more favourable financing conditions, as the green bonds can possess a lower yield. The goal of this dissertation is the derivation of a theoretic model that is capable to explain the mechanisms of the German green bond concept. In detail, it aims to provide a decomposition of the yield differential between both twins into its individual components. Namely, a green premium, a liquidity premium and the added value of the switch mechanism. A successful disentanglement of the yield differential can provide issuers as well as investors with crucial information for evaluating this concept. From an issuers’ perspective, this may answer the question if the framework is a viable approach to mitigate undesired illiquidity-induced costs and thus secure more favourable costs of capital. From an investors’ perspective, this information can also be relevant to correctly account for their exposure to potential liquidity risks. The added value of this dissertation is therefore viewed as a theoretical contribution to improve the understanding of the implications of the German twin bond approach with a focus on its most defining feature, the switch mechanism between green and conventional twin bonds. The remainder of this paper is structured as follows. First, an overview of relevant literature is provided. Then an evaluation of the German green bond concept and a comparison to their green sovereign peer bonds is conducted. In the following part, the theoretical model is derived and a calibration of the model parameters is performed. Finally, the model implications are evaluated, including a sensitivity analysis and a discussion of its limitations. 2. Literature Review 2.1. Green Premium There exists an increasing body of literature with the goal to explain and quantify the potential yield premium for green bonds. Such a green premium can be defined as the difference in yield to a conventional bond, which shares otherwise the same characteristics (Zerbib,2019). A negative premium implies that investors require a lower return when investing into a green asset. Fama and French (2007) find that the taste for an asset, expressed by additional utility from holding it beyond its financial payoff, can help to explain its prices. In line with this result, Dorfleitner et al. (2021) argue in the context of green bonds that a lower yield for holding a green asset may be compensated by a non-financial utility component. This is also supported by findings from Riedl and Smeets (2017) who observe that social preferences and signalling outweighs financial motives for explaining socially responsible investment decisions based on a survey conducted in 2011 with Dutch investors. The impact of non-pecuniary factors is also supported by Hartzmark and Sussman (2019), who evaluated the effect of the first introduction of sustainability ratings by Morningstar for the U.S. mutual fund market in 2016, which supported the evaluation of a funds’ sustainability. They found that fund flows for more sustainable T. F. Fauß /Junior Management Science 7(3) (2022) 668-689670 funds were positively affected, while the flows to less sustainable funds decreased. In light of this, there is a growing branch of studies evaluating the size of a potential green premium. Reviewing 15 publications that have been published in this area between 2007 and 2019, MacAskill et al. (2021) report a lack of consensus regarding the existence of such premium, which they attribute to different methodological approaches. Nevertheless, its presence is reported in the majority of the studies for both, the primary market (56%) and the secondary market (70%). For the latter market, the reviewed studies observe an average green premium of -1 to -9 basis points (bps). Further, MacAskill et al. (2021) highlight that the premium is generally more profound for green bonds that are “government issued, investment grade, and that follow defined green bond governance and reporting procedures”. For the latter, they provide recognized green bond certification standards and third-party verification for the use of proceeds as main drivers. Hachenberg and Schiereck (2018) argue that such an enhanced reporting is necessary to mitigate information asymmetries between the issuers and investors. This aligns with the results of a survey conducted by the CBI (2019) with 48 of the largest Europe-based fixed income asset managers, which showed that they view green credentials, next to pricing, as one of the most important factors for their investment decisions. This is also consistent with Dorfleitner et al. (2021), whose findings support the positive effect of external validation on the green premium. Moreover, Immel, Hachenberg, Kiesel, and Schiereck (2021) and Hachenberg and Schiereck (2018) find the issuer’s Environmental, Social and Governance (ESG) rating to influence the yield differential between green and conventional bonds. Finally, Kapraun, Latino, Scheins, and Schlag (2019) identify a bond’s “Green credibility” as a main driver for the green premium. The German Green Federal Securities, which are the focus of this dissertation, seem to fulfil the above-mentioned driving factors. However, the evidence for the existence of a green premium for sovereign green bonds considering both, the primary market and the secondary market, is not conclusive. For example, while Doronzo, Siracusa, and Antonelli (2021) find no definite evidence for such a premium based on bond data from 14 countries that have been issued between end-2016 and 2020, Kapraun et al. (2019) find a significant green premium between 5 and 18.5bp for bonds that are issued by government entities. A feasible methodological approach to evaluate the premium of green bonds is the comparison with a counterfactual brown (i.e., non-green) bond that otherwise exhibits the same characteristics (Bachelet, Becchetti, & Manfredonia,2019). As such a security is in general not available, one viable alternative is to find a proxy based on a matching method. For example, Bachelet et al. (2019) identify brown nearest-neighbours that have the same currency, issuer, rating coupon type and a similar maturity date, coupon rate and amount issued. Doronzo et al. (2021) and Zerbib (2019) also use a direct matching approach by constructing a synthetic brown bond based on other bonds that have similar properties. Alternatively, two-step matching procedures such as propensity score matching (e.g., Gianfrate and Peri, 2019) or coarsened exact matching (see Löffler, Petreski, and Stephan,2021) are applied to obtain estimates for the “untreated” brown bonds. However, in this study it is not necessary to rely on proxies for a counterfactual brown bond, as the German Green Federal securities are issued as twins to conventional bonds that share most of their characteristics. 