Appendix
A.1 Credit risk migration model
To measure the credit-standing migration risk to which each credit risk class is exposed, we propose a methodology with which to calculate perpetual annuities based on a theoretical through-the-cycle transition matrix to show the feasibility of the Debt Agency framework proposed. Given a point-in-time transition matrix \(TM_{t}\) at time t, its generic element \(a_{ji}\) represents the annual probability that an obligor of the credit risk class j in year t will pass to a credit risk class i in the following year. The matrix has dimension \(n\times n\) and the elements of row j, \(a_{j1},\ldots ,a_{jn}\) must sum to unity, since every obligor with rating j will certainly be assigned to some credit risk class \(z\in \) \(\left\{ 1,\ldots ,j,\ldots n\right\} \) from year t to \(t+1\); including the case of being reassigned to the same class j. As a convention, the rows and the columns of \(TM_{t}\) are ordered according to safety class, from the safest (conventionally labeled AAA) to the default (label D: default state). Following the standard diagonalization method for square matrices, we assume that the \(TM_{t}\) matrix can be decomposed in a Q matrix and a \(L_{t}\) diagonal matrix so that:
$$\begin{aligned} TM_{t}=QL_{t}Q^{-1}. \end{aligned}$$
(3)
The \(L_{t}\) matrix depends on t and shows correlations with the business cycleFootnote 14 in its elements \(l_{j}(t)\). In particular we assume that these values depend on shocks of the business cycle according to the following generalized logistic function:
$$\begin{aligned} l_{j}(y)=\frac{\theta _{j1}}{(1+\theta _{j2}\exp (\theta _{j3}y))} \end{aligned}$$
(4)
with \(y\ \)a single factor stochastic process and parameters \(\theta _{j,1,2,3}\) calibrated so that the expected value of the \(l_{j}\) is \(E(l_{j})=\lambda _{j}\), with \(\lambda _{j}\) being the element j of the eigenvalues diagonal matrix \(\Lambda \) in the decomposition:
$$\begin{aligned} TTC=Q\Lambda Q^{-1}. \end{aligned}$$
(5)
Since the decomposition is unique unless linear transformations, then Q represents the eigenvectors matrix of the above linear functional.
Proposition 3
Given the filtered probability space \(({\varvec{\Omega }},{\varvec{\Sigma }},{\varvec{\digamma }} _{t},{\mathbf {P}})\), the matrix TTC, interpreted as a through-the-cycle matrix,Footnote 15is the expectation of the stochastic process \(\left\{ TM_{t}\right\} _{t\ge 0}\) adapted to the natural filtration \({\varvec{\digamma }}_{t}\) generated by y.
Proof
Take the expectation of \(TM_{t}\) and substitute \(E(l_{j})\) with \(\lambda _{j} \):
$$\begin{aligned} TTC&=E(TM_{t})\\ &=E(QL_{t}Q^{-1})\\&=QE(L_{t})Q^{-1}\\ &=Q\Lambda Q^{-1} \end{aligned}$$
\(\square \)
Following this model, the expected cumulative default probability in the interval \(\left[ t,t+m\right] \) is the linear operator given by:
$$\begin{aligned} E\left[ {{\mathbf{cdp}}}(t+m)-{\mathbf{cdp}}(t)\right] =Q(\Lambda ^{t+m} -\Lambda ^{t})Q^{-1}{\mathbf {v}} \end{aligned}$$
(6)
where \({{\mathbf{cdp}}}(t)\) is an n-components stochastic process, the j-th element of which, \(cdp_{{\mathbf {j}}}(t)\), represents the cumulative default probability that an obligor of rating grade class \(j=1,\ldots ,n\) will have defaulted by time t, with \({{\mathbf{cdp}}}(0)=0\) and \({\mathbf {v}}\) a null vector apart its last element equal to 1.
Proposition 4
The process \({{\mathbf{cdp}}}(t)\) can be seen as a stochastic vector depending from the process y. Since the matrix \(L_{t}\) depends deterministically from y, this guarantees that \({{\mathbf{cdp}}}(t)\) is measurable given the filtration \({\varvec{\digamma }}_{t}\) generated by y.Footnote 16
A.2 Perpetual annuities and fundamental pricing
Since our primary goal is to price the fundamental risk of each obligor credit risk class j, henceforth we will use the matrix TTC as the reference risk metric to calculate obligors’ default probabilities under expectations. We assume that the valuation date occurs at time \(t=0\) unless otherwise specified, which simplifies our notation, but all the following mathematical expressions can be easily rephrased in order to include index \(t_{0}\) to fix any time reference in the interval \(\left[ 0,t\right] \).
