Are NPL-backed securities an investment opportunity?

Journal Pre-proof Are NPL-backed securities an investment opportunity? Enrica Bolognesi, Patrizia Stucchi, Stefano Miani

PII:

S1062-9769(18)30328-4

DOI:

https://doi.org/10.1016/j.qref.2019.10.007

Reference:

QUAECO 1317

To appear in:

Quarterly Review of Economics and Finance

Received Date:

19 December 2018

Revised Date:

17 October 2019

Accepted Date:

23 October 2019

Please cite this article as: Bolognesi E, Stucchi P, Miani S, Are NPL-backed securities an investment opportunity?, Quarterly Review of Economics and Finance (2019), doi: https://doi.org/10.1016/j.qref.2019.10.007

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Are NPL-backed securities an investment opportunity?

Enrica Bolognesi, corresponding author University of Udine. Department of Economics and Statistics. Via Tomadini 30, 33100 Udine; [email protected]

Patrizia Stucchi University of Udine. Department of Economics and Statistics. [email protected]

Stefano Miani

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University of Udine. Department of Economics and Statistics. [email protected]

Statistics. Via Tomadini 30, 33100 Udine; [email protected]

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We focus on the risk-return profile of the NPL-backed securities. We examine the junior tranche and observe its return distribution as a function of the portfolio recovery rate We evaluate the probability of achieving selected target returns using a value-at-risk approach. We test the impact of a state-backed guarantee on the senior note to support securitizations. We compare our results to the ones achieved by the same portfolio, sold straight to the market, and provide evidence that securitization is the most valuable deleveraging strategy from the perspective of both investors and banks. We show that junior notes could be an attractive new asset class requiring the development of a secondary market. .

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Highlights:

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Corresponding author: Enrica Bolognesi, University of Udine. Department of Economics and

Are NPL-backed securities an investment opportunity?

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ABSTRACT

This paper focuses on the risk-return profile of the asset-backed securities deriving from the securitization of non-performing loans (NPLs). We test several hypotheses concerning the portfolio sell price, tranche note size and use of a public guarantee supporting the senior notes. We observe the return distribution of junior notes as a function of the portfolio recovery rate, assuming a lognormal distribution. Moreover, we focus on the probability of achieving selected target returns using a value-at-risk approach. Our results provide evidence 1

that securitization is the most valuable deleveraging strategy from the perspective of both investors and banks. 1. Introduction Non-performing loans (NPLs) are usually defined as loans that are either more than 90 days past due or unlikely to be repaid in full. The global financial crisis and subsequent recession have left many European countries with elevated levels of NPLs, which are particularly high in the southern part of the euro area, as well as in several eastern and south-eastern European countries (Aiyar,

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2015). In December 2016, the stock of NPLs in the EU reached a historical peak, exceeding 1 trillion € and doubling the value reached in 2009. Looking at the EU region in aggregate, the

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weighted average NPL ratio was 5.1% of the gross book value (European Parliament, 2017), which is up to three times higher than in other global jurisdictions (EBA, 2016a). In December 2018, the stock and NPL ratios were reduced to 587 billion € and 3.7%, respectively (ECB, 2019), mainly

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due to disposal operations by European banks, half of which were attributable to Italian banks, which were the most involved banks, owning almost one-fourth of the aggregate EU stock of NPLs

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(148 billion €) and showing an NPL ratio equal to 8.37% of GBV (from 15.4% two years earlier). To better understand the importance of Italian NPLs in Europe, it is worth remembering that, in the

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European ranking, Italian banks are followed by French and Spanish banks, owning stocks of 126 billion € and 88 billion €, respectively, but showing more moderate NPL ratios of 2.7% and 3.6%, respectively.

A vast body of literature investigates the determinants and consequences of high NPL levels,

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focusing on both macroeconomic and bank-specific issues. In a nutshell, these studies demonstrate that high stocks of NPLs undermine a country's economic growth as they slow down the credit supply (see Messai and Jouini, 2013, for a review). From banks’ perspective, high levels of NPLs mean higher provisioning, higher funding costs, lower profitability and thus higher credit rationing. The threat to financial stability, combined with the aforementioned statistics on the huge amount

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of bad loans on European banks’ balance sheets, helps the understanding of why European supervisory authorities have stimulated the discussion about their disposal and its policy implications (EBA, 2016a; EBA, 2016b). In September 2016, the European Central Bank released a draft of a guidance to banks on NPLs (ECB, 2016), in force from March 2017 (ECB, 2017) and followed by an addendum in March 2018 (ECB, 2018). Furthermore, the European supervisory authorities encouraged the cleaning up of bank balance sheets, with particular emphasis on Italian banks. In turn, banks can proceed with the deleveraging following two alternative strategies: the bad loan portfolio straight sale to the market or its securitization. 2

In this paper, we focus on the second strategy. Securitization consists of selling the NPL portfolio by the originator bank to a special purpose vehicle that is financed by issuing NPL-backed securities to investors. Depending on the structural details of this operation, these securities have different ranks of seniority being repaid through a waterfall scheme that first rewards senior notes, then mezzanine notes (if issued) and, subordinately, junior notes. Subsequently, each class of security exhibits different risk profiles and rates of return, descending from senior to junior notes; in particular, the return of the junior tranche depends strictly on the effective recovery rate of the bad loan portfolio. The recovery rate is defined as the loan payback quota, and its estimation and

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distribution are of crucial importance for the pricing of the portfolio that must be disposed. The academic literature dealing with bank loan securitization focuses on both the possible

benefits of securitization for the originating bank and the main drivers and effects of the process.

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Theory provides four determinants of the decision to securitize bank loans: the need for new

sources of funding, the transfer of credit riskiness, the search for new profit opportunities and the

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role of capital (see, for a review, Affinito and Tagliaferri, 2010). Several studies, based mainly on US data, focus on the effects of informational asymmetries on the securitization process (DeMarzo,

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2005), arguing that banks tend to securitize lower-quality assets (Mian and Sufi, 2009; Dell’Ariccia et al., 2012). Moving to the securitization of bad loans, as mentioned, the main driver of securitization is regulatory constraints, and for this reason, the debate on the effects of NPLs’

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deleveraging process is fuelled by the national and European supervisory authorities. The Bank of Italy – the supervisory authority most involved in NPL management – has contributed significantly in this field, releasing useful statistics through recent notes. The high level

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of attention of Italian institutions towards NPL disposal makes the Italian banking system the perfect framework for the analysis of alternative deleveraging strategies. For this reason, this study is based on the characteristics of a country-specific banking system, although our findings can be generalized to the European level. Therefore, our pricing model relies on the statistics provided by the Bank of Italy. For example, Carpinelli et al. (2016) present the results of a survey on the

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efficiency of credit recovery procedures employed by 25 large banking groups between 2011 and 2014; they offer detailed statistics on the credit recovery rates and their timeframe. Moreover, Ciavoliello et al. (2016) focus on the gap between the book value of bad loans and the price that the market operators are willing to pay, demonstrating that the major determinants are the high rate of return demanded by investors. Furthermore, the note by Ciocchetta et al. (2017) and the following update notes of Conti et al. (2017) and Fischetto et al. (2018) aim to fill an information gap on the evaluations of bad loan recovery rates that materialize due to the scarcity of reliable public data on banks’ performance on this topic. Still focusing on the literature on recovery rates, a number of 3

academic papers have focused mainly on corporate bonds rather than bank loans due to the lack of data and issues relating to confidentiality (see Grunert and Weber, 2009 for a detailed review). Most contributions on NPLs are mainly descriptive and focused on the bank's perspective. If we turn to the investor’s perspective, as far as we know, the academic literature lacks contributions focused on the risk and return profile of NPL-backed securities. Regarding this issue, Bruno et al. (2018) design a securitization of a mixed NPL portfolio aimed at identifying the transfer price of the portfolio to the Special Purpose Vehicle and evaluating the returns offered by the different tranches. More recently, Lusignani et al. (2019) broaden this analysis by introducing a

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parsimonious top-down model for simulating the recoveries of NPL portfolios based on a logistic transformation of a discrete Vasicek model (Vasicek, 1977) and on expected recovery time

distribution curves. Their purpose is to offer a pricing model for NPL-backed securities as a

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contribution to a higher degree of transparency in the NPL market.