2.2. Liquidity Premium One property that differs is that German Green bonds have a lower issuance volume than their conventional counterparts (Finanzagentur GmbH,2021b). The Finance Agency (2021a) argues that a sufficiently high amount outstanding is necessary to ensure that they can be traded in large quantities and at any time. This is because a low volume can imply a lower liquidity due to less owners and thus higher search costs (Helwege, Huang, & Wang,2014). Therefore, investors may require a higher return to compensate for the additional risk of holding an illiquid asset (Kempf & Uhrig-Homburg, 2000). This understanding of liquidity is based on Fisher (1959), who views it as the ability to sell a bond quickly and without a discount on its value. The impact of illiquidity on bond prices in general is widely researched (e.g., Chen, Lesmond, and Wei (2007); Dick-Nielsen, Feldhütter, and Lando (2012); Helwege et al. (2014); Kempf and Uhrig-Homburg (2000); Schestag, Schuster, and Uhrig-Homburg (2016)). A main advantage of understanding and measuring illiquidity costs is that it can help investors to improve the management of their exposure to risks. For example, investors who hold a bond until maturity (i.e., no need to sell it early) are not affected by liquidity disadvantages and thus may favour a premium for holding an illiquid asset (Wegener, Basse, Sibbertsen, & Nguyen,2019). However, if the premium is attributed to other factors (e.g., credit risk) instead, this may not be the optimal investment alternative for such investors. Nevertheless, it is not straightforward to derive a feasible proxy for the liquidity, which can be attributed to the lack of a universal definition (Díaz & Escribano,2020). Díaz and Escribano (2020) provide an overview on the various dimensions of liquidity and the selection of proxies that measure its different characteristics. In the context of this dissertation, a proxy that indicates the size of illiquidity costs over time is required. One viable approach to estimate this liquidity premium is the comparison of yields of bonds that only differ in their degree of liquidity. While Schwarz (2019), Monfort and Renne (2014) and Schuster and Uhrig-Homburg (2012) compare the liquid German Federal Securities with less liquid bonds from the German state-owned investment and development bank, Kreditanstalt für Wiederaufbau (KfW), Kempf, Korn, and Uhrig-Homburg (2012) compare the German Federal Securities with less liquid Pfandbrief bonds (Covered Bonds) and Wegener et al. (2019) compare less liquid traditional Pfandbrief bonds with Jumbo Pfandbrief bonds that have a larger issuance volume. To relate the liquidity premium to different investment horizons, Kempf et al. (2012) T. F. Fauß /Junior Management Science 7(3) (2022) 668-689 671 model the premium based on the Nelson and Siegel (1987) approach, while Koziol and Sauerbier (2007) use the Svensson (1994) method. Both parametric models provide the term structure of the current spot rate for zero coupon bonds. The impact of differences in liquidity is also considered in the context of green bonds. For example, Zerbib (2019) remarks the explanatory power of a liquidity proxy for the yield differential between green bonds and counterfactual conventional bonds, when estimating the green premium. Further, Bachelet et al. (2019) find evidence for a higher liquidity of green bonds issued by public institutions in relation to their brown (i.e., conventional) counterparts. Finally, Wulandari, Schäfer, Stephan, and Sun (2018) find a negligible impact of liquidity risk on green bonds. Finally, liquidity risks can affect the financing costs for issuers of bonds. The costs of capital are determined by the primary market yields issuers can secure at issuance. However, the return investors require from holding a bond may be affected by its expected performance on the secondary market. For example, Goldstein, Hotchkiss, and Pedersen (2019) find evidence based on corporate bonds that the expected aftermarket liquidity at issuance can have an economically large impact on the financing costs. A viable explanation for this finding is that the initial investors have a lower perceived risk in case they need to sell the asset before its maturity, and are thus willing to pay a premium. From an issuer’s perspective, it can therefore be advantageous to ensure a liquid secondary market for its bonds. 2.3. Term Structure Models To derive a structural model for the effect of illiquidity on the German green bonds, we assume a stochastic model that can reflect the development of the liquidity premium until maturity. For this purpose, we apply a term structure model of the short rate, which provides their development over time. In other words, we use a stochastic process to model a sequence of interest rates (i.e., a liquidity premium) each for an infinitely small period of time. This type of models are widely applied to value interest rate derivatives, such as European bond options (Hull,2018). Further, they have also been used in studies that model bond illiquidity (e.g., Kempf and Uhrig-Homburg (2000); Koziol and Sauerbier (2007)). In general, the various approaches can be divided into equilibrium models and no-arbitrage models (Hull,2018). The drift of the short rate in equilibrium models is no function of the time, whereas the drift in no-arbitrage models is time-dependent. While the latter approach allows an exact fit to the current term structure, this is not required for the present application (Hull,2018). This is because the used liquidity proxy is not calibrated to the actual (il-)liquidity of German green bonds, which only allows a relative evaluation of the effects. Hull (2018) presents the Rendleman and Bartter (1980) model, the Vasicek (1977) model and the Cox, Ingersoll, and Ross (1985) model as possible equilibrium models. The Rendleman and Bartter model differs in a way from the other models that it does not assume a mean reversion for the short rate, while the Cox, Ingersoll, and Ross model excludes negative interest rates by construction. The Vasicek model assumes a mean-reverting process for the short rate and allows for negative rates. 