Under the condition (6), the survival probability vector in the interval \(\tau \in [0,t]\) can be written as:
$$\begin{aligned} {\mathbf{sp}}(0,t)=E\left[ {\varvec{1}}-{{\mathbf{cdp}}}(t)\right] ={\varvec{1}}-Q\Lambda ^{t}Q^{-1}{\mathbf {v}} \end{aligned}$$
(7)
where \({\varvec{1}}\) is the unit vector. The n-th element of vector \({\mathbf{sp}}\left( 0,t\right) \) corresponding to the default state is null.
The expected present value of a vector of unitary annuity maturing at time t can be written as:
$$\begin{aligned} {\mathbf {a}}(0,t)=\sum _{\tau =0,1,\ldots t}\frac{1}{(1+\pi )^{\tau }}{\mathbf{sp}} (0,\tau ) \end{aligned}$$
where \(\pi \) represents a common appropriate financial discount rate.Footnote 17 Note that the components of vector \({\mathbf {a}}(0,t)\) are ordered decreasingly, with the highest rating grades corresponding to higher annuity values since the present value of a unitary annuity is proportional to the survival probability of the corresponding credit risk class and a null value for the vector’s last component. Using the expression of \({\mathbf{sp}}(t)\) in Eq. (7), we can rewrite:
$$\begin{aligned} {\mathbf {a}}(0,t)=\sum _{\tau =0,1,\ldots t}\left[ \frac{1}{(1+\pi )^{\tau }}\left( {\varvec{1}}-Q\Lambda ^{\tau }Q^{-1}{\mathbf {v}}\right) \right] . \end{aligned}$$
Now, by letting \(\alpha =1/(1+\pi )\) and \(\beta _{j}=\) \(\lambda _{j}/(1+\pi )\), with \(\lambda _{j}\) the j-th eigenvalue in matrix \(\Lambda \), the above expression can be written:
$$\begin{aligned} {\mathbf {a}}(0,t)=\alpha \frac{1-\alpha ^{t}}{1-\alpha }{\varvec{1}} -QBQ^{-1}{\mathbf {v}} \end{aligned}$$
where B is a diagonal matrix whose j-th element is \(b_{j}=\) \(\beta _{j}\frac{1-\beta _{j}^{t}}{1-\beta _{j}}\). Since the terms in \(\Lambda \) are \(\lambda _{j}\le 1\), it follows \(\alpha ,\beta _{j}\in (0,1)\). By taking the limit for \(t\rightarrow \infty \) we obtain the following perpetual annuity formula:
$$\begin{aligned} {\mathbf {a}}(0)=\lim _{t\rightarrow \infty }{\mathbf {a}}(0,t) =\frac{\alpha }{1-\alpha }{\mathbf {1}}-QB^{\prime }Q^{-1}{\mathbf {v}} \end{aligned}$$
(8)
with \(B^{\prime }\) a diagonal matrix with j-th element \(b_{j}^{\prime } =\frac{\beta _{j}}{1-\beta _{j}}\). The vector \({\mathbf {a}}(0)\) in the Eq. (8) represents expected present values at \(t=0\) of an irredeemable mortgage annuity paid by each obligor according to its rating grade.
In order to consider the possibility to recover part of the credit if an obligor defaults, we should adjust the value of \({\mathbf {a}}(0)\) accordingly. To this end, Eq. (8) should be modified to take this effect into account. Introducing the loss-given-default (LGD), \((1-rr)\), and letting rr be the recovery rate,Footnote 18 the vector of expected values of the recovery rate by credit risk class for \(t\rightarrow \infty \) isFootnote 19:
$$\begin{aligned} {\mathbf {r}}(0)=rrQ\left[ (I-\Lambda ^{-1})B^{\prime }\right] Q^{-1}{\mathbf {v}}. \end{aligned}$$
(9)
Following a unitary-payment perpetual amortizing scheme and allowing for partial recovery of funds in case of default, the present value of an expected positive exposure \({\tilde{a}}_{j}\) must always satisfy the equivalence \({\tilde{a}}_{j}(1-r_{j})=a_{j}\), where \(r_{j}<1\) is j-th element of the vector \({\mathbf {r}}_{0}\). Bearing this in mind, the final expectation of a unitary perpetual annuity value at time \(t=0\) calculated for each obligor according to its rating grade j is then:
$$\begin{aligned} {\tilde{a}}_{j}=\frac{a_{j}}{1-r_{j}}. \end{aligned}$$
(10)
The vector \({\tilde{\mathbf {a}}}(0)\), whose elements are the values \(\tilde{a}_{j}\), can be interpreted as a set of perpetual annuities based on fundamental risk metrics inherent to obligors labelled with specific credit risk class.
