Abstract
The attractiveness of insurance saving products is driven by dividend payments to policyholders and participation in profits. These are mainly constrained by regulatory disposals on profit-sharing on the basis of statutory accounts. Moreover, since both prudential (Solvency II) and financial reporting (IFRS 17) regulations require market consistent best estimate measurement of insurance liabilities, cash-flow projection models have to be used in order to derive the underlying financial incomes. In most cases, such models are based on Monte-Carlo techniques, which simulate future accounting profit and losses needed for profit-sharing mechanisms. In this paper, we deal with modelling future impairment losses on equity securities for financial portfolios. As a matter of fact, if impairment losses are determined on an instrument-by-instrument basis, projection models deal with model points of financial instruments (e.g. groups of shares). Since individual depreciation mechanisms are non-linear, projecting is quite a challenge for model designers. Our motivation is to describe the joint distribution of market value and impairment provision of a book of equity securities, with regard to the French accounting rules for depreciation. The derived results can more effectively represent such an asymmetric mechanism by using our results. Formally, an impairment loss is recognized for an equity instrument if there has been a significant and prolonged decline in its market value below the carrying cost (acquisition value). Such constraints are formalized using an assumption about the dynamics of the equity, and lead to a complex option-like pay-off. Using this formulation, we propose analytical formulas for some quantitative measurements related to the impairment losses of a financial equities book. These are derived from a general framework and some tractable examples are illustrated. We also investigate the operational implementation of these formulas and compare theircomputational time to a basic simulation approach.
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Notes
Letβs note that there is a mistake in Corollary 2 (He et al. 1998) and t is sometimes missing
If Ο =β0, then
References
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This work has been supported by the BNP Paribas Cardif Chair βData Analitycs and Models in Insuranceβ. The views expressed in this document are the authors owns and do not necessarily reflect those endorsed by BNP Paribas Cardif.
Appendices
Appendix A Comonotonic securities
In order to prove our results, we shall need the joint distribution of a Brownian motion with drift and its supremum. This can be easily derived using Harrison (1985).
Proposition 6
Let Xt = ΞΌt + ΟBt, and Mt = supu β[0, t]Xu where B is a standard Brownian motion. Then, the probability \({\mathbb P}(X_{t}\leq y, M_{t}\geq x)\) is given by:
On the other hand, the c.d.f. \(\mathbb {P}\left (M_{t}\leq x \right )\) is given by:
where Ο is the c.d.f of a standard Gaussian r.v.
We easily deduce
Another useful result is the following :
Proposition 7
Let \(\alpha ,\beta , w,c\in {\mathbb {R}}\) and X is a standard Gaussian random variable, i.e. \(X\sim {\mathcal {N}}(0,1)\) with distribution function Ο. Then
where Ξ¦(U, V ; π) is the joint c.d.f. of two correlated Gaussian r.v.s U and V with a given correlation coefficient π.
Proof
If one considers two correlated random variables X1 and X2 with correlation coefficient Ο such that \(X_{i}\sim {\mathcal {N}}(\mu _{i}, \sigma _{i})\), i =β1, 2 then the distribution of X2 conditionally on X1 can be written as follows:
Thus, we can write the joint distribution of X1 and X2 in the following form
which can also be given in terms of the joint distribution Ξ¦ of two standard Gaussian r.v.βs with a given correlation coefficient. Therefore, we can write for any a, b, c and w
which is the desired result. β‘
1.1 A.1 Proof of Proposition 1
The cases x <β0 and x β₯ ai are obvious. Remark that on \(\{\tilde {S}^{i}_{t}\leq \alpha a_{i} \}\), \(a_{i}(1-\alpha )\leq a_{i}-{S^{i}_{t}}\). Hence, for x < ai(1 β Ξ±),
We then use the Markov property of Si and Proposition 6.