In our paper, we design several securitizations of a portfolio composed of a mix of bad loans in

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terms of underlying guarantees (secured and unsecured loans). We test different structures of securitization in terms of tranching between senior and junior notes. Furthermore, we test the

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impact of a state-backed guarantee on the senior note relying on the one promoted by the Italian government since 2016 (also known by the acronym GACS). Then, we focus on the junior notes, with the riskiest security being the most exposed to the uncertainty of the recovery procedure. In

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particular, we observe the pattern of the return on junior notes, which depends on the discrepancy between the expected recovery ratio (entering the pricing model) and the effective recovery ratio registered at the end of the recovery procedure. Then, we observe the return distribution

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characterizing the junior note as a function of the dynamics of the recovery rate. Furthermore, we analyze the risk-return profile of the junior notes using the value-at-risk (VaR) approach (Jorion, 2001 and Alexander, 2008).

To better understand our results, we use the price and the risk-return profile of the bad loan portfolio disposed through straight sale to the market as a benchmark. Straight sale is certainly the

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deleveraging strategy most frequently used by banks because it is the simplest and quickest strategy for disposal, as it is a bilateral agreement and does not need a proper market for NPLs. As a disadvantage, however, this strategy must take into account the high interest rates demanded by investors when buying the entire portfolio due to both the high risks surrounding the recovery of bad loans and the low demand compared to the oversupply of NPLs. Finally, we invert this perspective by comparing the junior notes and the portfolio sold by straight sale in terms of VaR. The aim here is to estimate the portfolio sell prices that allow

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equivalent investments in terms of the probability of achieving selected target rates of return using securitization and straight sale alternatively. With respect to previous contributions, this paper shows the risk level associated with any expected rate of return of the junior note (and vice versa) in a framework where the rate of return does not follow a normal distribution (meaning that the standard deviation is not a suitable risk measure). Furthermore, our analysis is the first aimed at comparing the risk-return profile of a junior tranche and of a bad loan portfolio sold straight to the market. Our findings demonstrate that securitization is the deleveraging strategy that permits the issue of

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securities showing the most attractive risk-return profile. In particular, comparing the distribution of returns of the junior note and of the portfolio sold by a straight sale, we provide evidence of the higher returns registered by the junior note with the same probability. Indeed, in the case of low

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portfolio sell prices, the junior note shows the dominance of its distribution of returns. Moving to the banks’ perspective, our findings reveal the opportunity to proceed with bad loan disposal

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through securitization, which significantly reduces the cost of deleveraging.

The policy implications of our results are straightforward: securitization should be promoted by

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governments and supervisory authorities. As demonstrated, the support of guarantees on securities deriving from securitizations allows significant benefits for both banks and investors. Recently, the Italian parliament introduced important innovations to the regulation of securitization1 with the aim

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of facilitating the disposal of NPLs. The changes to the existing regulatory framework, summarized by Albamonte (2017), widen the scope for maneuvering by special purpose vehicles, allowing them to grant additional loans to borrowers whose debts have been sold by banks, to acquire holdings

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derived from securitized NPLs through debt-to-equity swaps and to purchase and manage immovable (real estate) or other property placed as collateral on the securitized NPLs. Determining what is still missing is the development of a secondary market for NPL-backed securities is crucial to supporting securitization. A developed secondary market, accompanied by a greater number of investors, would have positive repercussions, first of all, for the liquidity of these

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securities. Furthermore, an interesting risk and return profile, combined with a low correlation of bad loans with other asset classes, makes NPL-backed securities extremely attractive in the portfolio asset allocation process. The paper is organized as follows: Section 2 describes the portfolio construction models.

Section 3 uses the VaR approach to analyze the risk and rewards associated with the portfolios generated by the disposal; Section 4 presents the results; the final section concludes the paper.

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Decree law n. 50/2017. 5

2. Portfolio Construction Models

2.1 The securitization scheme Securitization is a sale of a bad loan portfolio by a bank (the “Originator”) to an SPV. Operationally, the SPV buys the bad loan portfolio from the Originator at the agreed-upon price, the transfer price, and issues NPL-backed securities divided in tranches that are characterized by

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different levels of seniority (senior, mezzanine and junior). A servicer party is generally nominated by the junior subscribers to manage the recovery procedure for bad loans, and they receive a

servicing fee for this task. Cash flows generated by the recovery are used for payment of the ABS,

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following the typical waterfall scheme that covers, first, the operating costs of the SPV and the

servicer, then the interest and principal of the senior and the mezzanine notes and, on a residual

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basis, the nominal value and an interest premium of the junior note.

To determine the risk and reward of the junior note, we design a first securitization

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characterized by the following structure and related assumptions:

1) The SPV issues two types of notes, a senior tranche and a junior tranche; 2) The value of the portfolio is divided between the senior and junior tranches (%S - %J)

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considering three different tranching hypotheses (75% - 25%; 50% - 50% and 40% - 60%); 3) The annual coupon interest rate of the senior tranche is 2%2; 4) The transfer price of the portfolio is a percentage of the GBV and ranges in the [15% - 35%] in Italy;

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interval. This range includes the prices of securitizations that have been carried out in recent years 5) We do not consider any retention, which should be at least 5% of the GBV. Moreover, we design a further securitization based on a public guarantee on senior notes. We rely on the guarantee provided by the Italian government in 2016 to allow the placement of senior

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bonds, called the GACS (Garanzia Cartolarizzazione Sofferenze). In particular, in accordance with the GACS scheme, the SPV issues 3 classes of notes: 1) a senior investment grade tranche (%S=84% of the total issue) with a coupon of 3mEuribor + 50bp and rated by Moody’s and DBRS as BBB(high)/Baa1; 2) a mezzanine tranche (%M=10% of the total issue) with a coupon of 3mEuribor +6% rated by Moody’s and DBRS as B(high)/B2; and 3) a junior tranche (%S=6% of

This rate is rather conservative, as it reflects neither the use of guarantees nor the presence of a rating for the senior tranche. 6 2

the total issue) with a coupon of 3mEuribor +15%3. The cost of the GACS is based on credit default swap prices from Italian primary credit quality issuers, showing the same level of risk as the senior note. We assume a cost of GACS equal to 1%4 (calculated on the outstanding principal of the senior note) and 3mEuribor equal to 0%. Furthermore, to analyze the impact of positive dynamics of interest rates, we consider an increase in the 3mEuribor of 200 bps, determining a senior coupon rate of 2.5% and a mezzanine coupon rate of 8%. According to our assumptions, the value of the portfolio, transferred to the SPV, is