A.3 Portfolio pricing
Equation (10) allows to establish conditions for credit portfolio pricing and Debt Agency financial equilibrium, once the functional form for the portfolio default probability \(cdp_{_{W}}(t)\) for \(t\ge 0\) has been specified.
Proposition 5
Given the initial portfolio asset allocation \({\mathbf {w}}_{0}^{T}=\left[ w_{1},\ldots ,w_{j},\ldots w_{n}\right] \), with \(w_{j}=d_{j}/TD\), the Debt Agency will price the risks for \(t\ge 0\) using the portfolio expected default probabilities \(E(cdp_{_{W}}(t))={\mathbf {w}}_{0}^{T}Q\Lambda ^{t}Q^{-1}v\) and setting the overall annual payments \(I_{W}(t)\) at time \(t=0\) so that its financial equilibrium holds in expectations:
$$\begin{aligned} TD=I_{W}(0){\tilde{a}}_{W}(0)=I_{W}(0){\mathbf {w}}_{0}^{T}{\tilde{\mathbf {a}} }(0)=I_{W}(0)\sum _{j=1,\ldots ,n}w_{j}{\tilde{a}}_{j}. \end{aligned}$$
(11)
Proof
In analogy with formula (10), let be \({\tilde{a}}_{W}(0)\) such that \({\tilde{a}}_{W}(0)(1-r_{W}(0))=\sum _{j=1,\ldots ,n} w_{j}a_{j}\), with \(r_{W}(0)=\sum _{j=1,\ldots ,n}z_{j}r_{j}\) and \(\sum _{j=1,\ldots ,n}z_{j}=1\), we have:
$$\begin{aligned} {\tilde{a}}_{W}(0)\left( \sum _{j=1,\ldots ,n}z_{j}-\sum _{j=1,\ldots ,n}z_{j}r_{j}\right)&=\sum _{j=1,\ldots ,n}w_{j}a_{j}\\ \sum _{j=1,\ldots ,n}z_{j}{\tilde{a}}_{W}(0)(1-r_{j})&=\sum _{j=1,\ldots ,n}w_{j}a_{j}. \end{aligned}$$
A non trivial solution of this equation is \(z_{j}{\tilde{a}}_{W}(0)=w_{j} \frac{a_{j}}{(1-r_{j})}=w_{j}{\tilde{a}}_{j}\), for \(j=1,\ldots ,n\). Summing up for j, we obtain \({\tilde{a}}_{W}(0)\sum _{j=1,\ldots ,n}z_{j}=\sum _{j=1,\ldots ,n} w_{j}{\tilde{a}}_{j}\), i.e. \({\tilde{a}}_{W}(0)=\sum _{j=1,\ldots ,n}w_{j}{\tilde{a}} _{j}\). It also follows that \(z_{j}\) is a linear combination such that:
$$\begin{aligned} z_{j}=\frac{w_{j}{\tilde{a}}_{j}}{\sum _{j=1,\ldots ,n}w_{j}{\tilde{a}}_{j}}. \end{aligned}$$
\(\square \)
However, the solution \(I_{W}(0)\) is not the only one solving the Eq. (11). In fact, letting the vector \({\mathbf {c}}_{t}^{T}=\left[ c_{1},c_{2},\ldots ,c_{j},\ldots c_{n}\right] \) represent the overall payments due at time \(t=0\) by each credit risk class j, the financial equilibrium requires that the sum of their expected value, \({\mathbf {c}}^{T}{\mathbf {a}}(0)\), and the overall expected recovery value, \({\mathbf {d}}^{T}{\mathbf {r}}(0)\), be equal to the total credit portfolio holding TD:
$$\begin{aligned} TD=\sum _{j=1,\ldots ,n}d_{j}={\mathbf {c}}^{T}{\mathbf {a}}(0)+{\mathbf {d}}^{T} {\mathbf {r}}(0)=\sum _{j=1,\ldots ,n}c_{j}a_{j}+{\mathbf {d}}^{T}{\mathbf {r}}(0). \end{aligned}$$
(12)
The solution value for \(c_{j}\) (12) can be found for infinite arbitrary combinations of the other \(c_{i}\), \(i\ne j\). General financial criteria of course apply to determine “admissible” values to the \(c_{j}.\) A special solution, that we call here idiomatic fundamental pricing solution, consists in relating the payments of the j-th class to the corresponding debt level and riskiness:
$$\begin{aligned} c_{fj}=\frac{d_{j}(1-r_{j})}{a_{j}}=\frac{d_{j}}{{\tilde{a}}_{j}}. \end{aligned}$$
(13)
The specific feature of this solution is that each obligor pays for the risk inherent to the specific credit risk class to which it is assigned, without any form of solidarity or mutuality among obligors of different classes. Nevertheless, the solution (13) does not price in line with portfolio expected default probabilities, giving rise to the following straightforward proposition.