For ai(1 β Ξ±) β€ x < ai, we have
Using the Markov property of Si, we get
where B is a standard Brownian motion independent of W. We firstly use Proposition 6 to compute the conditional expectation and secondly we remark that \(S^{i}_{t-\frac {1}{2}}={S^{i}_{s}}e^{X_{\delta -\frac {1}{2}}}\) where \(X_{\delta -\frac {1}{2}}=\mu _{i}(\delta -\frac {1}{2})+\sigma _{i}\sqrt {\delta -\frac {1}{2}}X\) where X is a standard Gaussian random variable independent of \(\mathcal {F}_{s}\), with distribution function Ο(β ). We easily obtain
This result could be enhanced as the expectations involved in the formulas of p1 and p2 can be explicitly computed using Eq.Β 1.2.
1.2 A.2 Proof of Proposition 2
We remark that \({\mathbb P}_{s}({\Gamma }_{t}>0)=1-{\mathbb P}_{s}({\Gamma }_{t}=0)=1-{\mathbb P}_{s}(\tilde {S}_{t}>\alpha \max (a_{1}{\dots } a_{n}))\) and we conclude as in the proof of Proposition 1 that \(\mathbb {P}_{s}({\Gamma }_{t}>0)=1-p_{1}(\max (a_{1}{\dots } a_{n}), S_{s})\), which is the desired result.
1.3 A.3 Proof of Proposition 3
We notice that the expected impaired amount for a single share S with an acquisition cost ai can be computed as follows
The first term \({\mathbb P}_{s}({\widetilde {S}}_{t}\leq a_{i}\alpha )=1-p_{1}(a_{i}, S_{s})\). The second term can be expressed, using the Markov property, in the following form:
which needs the density \(\mathbb {P}\left (X_{t}\in dx, \sup _{u\in [0,t]}X_{u}\leq a\right )\). Now, using this density we can write that \(\mathbb {E}_{t-\frac {1}{2}}(\dots )\) equals
Finally, using the fact that \(S_{t-\frac {1}{2}}=S_{s}e^{\mu (\delta -\frac {1}{2})+\sigma \sqrt {\delta -\frac {1}{2}} X}\), with Ξ΄ = t β s and X a standard normal r.v. independent of \(\mathcal {F}_{s}\), together with the identity in Eq.Β 1.2 leads to the desired formula.
1.4 A.4 Proof of Proposition 4.1
To find the joint probability \(\mathbb {P}_{t}\left ({\Gamma }_{t+1}\leq y, {\Sigma }_{t+1}\leq x \right )\), we will need the following technical lemma (to prove it, we use Eq.Β 1.2).
Lemma A.1
Leta >β0, Ξ², Ξ³andΞ±be some constants, then fors, t β₯β0 we have the following
Similarly, we have
Let n β₯β1, we are concerned with the computation of the conditional joint c.d.f. of (Ξt+β1,Ξ£t+β1):
In the above equation, the probability conditional on \(\mathcal F_{t+\frac {1}{2}}\) can be decomposed in the following form:
It is easily seen that the term (1.3) can be written as
\({\mathbb P}_{t+\frac {1}{2}}\left ({\widetilde {S}}_{t+1}\leq a_{1}\alpha , \frac {1}{n}({\sum }_{i=1}^{n}a_{i}-y)\leq S_{t+1}\leq \frac {x}{n}\wedge a_{1}\alpha \right ),\)
and similarly the term (1.4) as
\({\mathbb P}_{t+\frac {1}{2}}\left (a_{j}\alpha < {\widetilde {S}}_{t+1}\leq a_{j+1}\alpha , \frac {1}{n-j}({\sum }_{i=j+1}^{n}a_{i}-y)\leq S_{t+1}\leq \frac {x}{n}\wedge a_{j+1}\alpha \right ).\)
Moreover, we use Markov property and Proposition 6 to write the term (1.5) in the following form
On \(\{\frac {1}{n}({\sum }_{i=1}^{n}a_{i}-y)\leq \frac {x}{n}\wedge a_{1}\alpha \}\), the term (1.3) is equal to:
On \(\{\frac {1}{n-j}({\sum }_{i=1+j}^{n}a_{i}-y)\leq \frac {x}{n}\wedge a_{j+1}\alpha \}\) the term (1.4) is equal to:
We conclude the proof using Theorem A.1 to compute the expectations conditional to \( {\mathcal {F}}_{t}\).