3. Green Sovereign Bonds The CBI reports that 22 national governments have issued sovereign Green, Social, and Sustainability (GSS) bonds until November 2020 with a total amount of 96 bn. USD (Harrison & Muething,2021). In the same study that was published in January 2021, Harrison and Muething (2021) report for the majority of GSS bonds a relatively higher imbalance between their supply and demand compared to their conventional counterparts, which was suggested by a mostly higher book cover (i.e., oversubscription). This indicates market growth potential for the green sovereign bonds. Moreover, they report that based on 23 issuances between 2017 and November 2020, ten bonds priced on the yield curve of conventional peers, nine priced below and four above. As the green bond issuance at a yield below the yield curve of conventional (i.e., vanilla) bonds implies more favourable financing costs, the observed sovereign bonds provide no clear evidence for such a potential yield advantage. In the following, we first provide an insight into the German green bond framework. Then we compare it to a small peer group of sovereign green bonds with a focus on how potential liquidity issues are addressed. 3.1. German Green Federal Securities Since September 2020, the German federal government issued Green federal securities with a total volume of 24 bn. EUR (see Table 1). In 2021, it issued 12.5 bn. EUR of Green bonds, which accounted for 2.6% of the total issuance volume (482.7 bn. EUR) of tradable government debt that year (Finanzagentur GmbH,2021d). For 2022, it anticipates a similar annual volume (Finanzagentur GmbH,2021a). The German Finance Agency (“Finanzagentur”), which administers the issuance of green bonds in the primary market, acknowledges the need to account for sustainability in financial decisions in light of economic risks as well as investment opportunities that result from climate change and transition towards a “more sustainable global ecosystem” (Finanzagentur GmbH,2020). It concludes that this requires an enhanced transparency and development of the market for green and sustainable investments, to which the Green Federal securities are a key driver. On the one hand, the enhanced transparency can be attributed to the chosen evaluation, selection, and reporting process. The criteria to identify eligible budget items align with the Green Bond Principles by the ICMA, the EU Charter of Fundamental Rights and consider elements of the draft EU Green Bond Standard (European Commission,2012;Finanzagentur GmbH,2020;International Capital Market Association,2021). In its Green Bond Investor Presentation (2021b), the Finance Agency provides an overview of the use of proceeds of the German Green Bonds. For example, it attributes the eligible expenditures in 2019 to the following sectors: Transport T. F. Fauß /Junior Management Science 7(3) (2022) 668-689672 Table 1: Overview of German Twin Federal Securities Name Issuance Maturity Date Outstanding Type ID 2021 (2050) Bund/g 18.5.2021 15.8.2050 6.0 bn. EUR Green G2050 2019 (2050) Bund 23.8.2019 15.8.2050 29.0 bn. EUR Conventional C2050 2021 (2031) Bund/g 10.9.2021 15.8.2031 6.5 bn. EUR Green G2031 2021 (2031) II Bund 18.6.2021 15.8.2031 26.5 bn. EUR Conventional C2031 2020 (2030) Bund/g 9.9.2020 15.8.2030 6.5 bn. EUR Green G2030 2020 (2030) II Bund 19.6.2020 15.8.2030 30.5 bn. EUR Conventional C2030 Bobl/g 6.11.2020 10.10.2025 5.0 bn. EUR Green G2025 Bobl 10.7.2020 10.10.2025 25.0 bn. EUR Conventional C2025 The data in this table is based on Finanzagentur GmbH (2021a) and Refinitiv Eikon (Accessed: 21.11.2021). Table 16 in the Appendix shows an extended version of this table. (57.9%), International cooperation (24.2%), Energy (9.7%), Research (5.1%) and Agriculture (3.1%). Moreover, it highlights amongst other eligible expenditures the upgrade and electrification of the railway between Ulm and Lindau with total costs of approx. 225 million (mn.) Euro (EUR) and the development loan for a renewable power plant (i.e., solar PV) in India amounting to 89.3 mn. EUR as examples from the infrastructure and international cooperation sector, respectively. The evaluation criteria are only applied to expenditures that are already accrued (Finanzagentur GmbH,2020). This process enhances the transparency of the use of proceeds, as the projects precede the issuance of the securities. However, it also restricts the issuance amount of green bonds as, for example pending expenditures are not eligible. In addition to the selection criteria, the agency provides a Second Party Opinion on the Green Bond Framework (see ISS ESG,2020), an external Third Party Verification of the Allocation Report by the auditing firm Deloitte, and impact reporting (Finanzagentur GmbH,2021c). On the other hand, the German Green Bonds aim to support the development of the European green fixed income market by establishing a new interest rate benchmark for such assets, a green yield curve (Finanzagentur GmbH, 2020). While conventional government bonds with a high credit rating can be used to serve as a benchmark return for risk-free investments only, the green yield curve can provide the term structure of interest rates for riskless and green assets as they are ranked pari-passu (i.e., equally) to the conventional bonds from the same issuer. This means that they could provide a reference for the required future payoff of a risk-free green investment with a specific time horizon. To provide this information to potential investors and quantify their preference for green investments, the German Green Bonds are issued based on a unique twin bond concept (Finanzagentur GmbH,2020). As summarized in Table 16 in the Appendix, the green bonds and their conventional twins share the same coupon rate and time to maturity, but differ in their issuance volume and are traded separately (i.e., they have different ISIN codes). In addition to the high credit quality, another requirement of German Federal securities to serve as benchmark is