Market Value Margin calculations under the Cost of Capital approach within a Bayesian chain ladder framework

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Market Value Margin calculations under the Cost of Capital approach within a Bayesian chain ladder

framework

Christian Robert

To cite this version:

Christian Robert. Market Value Margin calculations under the Cost of Capital approach within a Bayesian chain ladder framework. Insurance: Mathematics and Economics, Elsevier, 2013,

�10.1016/j.insmatheco.2013.05.003�. �hal-00618013�

Market Value Margin calculations under the Cost of Capital approach within a Bayesian chain ladder framework

Christian Y. ROBERT

Université de Lyon, Université Lyon 1, Institut de Science Financière et d’Assurances, 50 Avenue Tony Garnier, F-69007 Lyon, France

Abstract

In the Solvency II framework, insurance companies need to calculate the Best Estimate val- uation of Liabilities (BEL) and the Market Value Margin (MVM) for non-hedgeable insurance- technical risks. The Cost-of-Capital approach de…nes the MVM as the present value of the current and future Solvency Capital Requirement (SCR) of the non-hedgeable risks to protect against adverse developments in the run-o¤ of the insurance liabilities. However the SCR at time t itself depends on the increase in the MVM betweent andt+ 1. Hence there exists an intricate circularity dependency between both quantities. In this paper we present exact and accurate approximate analytic formulas for MVMs within a Bayesian log-normal chain ladder framework.

1 Introduction

Solvency II is creating a new approach to regulate capital requirements by quantifying risks and is giving incentives for companies to develop good risk management practices. It will be based on economic principles for the measurement of assets and liabilities and capital requirements will depend directly on this. In particular it introduces the market consistent economic (solvency) balance sheet and two points in time are considered to calculate the Solvency II Capital Requirement (SCR): the current balance sheet and the balance sheet at the end of the year. The main components of this balance sheet are the Market Value of Assets (MVA) and the Fair Value of Liabilities (FVL) consisting of the sum of two components: the Best Estimate valuation of Liabilities (BEL) and the Market Value Margin (MVM).

The BEL is the present value of expected future cash ‡ows using best estimate assumptions with no explicit margins incorporated. However, for the non-hedgeable insurance risks, since there exists the risk that actual experience will be more adverse than expected, there needs to be an additional “risk margin” component added to the BEL, the MVM. It can be interpreted as the cost of risk and uncertainty in the amount and timing of future payments needed to satisfy insurance liabilities.

For several decades, actuaries have used a variety of technical methods to consider the risk in the valuation of their insurance liabilities and risk margins have been implicitly or explicitly embedded in the assumptions or in the methods. Approaches for determining risk margins have been grouped into four families (see IAA position paper [9])

This work has been partially supported by the BNP Paribas Insurance Chair «Management de la modélisation».

The views expressed in this document are the authors own and do not necessarily re‡ect those endorsed by BNP Paribas Insurance.

the quantile methods that use risk measures such as the Value-at-Risk (VaR), the Conditional Tail Expectation (CTE) and the Tail Value-at-Risk (TVaR) and de…ne the risk margin as the di¤erence of the risk measure of the discounted ultimate future payments and the BEL;

the Cost-of-Capital (CoC) approach that de…nes the risk margin as the present value of the current and future SCRs for the non-hedgeable risks to protect against adverse developments in the run-o¤ of the insurance liabilities;

the discount related methods that de…ne the risk margin as the di¤erence of the discount expected cash ‡ows using the risk-free interest rate minus a selected risk adjustment and the BEL (probability distortions to take into account risk aversion are alternatively used in [18]);

the methods that use explicit assumptions: the risk margin results from selecting prudent explicit parameters and simpler methodologies.

Over the recent years, the CoC approach has been preferred to estimate market-consistent risk margins for insurance contracts. One of the reason is that CoC approach is commonly used as a conceptual framework in both non-life and life insurance valuation applications. Solvency II prescribes that the MVM is calculated using the following 3 steps:

1. determine the expected SCRs for non-hedgeable risks until the run-o¤ of the portfolio (remind that the SCR is de…ned as the amount of capital required to support the claims paid out and the increase in the sum of the BEL and MVM following a one-in-200-year event over the next year);

2. calculate the capital charge as the SCRs multiplied by the CoC charge and take the present value of the product;

3. take the sum of the present values for all years until the run-o¤ to arrive at the MVM.