B Correlated securities
In order to prove our results in the correlated case, we shall need the joint distribution of two correlated Brownian motions with drift and their suprema. The joint distribution is obtained by making the following change of variables x2 = M2 β Ο2r sin π and \(x_{1}=M_{1}-\sigma _{1}r(\rho \sin \theta +\sqrt {1-\rho ^{2}}\cos \theta )\) in Lemma 3 and Remark 4 from He et al. (1998).
Proposition 8
Let \(X^{i}=\mu _{i}t+\sigma _{i}{W^{i}_{t}}\), where \(\mu _{i}\in {\mathbb {R}}\), Οi >β0 and Wi two correlated Brownian motions with correlation coefficient Ο β] ββ1, 1[. For x1 β€ M1, x2 β€ M2 with M1, M2 β₯β0, we have :
\({\mathbb P}({X^{1}_{t}}\in dx_{1}, {X^{2}_{t}}\in dx_{2}, sup_{u\leq t}{X^{1}_{u}}\leq M_{1} , sup_{u\leq t}{X^{2}_{u}}\leq M_{2})=p(r,r_{0}, \theta , \theta _{0}, t ; M_{1},M_{2})drd\theta \)
\({\mathbb P}(\tilde {X^{1}}_{0\rightarrow t}\leq M_{1} , \tilde {X^{2}}_{0\rightarrow t}\leq M_{2})=q(r_{0}, \theta _{0}, t ; M_{1}, M_{2})\), wherep andq are given in Definition 4.1 and
For some special correlation values, the function q may be written in a simpler form (see Corollary 2 in He et al. 1998). We easily obtain the following resultsFootnote 1 :
Lemma B.1
Under the assumptions of Proposition 8, if\(\rho _{n}=-\cos \frac {\pi }{n}\), we have :
2.1 B.1 Proof of Proposition 5
We remark that
Markov property, Proposition 8 and writing \( S^{1}_{t-1/2}={S^{1}_{s}}e^{\mu _{1}(\delta -1/2)+\sigma _{1}\sqrt {\delta -1/2}X}\) with \(X\sim {\mathcal {N}}(0,1)\) lead to the desired formula.
We need some technical results in order to find the joint law of Ξ and Ξ£.
Lemma B.2
Letx >β0. The c.d.f of the aggregate dynamics Ξ£ of two correlated shares is given by
where X is a standard Gaussian r.v. We denote this probability by \(f({S^{1}_{t}}, {S^{2}_{t}}, x)=\mathbb {P}_{t}({\Sigma }_{t+1}\geq 0).\)
Proof
Since W1 and W2 are two correlated Brownian motions with correlation coefficient Ο, then \(W^{2}=\rho W^{1} +\sqrt {1-\rho ^{2}}W\) with W1 and W two independent BM. Hence :
where X and Z are two independent standard Gaussian r.v. We integrate with respect Z and the conclusion holds. β‘
Lemma B.3
Let\({X^{i}_{t}}=\mu _{i}t+\sigma _{i}{W^{i}_{t}}\), where\(\mu _{i}, \sigma _{i}\in {\mathbb {R}}\)andWitwo correlated BM withcorrelation coefficientΟ β] ββ1, 1[. Then forM1 >β0, \(x_{1}, x_{2}\in \mathcal {R}\)
where
Proof
Without loss of generality we consider Οi =β1, i =β1, 2 (we then replace x1, M1, x2 by \(\frac {x_{1}}{\sigma _{1}}, \frac {M_{1}}{\sigma _{1}}, \frac {x_{2}}{\sigma _{2}}\)). Since
and \(W^{2}=\rho W^{1} +\sqrt {1-\rho ^{2}}W\) with W1 and W two independent BM, we can write for Οβ β0Footnote 2
The derivative of the first term is 0. For the second term, we integrate with respect to the law of Wt and then we use Proposition 6. Finally the derivative with respect to x1 and x2 leads to the desired result. β‘
We easily deduce the following :
Corollary B.3.1
Under the assumptions of Lemma B.3 letR1, R2, x, w, M1 >β0. Then
where X is a standard Gaussian r.v.