sufficient liquidity (Finanzagentur GmbH,2021a). This is to mitigate risks for bondholders that can be induced by illiquidity, for example, the inability to sell the bond rapidly or only for a lower transaction price (e.g., see Kempf and Uhrig-Homburg, 2000). In the context of green bonds, their issuance can entail liquidity risks for both, themselves and conventional twins. This is because a sufficiently high amount outstanding of each type of bond is necessary to ensure that they can be traded in large quantities and at any time (Finanzagentur GmbH,2021a). While a low issuance volume of green bonds may impede their own liquidity, a high volume can have detrimental effects on the liquidity of conventional bonds, if the total outstanding volume of Federal securities is maintained (Finanzagentur GmbH,2021a). As a consequence, this potential trade-off has to be solved in order to provide an interest rate benchmark for both, the green as well as the conventional European fixed income market. The figures in Table 1indicate that the volume of each green bond is significantly smaller than its conventional counterpart. In fact, the average amount outstanding of a German green bond is approximately one fifth (22%) of the amount of the average conventional twin bond. This suggests that the green bonds may be less liquid than their conventional twins. To test this hypothesis, we evaluate the bid-ask spread (BAS) of the daily closing bond yields, as this measure is frequently used to derive a proxy for the liquidity of bonds (e.g., Dick-Nielsen et al. (2012); Kapraun et al. (2019); Zerbib (2019)). A higher BAS represents higher transaction costs, which can indicate a lower liquidity. Figure 1shows that since the issuance of the first German green bond, the average monthly BAS of the green bonds is almost consistently larger than the spread of the conventional counterparts. To verify this visual impression, a paired t-test is performed whether the average bid-ask spread (BAS) since issuance of each green bond BASGcoincides with the same measure for the respective conventional twin BASC. The test results in Table 2show that we can reject the null hypothesis T. F. Fauß /Junior Management Science 7(3) (2022) 668-689 673 Sep-20 Nov-20 Jan-21 Mar-21 May-21 Jul-21 Sep-21 Nov-21 0 0.5 1 1.5 2 Avg. Closing Bid-Ask Spread [in bp] G2025 B2025 (a) Maturity in 2025 Sep-20 Nov-20 Jan-21 Mar-21 May-21 Jul-21 Sep-21 Nov-21 0 0.2 0.4 0.6 0.8 1 1.2 Avg. Closing Bid-Ask Spread [in bp] G2030 B2030 (b) Maturity in 2030 Sep-20 Nov-20 Jan-21 Mar-21 May-21 Jul-21 Sep-21 Nov-21 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Avg. Closing Bid-Ask Spread [in bp] G2031 B2031 (c) Maturity in 2031 Sep-20 Nov-20 Jan-21 Mar-21 May-21 Jul-21 Sep-21 Nov-21 0 0.2 0.4 0.6 0.8 1 Avg. Closing Bid-Ask Spread [in bp] G2050 B2050 (d) Maturity in 2050 Figure 1: Average monthly closing bid-ask spreads for German twin bonds The figure is based on data from Refinitiv Eikon (Accessed: 10.11.2021) and shows the average monthly closing bid-ask spread in basis points (bp) for German green bonds and their conventional twins displayed in Table 1for the period from 09.09.2020 to 10.11.2021. Summary statistics are displayed in Table 11 in the Appendix. H0:BASG=BASCon a significance level α=0.05 for all twins, in favour of the alternative hypothesis HA:BASG6= BASC. Therefore, we can conclude that the green bonds are traded during the observed period from 09.09.2020 to 10.11.2021 on average at a statistically significant wider spread than the conventional twin bonds. The absolute difference between the average spreads ∆ ranges from 0.1 to 0.7 basis points. To evaluate the economic significance of this difference, we assume values that align with the twin bonds that mature in the year 2031. Assuming a bid yield ybid =−40bp for both twins, a narrower spread for the conventional bond with yC ask =−39.7bp and a wider spread for the green bond yG ask =−39.3bp, we have ∆=0.4bp representing the additional transaction costs T C. Further, we assume a time to maturity of T=10 years. To compute the present value Pof a zero-coupon bond, we assume continuous compounding to discount its nominal value Nand thus use P=N·e−yT . We obtain the trading costs as T C =N·(Pask −Pbid ). Based on this specification, an investor, which acts as a price taker and executes a round trip trade with both, a German green bond and its conventional twin, by buying at the ask price Pask and selling at the bid price Pbid for an investment of N=1 mn. EUR and T=10 years, would incur additional trading costs for the green bond amounting to ∆T C =T CG−T CC=N·(PG ask −PC ask) = 385 EUR or about 4bp that are caused by its wider spread. From an economical perspective, this amount is relatively small, which aligns with the objective of the German approach to address liquidity risks. However, in relative terms, BASGis on average almost twice the size (+94%) of BASCfor the observed data. It should be noted that these transaction costs are different to the liquidity premium in the model that de- T. F. Fauß /Junior Management Science 7(3) (2022) 668-689674 Table 2: Closing bid-ask spreads for German twin bonds BAS2050 BAS2031 BAS2030 BAS2025 BASC[in bp]0.573 0.294 0.290 0.572 BASG[in bp]0.669 0.570 0.703 1.273 ∆=BASG−BASC0.096 0.276 0.413 0.701 t-statistic 2.9223 5.5221 17.8084 12.4163 p-value 0.0041 0.0000 0.000 0.0000 N131 47 305 262 The table shows the results of a paired t-test to determine whether the mean bid-ask spread for the closing yields of German green bonds BASGcoincides with the same measure for the conventional twins BASC. The null hypothesis H0:BASG=BASC is tested against the HA:BASG6=BASC. The data is retrieved from Refinitiv Eikon (Accessed: 10.11.2021) and covers the period from 09.09.2020 to 10.11.2021. Summary statistics are displayed in Table 11 in the Appendix. scribes a premium to the yield (i.e., a higher yield) for illiquid assets instead. The German green bond concept is designed to mitigate possible liquidity disadvantages for both twins. To ensure the liquidity of the conventional bonds, the Finance Agency issues the same amount as the green counterparts in its own stock, which can be used on the secondary market for repurchase agreements (i.e., repo transactions) and lending activities (Finanzagentur GmbH,2021a). Therefore, the