As can be read in this approach, estimating the SCRs for each future year in a theoretically correct way is far from straightforward because the SCRs appear to depend on the MVMs, and the MVMs depend on the SCRs. Hence there exists an intricate circularity dependency between both quantities. To mitigate the circularity issue, several simpli…cations have been suggested (from the more complex to the simpler):

exclude the risk margin in the FVL within the calculations for the SCRs (see e.g. the Swiss Solvency Test);

approximate the SCRs for the future years by using a ‘proportional proxy’: for example the current SCR is calculated, but the future SCRs are approximated by multiplying the current SCR by the ratio of the future BEL to the current BEL;

approximate the current MVM by using a ‘duration approach’: the MVM is calculated as the current SCR multiplied by a modi…ed duration of the insurance liabilities (see e.g. [13]);

approximate the current MVM by considering it as a percentage of the current BEL.

The problem of calculating the MVM in a theoretically correct way can only be solved by starting at the …nal time period and working backwards recursively. The problem is in general intractable without simpli…cations.

In [7], Haslip considers the liabilities of a non-life insurance company and assumes that the outstanding claims reserve at the end of each year is subject to uncertainty through a scaling factor drawn from a log-normal distribution with mean 1 and several sets of coe¢cients of variation. He makes some comparisons with proxies derived from the proportional method, the duration method or by considering the MVM as a percentage of the current BEL. The results are: the duration method understates the risk margin, the proportional method provides a reasonable approximation but sensitive to the coe¢cients of variation of the log-normal distributions, the ‘percentage of BEL’

proxy is more conservative than the others.

In [1] and [4], Bonnard, Daya and Margetts alternately assume that the cumulative paid claims (for all the accident years) at the end of each year are equal to the cumulative paid claims at the beginning of the year multiplied by a scaling factor drawn from a log-normal distribution with deterministic but path dependent mean and volatility. They …nd that the …rst approximation (i.e.

excluding the risk margin in the FVL within the calculations for the SCRs) gives higher results than the exact solution. The second and the third approximations (proportional and duration proxies) tend to underestimate the exact solution. The fourth approximation can be quite larger than the analytical solutions specially for the …rst development years.

In [14], Salzmann and Wüthrich derive analytical formulas for the risk margin and compare di¤erent proxies but under the assumption that the capital requirements are de…ned through the standard error of the sum of the claims paid out and the increase in the BEL and the risk margin instead of the 99.5th percentile of the change as needed for the Solvency II approach.

In this paper, we consider a Bayesian log-normal chain ladder model for a non-life insurance company and extend the results of Bonnard, Daya and Margetts to the case (i) where recent information is immediately absorbed by the Bayesian model, (ii) where the cumulative paid claims of each accident year follow a multiplicative log-normal model and/or (iii) where the cash ‡ows and the SCRs are discounted within the calculation of the BEL and the MVM. We give an exact analytical formula for the MVM of the liabilities of a speci…c accident year when reserves are not discounted. We propose accurate approximated analytic formulas for the MVM of the liabilities of all accident years (when reserves are discounted or not) by considering convex order techniques and approximations as used in [5], [6], [15], [16].

The paper is organized as follows. In Section 2 we de…ne the Bayesian log-normal chain ladder model for claims reserving and calculate the BEL depending on whether or not the cash ‡ows are discounted. We then give general results concerning convex order approximations of sums of log-normal random variables in Section 3. In Section 4 and 5 we provide respectively recursive analytical formulas for the calculation of the MVM when cash ‡ows are not discounted or when they are subject to a constant discount rate. Finally, in Section 6 we provide a real data example that is based on liability insurance data. The proofs are deferred to Section 7.