Proposition 6
Be \(I_{F}(0)=\sum _{j=1,\ldots ,n}c_{fj}=\sum _{j=1,\ldots ,n}\frac{d_{j}}{{\tilde{a}} _{j}}\), if we price the portfolio \({\mathbf {d}}^{T}\) using portfolio expected default probabilities, \(E(cdp_{W}(t))={\mathbf {w}}_{0}^{T}Q\Lambda ^{t} Q^{-1}{\mathbf {v}}\) for \(t>0\), the total payments at \(t=0\) are such that \(I_{F}(0)\ge I_{W}(0)\).Footnote 20
Proposition 7
If for \(t=0\) the Debt Agency computes the fair value of the portfolio pv(0) using portfolio default probabilities but charges obligors individually by using the idiomatic fundamental pricing formula (13), so that \(pv(0)=I_{F}(0){\tilde{a}}_{W}(0)\), then:
-
the portfolio fair value will be greater than the TD, thus generating for the agency a positive economic value of equity \(eve(0)=pv(0)-TD\)
$$\begin{aligned} pv(0)\ge TD \end{aligned}$$
-
the economic value of equity can be remunerated at a positive interest rate:
$$\begin{aligned} fc\le \left[ (I_{F}(0)-I_{W}(0))/eve_{0}\right] . \end{aligned}$$
Proof
From Eq. (11) it follows that
$$\begin{aligned} pv(0)=I_{F}(0){\mathbf {w}}_{0}^{T}{\tilde{\mathbf {a}}}_{{\mathbf {0}}}\ge I_{W}(0){\mathbf {w}}_{0}^{T}{\tilde{\mathbf {a}}}_{0}=TD. \end{aligned}$$
\(\square \)
Remark 8
The total payment \(I_{W}(0)\) obtained under the equilibrium condition (11) is structurally lower than the amount due in the idiomatic fundamental pricing configuration. This is attributable to a “pooling effect”, i.e., in our case, to the fact that, within a portfolio approach, the transition probabilities among credit risk classes entail a risk mitigation benefit, since they imply, in each year of observation, potential improvements for the worst credit risk classes.
The Proposition 5 is relevant because it shows that the Debt Agency generates value for potential investors, collecting additional risk capital that can be used for solvency purposes. Also relevant is the question of overall cost distribution among obligors. Here we only want to emphasise that the overall cost of the portfolio obviously depends on the cost distribution rule that we adopt to charge each risk class.
Proposition 9
Under the pricing rule defined by (11), if we charge each credit risk class j in proportion to its debt, i.e. using the weight \(wj=dj/TD\), then we equalize the price of risk uniformly, thus ending up mutualizing part of the debts among classes.
Proof
Under (11) set the portfolio unit cost of the risk equal \(ucr_{_{W}}=I_{W}(0)/TD\). Now consider the idiomatic fundamental risk pricing rule under (13) and set the unit cost of the risk for the credit risk class j equal \(ucr_{j}=\) \(c_{fj}/d_{j}\). For an obligor i of credit risk class j and nominal debt \(\delta _{ji}\), if \(ucr_{j} >ucr_{_{W}}\) then its equivalent nominal debt \(\delta _{ji}^{\prime }\) is
$$\begin{aligned} \delta _{ji}^{\prime }<\frac{ucr_{_{W}}}{ucr_{j}}\delta _{ji}. \end{aligned}$$
\(\square \)
From Proposition 5, the excess cost of \(\left( I_{F}(0)-I_{W}(0)\right) \) \(\ge 0\) could possibly be allocated among credit risk classes without compromising the agency’s financial equilibrium.