total amount of conventional securities and thus their liquidity remains unchanged. To ensure the liquidity of the green bonds, the German Finance Agency declares to engage in secondary market activities (Finanzagentur GmbH,2020). In their Investor Presentation from September 2021, the Finance Agency categorises them as (1) Outright (“one-way”) sales and purchases, (2) Repurchase agreements and securities lending, using the Federal Government’s own stock of green bonds and (3) Combined and debt-neutral sale-andpurchase (switch) transactions conducted with banks that are members of the Bund Issues Auction Group (Finanzagentur GmbH,2020,2021b). This means that it can influence the supply and thus the price of the green bonds on the secondary market. From the issuer’s perspective, green bonds are more valuable than the conventional twins. Although both zerocoupon bonds have the same face value and thus the same cash flows, this is because the green bonds provide an additional documentation for the usage of their proceeds. Even in situations, where investors would not attribute a higher value to the green bonds, this would still hold for the issuer, who sustains the associated added costs and more limited use of proceeds. Therefore, the switch, which is the simultaneous and debt-neutral sale of a conventional bond and purchase of a green bond, would be economically viable for the issuer at a yield spread of zero. Further, it can execute such transactions until the green securities are completely in their own holdings. In this case, the respective amount of conventional twins that was initially held back by the Finance Agency is then traded in the secondary market. Figure 2shows all available closing ask yields until 04.11.2021 of the German twin bonds displayed in Table 1retrieved via Refinitiv Eikon. It shows that the yields of the respective twins are closely related for the observed period of time. Further, the data supports an upward sloping yield curve for both bond types. This means that investors with a longer time to maturity require a higher rate of return, ceteris paribus. Figure 3shows the yield differential between German green bonds yGand their respective conventional twin yC(i.e., ∆y=yG−yC). For most of the observed period, the data shows a negative trending spread with an average of around -5bp. This implies that investors are increasingly willing to sacrifice return in favour of investing into the German Green Federal securities. However, it should be noted that the historical data covering a period of one year is relatively scarce and the future size of the spread may change. 3.2. Addressing Liquidity Risks This section aims to provide a brief insight into how selected sovereign issuers different to Germany address the possible risk of illiquidity. These issuers are France, the Netherlands, and Belgium. France issued its first green sovereign bond in 2017 for 7 bn. EUR, which was since then increased to a total amount outstanding of 28.9 bn. EUR République Française (2021). In the French framework for green Obligations assimilables du Trésor (OAT) (2017), their liquidity is emphasized on its first page. Also, the respective investor presentation covers the liquidity as one of six main topics (République Française,2021). This underlines the relevance of liquidity concerns. In the same document, they show that the average monthly bid-ask spread is consistently lower for their green bond (RIC: FR0013234333=) than a conventional bond (RIC: FR0013515806=) which matures one year later in 2040. Further, both bonds show a similar ownership structure with a share of long-term investors of 37% and 38%, respectively. They also highlight that its amount outstanding is with 31 bn. EUR similar to neighbouring (in terms of time to maturity) conventional bonds and argue that this supports its liquidity (République Française, T. F. Fauß /Junior Management Science 7(3) (2022) 668-689 675                           Figure 2: Yield of German government bonds and green twins The figure is based on data from Refinitiv Eikon (Accessed: 04.11.2021) and shows the closing ask yield in basis points for the German Federal securities displayed in Table 1. The summary statistics are displayed in Table 12 in the Appendix.                    Figure 3: Spread between German government bonds and green twins The figure is based on data from Refinitiv Eikon (Accessed: 04.11.2021) and shows the yield differential of German green bonds yGand their respective conventional twin yC(i.e., yG−yC) in basis points. The summary statistics are displayed in Table 13 in the Appendix. 2017,2021). The Dutch State Treasury Agency (2019) justifies the liquidity of the Dutch sovereign green bonds with a minimum issuance volume of 10 bn. EUR within several years, a quotation obligation for Primary Dealers to ensure the availability of tradable prices and a Repo facility available to Primary Dealers that serves as a lender of last resort. Belgium reports that its green bonds have no liquidity disadvantages and a similar issuance volume as conventional government bonds (The Kingdom of Belgium,2018). So far, there exists only one other country that decided to adopt the German approach that was introduced in 2020. The national bank of Denmark 2022 reported following the T. F. Fauß /Junior Management Science 7(3) (2022) 668-689682 Table 4: Summary of ML estimates for ˜ LP Vasicek model parameters LP2050 LP2031 LP2030 LP2025 Mean-reversion rate a 66.588 25.693 24.176 11.919 Long-term mean b [in bp]93.1 48.7 48.5 48.7 Instantaneous volatility σ0.0111 0.0046 0.0044 0.0031 Sample size N 548 548 548 548 The table shows the estimation results for the process of the liquidity premium. The data is based on published yield curves by the Deutsche Bundesbank (2021) and covers the period from 02.09.2019 until 01.11.2021. -5 0 5 10 15 20 LP (p.a.) [in bp] 0.936 0.938 0.94 0.942 Price P0 PC PIG PG (a) Bond prices -5 0 5 10 15 20 LP (p.a.) [in bp] 185 190 195 200 205 210 Y T M (p.a.) [in bp] yC yIG yG (b) Bond yields Figure 7: Model results for different LP The model results displayed in the figures above are based on a green premium of GP =8bp, a risk-free rate of rf=200bp, σ=0.0031, a=11.919, T=3.1 years and a trinomial tree length of N=791. of the zero coupon bonds (i.e., FV=1), a lower price Pt, ceteris