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𝑃𝑡𝑓 𝑉𝑎𝑙𝑢𝑒 = 𝐺𝐵𝑉 ∙ 𝑇𝑃

mezzanine (M) and junior (J) tranches are calculated as

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𝐹𝑎𝑐𝑒 𝑣𝑎𝑙𝑢𝑒𝑆 = 𝑃𝑡𝑓 𝑉𝑎𝑙𝑢𝑒 ∙ %𝑆

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where TP is the transfer price of the portfolio to the SPV. Hence, the face values of the senior (S),

𝐹𝑎𝑐𝑒 𝑣𝑎𝑙𝑢𝑒𝑀 = 𝑃𝑡𝑓 𝑉𝑎𝑙𝑢𝑒 ∙ %𝑀

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𝐹𝑎𝑐𝑒 𝑣𝑎𝑙𝑢𝑒𝐽 = 𝑃𝑡𝑓 𝑉𝑎𝑙𝑢𝑒 ∙ %𝐽

recovered amount (GRA) is

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Because the expected recovery rate (RR) is defined as a percentage of the GBV, the global gross

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𝐺𝑅𝐴 = 𝐺𝐵𝑉 ∙ 𝑅𝑅

and the gross cash flow (GCFh) is distributed over n years. At the end of each year h, GCFh is given by

𝐺𝐶𝐹ℎ = 𝑤ℎ ∙ 𝐺𝑅𝐴 ∀ ℎ = 1. . 𝑛

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where the sum of the weights wh is 100% and each wh is the percentage of the expected amount that will be recovered during the h-th year.

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The pre-enforcement waterfall allowed for interest on the mezzanine to be paid on the repayment of the senior note’s principal. 4 The average 3-year CDS of the BBB Italian basket registered at the first use of the public guarantee (January 2017) was 91bps. In December 2018, this value reached a maximum of 150bps due to increased sovereign risk (source: Bloomberg). We set a value of 100bps to avoid considering the volatility of the Italian political risk. 7

Considering the servicing fees (c) as the only cost imposed on the recovery process, the net cash flow generated at the end of each year h (NCFh) is 𝑁𝐶𝐹ℎ = 𝐺𝐶𝐹ℎ ∙ (1 − 𝑐) ∀ℎ = 1. . 𝑛

In accordance with the statistics provided by Quaestio Capital Management (2016), we assume servicing fees for portfolio management equal to 8%. Focusing on the stream of cash flows that characterizes the portfolio, we describe the securitization

𝑁𝐶𝐹1 1

𝑁𝐶𝐹2 2

…………… ……………

𝑁𝐶𝐹𝑛 𝑛

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−𝑃𝑡𝑓 𝑉𝑎𝑙𝑢𝑒 0

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as follows:

Furthermore, we follow a waterfall scheme based on our assumptions. The scheme is designed so

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that payments are addressed to cover the operating costs of the SPV and the Servicer first, followed by the senior coupon, the mezzanine coupon, the senior principal, the mezzanine principal and, on

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a residual basis, the principal and the interest premium of the junior.

Let rS be the annual interest rate of the senior tranche and rM of the mezzanine; DSh is the senior

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outstanding debt at any time h and DMh is the mezzanine outstanding. If DSh is greater than zero, the new senior outstanding debt DSh+1 is given by

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𝐷𝑆ℎ+1

𝐷𝑆ℎ − (𝑁𝐶𝐹ℎ − 𝑟𝑆 ∙ 𝐷𝑆ℎ − 𝑟𝑀 ∙ 𝐷𝑀ℎ ) 𝑖𝑓 𝑁𝐶𝐹ℎ ≥ 𝑟𝑆 ∙ 𝐷𝑆ℎ + 𝑟𝑀 ∙ 𝐷𝑀ℎ = { 𝐷𝑆ℎ − (𝑁𝐶𝐹ℎ − 𝑟𝑆 ∙ 𝐷𝑆ℎ ) 𝑖𝑓 𝑟𝑆 ∙ 𝐷𝑆ℎ ≤ 𝑁𝐶𝐹ℎ < 𝑟𝑆 ∙ 𝐷𝑆ℎ + 𝑟𝑀 ∙ 𝐷𝑀ℎ 𝐷𝑆ℎ + (𝑟𝑆 ∙ 𝐷𝑆ℎ − 𝑁𝐶𝐹ℎ ) 𝑖𝑓 𝑁𝐶𝐹ℎ < 𝑟𝑆 ∙ 𝐷𝑆ℎ

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When DSh is null, the mezzanine starts to be repaid: 𝐷𝑀 − (𝑁𝐶𝐹ℎ − 𝑟𝑀 ∙ 𝐷𝑀ℎ ) 𝑖𝑓 𝑁𝐶𝐹ℎ ≥ 𝑟𝑀 ∙ 𝐷𝑀ℎ 𝐷𝑀ℎ+1 = { ℎ 𝐷𝑀ℎ + (𝑟𝑀 ∙ 𝐷𝑀ℎ − 𝑁𝐶𝐹ℎ ) 𝑖𝑓 𝑁𝐶𝐹ℎ < 𝑟𝑀 ∙ 𝐷𝑀ℎ

Thus, if NCFh is greater than the coupon paid to both the senior and mezzanine subscribers, then the exceeding amount reduces the senior outstanding note. Once the principal of the senior note is fully repaid (only if NCFh is greater than the mezzanine coupon), the principal of the mezzanine starts to be repaid. When the mezzanine note is fully repaid, the cash flows start to repay the junior subscribers, meaning that the first cash flow is equal to NCFh-DMh and the following repayments are all equal to NCFh. 8

Focusing on the rate of return of the three notes, the reward of the senior note is equal to the coupon rate rS; the reward of the mezzanine note is equal to rM (in the case of a total repayment), and the reward of the junior note is calculated using the discount cash flow approach. While the senior note is deemed a safe investment, which can suffer a loss only in cases of extremely low effective recovery rates, the junior note is considered to bear almost all the risk arising from the uncertainties dominating the recovery process. In other words, the junior subscribers register losses when the effective recovery rate is significantly lower than its estimation.

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2.2 The “straight sale” portfolio Finally, to analyze the risk-return profile, we consider the disposal of the bad loan portfolio

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through straight sale, which is the most frequently used alternative deleveraging strategy. Focusing on the issue of the portfolio pricing, the price offered by investors is subject to their own

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assessment of the amount and timing of the cash flows generated by the recovery procedure, net of the servicing costs (c), and considering the target return on the investment. Thus, relying on the value of the NCF and calculated as follows: 𝑁𝐶𝐹ℎ (1+𝐼𝑅𝑅𝑆𝑎𝑙𝑒 )ℎ

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𝑂𝑃 = ∑𝑛ℎ=1

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same assumptions used to design the securitizations, the offer price (OP) is the sum of the present

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where IRRSale is the target investor’s internal rate of return in the case of straight sale. It follows that the offer price is positively related to the amount of NCF and negatively to the expected return asked by the investors, to the servicing costs and to the recovery time.