A.4 Intertemporal equilibrium
Thus far, we have supposed that the Debt Agency (1) determines its periodic cash flow according to an irredeemable amortization scheme; (2) prices risks using the metrics (5); and that (3) no extra provisions or capital charges for unexpected losses or other risks are needed (in the Appendix on solvency capital we will challenge and supersede this assumption). Since at time \(t\ge 0\) the proposition (4) always holds true, we illustrate the financial equilibrium of the Debt Agency by using the pricing formula given by the Eq. (11) and we leverage on a fundamental characterization of our agency institutional framework, which will be fully explained in the Appendix on solvency capital. Given the agency asset allocation, \({\mathbf {w}}_{t}^{T}=\left[ w_{1},\ldots ,w_{j},\ldots w_{n}\right] \) , and the portfolio intertemporal default probability \(cdp_{_{W}}(t)\) up to time t, we characterize the proportion of defaults at portfolio level, \(w_{n}\), as represented by the following process:
$$\begin{aligned} w_{n}=E[cdp_{_{W}}(t)]={\mathbf {w}}_{t-1}^{T}Q\Lambda Q^{-1}{\mathbf {v}} \end{aligned}$$
(14)
with \(\sum _{j=1,\ldots n}w_{j}=1\) and \({\mathbf {v}}\) a null vector apart from its last element equal to 1.
Remark 10
The formula (14) states that, although the asset allocation can evolve considering erratic rating grade migration and that the non-default classes can diverge considerably from their expected value, on the contrary, the default class always must evolve according to its expected value. In the Appendix on solvency capital, we will show that this setting is consistent with the particular meaning of default that works effectively for Member States in the institutional context of the Eurozone. This device allows to cumulate the share of debtors that have “theoretically” defaulted at every time t, which is necessary to correctly calculate the agency intertemporal equilibrium. Note that future portfolio default probabilities will be the result of actual rating grade migrations.
In developing the agency intertemporal equilibrium, we resort to the following statements:
-
1.
the appropriate average loss-given-default (LGD) rate is \((1-rr_{_{W}} )\), with \(rr_{_{W}}\) representing percentage of nominal debt recovered in the case of default (i.e. the portfolio recovery rate)
-
2.
the agency’s annual discount rate used to compute present values is \(\pi _{t}\)
-
3.
the agency’s reserves deposit is remunerated at an annualized interest rate \(\pi _{t}^{\prime }\) set by the Central Bank, which is conceived as the long term equilibrium rate of its monetary policy
-
4.
the agency’s liability equals \(TD=\sum _{j}d_{j}\) and it is rolled over for an infinite span of time, issuing and renewing at pair indexed bonds of unitary maturity (e.g. 1 year) of overall face value equal TD
-
5.
the agency’s funding cost \(FC=\pi _{t}^{\prime \prime }TD\) is determined using an annual interest rate of \(\pi _{t}^{\prime \prime }\).
If we suppose that the Debt Agency can reprice obligors’ funding costs by using the Eq. (11), then its net exposure at time t is subject to the following constraint, which has to be solved for \(I_{W}(t)\):
$$\begin{aligned} TD-rs(t)=pv_{W}(t)=I_{W}(t){\tilde{a}}_{W}(t)=I_{W}(t){\mathbf {w}}_{t} ^{T}{\tilde{\mathbf {a}}}_{t} \end{aligned}$$
(15)
where \(pv_{W}(t)\) is the present value of future cash flow and rs(t) is the total cumulated reserve deposit at the Central Bank, given by:
$$\begin{aligned} rs(t)&=rs(t-1)(1+\pi ^{\prime })+I_{W}(t-1)[1-(cdp_{_{W}}(t)-cdp_{_{W} }(t-1))]+ \\ &\quad +rr_{_{W}}TD(cdp_{_{W}}(t)-cdp_{_{W}}(t-1))-FC \end{aligned}$$
(16)
with \(rs(0)=0.\) Given the portfolio share of defaulted obligors \((cdp_{_{W} }(t)-cdp_{_{W}}(t-1))\) in the interval \([t-1,t]\) , \(rr_{_{W}}TD(cdp_{_{W} }(t)-cdp_{_{W}}(t-1))\) represents the cash inflow due to recovery from defaulted obligors. Note that rs(t) is a stochastic realization of the process \(cdp_{W}(\tau )\) for \(\tau \) \(\le t\), which is known at time t. This equilibrium implies that in the case of adverse risk migrations in the interval \([t-1;t]\), the surviving obligors will bear a greater cost for \(I_{W}(t)\) in order to assure the agency equilibrium over time, according to (15).