paribus, implies a higher yield to maturity yt, and vice versa. The conventional bond Cis assumed to be liquid, and thus not affected by changes in the premium that compensates for illiquidity of the asset. Therefore, the price PC(see Figure 7a) and yield yC(see Figure 7b) are unaffected by changes in the expected liquidity premium LP. On the other hand, the value of the illiquid green bond IG is affected as investors require a higher compensation for their liquidity risk and are thus only willing to pay lower prices. The German green bond Gdiffers from the bond IG by having the additional switch option ST. This prevents the bond price PG from assuming values lower than PC. In the same fashion, the yields yGcannot assume values higher than yC. When the green premium outweighs the liquidity premium, the model yield yGis smaller than the yield of the conventional bond yC. This implies for the secondary market that a negative yield differential ∆y(i.e., ∆y=yG−yC) is observed. Finally, it should be noted that the green premium GP is equal to 8bp for all scenarios in the figure. For the German green bonds, this means that the yield difference ∆ycan be equal to zero, although there exists a green premium GP larger than zero. In such cases, liquidity effects dominate and the value of the upper threshold for the yield, yC, is assumed. Moreover, in the case of bonds without a switch option, IG, the yield difference to a conventional twin can even assume positive values. This means that liquidity effects of bonds can potentially compensate the green premium. The model suggests that issuers and investors should therefore incorporate the bonds’ exposure to illiquidity in their emission and valuation decision, respectively. This finding aligns with the published investor presentations or Green Bond Frameworks from France, Netherlands and Belgium, who all address liquidity aspects of their bonds (see section 3.2). Moreover, the model also shows that the German approach can prevent the yield spread from becoming positive. Therefore, it can be a viable method for issuers to mitigate by illiquidity induced risks. Figure 8displays the yield of a German green bond for different degrees of illiquidity and a decomposition of its yield premium that exists relative to its conventional twin. The green premium GP is constant with GP =8bp for all values of the expected liquidity premium LP. The figure demonstrates that the switch option prevents the yield yGto become larger than the yield of the conventional bond yC. Further, its value (in bp) reflects the payoff structure of a short call T. F. Fauß /Junior Management Science 7(3) (2022) 668-689 683 -5 0 5 10 15 20 LP (p.a.) [in bp] 180 190 200 210 220 Y T M (p.a.) [in bp] yG (a) Yield of a German green bond -5 0 5 10 15 20 LP (p.a.) [in bp] -20 -10 0 10 20 Premium [in bp] LP GP ST (b) Decomposition of the yield Figure 8: Model results for German green bonds The model results displayed in the figures above are based on a green premium of GP =8bp, a risk-free rate of rf=200bp, σ=0.0031, a=11.919, T=3.1 years and a trinomial tree length of N=791. option on the illiquidity of the bond. Using this analogy, the strike would coincide with the value of GP. If LP assumes a value larger than GP, the switch option ST needs to compensate this difference. The time series of the yield spreads displayed in Figure 3indicate that a value of -5bp can be a realistic value for German green bonds. Based on the model results displayed in Figure 7b and Figure 8b, this would imply LP =3bp and ST =0bp, assuming GP =8bp. In words, this model specification indicates that the greenium is sufficiently larger than the liquidity premium so that market intervention by the Finance Agency is very unlikely to be necessary and thus the value of the switch option is equal to zero. 5.2. Maximum Switch Option Value Figure 9shows that the yield of the German green bond yGis capped by yCat an expected liquidity premium LP that is larger than the green premium GP. This is because the stochastic liquidity premium might still assume a lower value, in which case the execution of the switch option, (i.e., the execution of switch transactions) is not optimal. Based on Equation 1, we know that at this point the difference LP −GP coincides with the maximum value of the switch option STmax, as ∆yis equal to zero. This maximum value is relevant as it indicates how much additional liquidity costs in excess of a greenium the holders of a German-type green bond can bear until they assign the same value to it as to a conventional government bond. In comparison, in the case of an illiquid green bond without the switch option, this value would be zero. In light of the above, the maximum value of the switch option STmax can be defined as, STmax =max LP {ST |yG≤yC}.(24) In the following, we provide an overview of how this measure changes for different model specifications and an estimation precision of 0.01bp. Table 5shows the value of the switch option at execution, STmax for different levels of GP. The model results show that STmax is unaffected by the size of GP, ceteris paribus. This is because a higher GP increases the expected illiquidity LP that can be tolerated before the switch option is executed. From an issuers’ perspective, this implies that by adopting the German approach, they can compensate an additional liquidity premium of 4.1bp compared to conventional green bonds until the yield differential ∆y assumes a value equal to zero. To put the value of 4.1bp into perspective, we assume a total issuance volume of 5 bn. EUR which equals the size of the smallest currently issued German green bond. This implies a potential maximum value of approximately 2 mn. EUR for the switch option, given an issuance volume of 5 bn. EUR. However, the Green bond from this example currently (01.11.2021) trades at a spread ∆yof −8bp. Based on the model calibration displayed in Figure 7b, this would imply aLP <5bp, for which the value of the switch option ST is equal to zero (see Figure 8b). Table 6shows the value of STmax for different local volatilities of the underlying process for the liquidity premium. The model results indicate that a higher σincreases the maximum value of the switch option STmax. This is plausible, as a higher volatility of the stochastic liquidity premium