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2.3 Internal rate of return as function of the recovery rate

The internal rate of return of an investment on NPLs strictly depends on the effective recovery

rate of the bad loan portfolio. Therefore, the recovery rate is a crucial independent variable in our model specification: in particular, we assume that the recovery rate is a stochastic variable that follows a lognormal distribution. This assumption relies on the structural model by Merton (1974) based on the Black and Scholes (1973) option pricing theory. The main ideas by Merton are that the market value of a firm follows a geometric Brownian motion and that default occurs if the firm’s market value becomes smaller than that of its liabilities. In the case of default, the recovery 9

rate is the value of the firm divided by its debts at the time of default. Therefore, the lognormality of the value gives rise to a lognormal (truncated) distribution of recovery rates in the case of default. As highlighted by Altman et al. (2004), the main critiques of Merton’s model are the following: 1) the firm defaults only at maturity of the debt; 2) the firm usually has more than one class of debt in its capital base; and 3) the use of a lognormal distribution tends to overestimate recovery rates in the event of default (with respect to a fat-tailed distribution). However, here, we keep the distribution of the global (average) recovery rate of the NBV of Italian mixed credit portfolios as

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the starting point. Estimations of value evolution parameters of a single firm or with a single firm capital structure are not included, whereas the parameters of the average recovery rate are included. The main problem is the adoption of lognormality because of its thin tails. Moreover, we need to

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justify the lognormality of a mixture of lognormal recovery rates. While it is impossible to show

analytically that lognormal mixtures are lognormal themselves, many studies have focused on this

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problem. In the paper by Cobb et al. (2012), the authors describe a procedure to find the parameters for a single lognormal distribution that represents a good proxy for the sum of lognormal random

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variables.

Moving to our assumptions on the expected recovery rates and cash flow distribution, we rely on the following statistics:

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1) In accordance with the evidence provided by Carpinelli et al. (2016), cash flows are distributed over 12 years (<1 year = 16%; 1-3 yrs = 40.5%; 3-5 yrs = 22.5%; 5-8 yrs = 14%; 8-10

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yrs = 2.2%; >10 yrs = 4.6%); formally, we have

16% ∙ 𝐺𝑅𝐴 ℎ = 1 20.25% ∙ 𝐺𝑅𝐴 ℎ = 2 𝑎𝑛𝑑 3 11.25% ∙ 𝐺𝑅𝐴 ℎ = 4 𝑎𝑛𝑑 5 5% ∙ 𝐺𝑅𝐴 ℎ = 6 𝐺𝐶𝐹ℎ = 4.6% ∙ 𝐺𝑅𝐴 ℎ = 7 𝑎𝑛𝑑 8 1.1% ∙ 𝐺𝑅𝐴 ℎ = 9 𝑎𝑛𝑑 10 0% ∙ 𝐺𝑅𝐴 ℎ = 11 { 4.6% ∙ 𝐺𝑅𝐴 ℎ = 12

2) As demonstrated by Ciocchetta et al. (2017), the average recovery rate is equal to 46.9%,

which is the average recovery ratio registered by Italian banks over the 2006-2015 period5; 3) Still relying on Ciocchetta et al. (2017), the recovery rate standard deviation is equal to 7.5%, being the dispersion of the recovery rates across banks, in the range ±15% from the average, 5

The recovery rates registered in 2016 and 2017 are equal to 43.5% and 44.3%, respectively (Conti et al., 2017 and Fischetto et al., 2018). 10

depending on the bank’s recovery capacity. Thus, we assume that the portfolio recovery rate is in the [31.9%; 61.9%] range with a probability of 95.7%. Our last assumption is related to the possibility for banks to indifferently implement a deleveraging strategy based on securitization or straight sale, which allows us to use the characteristics of the portfolio sold by a straight sale as a benchmark for our analyses.

3. Risk and reward of NPL portfolios

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3.1 The internal rate of return distribution

To observe the IRR distribution of the junior note and of the straight sale portfolio (IRRJ and

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IRRSale), we vary the recovery rate across the range [31.9%%; 61.9%] for each selected price (i.e., the transfer price in the case of securitization and the offer price in the case of straight sale). The

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overall probability that the recovery rate is within this range, that is, Prob{31.9% ≤RR<61.9%}, is 95.7%. Proceeding in this direction, we consider 1501 values of RR within the selected range, and

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we calculate the probability Prob{RR-1bp≤ RR
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In more detail, starting from the assumed values of parameters E(RR)=mRR=46.9% and

(RR)= RR=7.5%, we find the corresponding parameters of the normal distribution mN and  N:

2 𝑚𝑅𝑅

ur na 𝑚𝑁 = 𝑙𝑛 (

2 2 √𝑚𝑅𝑅 + 𝜎𝑅𝑅

) ; 𝜎𝑁 = √𝑙𝑛 (

2 2 𝑚𝑅𝑅 + 𝜎𝑅𝑅 ) 2 𝑚𝑅𝑅

The probability in the range [a=RR-1bp,b=RR+1bp] in our lognormal framework is

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Prob{a≤ RR
(mN,N), that is,

𝐹𝑙𝑜𝑔𝑛 (𝑥, 𝑚𝑁 , 𝜎𝑁 ) =

1 𝜎𝑁 √2𝜋

+∞

∫ 0

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1 𝑥

𝑒𝑥𝑝 (−

(𝑙𝑛𝑥 − 𝑚𝑁 )2 2𝜎2𝑁

) 𝑑𝑥

Furthermore, we calculate the mean, standard deviation, skewness and kurtosis of IRRJ and IRRSale for each distribution, defined as follows: 𝑁

𝑚=∑

𝑗=1

𝑁

𝜎 = [∑

𝑟𝑗 ∙ 𝑝𝑗 1/2

2

𝑗=1

(𝑟𝑗 − 𝑚) ∙ 𝑝𝑗 ] 3

𝑠𝑘 =

∑𝑁 𝑗=1(𝑟𝑗 − 𝑚) ∙ 𝑝𝑗 𝜎3 4

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∑𝑁 𝑗=1(𝑟𝑗 − 𝑚) ∙ 𝑝𝑗 𝑘𝑢 = 𝜎4

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where rj is the observed IRR corresponding to the value of the recovery rate RRj (1501 values) and

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pj is the probability Prob{RRj-1bp≤ RRj
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3.2 The value-at-risk approach

To analyze portfolio risk, we focus on each IRR distribution and calculate the VaR for any level of probability or, vice versa, the probability corresponding to a target level of VaR. It is worth

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remembering that VaR is defined as the potential loss on a security (or portfolio) over a specified time horizon, with a fixed probability  or confidence level (see Jorion, 2001 and Alexander, 2008).

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With L as the random variable loss, VaR is implicitly defined by the following condition: 𝑃𝑟𝑜𝑏{𝐿 ≤ 𝑉𝑎𝑅} = 𝜋

To calculate VaR, distributional assumptions on L are required. Usually, information on portfolio

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returns is available, which suggests rewriting the condition as follows:

𝑃𝑟𝑜𝑏{𝐿 = −𝑤𝑅 ≤ 𝑉𝑎𝑅} = 𝜋 ↔ 𝑃𝑟𝑜𝑏 {𝑅 < −

𝑉𝑎𝑅 = 𝑟∗} = 1 − 𝜋 𝑤

where R is the random return, w is the initial value of the position and 𝑟 ∗ is the percentage VaR or the maximum return attainable with probability 1 − 𝜋 or, otherwise, the minimum return attainable

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with probability 𝜋. If the cumulative distribution function of returns is regular and invertible, which gives 𝑉𝑎𝑅 = −𝑤𝐹𝑅−1 (1 − 𝜋) = −𝑤𝑟 ∗

In our analysis, we focus on the potential return 𝑟 ∗ , or the opposite of the VaR percentage:

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𝑟 ∗ = 𝐹𝑅−1 (1 − 𝜋) = −%𝑉𝑎𝑅1−𝜋

The potential return is somewhat antithetic to %VaR in the sense that it can be interpreted as a measure of performance (minimum return with probability ) instead of as a measure of risk.