2 Bayesian log-normal chain ladder model

We consider the liabilities of a non-life insurance company. The random variables C_i;j >0denote the cumulative payments of accident year i 2 f1; : : : ; Ig after development year j 2 f0; : : : ; Jg, meaning that the incremental claims, X_i;j = C_i;j C_{i;j 1} for j = 1; : : : ; J, are paid in calendar (accounting) yeark =i+j. We assume that all claims are settled after development J and that I J + 1. We de…ne the individual claims factor F_i;j = C_i;j=C_{i;j 1} for j = 1; : : : ; J and let

i;j = log(Fi;j). The …rst payment Ci;0 is the initial value of the process (Ci;j)_j=0;:::;J and Fi;j are the multiplicative changes.

The time unit corresponds to years, the current year is I and we write Tt= fXi;j :i+j t; i= 1; : : : ; I; j = 0; : : : ; Jg fort=I; : : : ; I+J.

We consider a Bayesian log-normal chain ladder model to complete the run-o¤ trapezoid. For the claims reserving problem, Bayesian methods are now well investigated (see e.g. [17], [2]) and they provide an interesting approach for a successive information update in each accounting year.

Log-normal chain ladder models have been introduced by Hertig [8] and their Bayesian versions have been recently used by Merz and Wüthrich [12] to model paid-incurred chain claims. We restrict ourselves to this model because it allows for explicit analytical formulas for the MVM of the liabilities of a speci…c accident year when reserves are not discounted.

We assume that

given = ( ₀; : : : ; _{J 1}) and = ( ₀; : : : ; _{J 1}), the X_i;j are independent for di¤erent accident years i, and for j = 0; : : : ; J 1, the _i;j+1 are independent and satisfy _i;j+1 N( j; ²_j);

>0is deterministic and the _j, forj= 0; : : : ; J 1, are independent normally distributed,

j N( _j; s²_j), with prior parameters _j and s²_j >0;

(C1;0; : : : ; CI;0) and are independent.

This model is close to the normal-normal Bayes chain ladder model and belongs to the expo- nential dispersion family with associate conjugate priors (see e.g. [3]). It leads to exact formulas for the calculation of the BEL but the Bayesian estimators do not coincide with the linear credibility estimators.

At time twe have information T_t and we need to predict the liabilities that correspond to the random variables X_i;j+1 for(i; j+ 1)2D_twhere

D_t=f(i; j+ 1) :i=t J+ 1; : : : ; I; j=t i; : : : ; J 1g

or equivalently to the random variables _i;j+1for(i; j+1)2Dt. The following proposition gives the update formulas for the parameters of the posterior distribution of and derives the distributions of the random variables _i;j+1,(i; j+ 1)2D_t, givenTt.

Proposition 1 GivenTt, for j= 0; : : : ; J 1, the _j are independent normally distributed random variables such that

jj_T

t N( ^(t)_j ;(s^(t)_j )²) with posterior parameters

(t)

j = (s^(t)_j )² 2 4 ^j

s²_j + 1

2j

(t j 1)^I

X

i=1

i;j+1

3

5; (s^(t)_j )² = 1

s²_j +(t j 1)^I

2j

! 1

:

Given T_t, _i;j+1 (i;j+1)2Dt is a multivariate normally distributed random vector with E[ _i;j+1jT_t] = ^(t)_j ; (i; j+ 1)2D_t

Cov( _i;j+1; _k;l+1jT_t) = I_fj=lg (s^(t)_j )²+I_fi=kg ²_j ; (i; j+ 1) (k; l+ 1)2D_t²:

In practice, we update at each time the posterior parameters according to the information generated by the new observations. Let

(t+1)

j = (s^(t+1)_j )²

2j

; ^(t+1)_j = 1 ^(t+1)_j = (s^(t+1)_j )²

(s^(t)_j )² ; ^(t+q:t+l)_j = Yt+l

i=t+q (i) j ;

with ^(t+q:t+l)_j = 1 if q > l. The following proposition gives the recursive update formulas from which we can deduce that, forj = 0; : : : ; J 1,( ^(t)_j )t=I;:::;I+j is aTt-martingale:

Proposition 2 We have, for t j I,

(t+1)

j = ^(t+1)_j ^(t)_j + ^(t+1)_{j t j;j+1} (2.1)

and, for l= 1; : : : ;1 +I+j t,

(t+l)

j = ^(t)_j ^(t+1:t+l)_j + Xl

q=1

(t+q+1:t+l) j

(t+q)

j t j+q 1;j+1: (2.2)