increases the chance of realizing very low values, while larger values do not change the outcome once the threshold is yCis reached. This means that the switch option is executed later, which implies a higher value for LP and STmax. The model also accommodates a special case assuming a non-stochastic liquidity premium. In this case, the option is executed for LP =GP. As the liquidity premium cannot change over time, the option is executed as soon as liquidity effects and the green premium cancel each other T. F. Fauß /Junior Management Science 7(3) (2022) 668-689684 4 6 8 10 12 LP (p.a.) [in bp] 194 196 198 200 202 204 206 Y T M (p.a.) [in bp] yC yIG yG GP Figure 9: Bond yields for different LP The model results displayed in the figures above are based on a green premium of GP =8bp, a risk-free rate of rf=200bp, σ=0.0031, a=11.919, T=3.1 years and a trinomial tree length of N=791. Table 5: Option value at execution for different GP GP LP STmax 0 4.06 4.06 10 14.06 4.06 20 24.06 4.06 30 34.06 4.06 The table shows the values of STmax for different GP based on a risk-free rate of rf=200bp, σ=0.0031, a=11.9, T=3.1 years and a trinomial tree length of N=791. out. The resulting maximum value of the switch option before execution, STmax, is equal to zero in this scenario. For LP <GP, there is no chance that the option is executed as it implies a certain negative yield differential ∆y. Therefore, the value of the option is equal to zero in this case as well. Finally, Table 7shows STmax for different times to maturity T. In the model, this increases the length of the trinomial tree because ∆t=T N=1 250 is held constant. The results indicate a lower maximum value of the switch option STmax for longer maturities T. This is explained by the decreasing likelihood of the stochastic liquidity premium realizing an outcome lower than LP. Therefore, the switch option is executed for a lower expected liquidity premium LP reducing its maximum value STmax. 5.3. Sensitivity Analysis of Bond Yields The sensitivities of the initial yield to maturity to changes in the model parameters are estimated using finite differences that is motivated by a Taylor approximation. This approximation is required because a closed-form solution is not available due to the non-closed form of the model. Following Brandimarte (2006), a symmetric approximation of the first partial derivative of the yield y0with regard to the model parameters is computed, as this approach yields a lower order truncation error compared to forward or backward approximation. In its general form, the first derivative can be estimated using, ∂y0(x) ∂x≈y0(x+h)−y0(x−h) 2h,(25) where hdenotes a small and constant value and xthe parameter of interest, while the other model parameters are hold constant. The resulting sensitivities are displayed in Figure 10. The figures indicate that the sensitivity of the German green bond Ghas a continuous part, and a discontinuous part with jumps when LP assumes values above a certain threshold. The number of observed jumps in the figures for Gcoincide with jmax =4 (or −jmin) of the calibrated model. One viable explanation might be that nodes in the tree switch to the value of the conventional bond, if the liquidity premium assumes a high enough value so that PG<PC(see Equation 15). This also explains the continuous part on the left-hand side of the figures, as a switch scenario does not occur for low values of LP. Figure 10a describes how much units the yield changes, if LP changes by one unit. The yield of the illiquid green bond yIG changes by one basis point, if LP increases by one basis T. F. Fauß /Junior Management Science 7(3) (2022) 668-689 685 Table 6: Option value at execution for different σ σLP STmax 0 8 0 0.002 10.53 2.53 0.004 13.28 5.28 0.008 18.81 10.81 0.010 21.26 13.26 The table shows the values of STmax for different σbased on GP =8bp, a risk-free rate of rf=200bp, a=11.9, T=3.1 years and a trinomial tree length of N=791. Table 7: Option value at execution for different T T LP STmax 1 12.35 4.35 5 12.06 4.06 10 11.51 3.51 20 10.76 2.76 30 10.31 2.31 The table shows the values of STmax for different T(constant ∆t) based on GP =8bp, a risk-free rate of rf=200bp, σ=0.0031 and a=11.9. Changes in Taffect the tree length N, as ∆tis hold constant with ∆t=T N=1 250 . point, while yCis unaffected by changes in LP. The sensitivity of yGranges between 1 and 0. This aligns with the notion that the German green bond is valued as a conventional bond if LP is sufficiently high and valued as a counterfactual bond without switch option, if LP is sufficiently low, assuming a constant GP. In those cases, the stochastic process for LP either cannot assume values where yGis lower than yC, or where the switch option is executed. Figure 10b implies that a higher instantaneous volatility σdecreases yG. This is because the downside potential is restricted by the switch option, while a lower realized liquidity premium reduces yG. The parameter adescribes the mean reversion rate of the stochastic process. Therefore, this sensitivity is inversely related with the sensitivity of yGto σ. Finally, an increase in T, increases the yield yGas well. Based on the absolute size of the sensitivities, the evaluation suggests that changes in σ and LP have the strongest impact on the model results. In light of the evaluation, it should be noted that the sensitivities only reflect the impact of small changes in the parameters. Further, their changes and thus the effect on the model results is restricted by their plausible range. Nevertheless, the model outcome might be significantly larger or smaller, if different estimates for those parameters are chosen. 