Photoinduced anticancer effect evaluation of ruthenium(II) polypyridyl complexes toward human lung cancer A549 cells

C. R. Physique 20 (2019) 364–370 Contents lists available at ScienceDirect Comptes Rendus Physique www.sciencedirect.com From statistical physics to social sciences / De la physique statistique aux sciences sociales Lost in diversiﬁcation Diversiﬁcation inﬁdèle Marco Bardoscia a,1 , Daniele d’Arienzo b , Matteo Marsili c,d,∗ , Valerio Volpati e a Bank of England, Threadneedle St., London EC2R 8AH, UK Bocconi University, Department of Finance, Via Roentgen 1, 20136 Milan, Italy c The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy d Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Trieste, Italy e Capital Fund Management, 23, rue de l’Université, 75007 Paris, France b a r t i c l e i n f o Article history: Available online 27 June 2019 Keywords: Securitization Asset-backed securities Structured ﬁnance Information theory Mots-clés : Titrisation Titres adossés à des actifs Finance structurée Théorie de l’information a b s t r a c t As ﬁnancial instruments grow in complexity, more and more information is neglected by risk optimization practices. This brings down a curtain of opacity on the origination of risk, which has been one of the main culprits in the 2007–2008 global ﬁnancial crisis. We discuss how the loss of transparency may be quantiﬁed in bits, using information theoretic concepts. We ﬁnd i) that ﬁnancial transformations imply large information losses, ii) that portfolios are more information sensitive than individual stocks only if fundamental analysis is suﬃciently informative on the co-movement of assets, iii) that securitisation, in the relevant range of parameters, yields assets that are less information sensitive than the original stocks, and iv) that, when diversiﬁcation (or securitisation) is at its best (i.e. when assets are uncorrelated), information losses are maximal. We also address the issue of whether pricing schemes can be introduced to deal with information losses. This is relevant for the transmission of incentives to gather information on the risk origination side. Within a simple mean variance scheme, we ﬁnd that market incentives are not generally suﬃcient to make information harvesting sustainable. © 2019 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. r é s u m é À mesure que la complexité des instruments ﬁnanciers augmente, de plus en plus d’informations sont négligées par les méthodes d’optimisation des risques. Cela obscurcit l’origine du risque, ce qui a été l’une des causes principales de la crise ﬁnancière mondiale de 2007–2008. Nous discutons la manière dont la perte de transparence peut être quantiﬁée en bits, à l’aide de concepts de théorie de l’information. Nous constatons i) que les transformations ﬁnancières impliquent d’importantes pertes d’information, ii) que les portefeuilles sont plus sensibles que les stocks à l’information tant que si l’analyse fondamentale est suﬃsamment informative quant aux mouvements conjoints des actifs, iii) que la titrisation, dans la gamme pertinente de paramètres, produit des actifs moins * Corresponding author. E-mail address: marsili@ictp.it (M. Marsili). Any views expressed are solely those of the author(s) and so cannot be taken to represent those of the Bank of England or to state Bank of England policy. 1 https://doi.org/10.1016/j.crhy.2019.05.015 1631-0705/© 2019 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. M. Bardoscia et al. / C. R. Physique 20 (2019) 364–370 365 sensibles à l’information que ceux initiaux, iv) que, lorsque la diversiﬁcation (ou titrisation) est à son meilleur (c’est-à-dire lorsque les actifs sont non corrélés), les pertes d’information sont maximales. Nous abordons également la question de savoir si des systèmes de valorisation peuvent être mis en place pour faire face aux pertes d’information. Ceci est pertinent pour inciter les initiateurs à collecter des informations sur l’origine du risque. Dans le cadre simple d’une approximation moyenne–variance, nous constatons que les incitations de marché ne sont généralement pas suﬃsantes pour rendre la collecte d’information durable. © 2019 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. 