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With the aim of evaluating the probabilities of achieving a minimum target performance on the investment in junior notes, we set two return targets: 6% and 15%. We justify these levels

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considering that 6% is the target return declared for junior notes by the Atlante6 fund and 15% is the level of reward demanded by professional market investors in this field (Ciavoliello et al.,

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2016). In this analysis, our target returns are the potential returns r* for junior subscribers (rJ*). Moreover, to compare the reward of the junior note and the straight sale portfolio, we assume the equivalence between the transfer price and the offer price, which means that the two disposal

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strategies have the same cost of deleveraging and, for this reason, securitization and straight sale are considered equivalent from the bank’s shareholder perspective. Thus, we calculate the r Sale* of the straight sale portfolio distribution showing the same confidence level of the rJ*. In other words,

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our aim is to calculate the reward of the straight sale portfolio, which has the same probability of the target return of the junior note.

The next step is to reverse the analysis and to search the equivalent portfolio in terms of VaR. In this case, we aim to calculate the portfolio price showing the same risk profile in terms of VaR. We determine the transfer price and offer price depending on the recovery rate. As previously

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described, we can associate the probabilities of different recovery rates, remembering that the recovery rate is assumed to have a normal distribution with a mean of 46.9% and a standard deviation of 7.5%. We find different IRR distributions when varying the transfer price and the offer price; however, it is worthwhile to compare the different prices showing the same value of (1cumulative probability) and the same IRR. We test our targets IRR (6% and 15%) and add the

6

Atlante is an Italian fund launched by Quaestio Capital Management SGR SpA, which was established to invest in junior or mezzanine tranches of NPLs. The target return declared is 6% (Quaestio Capital Management, 2016). 13

target return of 20% to stress the analysis when considering higher returns demanded by the market players. We focus on the 95% confidence level that is generally used in portfolio evaluation, corresponding to a recovery rate equal to 27.7%.

4. Results

The main risk when investing in NPLs arises from the misevaluation of the portfolio in terms of cash flows and their timing, leading to portfolio mispricing. Therefore, we first present the

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dynamics of the internal rate of return of the junior note and of the straight sale portfolio, assuming different values of the ex post recovery rate (registered at the end of the recovery procedure) with

respect to its ex ante evaluation. In other words, we observe the impact on the IRR of a discrepancy

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between the expected recovery rate (coming from the portfolio due diligence and entered into the

pricing model) and its effective value. Figure 1a shows the patterns of the IRR: the continuous line

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represents the IRR of the junior note, while the dashed line represents that of the straight sale portfolio. Our aim is to observe the impact on the rates of return of a misevaluation of the bad loan

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portfolio. Given the expected recovery rate equal to 46.9%, this analysis is focused on securitization without guarantees based on the 75%-25% tranching hypothesis and on an expected IRR of the junior note (IRRJ) equal to 15%. Entering these assumptions into the securitization

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pricing model, we obtain a transfer price of the portfolio equal to 31.7% of the GBV. Observing the graph (and its zoom, Figure 1b), point A, on the continuous line, represents the case in which the expected and the effective recovery rates coincide. As the effective recovery rate declines, IRRJ

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also decreases until it reaches zero at the recovery rate equal to 35.8% (point B). Below a recovery rate equal to 27.9%, the IRRJ falls sharply, and the total loss of capital is recorded for effective recovery rates below 27.8%. This threshold also represents the attachment point of the senior note. Focusing on the IRR in the case of straight sales, the dotted line represents the pattern of IRRSale assuming an offer price of 31.7% (equal to the transfer price). Observing this line, when the

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recovery rate is 46.9%, the IRRSale is equal to 8.8% (point C); as the effective recovery rate declines, also IRRSale decreases but in a smoother way with respect to IRRJ. In the case of straight sale, the decline is gradual: IRRSale is null when the recovery rate is equal to 34.4% (point D) and only in the case of an effective recovery rate close to zero does a loss of capital occur. This graph shows that, in the case of a reliable portfolio assessment, IRRJ is higher than IRRSale. In particular, as long as the discrepancy between the expected and the effective recovery ratio is below 9.8% (point E), IRRJ is higher than IRRSale.

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- Insert Figure 1a and 1b about here -

Moving to the IRR distributions, Figure 2 shows the graphs of the densities and cumulative distribution functions of the junior note and the straight sale portfolio for each price level (i.e., the transfer price in the case of securitization equal to the offer price in the case of a straight sale), and considering the tranching assumption of 75%-25% between senior and junior (the graphs related to the further tranching hypotheses, 50%-50% and 40%-60%, are presented in Appendix A). In particular, we design the IRR distribution of the junior note (IRRJ) and of the straight sale portfolio

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(IRRS), assuming five price hypotheses varying in the range [15%; 35%] with a step equal to 5%.

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- Insert Figure 2 about here -

These distributions and their statistics show that 1) in the case of low price levels, the two

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distributions present a significant difference in terms of averages, which narrows with increasing price levels; 2) the values of the standard deviation of IRRJ are consistently higher than IRRSale; 3)

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when the price is equal to 35%, IRRJ shows a high level of negative skewness (-7.6%) and fat left tails (kurtosis equal to 2.61); 4) the mean of IRRJ ranges between 75.8% and 8.2%, while the mean of IRRSale lies between 40.8% and 5.3%; 5) the cumulative distribution function shows a stochastic discussion).

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dominance of IRRJ distribution on IRRSale for each transfer price below 20% (see Appendix B for a Comparing the distributions under the three tranching hypothesis, we observe that, for an

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increase in the junior percentage: 1) the two distributions approach each other, showing that the structure of the securitization in terms of tranching affects the riskiness of the junior note (i.e., deriving from the fact that the cash flows exceeding the coupon and nominal value of the senior note generate a greater impact on the performance of the junior note with a lower face value); 2) the kurtosis decreases (from a maximum of 2.61 in the case of 75%-25% to a maximum of 2.44 in

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the case of 40%-60%).

Focusing on the securitization model with GACS, Figure 3 shows the densities and cumulative

distribution functions of the straight sale portfolio and junior note (considering both hypotheses on the Euribor, which are equal to 0% and 2%) and for each sell price. In this case, the junior note profile is even more attractive: the mean of the IRRJ ranges between 153.8% and -12.9%. The impact of the Euribor increase of 200 bps appears to consider only the highest price tested. In particular, in the case of a price equal to 30%, the risk-return profile of the junior note deteriorates;

15

when the price is equal to 35%, the expected return becomes negative, which means that the interest rise undermines the attractiveness of using securitization in the case of high sell prices. - Insert Figure 3 about here –

Table 1 provides the probability that IRRJ is higher than our target returns (rJ*) when the transfer price ranges between 15% and 35%7 and under the three tranching hypotheses. For example, relating to 75%-25% and the price is equal to 30%, the probability of registering an IRRJ

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greater than 6% is 89.4% and that of registering one greater than 15% is 59.9% (Panel A). The >99.1% figure in this table means that the lowest return (corresponding to the recovery rate equal

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to 31.9%) is higher than r*.