Considering the three mentioned rates \(\pi _{t},\pi _{t}^{\prime }\ \)and \(\pi _{t}^{\prime \prime },\) to be noted is that:
-
1.
they are expectations conditional on information available at time t;
-
2.
by institutional design, the agency will fix the \(\pi _{t}\) rate equal to the Central Bank’s long-term rate \(\pi _{t}^{\prime }\);
-
3.
by design, also the agency issuances are indexed to \(\pi _{t}^{\prime }\);
-
4.
the rate \(\pi _{t}^{\prime \prime }\) will always be such that \(\pi _{t}^{\prime \prime }\le \) \(\pi _{t}^{\prime }\), since we make the assumption that the Central Bank can always buy the residual issuance in order to ensure the alignment of the yield to \(\pi _{t}^{\prime }\) (see the institutional role to respect to the Debt Agency assigned to the ECB in Sect. 3).
Proposition 11
For \(t\ge 0\), if \(\pi _{t}=\pi _{t}^{\prime }=\pi _{t}^{\prime \prime }\) and the agency reprices the obligors cost at every t using Eq. (11), then \(TD=pv_{W}(t)+rs(t)\): the agency balance sheet asset-side always equates the liability-side, and the agency equilibrium is assured over time.
Proof
The proposition follows directly from Eq. (15), which states that for the portfolio fair value at time t we have \(pv_{W} (t)=I_{W}(t){\tilde{a}}_{W}(t)\) , then:
$$\begin{aligned} pv_{_{W}}(t)+rs(t)=TD \end{aligned}$$
with \(pv_{W}(0)=TD.\) \(\square \)
Remark 12
Note that, for \(t\longrightarrow \infty \) , since \(pv_{_{W}}(t)\longrightarrow 0\) then \(rs(t)\longrightarrow TD\) and the agency will have piled up enough reserves to repay the nominal value of its bonds.
Proposition 13
For \(t\ge 1\), if \(\pi _{t}=\pi _{t}^{\prime }=\pi _{t}^{\prime \prime }\) then returns from the Debt Agency’s asset side are expected to remunerate the liability side, i.e. \((pv_{W}(t)+rs(t))(1+\pi )=TD+FC.\)Footnote 21
We have shown that, since the Debt Agency is able to align:
-
1.
interest rates used to compute revenues and present values
-
2.
returns on reserves in form of deposits at the Central Bank
-
3.
the cost of funding,
then, by the Propositions 6 and 7 the Debt Agency will always be able, in expectation, to back liabilities with the fair value of its assets. This entails that, in an “arm’s length transaction”, as prescribed by standards such as Solvency II and Basel III, the agency will always be able to repay its overall debt of TD whenever requested.
A.5 Solvency capital
So far, we have considered that the Debt Agency prices its own risks using the expected default probabilities term-structure with infinite granularity of obligors in each credit risk class of the underlying portfolio.
Within a “closed portfolio” irredeemable framework, default outcomes deal only with the “when” of their occurrence, since the portfolio cumulative default probability over an infinite time span always equals 1, given that the TTC matrix is recursive and has one absorbing state, coinciding with the default state. As a consequence, when the default-term-structure evolves, the pricing operated by the Debt Agency will allow it to accumulate enough reserves or adjust pricing to maintain its financial equilibrium according to Eq. (15).
What if, however, the events of default anticipate and are much more concentrated? Such an eventuality would violate the assumed hypothesis of infinite granularity of obligors, thus causing the Debt Agency to remain with insufficient accumulated reserves and with a lack of revenues to cover its future expected liabilities.
First of all, we should speculate what would be a default of a Member State once the Debt Agency has been established. Should a Member State incur a state of insolvency, it is nonetheless likely that, under a suitable “ex-ante budget control regime” necessary to assure the agency‘s ongoing correct operational course, the failing Member State will soon or later be able to restore its ability to pay its future instalments. This means that the Member State would undergo a period of restructuring before full recovery and that, in reality, the default of a Eurozone Member State should be considered more properly a state of “forbearance”. Moreover, to be noted is that, during this finite period, the distressed asset corresponding to Member State‘s debt does not need to be written off but will remain “frozen” on the Debt Agency balance sheet. Consequently, because of the reduced ability of the Member State to pay its overdue instalments, the Debt Agency will not be able to maintain its equilibrium and to finance its funding costs.