It follows that( ^(t)_j )t=I;:::;I+j is a Tt-martingale, i.e. E[ ^(t+l)_j jTt] = ^(t)_j . Moreover log E[e ^(t+l)j jTt] = ^(t)_j + 0:5((s^(t)_j )²+ ²_j)

Xl

q=1

( (t+q+1:t+l) j

(t+q)

j )²

+(s^(t)_j )² X

1 p<q l

(t+p+1:t+l) j

(t+q+1:t+l) j

(t+p) j

(t+q)

j :

Solvency II requires from non-life insurance companies a market-consistent valuation of their insurance liabilities. This implies that the outstanding loss liability cash ‡ows need to be discounted with time values and should be determined given the latest information available. In old accounting tradition, insurance companies have estimated nominal (i.e. not discounted) claims reserves for their outstanding loss liabilities because undiscounting includes in fact a certain risk margin (depending on the level of the discount rate).

We de…ne the undiscounted reserve at time tfor accident year i=t J+ 1; : : : ; I by R_i;t =

XJ

j=t i+1

X_i;j =C_i;J C_{i;t i}; and the discounted reserve by

R^(d)_i;t = XJ

j=t i+1

X_i;j (1 +r)^{j t+i} =

XJ

j=t i+1

C_i;j C_{i;j 1} (1 +r)^{j t+i}:

We restrict ourselves to a constant discount rate case for the sake of avoiding unnecessary compli- cations but the case of a family of forward rates could also have been considered.

Let

M_l:m^(t) = Xm

j=l (t)

j and S_l:m^(t) =

Xm

j=l

(s^(t)_j )²+ ²_j

with the convention that M_l:m^(t) = 0 and S_l:m^(t) = 0 if l > m. The Best-Estimate outstanding loss Liabilities (BEL) are given in the following proposition.

Proposition 3 For i=t J+ 1; : : : ; I,

BEL_i;t = E[R_i;tjTt] =C_{i;t i} e^{Mt(t)i:J 1+0:5St(t)i:J 1} 1 BEL^(d)_i;t = E[R^(d)_i;tjTt] =C_{i;t i}

0

@

J 1

X

j=t i

1

(1 +r)^{j t+i+1} e^{Mt(t)i:j+0:5St(t)i:j} e^{Mt(t)i:j 1+0:5St(t)i:j 1} 1 A: The undiscounted one year loss deterioration describes the deterioration of the expected reserves over the next year. It is de…ned at time tfor accident year i=t J+ 1; : : : ; I by

L⁽¹⁾_i;t =X_{i;t+1 i}+BEL_i;t+1 BEL_i;t =E[C_i;JjTt+1] E[C_i;JjTt]:

The discounted one year loss deterioration is de…ned by L^(1;d)_i;t = X_{i;t+1 i}

1 +r + 1

1 +rBEL^(d)_i;t+1 BEL^(d)_i;t:

The one year deteriorations are centered, i.e. E[L⁽¹⁾_i;tjTt] = 0 and E[L^(1;d)_i;t jTt] = 0. Note that the undiscounted claims development results (CRD) introduced by [11] are just the opposite of the undiscounted one year deteriorations.

3 Approximation for quantiles of sums of log-normal random vari- ables

Many problems in …nance, insurance and engineering involve the evaluation of the distribution function of a random variable S of the form

S= Xn

i=1 ie^Zi

where the _i are non-negative real numbers and (Z₁; : : : ; Z_n) is a multivariate normal distributed random vector. In this paper we are mainly interested in the quantiles ofS. Because it is impossible to obtain analytical expressions for the distribution function, Kaas, Dhaene and Goovaerts [10]

propose to approximateS by

S^l=E[Sj ]

for an appropriate choice of the conditioning . The multi-dimensionality of the problem, caused by(Z₁; Z₂; : : : ; Z_n), is then transformed to a single dimension leading to the comonotonicity of the vector(E[e^Z1j ]; : : : ;E[e^Znj ]). From Jensen’s inequality one can prove thatS^lis smaller in convex order than S. In literature a convex upper bound for S has also been proposed (see e.g. [5] and [6]).