1. Introduction Financial innovations have been seen as a formidable tool to increase the eﬃciency of the market, by controlling the risk of ﬁnancial assets, thus easing resource allocation between investors and the real economy. Yet, several authors [1–3] have suggested that the increasing complexity of ﬁnancial products may trigger the emergence of instabilities and systemic risks. The most commonly believed determinant of the 2007–2008 global ﬁnancial crisis is the rise of structured ﬁnancial products [4]. The formidable complexity of these types of products effectively brought down a curtain of opacity between the risk originators and the buyers of the ﬁnancial products, that hid the true risks of the underlying assets (e.g., mortgages, loans, credits) [5,6]. While the dangers of these instruments had been highlighted well before the crisis [7], most of the response to the crisis did not address the core issue of the transparency loss implicit in ﬁnancial transformations, but rather focused on ring fencing the ﬁnancial system with various measures [8]. An exception to that is the proposal [9] to build an eﬃcient and standardized system, or a common language, through which information on the origin of risks should be easily available to all market participants. Such a ﬁnancial barcode, which might be attached to any ﬁnancial product, should contain all the information that is relevant in order to make realistic estimates about the return and risk of the product, from the risk proﬁles of the building blocks to market fundamentals. Yet, it is not clear how such barcodes should be constructed, which information they should contain, and whether they should be statically or dynamically updated, when new information is available. In particular, an interesting open question is whether demand for such barcodes may “naturally” arise and how barcodes should be priced, since without a barcode price the sellers would have no incentive for sharing the information. Apparently, within the prevailing market eﬃciency hypothesis paradigm, according to which prices of any stock exchanged in the market reﬂects faithfully any relevant information [10,11], these barcodes would be worthless. Indeed, for example, the price of Asset Backed Securities (ABS) were computed only on the basis of default probabilities of the underlying assets (e.g., mortgages). Even though, in principle, all the documentation about the underlying assets was available to buyers, the prices of ABS did not depend at all on it, with the consequence that incentives for due diligence in collecting information on the underlying by issuers were lacking [7,6]. Yet, market information eﬃciency resides in the balance between traders seeking information (fundamental analysis) and traders exploiting it (technical analysis), as shown by a wealth of results in agent-based modelling of ﬁnancial markets (see, e.g., [12]). The former proﬁt by the fact that the information they gather grants them an excess return. Here the proﬁts of collecting information accrue to the individual trader, while in the case of a structured ﬁnancial product these are passed over to the buyer. Accordingly, they should be reﬂected in the price. This simple logic is the basis of the present paper. As a ﬁrst step, we quantify the transparency loss by the amount of bits of information lost in diversiﬁcation. Secondly, we address the issue of deriving the optimal barcode, the one that contains the maximal information on the return of the ﬁnancial instrument. Then we compute the price associated with the barcode as the value of the information within a simple mean variance framework. The information loss and the barcode price are then quantiﬁed within a model system based on Gaussian variables (see [13] for an extension to binary variables). Within this framework, we ﬁnd that ﬁnancial transformation implies large information losses and that market incentives are not generally suﬃcient to make information harvesting sustainable. The remaining sections are organized as follows: in Section 2 we discuss the general setting, information and ﬁnancial transformation. Then we quantify information losses for a simple model of Gaussian log-returns and address the issue of information pricing. We conclude with some general remarks. 