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- Insert Table 1 about here -

Moreover, to compare the junior note and straight sale portfolio, we calculate the r* of the

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straight sale portfolio distribution (last column of Table 1), showing the same confidence level of the rJ*. In other words, our aim is to calculate the IRRSale, which shows the same probability of the

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target IRRJ. For example, when rJ* is equal to 6%, the rSale* is equal to 4.0%, with a probability greater than 99.1% when the price is set under 25% and 89.4% and 61.0% when the price is equal to 30% and 35%, respectively. These results confirm that low price levels increase the appeal of

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investing in NPLs because of the low risk of registering low returns. Moreover, we show that the investment in the junior note offers higher expected returns than buying the entire portfolio, considering the return targets equal to 6% and 15%. Figure 4 presents graphical explanations for two of our results. We focus on the cumulative distribution functions based on the first tranching hypothesis (75%-25%) and assuming a price equal to 30% and a target return of 6% (Figure 4a); moreover, we assume a price equal to 25% and

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a target return of 15% (Figure 4b). Focusing on Figure 4a and zooming the cumulative distribution function at approximately 6% on the horizontal axis, we can verify that for the same probability level of 10.6% (=1-89.4%), the inverse function indicates rSale* = 4% and rJ* = 6%. In other words, these r* (-%VaR) are the minimum IRRs for a confidence level equal to 89.4%. Likewise, observing the cumulative distribution of Figure 4b, the inverse, corresponding to a probability 8.1% (=1-91.9%), gives rSale* = 8.8% and rJ* = 15%. 7

In cases of very low P (i.e., 15%), the lowest IRRJ (𝑟 ∗ or -%VaR) is always greater than 15%. 16

- Insert Figure 4 about here -

Table 2 presents the results of the same analysis under securitization with GACS. Our results show that the probability of achieving the target IRRJ (6% and 15%) is higher when GACS is available. When we consider the 200 bps rise in the Euribor, the probability of achieving the target return is lower due to the higher costs of the securitization. As a consequence, the r*Sale figures associated with each probability are higher than when Euribor is equal to 0%. Even so, r*Sale is

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lower than rJ* for each target return.

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- Insert Table 2 about here -

Our last analysis aims to provide the portfolio sell price that equals the risk of the junior note

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and of the straight sale portfolio in terms of VaR. Table 3 summarizes the results.

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- Insert Table 3 about here -

Considering the tranching hypothesis (75%-25%), the transfer price of the NPL portfolio, with 95%

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confidence, produces IRRJ values higher than 6% and 15%: 21.9% and 18.7%, respectively. The offer prices of the same portfolio in the case of a straight sale are 20.2% and 15.5% of the GBV, respectively. Higher prices mean resource savings for the bank shareholders: in other words, a

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positive difference between the transfer price and the offer price indicates the advantage of adopting the securitization rather than the straight sale as the disposal strategy. This difference ranges between 0.6% and 1.7% in the case of a 6% target return and between 1.0% and 3.2% in the case of a 15% target return. If we assume an even higher target return of 20%, this difference widens. It is worth highlighting that the transfer price is higher than the offer price for each

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tranching hypothesis.

In the case of securitization with GACS, the outcomes are amplified with respect to the standard

securitization due to the smaller size of the junior note. The more the target return increases, the greater the convenience of using securitization as the difference between the transfer price and the offer price widens. These results confirm the superiority of securitization over the straight sale strategy from the bank’s perspective. In fact, assuming the same risk-return profile of the junior note and of the

17

straight sale portfolio, the sell price is higher in the case of securitization for each target return, improving the bank’s deleveraging.

5. Conclusions

The reduction in the large stock of NPLs for many European countries is a supervisory priority for the single supervisory mechanism in order to help promote financial stability in the Eurozone. Suffice it to consider that the disposal of impaired loans is a prerequisite for the concrete

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establishment of the European Deposit Insurance Scheme (EDIS), meaning that the “NPL issue and the EDIS are interlinked” (Committee on economic and monetary affairs, 2017). As a consequence, European banks are forced to implement a deleveraging strategy in tight times and during a period

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characterized by banking system weakness.

Banks can sell their portfolios straight to the market or establish securitization that involves the

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issue of NPL-backed securities. From investors’ perspective, the riskiest security is the junior note, as it is repaid last; for this reason, it is the focus of our analysis.

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We base our analysis on the Italian banking system because it is the most exposed to this problem and, for this reason, is strictly monitored by Bank of Italy; additionally, research papers provide reliable statistics needed for the pricing of a bad loan portfolio. It is worth noting that our

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results can be easily generalized within the European banking system. We observe the dynamic of the internal rate of return when investing in the junior note issued by a securitization and when investing in the straight sale portfolio to an incorrect assessment of the

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portfolio recovery rate. We highlight that the junior note shows significantly higher rates of return, except in the case of an unlikely wide discrepancy between the expected and effective recovery rate.

Comparing the rate of return distributions of the two investments, we provide evidence of the higher probability of achieving selected target returns in investing in the junior note. We test the

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securitization model with GACS, the public guarantee scheme approved by the Italian government to support the NPL deleveraging process, and demonstrate that the support of this type of public guarantee improves the risk-return profile of the junior notes even assuming an increase in the interest rates.

It should be emphasized that our results depend on the reliability of the estimates of the variables used in the model. In particular, higher or lower levels of the recovery rate or a different dispersion around the average may affect our results. The same criticality also concerns the

18

hypotheses about the cost of GACS, which is a market-based fee influenced by the country risk of the issuer. In this case, a riskier perception of Italy increases the cost of the guarantee. Relative to the existing literature, the main contribution of this paper is to show the superiority of the securitization strategy in the deleveraging process using a VaR approach, a measure widely recognized and used since the adoption by the Basel Committee (e.g., see BIS 2006). The results demonstrate that securitization is the most convenient strategy for banks to clean up their portfolios because of the lower deleveraging cost with respect to the straight sale, assuming that the same risk-return profile characterizing the investment opportunities (i.e., the same

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probability of achieving target returns) is offered. This result clashes with the absence of a structured and developed market for NPL-backed securities, which is extremely useful for

European banks to avoid a significant transfer of value from their shareholders to the market

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(Jassaud and Kang, 2015; Aiyar et al., 2015; Fell et al., 2017). Furthermore, our results show the significant impact of public support on the cost of NPL disposals with clear benefits for the

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banking system, which is already heavily affected by increasingly stringent capital requirements and high costs of compliance. In fact, the granting of a state guarantee on the senior tranche reduces

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the cost of deleveraging and, at the same time, increases the feasibility of the securitization, with the senior note being more attractive for institutional investors. Focusing on the Italian experience, the attractiveness of this support is evidenced by the fact that since its approval in 2016, the GACS

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has been subsequently extended and renewed in 20198. Finally, our research shows the risk-return profile of the junior note under different assumptions of the securitization structure, highlighting their attractiveness as a potential new asset class in the asset management industry.

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Future research should continue to pay attention to this asset class by following two lines of research. The first is the monitoring of the recovery rates of NPLs, expanding the range of analysis to the different European countries. Different countries, in fact, are characterized by different times of justice and, therefore, different times to recovery. Furthermore, they are characterized by different phases of the real estate market, a critical variable that influences the recovery rate of

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secure loans. The second line of research focuses on institutional investors specializing in the purchase of these NPL-backed securities. Observing their performance over time would be useful for assessing the actual correlation of these securities with other asset classes, both alternative and traditional. 8

GACS was envisaged for the first time with the decree-law February 14, 2016, n. 18, which was extended until March 6, 2019. With the decree-law March 25, 2019 n. 22 (so-called "Financial stability"), GACS was renewed, with some modifications, for 24 months (extendable for a further 12 months) starting from the positive decision of the European Commission. 19

Declarations of interest: none Acknowledgments

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We thank the participants at the 2017 ADEIMF Conference in Rome, 2017 Finest Conference in Trani and 2019 FEBS Conference in Prague for constructive comments. Moreover, we thank Luigi Bussi and Paolo Gabriele of Finanziaria Internazionale Securitization Group for support in the construction of the securitization models.