5.4. Limitations The above discussed model for the green bond yields provides a first insight into the potential effects of the switch option between green and conventional bonds, which was pioneered by the German twin bond approach. However, the model is subject to some limitations that are discussed in the following. First, the model cannot decompose observed green bond yields ˆ yGinto the different components suggested by the model. Namely, the observed yield of the respective conventional twin ˆ yC, the liquidity premium LP, the green premium GP and the added-value of the switch option ST. This means that a calibration of the model parameters is not straightforward and proxies need to be applied instead. Moreover, this impedes the validation of the model results based on actual observations. Another possible limitation can be the assumed process for the liquidity premium and its translation into a trinomial tree representation. For example, the Vasicek process in Equation 2assumes a constant volatility and is, in addition to a mean-reversion parameter, defined by its first two moments. This means that it cannot accommodate possible volatility clusters or skewness that is introduced by jumps in the liquidity premium, as shown in Figure 6. Moreover, deriving the trinomial tree representation, we assume a maximum range from LPjmin to LPjmax for the liquidity premium to ensure positive tree probabilities. This creates an upper and lower threshold that the liquidity premium cannot exceed. However, increasing the volatility of the process may provide a first idea of the possible implications when accounting for these effects, as it increases the overall dispersion of the stochastic premium. Finally, the model assumes a constant risk-free rate rand green premium GP. While adding additional complexity to the model by introducing more flexible (e.g., stochastic or T. F. Fauß /Junior Management Science 7(3) (2022) 668-689686 0 5 10 15 LP (p.a.) [in bp] 0 0.2 0.4 0.6 0.8 1 1.2 @ Y T M @ LP yC yIG yG GP (a) Sensitivity to changes in LP 0 5 10 15 LP (p.a.) [in bp] -0.01 -0.008 -0.006 -0.004 -0.002 0 @ Y T M @< yC yIG yG GP (b) Sensitivity to changes in σ 0 5 10 15 LP (p.a.) [in bp] 0 1 2 3 4 5 @ Y T M @ a #10!6 yC yIG yG GP (c) Sensitivity to changes in a 0 5 10 15 LP (p.a.) [in bp] -2 0 2 4 6 8 10 @ Y T M @ T #10!6 yC yIG yG GP (d) Sensitivity to changes in T Figure 10: Model sensitivities The model results displayed in the figures above are based on a green premium of GP =8bp, a risk-free rate of rf=200bp, σ=0.0031, a=11.9, T=3.1 years, a trinomial tree length of N=791 and h=0.00001 time-dependent) components might improve the calibration to observed yield spreads, this is not relevant for the main objective of this dissertation to better understand the potential impact of the switch option. 6. Conclusion The goal of this dissertation is to provide a theoretical model for the pricing of green bonds that are based on the German twin bond approach. The focus here is on improving the understanding of the potential effects of introducing a switch mechanism between green bonds and their conventional counterparts. For this purpose, a non-closed form solution was derived that decomposes the yield differential into three effects: A liquidity premium, a green premium and the added value of the switch option. The model assumes a stochastic liquidity premium that follows a Vasicek process in discrete time, a constant green premium as well as a constant risk-free rate. The switch mechanism is modelled by assuming the theoretical value of conventional bonds as a lower limit for the green bond prices. For the model calibration the term structures of German Bundesanleihen and Pfandbriefen are used to obtain a proxy for the stochastic liquidity premium. The main learning from the model is that the switch option can in certain conditions increase the value of the green bonds, which corresponds to a lower yield. Based on the calibration of the model, a maximum added-value of 4.1 bp before the execution of the option was identified. This translates to a maximum value of about 2 mn. EUR assuming a green bond with a 5 bn. EUR issuance volume. This means that issuers adopting the twin bond concept may be able to secure lower costs of capital compared to a traditional green bond concept that does not provide the switch option. For investors the concept reduces their exposure to potential liquidity risks by using the liquid conventional bonds to create a T. F. Fauß /Junior Management Science 7(3) (2022) 668-689 687 lower limit for the green bond price. The model improves the understanding of the twin bond concept and thereby fills a gap in the literature. From a practical perspective, the model implications may assist issuers in the design choice of their green bond framework. For example, Denmark decided to adopt the twin bond concept, including a switch mechanism, which supports the potential benefits of this approach. Green bonds are one important instrument to finance the transition to a more sustainable economy. In light of the significant growth of the green bond market in recent history and the competing frameworks, it is crucial to elaborate on their respective advantages and disadvantages. While this work contributes to the understanding of the twin bond switch mechanism, the current model can be further developed. On the one hand, an improved proxy for the liquidity premium and a larger sample of historic data may affect the calibration results, which can impact the size of the evaluated effects. On the other hand, a more sophisticated stochastic process for the liquidity premium and less restrictive assumptions in its discrete representation may increase the precision of the model results. In a broader context, one should evaluate if a high issuance volume of green bonds can affect the liquidity of similar conventional bonds, and whether a potential effect vanishes for lower volumes. If such effects are found, this would support the relevance of the twin bond approach with switch option to mitigate liquidity risks, as lower overall issuance volumes may be required. Otherwise, ensuring a critical volume that is high enough to avoid liquidity costs may be a viable alternative to this concept. T. F. Fauß /Junior Management Science 7(3) (2022) 668-689688 References Almeida, M., Filkova, M., Harrison, C., & Sette, P. (2019). Green bond european investor survey. Retrieved from https://www.climatebonds.net/resources/reports/ green-bond-european-investor-survey-2019 (Last accessed: 19.01.2022) Bachelet, M. J., Becchetti, L., & Manfredonia, S. (2019). The green bonds premium puzzle: The role of issuer characteristics and third-party verification. Sustainability,11(4), 1098. Bongaerts, D., & Schoenmaker, D. (2020). Green certificates: a better version of green bonds. Policy Contribution 2020/20, Bruegel. Brandimarte, P. (2006). Numerical methods in finance and economics. 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