2. The general framework = ( X 1 , . . . , Xn ) be the associated Let us suppose we have a pool of n assets, e.g., stocks, loans or mortgages, and let X vector of (log-)returns. The values X are unknown to the investor, so we shall treat them as a vector of random variables, ). We consider a situation where some side information related to the stock described by a probability distribution p ( X X i , e.g., the income of the borrower of the loan or information on the fundamentals of asset i, is possibly available. This information is captured by a random variable Y i , which, inspired by Ref. [9], we shall call the barcode associated with asset i. Barcodes allow the investors who bought the asset to retrieve all information that is relevant to estimate the return of the 366 M. Bardoscia et al. / C. R. Physique 20 (2019) 364–370 |Y ) instead of p ( X ). We shall asset, in the sense that, given the barcodes Y , they can use the conditional distribution p ( X use the mutual information [14]: I (U , V ) = E log p (u | v ) p (u ) (1) to quantify the amount of information that the knowledge of a variable V provides on the random variable U . In Eq. (1), p (u | v ) is the conditional distribution of U given that V = v and p (u ) is the unconditional one.2 Hence, I ( X i , Y i ) measures , Y ) measures the total amount of information that the barcodes Y in bits the information that Y i provides on X i and I ( X . provide on the returns X 2.1. Financial transformations We consider ﬁnancial transformations → F ( X ), X X= n Xi (2) i =1 that entail pooling the n assets into a single portfolio X and applying a transformation F ( X ). This generates a new ﬁnancial asset with log-return F ( X ). The simplest such transformation is the portfolio itself that delivers the average log-return F X̄ ( X ) ≡ X = X /n (3) can be the log-returns of individual stocks. In this case, Y would encode information on fundamentals (e.g., corpoHere, X rate structure of the ﬁrm, analysis of the sector they operate etc) for each stock. X corresponds to the most basic diversiﬁcation techniques, which entails investing a fraction 1/n in each of the n assets, instead of investing in a single asset X i . The beneﬁt of diversiﬁcation is that it reduces the risk. For example, for n i.i.d. stocks, the variance V ( X ) = V ( X i )/n is reduced by a factor of n, w.r.t. that of individual stocks. Another class of products we consider are Asset Backed Securities (ABS), the typical products of structured ﬁnance [4], whose return function is based on a prioritized structure of claims. In these products, the claims over the cash ﬂow of the returns of the underlying assets X i are structured in such a way that ABS yields a positive return when the total return is larger than a given threshold k. The return of these products is F k ( X ) = θ ( X − k) (4) where θ(x) = 1 when x ≥ 0 and θ(x) = 0 otherwise. Different tranches correspond to different risk proﬁles that can be → F k ( X ) is an example of securitisation and the advantage of it is obtained with different values of k. The transformation X that it turns a set of risky assets X i into assets with a controlled risk proﬁle. Suﬃciently small values of k yield assets that are very safe, i.e. for which F k ( X ) = 1 with high probability. As an example, mortgage backed securities (MBS) [4] are based of mortgages granted to n households, where X i = +1 if household i repays the mortgage and X i = 0 if on a portfolio X i defaults. In this case, Y i may encode the occupational status of i, the characteristics of the neighbourhood of the house bought with the ith mortgage. In this case, i’s default may occur for idiosyncratic reasons, or for systemic ones (e.g., crisis in the sector of the economy of the company where Mr. i works or a natural disaster in that region) that may affect different households in the same way. Investors can transmit all the information Y about the individual assets to the buyers of the engineered asset F . Yet, some of this information may not be relevant to estimate the return F ( X ), i.e. all the information relevant to estimate the return of F may be compressed in a single variable G F that we call the barcode of F . Clearly, G F (Y ) has to be a function of Y , and, ideally, the barcode G F should be the simplest3 among all possible variables V (Y ) such that I F ( X ), V (Y ) = I F ( X ), Y . A general result can be obtained by invoking the data processing inequality [14]. This states that in any transformation → F ( X ), some information may or may not be lost, but for sure no information can be gained. In terms of the mutual X information, this reads , Y ) I ( F ( X ), G F (Y )) ≤ I ( F ( X ), Y ) ≤ I ( X , Y ) ≤ I ( X (5) The term on the right end of this chain of inequalities quantiﬁes the total amount of information in bits that barcodes Y . In the typical case of weakly dependent assets, this is proportional to the number n of assets. provide on the log-returns X For continuous variables p (u | v ) and p (u ) are probability density functions, see, e.g., [14]. Here simplest, in information theoretic terms implies the one requiring less bits for its description. For discrete variables, this corresponds to the variable V with the smallest entropy H [ V ] = − V p ( V ) log p ( V ). For continuous variables, it is necessary to resort to the relative entropy D K L ( p || p 0 ) = dV p ( V ) log[ p ( V )/ p 0 ( V )], where p 0 ( V ) is a baseline distribution. In the cases we shall discuss in the following, the notion of simplicity is rather intuitive, so we shall not discuss these details further. 2 3 M. Bardoscia et al. / C. R. Physique 20 (2019) 364–370 367 By contrast, the second term from the right is upper bounded by the entropy of the random variable X , which grows at most as log n. Hence, generally, ﬁnancial transformations imply information losses. The choice of the optimal barcode G F (Y ) can only mitigate further information losses, and at most saturate the leftmost inequality in Eq. (5). In the next section, we shall turn to the quantitative analysis of a representative case. 3. Barcoding ﬁnance in a Gaussian world Let us assume that X i = μ + ξ i + a ξ0 + J Y i (6) Y i = η i + c η0 (7) where μ > 0 is a positive constant, and ξi ’s and ηi ’s are i.i.d. Gaussian random variables with mean zero and variance one for i = 0, 1, . . . , n. This corresponds to a one-factor model, where the covariance E[( X i − μ)( X j − μ)] = a2 + J 2 c 2 between assets can partly be explained by the barcode variables (c = 0). Notice also that all assets are equivalent, i.e. the distribution , Y ) is invariant under permutations of the assets. of ( X The mutual information on individual assets is given by4 I ( Xi , Y i ) = 1 2 log 1 + J 2 1 + c2 (8) 1 + a2 is whereas the total information that barcodes provide on X , Y ) = I(X n−1 2 log(1 + J 2 ) + 1 2 log 1 + J 2 1 + nc 2 1 + na2 (9) Since X = nμ + n ξi + naξ0 + J Y , Y= i =1 n Yi (10) i =1 then the optimal barcode for any F ( X ) is G F (Y ) = Y . Indeed, I ( F ( X ), Y ) = I ( F ( X ), Y ), which saturates the leftmost inequality in Eq. (5). The upper bound on the information content of the barcode is given by I(X, Y ) = 1 2 log 1 + J 2 1 + nc 2 1 + na2 (11) We notice that: 1) the barcode’s information on the portfolio log-return X is larger than that on individual assets (i.e. I ( X , Y ) > I ( X i , Y i )) only if c > a, i.e. if barcodes are suﬃciently informative on the co-movement of assets; 2) I ( X , Y ) equals the second term in Eq. (9); therefore, the total loss of information is upper bounded by the ﬁrst term on the right hand side of Eq. (9), which increases linearly with n; , Y ) − I ( X , Y ) is independent of a and c, because all the available information on the 3) the total loss of information I ( X co-movement of stocks is captured by Y ; 4) when barcodes are not informative about the correlated variation of assets, i.e. for c = 0, the information content of the barcode Y vanishes, I ( X , Y ) → 0 as n → ∞. It is instructive to observe that, when c = 0, the barcode Y j provides also information on the return of asset i.5 The case when barcodes are independent (c = 0) may be appropriate for a portfolio of stocks where Y i accounts only for the fundamental analysis of stock i. In this case, very large portfolios become insensitive to information on the fundamentals of individual stocks (I ( X , Y ) → 0 as n → ∞). This is because X is dominated by the common component aξ0 on which barcodes Y i provide no information. The behaviour of I ( X , Y ) for the portfolio is summarized in the left panel of Fig. 1. The mutual information can be computed also for the ABS as follows. In a model of Gaussian log-returns and information, the threshold parameter k of a tranche F k can be related to