However, if adequately complemented with a suitable insurance scheme, this “restructuring nature” of a failing Member State allows us to maintain some essential features of our model, even outside our hypothesis of infinite granularity of obligors. We are then entitled to transform the initial hypothesis into an equivalent one, according to which the debt of each Member State is assumed to be infinitely divisible into infinitesimal parts, supposed mutually independent only for mathematical convenience. This methodological assumption enables us to proceed in our calculations as if only a portion of a Member State s debt was recorded as a loss, according to the expected default probability of the rating grade class of the Member State in the interval \((t;t+1)\). As a consequence, the portfolio default probability is also assumed to follow the process under (14).
Thanks to this “fiction”, the cost of debt would be re-estimated in each period only on the basis of the following risk factors:
-
1.
the migration risk, when the agency asset allocation by credit risk classes is changed due to effective Member State risk up-grade or down-grade, and
-
2.
changes in the expected monetary policy rates of the Central Bank.
On this line of reasoning, the repricing mechanism of the Eq. (15) will ensure that the pricing applied by the agency allows the maintenance of its financial equilibrium. Moreover, and in order to fully implement that setting, we need:
-
1.
to adjust the annual pricing to take account of the theoretical loss as mentioned,
-
2.
to allow for an insurance scheme designed to provide financial support in the form of a capital equivalent to the present values of annual payments lost during the forbearance period of a Member State.
As regards the first point, our fiction consists in imagining that a single Member State i with initial debt equal \(D_{ij}\), representing a significant fraction of a specific credit risk class j, will pay only an amount determined using the partition rule under (13) adjusted in proportion to the theoretical default for each given interval \([t,t+1]\) as follows:
$$\begin{aligned} {\tilde{c}}_{ij}(t)=\left[ {\tilde{c}}_{ij}(t-1)-rec_{ij}(t-1)\right] (1-E\left[ cdp_{j}(t)-cdp_{j}(t-1)\right] )+rec_{ij}(t) \end{aligned}$$
where \({\tilde{c}}_{ij}(0)=\frac{D_{ij} }{{\tilde{a}}_{j}}\) according to formula (13), \(\tilde{a}_{j}\) and \(cdp_{j}(t)\) are credit risk class characteristic quantities the meaning of which we have widely discussed in the previous sections, and \(rec_{ij}(t)\) the recovery proportion such that:
$$\begin{aligned} rec_{ij}(t)=rr_{_{W}}D_{ij}E\left[ cdp_{j}(t)-cdp_{j}(t-1)\right] \end{aligned}$$
with \(rr_{_{W}}\) the theoretical portfolio recovery rate. Note that \(\lim _{t\rightarrow \infty }\) \({\tilde{c}}_{ij}(t)=0\), in line with the amortizing nature of the plan developed within this framework.
As for the second point, the Debt Agency will only need to relay upon an available capital endowment to cover the temporary shortage of cash inflow due by an Member State in state of forbearance. This capital allowance, which takes here the form of an insurance, should then be proportional to the total annual cash flows at risk for a period of time, fi, that we call forbearance interval. The present value at time t of future payments due during the forbearance in the interval \([t,t+fi]\) is:
$$\begin{aligned} fp_{ij}(t)= {\textstyle \sum \limits _{h=t}^{t+fi}} \frac{{\tilde{c}}_{ij}(h)}{(1+\pi )^{h-t}}. \end{aligned}$$
The periodic premium that should be paid is then given by:
$$\begin{aligned} prem_{ij}(t)=\frac{ {\textstyle \sum \limits _{\tau =t}^{+\infty }} \left( \frac{fp_{ij}(\tau )}{(1+\pi )^{t}}\right) (cdp_{stress,j} (\tau +1)-cdp_{stress,j}(\tau ))-rm_{i}(t)}{a_{j}} \end{aligned}$$
(17)
where \(a_{j}\) is the j-th element of the vector \({\mathbf {a}}(t)\), \(\pi \) is a suitable discounting rate, and the \(cdp_{_{stress,j}}\) represents a stressed default probability obtained through the Eq. (4) by using a suitable confidence interval such that:
$$\begin{aligned} P_{cdp_{w}}(x\le cdp_{stress})=\alpha \end{aligned}$$
with \(\alpha \) a prudential probability threshold. The \(rm_{i}(t)\) represents the cumulated mathematical reserve at time t given by:
$$\begin{aligned} rm_{i}(t)=rm_{i}(t-1)(1+\pi )+prem_{ij}(t)-{\varvec{1}}_{[fi]} (t)fp_{ij}(t) \end{aligned}$$
where \({\varvec{1}}_{[fi]}(t)\) is equal to one when t is the first year of a forbearance period. Note that this reserve is different from the reserve deposit rs(t). To include the insurance premium, a suitable mark up should be considered by adjusting overall payments accordingly. The final cost for the Member State i of rating grade j is then equal to:
$$\begin{aligned} {\hat{c}}_{ij}(t)={\tilde{c}}_{ij}(t)+prem_{ij}(t). \end{aligned}$$
If we define \({\hat{I}}_{F}(t)=\sum _{j=1,\ldots ,n}[{\tilde{c}}_{ij}(t)+prem_{ij}(t)]\) then the following inequalities hold:
$$\begin{aligned} {\hat{I}}_{F}(t)\ge I_{F}(t)\ge I_{W}(i). \end{aligned}$$
This shows that the economic equilibrium of the Debt Agency is assured.