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References Affinito M, Tagliaferri E. (2010). Why do (or did?) banks securitize their loans? Evidence from Italy. Journal of Financial Stability, 6(4), 189-202. Aiyar, S., Berghthaler, W., Garrido, J.M., Ilyina, a., Jobst, A., Kang, K., Koutun, D., Liu, Y., Managhan, D., Moretti. M. (2015). A Strategy for Resolving Europe’s Problem Loans, IMF Staff Discussion Notes, No.19, September. Albamonte D. (2017). Recent changes to Law 130/1999 on securitization of loans. Notes on Financial Stability and Supervision. Bank of Italy, No.10, July.

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Alexander, C. (2008). Market Risk Analysis, Volume IV Value-at-Risk Models. John Wiley & Sons Ltd.

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Altman E., Resti A., Sironi A. (2004). Default Recovery Rates in Credit Risk Modeling: A Review of the Literature and Empirical Evidence, Economic Notes, 33, 183-208.

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BIS (2006). International Convergence of Capital Measurement and Capital Standards. A Revised Framework. June. Black, F., Scholes M. (1973). The Pricing of Options and Corporate Liabilities, Journal of Political Economics, May, 637-659.

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Bruno, B., Lusignani, G., Onado, M. (2018). A securitisation scheme for resolving Europe's problem loans, in: Finance and investment: the European case, Oxford University Press, 157-167.

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Carpinelli, L., Cascarino, G., Giacomelli, S., Vacca, V. (2016). The management of nonperforming loans: a survey among the main Italian banks. Bank of Italy Occasional papers, No. 311, February. Ciavoliello, L.G., Chiocchetta, F., Conti, F.M., Guida, I., Rendina, A., Santini, G. (2016). What’s the value of NPLs?, Notes on Financial Stability and Supervision. Bank of Italy, No.3, April.

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Ciocchetta, F., Conti, F.M., De Luca, R., Guida, I., Rendina, A., Santini, G. (2017). Bad Loan recovery rates, Notes on Financial Stability and Supervision. Bank of Italy, No.7, January. Cobb, B.R., Rumi R., Salmeròn, R.R.A. (2012). Approximating the Distribution of a Sum of Lognormal Random Variables, Statistics and Computing 16(3), 293-308. Committee on economic and monetary affairs. (2017). Monetary dialogue with Mario Draghi, President of the ECB. Brussels, November 20th.

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Conti, F.M., Guida, I., Rendino, A., Santini, G. (2017). Bad loan recovery rates in 2016, Notes on Financial Stability and Supervision. Bank of Italy, No.11, November. Dell’Ariccia, G., Igan, D., Laeven, L. (2012). Credit Booms and Lending Standards: Evidence from the Subprime Mortgage Market, Journal of Money, Credit and Banking, 44, 367–84. De Marzo, P. M. (2005). The pooling and tranching of securities: a model of informed intermediation, Review of Financial Studies, 18, 1-35. EBA (2016a). EBA report on the dynamics and drivers of non-performing exposures in the EU banking sector, June. EBA (2016b). Risk assessment of the European Banking System, December. ECB (2016). Draft guidance to banks on non-performing loans, September. 21

ECB (2017). Guidance to banks on non-performing loans, March. ECB (2018). Addendum to the ECB Guidance to banks on non-performing loans: supervisory expectations for prudential provisioning of non-performing exposures, March. ECB (2019). Supervisory banking Statistics, Q42018. European Parliament (2017). Non-performing loans in the Banking Union: state of play, April 5th. Fell, J., Moldovan, C., O’Brien, E. (2017). Resolving non-performing loans: a role for securitization and other financial structures?, Financial Stability Review, ECB, May. Fischetto, A.L., Guida, I., Rendina, A., Santini, G., Scotto di Carlo, M. (2018). Bad loan recovery rates in 2017, Notes on Financial Stability and Supervision. Bank of Italy, No.13, December.

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Grunert J., Weber, M. (2009). Recovery rates of commercial lending: empirical evidence for German companies. Journal of Banking and Finance, 33, 505-513.

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Jassaud, N, Kang, K. (2015). A Strategy for Developing a Market for Nonperforming Loans in Italy. IMF WP/15/24.

Jorion, P. (2001) Value at Risk: The New Benchmark for Managing Financial Risk, McGraw-Hill. Lusignani G., Pettinari, L., Tedeschi, R. (2019). NPLs: a new asset class? Prometeia WP.

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Merton, R. C. (1974). On the Pricing of Corporate Debt: The Risk Structure of Interest Rates, The Journal of Finance, 29(2), 449-470.

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Messai, A.S., Jouini, F. (2013). Micro and Macro Determinants of Non-performing Loans. International Journal of Economics and Financial Issues. 3(4), 852-860. Mian, A., Sufi, A. (2009). The Consequences of Mortgage Credit Expansion: Evidence from the U.S. Mortgage Default Crisis. Quarterly Journal of Economics, 124, 1449-96.

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Quaestio Capital Management (2016). Presentazione Fondo Atlante. April 29th.

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Vasicek, O. (1977). An equilibrium characterization of the term structure, Journal of Financial Economics, 5(2), 177-188.

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APPENDIX A

mean 53.2% 40.8%

std deviation 10.7% 8.1%

skewness 0.17 0.10

kurtosis 2.52 2.40

Junior Sale

mean 21.4% 16.6%

std deviation 7.2% 5.4%

skewness 0.11 0.10

kurtosis 2.38 2.40

Junior Sale

mean 6.4% 5.3%

Junior Sale

mean 33.8% 26.0%

std deviation 8.5% 6.4%

skewness kurtosis 0.07 2.38 0.10 2.40

std deviation 6.2% 4.7%

skewness kurtosis 0.11 2.42 0.09 2.40

Junior Sale

mean 12.7% 10.1%

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Junior Sale

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IRR distributions of the junior note and straight sale portfolio in the case of 50%-50% tranching

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Figure A

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std deviation 5.5% 4.1%

skewness 0.07 0.09

kurtosis 2.41 2.40

Note: Straight sale portfolio (dashed line); junior note (continuous line).

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Figure B

IRR distributions of the junior note and straight sale portfolio in the case of 40%-60% tranching

mean 49.2% 40.8%

std deviation 10.3% 8.1%

skewness 0.19 0.10

kurtosis 2.39 2.40

Junior Sale

Junior Sale

mean 19.7% 16.6%

std deviation 6.5% 5.4%

skewness 0.08 0.10

kurtosis 2.39 2.40

Junior Sale

Junior Sale

mean 6.0% 5.3%

std deviation 5.0% 4.1%

skewness 0.09 0.09

mean 31.0% 26.0%

std deviation 7.7% 6.4%

skewness kurtosis 0.10 2.44 0.10 2.40

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Junior Sale

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mean 11.9% 10.1%

kurtosis 2.41 2.40

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Note: Straight sale portfolio (dashed line); junior note (continuous line).