2.2 B.2 Proof of Proposition 4.2
The joint c.d.f
can be decomposed in the following form :
- Step 1 - Probabilities decomposition :
-
-
The first term \(C_{1}({S^{1}_{t}}, {S^{2}_{t}})={\mathbb P}_{t}\left (S^{1}_{t+1} +S^{2}_{t+1}\leq x, \tilde {S^{1}}_{t+1}>\alpha a_{1}, \tilde {S^{2}}_{t+1}>\alpha a_{2} \right )\) may be written as :
Recall that \(S^{1}_{t+1} +S^{2}_{t+1}={\Sigma }_{t+1}\). Note that on {x β₯ Ξ±(a1 + a2)},
$${\mathbb P}_{t}\left( S^{1}_{t+1} +S^{2}_{t+1}\leq x, \tilde{S^{1}}_{ t+1}\leq\alpha a_{1}, \tilde{S^{2}}_{t+1}\leq\alpha a_{2} \right)=\mathbb{P}_{t}\left( \tilde{S^{1}}_{t+1}\leq\alpha a_{1}, \tilde{S^{2}}_{t+1}\leq\alpha a_{2} \right).$$Hence,
-
The second term \(C_{2}({S^{i}_{t}}, {S^{j}_{t}}, a_{i},a_{j})={\mathbb P}_{t}\left (S^{1}_{t+1} +S^{2}_{t+1}\leq x, \tilde {S^{i}}_{t+1}\!\leq \!\alpha a_{i}, \tilde {S^{j}}_{t+1}\!>\!\alpha a_{j}, S^{i}_{t+1}\geq a_{i}-y \right )\) equals to :
-
The last term \(C_{3}({S^{1}_{t}}, {S^{2}_{t}})={\mathbb P}_{t}\left (a_{1}+a_{2}-y\leq S^{1}_{t+1} +S^{2}_{t+1}\leq x, \tilde {S^{1}}_{t+1}\leq \alpha a_{1}, \tilde {S^{2}}_{ t+1} \leq \alpha a_{2} \right )\) may be written as :
-
- Step 2 - Grouping of terms :
-
Remark that for the term \( C_{2}({S^{i}_{t}}, {S^{j}_{t}}, a_{i},a_{j})\), we may use Markov property and Proposition 8. Hence we can write (consider for example the case i =β1 and j =β2) :
where \({X^{i}_{t}}=\mu _{i}t+\sigma _{i}{B^{i}_{t}}\) where Bi are two correlated BM independent of \(\mathcal {F}_{t+\frac {1}{2}}\), and D1 given before.
The same reasoning allows to write \({\mathbb P}_{t}({\Sigma }_{t+1}\leq x, \tilde {S^{1}}_{t+1}\leq \alpha a_{1}, \tilde {S^{2}}_{t+1}\leq \alpha a_{2}, S^{2}_{t+1}\geq a_{2}-y)\), \({\mathbb P}_{t}({\Sigma }_{t+1}\geq a_{1}+a_{2}-y, \tilde {S^{1}}_{t+1}\leq \alpha a_{1}, \tilde {S^{2}}_{t+1}\leq \alpha a_{2})\), \(\mathbb {P}_{t}({\Sigma }_{t+1}\leq x, \tilde {S^{1}}_{ t+1}\leq \alpha a_{1}, \tilde {S^{2}}_{t+1}\leq \alpha a_{2})\) and \(\mathbb {P}_{t}(a_{1}+a_{2}-y\leq {\Sigma }_{t+1}\leq x, \tilde {S^{1}}_{t+1}\leq \alpha a_{1}, \tilde {S^{2}}_{t+1}\leq \alpha a_{2})\) in a similar form with D2, D3(a1 + a2 β y), D3(x) and D3(x) β D3(a1 + a2 β y).