the default probability 4 5 These results can be derived straightforwardly using textbook formulas (see, e.g., [14]). A trite calculation shows that I ( Xi , Y j ) = − 1 2 log 1 − J 2 c2 (1 + c 2 )[1 + a2 + J 2 (1 + c 2 )] (12) 368 M. Bardoscia et al. / C. R. Physique 20 (2019) 364–370 Fig. 1. Mutual information I ( X , Y ) and I ( F k , Y ) of barcodes for portfolio (left) and ABS (right) generated from Gaussian underlying assets ( J = 0.5), as a function of n. pkd = p ( X < nμ + k) = H z √ k V (X) where H ( z) = −∞ √dz e−z /2 is the cumulative normal distribution function. When information Y is revealed, this default 2π probability changes into 2 pkd (Y ) = p ( X ≤ nμ + k|Y ) = H k − JY √ V ( X |Y ) using this expression for pkd (Y ) and using Eq. (1), I (Fk, Y ) = E pkd (Y ) log pkd (Y ) pkd + (1 − pkd (Y )) log 1 − pkd (Y ) 1 − pkd (13) where the expectation is taken on the distribution of Y . In the right panel of Fig. 1, we plot the behaviour of the mutual information for the ABS. I ( F k , X ) follows the same qualitative behaviour of I ( X , Y ), although its value is considerably smaller (more than tenfold in the example of Fig. 1). In addition, I ( F k , Y ) decreases for safer and safer assets (i.e. as pd decreases), showing that most senior tranches of ABS tend to be remarkably information insensitive. 3.1. The cost of information: pricing barcodes Let us now address the issue of quantifying the value of the information conveyed by the barcodes. The key question we want to address is whether the demand for barcodes can endogenously arise in a market. This is possible if barcodes can be priced in such a way that the value of the barcode of a ﬁnancial instrument provides enough incentives for gathering information on the individual assets. We address this question within a mean-variance pricing scheme. Hence the setting we consider is that of a portfolio manager that gathers information on n assets, and sells n shares of the resulting portfolio, charging an additional amount related to the price of the information contained in the portfolio’s barcode. Investors exploit their available information about asset payoffs to price the ﬁnancial product Z = F Z ( X ). When they have access to the barcode, they use the probability p ( Z |Y ) to assess future performances of the assets, otherwise they use p ( Z ). In a mean variance framework, the price of X depends on the ﬁrst two moments of Z . When no barcode is provided, the mean variance price reads: p Z = E[ Z ] − α V [ Z ] (14) where α > 0 is the relative risk aversion coeﬃcient. A micro-foundation of the previous formula is discussed in Appendix A. As shown there, conditionally on knowing the barcode’s value y, the price is: p Z |Y = y = E[ Z |Y = y ] − α V [ Z |Y = y ] (15) Depending on the realized information y, the price difference between having or not the barcode can be positive or negative. The price of the barcode should be computed before the realized value of Y is known, therefore it is given by M. Bardoscia et al. / C. R. Physique 20 (2019) 364–370 δ p Z := E[ p Z |Y ] − p Z = α {V ( Z ) − E [V ( Z |Y )]} = α V (E[ Z |Y ]) 369 (16) where V ( Z |Y ) is the variance of Z on the distribution p ( Z |Y ) and E[ Z |Y ] is the expected value of Z conditional on the value Y . Eq. (16) takes the expected value over Y of V ( Z |Y ) and the variance of E[ Z |Y ] over the distribution of Y . This result reﬂects the fact that the knowledge of the distribution of Y does not change the unconditional expected log-return of the asset, but produces a reduction in variance, which is equal to the variance of the conditional expected return. To assess the presence of incentives for barcodes, we shall compare expected revenues from the barcode with the cost of gathering information, which is given by the cost of the barcodes of the original assets. When log-returns are given by Eq. (6), the cost of gathering information for a single asset is δ p X i = α J 2 (1 + c 2 ) The additional log-return that the optimal barcode Y = (17) i ηi + nc η0 yields is δ p X = α J 2 (1/n + c 2 ) (18) Considering this as the price of the barcode that the portfolio manager can charge when selling X , together with the barcode, we ﬁnd that the budget’s balance for the portfolio manager for selling n shares of X is nδ p X − n δ p X i = −α (n − 