To get accurate approximations, should be chosen such that it is close to S. Consider the conditioning random variable as the linear combination ofZ₁; Z₂; : : : ; Z_n determined by

= Xn

i=1 iZ_i: Then S^l can be written

S^l= Xn

i=1

ie^{E[Zi]+0:5(1 ri2) 2Zi+ri Zi( E[ ])=}

where ²_Z

i and ² are respectively the variances ofZ_i and , and r_i=r_i( ₁; : : : ; _n) = cov(Z_i; )

Zi

: (3.3)

Note that the expected values of the random variables S and S_l are equal and that, in case all r_i 0,

q_99:5% S^l = Xn

i=1

ie^{E[Zi]+0:5(1 ri2) 2Zi+ri Zi'} where'is the VaR at the 99:5%level of the standard Gaussian distribution.

In [15],Vandu¤el, Hoedemakers and Dhaene propose to choose such that a …rst-order approx- imation of the variance of S^l is as large as possible and therefore the closest to the variance of S.

The choices of the parameters _i are then given by

i= _ie^{E[Zi]+0:5 2Zi}: (3.4)

In [16], Vandu¤el, Chen, Dhaene, Goovaerts, Henrard Kaas alternately propose to choose such that a …rst-order approximation of the p-level Conditional Tail Expectation of S^l is as large as possible and therefore the closest to thep-level Conditional Tail Expectation ofS. The choices of the parameters _i are then given by

i = ie^{E[Zi]+0:5 2Zi 0} ri( 1e^{E[Z1]+0:5 2Z1}; : : : ; ne^{E[Zn]+0:5 2Zn}) _Zi ¹(p)

where is the probability distribution function of a standard Gaussian random variable. As one can see, the parameters only di¤er up to proportional coe¢cients from the parameters that maximize the …rst-order approximation of the variance ofS^l.

Since Monte-carlo simulations in [15] show that the approach by Vandu¤el, Hoedemakers and Dhaene gives very accurate approximations for the0:995-quantile ofS, we keep their choice for the parameters _i in the remainder of the paper.

4 MVM for no discounted cash-‡ows

We …rst consider the simple case where the cash ‡ows are not discounted, although it is not coherent with the Solvency II framework. The case of discounted cash ‡ows is provided in the next section.

The Fair-Value of Liabilities for accident year i at time t is de…ned as the sum of the Best Estimate valuation of Liabilities (BEL) and the Market Value Margin (MVM) of this accident year

F V L_i;t =BEL_i;t+M V M_i;t:

Let us remind that the Solvency Capital Requirement (SCR) is de…ned as the amount of capital required to support the claims paid out and the increase in the FVL following a one-in-200-year event over the next year, and that the Market Value Margin is de…ned as the sum of the current and future SCRs. For accident yeariat timet, the SCR and the MVM are then given by the two simultaneous equations

SCRi;t = q_99:5%(F V Li;t+1 F V Li;t+Xi;t+1 ijTt) (4.5)

= q_99:5% M V M_i;t+1 M V M_i;t+L⁽¹⁾_i;tjTt (4.6) M V M_i;t = c

i+JX1

j=t

E[SCR_i;jjTt] (4.7)

where c is the cost of capital rate (taken as a …xed 6%). Note that SCR_i;t and M V M_i;t are Tt- measurable random variables and thatSCR_i;i+J = 0since all liabilities are completely extinct for this accident year after the accounting yeari+J: Ci;J =Ci;J+1=Ci;J+2 =: : :.

The Fair-Value of Liabilities for all accident years with outstanding liabilities is given by F V L_t=

XI

i=t J+1

BEL_i;t+M V M_t:

The SCR_t and theM V M_t are then given by the two simultaneous equations SCR_t = q_99:5% F V L_t+1 F V L_t+

XI

i=t J+1

X_{i;t+1 i} Tt

!