Hence, we have
- Step 3 - Final calculation :
-
Remark that \({\mathbb P}_{t}(\tilde {S^{1}}_{t+1}\leq \alpha a_{1}, \tilde {S^{2}}_{t+1}\leq \alpha a_{2})=Q({S^{1}_{t}}, {S^{2}_{t}}, 1)\) with Q given in Proposition 5. Also, the conditional law of Ξ£, i.e. \(\mathbb {P}_{t}({\Sigma }_{t+1}\leq x)\), is given in Theorem B.2. Since the calculation method is the same for all the other terms, we only detail here the probability \(\mathbb {P}_{t}({\Sigma }_{t+1}\leq x, \tilde {S^{1}}_{t+1}<\alpha a_{1})\). Thus, Markov property gives
where \(X^{i}_{\frac {1}{2}}=\frac {\mu _{i}}{2}+\sigma _{i}B^{i}_{\frac {1}{2}}\), Bi two correlated BM independent of \(\mathcal {F}_{t+\frac {1}{2}}\). Next, using Corollary B.3.1, we find that \(\mathbb {P}_{t+\frac {1}{2}}\left (S^{1}_{t+\frac {1}{2}} e^{X^{1}_{\frac {1}{2}}}+S^{2}_{t+\frac {1}{2}} e^{X^{2}_{\frac {1}{2}}}\leq x, \tilde {X}^{1}_{ \frac {1}{2}}<\ln \frac {\alpha a_{1}}{S^{1}_{t+\frac {1}{2}}} \right )\) equals
where Z is a standard Gaussian r.v. independent of \(\mathcal {F}_{t+\frac {1}{2}}\).
We conclude writing \(S^{1}_{t+\frac {1}{2}}={S^{1}_{t}}+\frac {\mu _{1}}{2}+\sigma _{1}\sqrt {0.5}X\) (resp. \(S^{2}_{t+\frac {1}{2}}={S^{2}_{t}}+\frac {\mu _{2}}{2}+\sigma _{2}\sqrt {0.5}Y\)) with X, Y two correlated standard Gaussian r.vβs, that:
We also find
Remark that on I1, x > Ξ±a1, so :
$$ \begin{array}{@{}rcl@{}} &&{\mathbb P}_{t}({\Sigma}_{t+1}\leq x, \tilde{S^{1}}_{t+1}<\alpha a_{1})+ {\mathbb P}_{t}({\Sigma}_{t+1}\leq x, \tilde{S^{1}}_{t+1}\leq\alpha a_{i}, S^{1}_{t+1}\geq a_{1}-y)\\ &&= {\mathbb{E}}\left[ J_{1}(s_{1},s_{2},X,Z)\phi_{12}(s_{1}, s_{2},X,Y,Z)\right]\\ &&\quad-\left( \frac{\alpha a_{1}}{{S^{1}_{t}}}\right)^{\frac{2\mu_{1}}{{\sigma^{2}_{1}}}}e^{-\frac{{\mu^{2}_{1}}}{{\sigma^{2}_{1}}}}\mathcal{E}\left[ J_{2}(s_{1},s_{2},X,Z)\tilde{\phi}_{12}(s_{1}, s_{2},X,Y,Z)\right]. \end{array} $$This completes the proof.
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Dorobantu, D., Salhi, Y. & ThΓ©rond, PE. Modelling Net Carrying Amount of Shares for Market Consistent Valuation of Life Insurance Liabilities. Methodol Comput Appl Probab 22, 711β745 (2020). https://doi.org/10.1007/s11009-019-09729-1
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DOI: https://doi.org/10.1007/s11009-019-09729-1