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std deviation 5.7% 4.7%

skewness kurtosis 0.10 2.39 0.09 2.40

APPENDIX B

Stochastic dominance and value-at-risk

The random returns IRRJ and IRRSale and their cumulative distribution function are examined below. Commonly, a decision maker should (weakly) prefer the stochastic IRRJ to IRRSale (in other words, the investor will prefer the junior note instead of the sale portfolio) if the following

𝑃𝑟𝑜𝑏{𝐼𝑅𝑅𝑗 > 𝑥} ≥ 𝑃𝑟𝑜𝑏{𝐼𝑅𝑅𝑆 > 𝑥} ∀ 𝑟𝑒𝑎𝑙 𝑥

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condition holds:

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In fact, rather intuitively, for any x, the probability that IRRJ will be greater than x is always higher than the probability that IRRSale will be greater than the same x, meaning that IRRJ dominates

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IRRSale for the first stochastic dominance criterion (FSD):

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The inequality can be written as follows:

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IRRJ ≿𝐹𝑆𝐷 IRRS

1 − 𝑃𝑟𝑜𝑏{𝐼𝑅𝑅𝑗 ≤ 𝑥} ≥ 1 − 𝑃𝑟𝑜𝑏{𝐼𝑅𝑅𝑆 ≤ 𝑥} ∀𝑥

That is,

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𝐹𝐼𝑅𝑅𝐽 (𝑥) = 𝑃𝑟𝑜𝑏{𝐼𝑅𝑅𝑗 ≤ 𝑥} ≤ 𝐹𝐼𝑅𝑅𝑆𝑎𝑙𝑒 (𝑥) = 𝑃𝑟𝑜𝑏{𝐼𝑅𝑅𝑆 ≤ 𝑥} ∀𝑥

Thus, IRRJ dominates IRRSale if its cumulative distribution function is always less than the cumulative function of IRRSale or, geometrically, if 𝐹𝐼𝑅𝑅𝐽 (𝑥) lies always below 𝐹𝐼𝑅𝑅𝑆 (𝑥). Observing the graphs in Figure 1, there is always dominance when P≤25%. The dominance is

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confirmed in the 50%-50% and 40%-60% tranching hypotheses when we assume different recovery rate distributions. Finally, we recall that the inverse of the cumulative distribution function represents the potential return r* (or -%VaR) (see Figure 3 in Section 5). When IRRJ dominates IRRSale, for any fixed level of probability, the inverse will give a 𝑟𝐽∗ greater than 𝑟𝑆∗ . For low probabilities, this means that the junior note shows a high confidence level to register a reward higher than that of the straight sale portfolio.

25

FIGURES

The IRR pattern of alternative investments in NPLs in the case of the 75%-25% tranching hypothesis

Zoom for 33%≤RR≤47%

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Figure 1b

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Figure 1a

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Figure 2

IRR distributions of the junior note and straight sale portfolio in the case of the 75%25% tranching hypothesis

mean 75.8% 40.8%

std deviation 15.9% 8.1%

skewness 0.07 0.10

kurtosis 2.36 2.40

Junior Sale

Junior Sale

mean 29.6% 16.6%

std deviation 9.9% 5.4%

skewness 0.04 0.10

kurtosis 2.38 2.40

Junior Sale

mean 46.9% 26.0%

std deviation 12.1% 6.4%

skewness 0.21 0.10

kurtosis 2.46 2.40

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Junior Sale

std deviation 8.8% 4.7%

Junior Sale

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mean 17.5% 10.1%

mean 8.2% 5.3%

std deviation 8.1% 4.1%

skewness -0.08 0.09

kurtosis 2.61 2.40

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Note: Straight sale portfolio (dashed line); junior note (continuous line).

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skewness 0.07 0.09

kurtosis 2.40 2.40

Junior _ _ _

mean 153.8%

std deviation 35.9%

skewness 0.21

kurtosis 2.24

148.7% 41.4%

35.1% 8.2%

0.24 -0.05

2.32 2.38

Junior _ _ _ Junior Sale

___

mean 94.4%

std deviation 23.7%

skewness -0.18

kurtosis 2.41

90.1% 26.5%

24.0% 6.5%

-0.18 -0.06

2.31 2.38

std deviation

skewness

kurtosis

59.5%

20.5%

0.01

2.36

Junior _ _ _

Junior _ _ _ Sale

54.7%

20.7%

-0.02

2.46

17.1%

5.4%

-0.06

2.38

Junior _ _ _ Sale

mean

std deviation

skewness

kurtosis

32.9%

25.2%

-2.71

15.29

24.1%

34.9%

-2.48

9.48

10.5%

4.7%

-0.07

2.38

Junior _ _ _ Junior _ _ _ Sale

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mean Junior _ _ _

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Junior Sale

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IRR distributions of the junior note and sale portfolio in the case of securitization under GACS (84%-10%-6% tranching)

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Figure 3

mean

std deviation

skewness

kurtosis

2.9%

44.4%

-1.66

4.31

-12.9%

52.8%

-0.94

2.10

5.7%

4.2%

-0.07

2.38

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Note: Sale (dashed line); Junior assuming Euribor equal to 0% (black continuous line); Junior assuming Euribor equal to 2% (red continuous line).

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Figure 4

Comparison between IRRSale and IRRJ in the case of a 6% target return and 30% sell price and in the case of a 15% target return and 25% sell price (75%-25% tranching hypothesis) b)

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Note: Straight sale portfolio (dashed line); junior note (continuous line).

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a)

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Table 1

Confidence levels for different transfer prices

Panel A (75%-25%) Panel B (50%-50%)

TP=15 % >99% >99% >99% >99% >99% >99%

TP=20 % >99% >99% 98.7% >99% 98.1% >99%

TP=25 % 91.9% >99% 79.4% 98.5% 74.9% 98.3%

TP=30 % 59.9% 89.4% 75.7% 85.0% 31.7% 83.4%

TP=35 % 23.6% 61.0% 9.7% 52.6% 7.4% 50.0%

r* (Sale) 8.8% 4.0% 11.9% 5.0% 12.7% 5.3%

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Panel C (40%-60%)

r* (Junior) 15% 6% 15% 6% 15% 6%

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Table 2

Confidence levels for different transfer prices in the case of GACS

Panel A (84%-10%-6%) Senior rate 0.5% - Mezzanine rate 6%

Panel B (84%-10%-6%) -

TP=15% TP=20% TP=25% TP=30% TP=35% >99% >99% 96.9% 84.1% 54.9% >99% >99% >99% 91.3% 70.6% >99% >99% 97.4% 76.9% 42.7% >99% >99% 95.1% 85.9% 58.4%

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Senior rate 2.5% - Mezzanine rate 8%

r* (Junior) 15% 6% 15% 6%

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r* (Sale) 5.2% 3.1% 6.6% 4.7%

Table 3 level

Comparison between the transfer price and the offer price related to a minimum of IRR (r*) with a 95% confidence level

r*=6%

r*=15%

r*=20%

Split hypothesis TPOP 1.7% 0.8% 0.6%

TP 18.7% 17.0% 16.5%

OP 15.5% 15.5% 15.5%

22.6%

20.2%

2.4%

21.0%

15.5%

21.4%

20.2%

1.2%

20.0%

15.5%

TPOP 3.2% 1.5% 1.0%

TP 17.3% 15.3% 14.8%

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OP 13.6% 13.6% 13.6%

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OP 20.2% 20.2% 20.2%

TPOP 3.7% 1.7% 1.2%

5.5%

20.1%

13.6%

6.5%

4.5%

19.3%

13.6%

5.7%

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75%-25% 50%-50% 40%-60% 84%-10%-6% (Euribor 0%) 84%-10%-6% (Euribor 2%)

TP 21.9% 21.0% 20.8%