Remark 14
Following Proposition 5, we have shown that under the pricing configuration (13) the agency has the potential to generate a positive economic value of equity, which in our numerical elaborations turns out to be of the same order of magnitude as the present value of future insurance premiums, partly mitigating the costs required to finance the aforementioned insurance scheme.
Another important source of risk for the agency is interest rate volatility. We argue that the agency architecture hereby proposed can be thought of as a shield to protect Member States against “liquidity spread risk”, thanks to the link with the Central Bank and the insurance scheme to cope with unexpected sovereign default risk. Nonetheless, interest rate asymmetric movements can cause repricing risk. We supposed that the agency will roll over its debt by issuing bonds indexed to prevailing Central Bank fund rates, but there is always the possibility that alignment will fail to be effective. In this case the agency will always have the ability to reprice the total instalment needed to restore its financial equilibrium, by applying the equilibrium formula under Eq. (15).
A.6 Rebalancing of the instalments
Proof
Define the Lagrangian:
$$\begin{aligned} L(w_{1}^{\prime },\ldots ,w_{j}^{\prime },\ldots w_{n}^{\prime };\lambda )=\sum _{j=1,\ldots ,n}I_{W}(t)^{2}{\tilde{a}}_{j}^{2}\left( w_{j}^{\prime }-\frac{c_{f,j} }{I_{W}(0)}\right) ^{2}-\lambda \left( \sum _{j=1,\ldots ,n}w_{j}^{\prime }-1\right) . \end{aligned}$$
First order conditions give:
$$\begin{aligned} \frac{\partial L}{\partial w_{j}^{\prime }}&=2I_{W}(t)^{2}{\tilde{a}}_{j} ^{2}(w_{j}^{\prime }-\frac{c_{f,j}}{I_{W}(t)})-\lambda =0\\&=w_{j}^{\prime }-\frac{c_{f,j}}{I_{W}(t)}-\frac{\lambda }{2I_{W}(t)^{2} {\tilde{a}}_{j}^{2}}=w_{j}^{\prime }-\frac{c_{f,j}}{I_{W}(t)}-\lambda ^{\prime }{\tilde{\pi }}_{j}^{2}\text{, with }\lambda ^{\prime }=\frac{\lambda }{2I_{W}(t)^{2}}\\ \frac{\partial L}{\partial \lambda }&=\sum _{j=1,\ldots ,n}w_{j}^{\prime }-1=0. \end{aligned}$$
Summing up by j and considering the constraint, we have:
$$\begin{aligned} \lambda ^{\prime }\sum _{j=1,\ldots ,n}{\tilde{\pi }}_{j}^{2}&=\sum _{j=1,\ldots ,n} w_{j}^{\prime }-\frac{I_{F}(t)}{I_{W}(t)}=\frac{I_{W}(t)-I_{F}(t)}{I_{W}(t)}\\ w_{j}^{\prime }I_{W}(t)&=c_{f,j}-\left( I_{F}(t)-I_{W}(t)\right) \frac{{\tilde{\pi }}_{j}^{2}}{\sum _{j=1,\ldots ,n}{\tilde{\pi }}_{j}^{2}}. \end{aligned}$$
\(\square \)