(4.8)

= q_99:5% M V Mt+1 M V Mt+L⁽¹⁾_t jTt (4.9) M V M_t = c

I+JX1

j=t

E[SCR_jjTt] (4.10)

where

L⁽¹⁾_t =E

" _I X

i=1

C_i;J Tt+1

# E

" _I X

i=1

C_i;J Tt

# : Note that there is no reason M V M_t and SCR_t are respectively equal to PI

i=t J+1M V M_i;t and PI

i=t J+1SCRi;t because the quantile function is nonlinear. MoreoverSCRI+J = 0.

Throughout this section, we will use the following notation: for I t < l i+J, log( _i;l;t) = 1

2

l iX1

j=t i

(s^(t)_j )²+ ²_j +1 2

J 1

X

j=l i

(s^(t)_j )²+ ²_j Xl t

q=1

(t+q+1:l) j

(t+q) j

2

+

J 1

X

j=l i

(s^(t)_j )² X

1 p<q l t

(t+p+1:l) j

(t+q+1:l) j

(t+p) j

(t+q) j

and 0otherwise.

4.1 MVM for one accident year

By using(4:6)and(4:7), the SCR expression for accident yearican be developed as follows SCR_i;t = q_99:5%

0

@c

i+JX1

j=t+1

E[SCR_i;jjT_t+1] c

i+JX1

j=t

E[SCR_i;jjT_t] +E[C_i;JjT_t+1] E[C_i;JjT_t] T_t 1 A

= q_99:5%

0

@c

i+JX1

j=t+1

E[SCR_i;jjT_t+1] +E[C_i;JjT_t+1] T_t 1 A c

i+JX1

j=t

E[SCR_i;jjT_t] E[C_i;JjT_t]:

Since E[SCRi;tjTt] =SCRi;t, we derive the recursive equations for the SCRs:

SCR_i;t = 1 1 +c

0

@q_99:5%

0

@c

i+JX1

j=t+1

E[SCR_i;jjTt+1] +E[C_i;JjTt+1] Tt

1 A c

i+JX1

j=t+1

E[SCR_i;jjTt] E[C_i;JjTt] 1 A: (4.11)

These equations have to be solved by starting at the …nal time period (i+J 1) and working backward recursively. The following proposition gives the analytical recursive formulas for the SCR. The MVM is then deduced by using(4:7).

Proposition 4 For t=I; : : : ; I+J and i=t J+ 1; : : : ; I, SCR_i;t=C_{i;t i}e^{Mt(t)i:J 1}a_i;t where, for t i+J, a_i;t = 0 and, for t < i+J,

a_i;t= 1 1 +c

"

c

i+JX1

l=t+1

a_i;l( _{i;l;t+1 i;t i;l;t}) +e^{0:5St+1(t+1)i:J 1} _i;t e^{0:5St(t)i:J 1}

#

with

log( i;t) =' vu

ut((s^(t)_{t i})²+ ²_{t i}) +

J 1

X

j=t+1 i

( ^(t+1)_j )²((s^(t)_j )²+ ²_j):

Moreover, at the current date t=I, we have

M V M_i;I =cC_{i;I i}e^{MI(I)i:J 1}

"_{i+J 1} X

l=I

a_{i;l i;l;I}

# :

It is worth noting that, given C_{i;t i} and M_{t i:J}^(t) ₁ = PJ 1 j=t i

(t)

j , the sum of the posterior expected values of the random variables _j given Tt, the expressions forSCR_i;t and M V M_i;t are obtained by only calculating recursively the constants a_i;l and _i;l;t for l = t; : : : ; i+J 1. In contrast to Bonnard, Daya and Margetts’ results, the parameters of the log-normal distributions have to be updated given the latest information available.

4.2 MVM for aggregated accident years

In the previous subsection we have studied the MVM for one single accident yeari. But in practice we want to calculate the MVM for all insurance liabilities, i.e. over all accident years. By using the same arguments as previously, we derive the recursive equations for the SCRs for all accident years: fort=I; : : : ; I+J 1,

SCR_t = 1

1 +cq_99:5%

0

@c

n 1

X

j=t+1

E[SCR_jjT_t+1] +E

" _I X

i=1

C_i;J T_t+1

# T_t

1 A

1 1 +c

0

@c

n 1

X

j=t+1

E[SCRjjTt] +E

" _I X

i=1

C_i;J Tt

#1

A: (4.12)

The MVM is then given by

M V M_t=c

I+JX1

j=t

E[SCR_jjTt]:

Contrary to the previous case, there is no exact analytical recursive formula for the SCR because the SCRs are quantiles of sums of log-normal random variables over the di¤erent accident years.