1) J 2 (19) i =1 which is negative. In other words, this pricing mechanism does not provide incentives to gather information on individual assets. Interestingly, when barcodes provide information on the co-movement of the assets (c > 0), the value of information on the whole portfolio, instead, is larger than the sum of the cost of information on individual assets, i.e. δpX − n δ p X i = αn(n − 1) J 2 c 2 (20) i =1 This is a consequence of the non-linearity of the pricing mechanism and of the fact that, for c > 0, the barcode Y i of asset i provides information also on other asset log-returns X j . Indeed, there is a minimal share size, above which the barcode associated with the log-return X /m provides enough incentives to gather information on individual assets in the sense that mδ p X /m − nδ p X i ≥ 0. A simple calculation shows that m≤ 1 + J 2 (1 + nc 2 ) (21) 1 + J 2 (1 + c 2 ) The same calculation can be extended to ABS in a straightforward manner, yet the results depend on the way in which the portfolio X is divided into tranches X= fk Fk( X ) (22) k for some positive constants f k . Then, we show in Appendix B that δpX > k δ p fk Fk ( X ) (23) This, together with the fact that, for c = 0, Eq. (20) implies that δ p X = i δ p X i , shows that if barcodes do not provide information on the correlated defaults, securitisation cannot provide incentives to gather information on individual assets (within the present mean variance framework). This suggests that, unless information on the correlated defaults of individual assets is accounted for, securitisation decreases the value of information contained in barcodes. 4. Conclusion In this paper, we exploit information theoretic concepts to investigate the lack of transparency associated with ﬁnancial transformations. We discuss a setting where side information about the returns of assets is modelled with an associated random variable, and the information content is quantiﬁed using the mutual information. In this setting, we show that every ﬁnancial transformation implies information losses. In a model of Gaussian log-returns, we ﬁnd that when fundamental analysis on individual assets is not informative on the co-movement of assets, the information is totally lost in the limit of very large portfolios. In addition, we show that, within a mean variance framework, the value of information also decreases, which suggests that incentives to gather information on individual assets cannot be transmitted across ﬁnancial transformation. This puts serious doubts on whether market incentives alone are enough to make the introduction of a system of barcodes, as advocated in Ref. [9], sustainable. 370 M. Bardoscia et al. / C. R. Physique 20 (2019) 364–370 These result generalise to a model of assets with binary returns, which is more appropriate for credit derivatives (see Ref. [13]). The aim of the present paper is that of suggesting ways forward to quantify transparency losses in ﬁnance and to raise a few key issues. As such, it might be a benchmark for more complex and realistic theoretical models, or for more appropriate schemes to value information in order to overcome these issues. Appendix A. Mean variance pricing In order to do this, we adopt a standard mean-variance framework.6 Consider a representative agent with an initial wealth W , that is facing the decision of buying W units of wealth of an asset with return Z . If her utility function is given by U (·), the certainty equivalent w of this investment is deﬁned as that value for which the investor is indifferent between investing in the asset or receiving w units of wealth, i.e. U ( W + w ) = E [U ( W + W Z )] (24) where E[. . .] stands for the expectation on the random variable Z . We take w as a measure of the value of the investment that incorporates the risk premium. Assuming that 1 and w W , we can expand both sides and derive, to leading order, the price of Z as the value per unit of investment. pZ ≡ w ≃ E[ Z ] − α V ( Z ), αW α = − U (W )W 2U ( W ) (25) If we further assume investors with constant relative risk aversion (CRRA), then α is a constant, that we assume can be estimated from market data. Appendix B. Pricing ABS From Eq. (22) E[ X |Y ] = f k E[ F k ( X )|Y ] (26) k We assume that X ≥ 0 and that f k > 0. 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