We need to approximate these quantiles such that they are equal to the sum of quantiles of some log-normal random variables but we have to choose these random variables in such a way that we are able to construct analytical recursive formulas for the SCRs.

Let us …rst introduce some notation. Let, for i=t J + 1; : : : ; I;

r^(t)_i = 1

&^(t)_i &^(t)_S

i 1

X

l=t J+1 (t)

l (&^(t)_l )²+ (&^(t)_i )² XI

l=i (t) l

!

where

(&^(t)_i )² = ((s^(t)_{t i})²+ ²_{t i}) +

J 1

X

j=t+1 i

( ^(t+1)_j )²((s^(t)_j )²+ ²_j)

(&^(t)_S )² =

XI

i=t J+1 (t) i

2(&^(t)_i )²+ 2 X

t J+1 i<l I (t) i

(t) l (&^(t)_i )² and

log( ^(t)_i ) = log(B_i;t⁽²⁾) + log(C_{i;I i}) +M_{I i:t i}^(I) ₁+ 0:5S_{I i:t i}^(I) ₁ +

J 1

X

j=t+1 i (t+1) j

2

4 ^(I)_j + 0:5 ^(t+1)_j ((s^(I)_j )²+ ²_j) Xt I

q=1

( ^(I+q+1:t)_j ^(I+q)_j )² 3 5

+

J 1

X

j=t+1 i

( ^(t+1)_j )²(s^(I)_j )² X

1 p<q t I

(I+q+1:t) j

(I+q) j

(I+p+1:t) j

(I+p) j

+ 0

@ ^(I)_{t i}+

J 1

X

j=t+1 i (t+1) j

(I) j

1 A+ 0:5

0

@((s^(I)_{t i})²+ ²_{t i}) +

JX1

j=t+1 i

( ^(t+1)_j )²((s^(I)_j )²+ ²_j) 1 A

with

B_i;t⁽²⁾=

i+JX1

l=t+1

cb_{i;l i;l;t+1}+e^{0:5S(t+1)t+1 i:J 1}

andb_i;lde…ned in(4:13). The parametersr_i^(t), ^(t)_i ,(&^(t)_i )²and(&^(t)_S )² are respectively the equivalents of the correlation coe¢cients r_i in (3:3), the constants _i in (3:4), the variances ²_Z

i and ² of Z_i

and in Section 3.

The following proposition gives the recursive approximated formulas and derive the approxi- mated expression for the MVM.

Proposition 5 For t=I; : : : ; I+J, SCR_t'

XI

i=t J+1

C_{i;t i}e^{Mt(t)i:J 1}b_i;t where b_i;t is de…ned recursively by

- for t i+J, b_i;t = 0, - for t < i+J

b_i;t= 1 1 +c

"

c

i+JX1

l=t+1

b_{i;l i;l;t+1} ^S_{i;t i;l;t} +e^{0:5St+1(t+1)i:J 1 S}_i;t e^{0:5St(t)i:J 1}

#

(4.13)

with

log( ^S_i;t) = 0:5(1 (r^(t)_i )²)(&^(t)_i )²+r_i^(t)&^(t)_i ':

Moreover, at the current date t=I, we have M V M_I 'c

I+JX1

l=I

XI

i=l J+1

C_{i;I i}b_{i;l i;l;I}e^{MI(I)i:J 1}:

It can be seen that ^S_i;t 6= i;t even if the losses of the accident years are independent. It follows thatb_i;t 6=a_i;t and thatM V M_I6=PI

i=I J+1M V M_i;I.

5 MVM for discounted cash-‡ows

We now consider the case where the cash ‡ows are discounted for the calculations of the reserves, the SCRs and the MVM, following the Solvency II framework.

The Fair-Value of Liabilities for accident year i at time t is still de…ned as the sum of the Best Estimate valuation of Liabilities (BEL^(d)_i;t) and the Market Value Margin (M V M_i;t^(d)) of this accident year, but the SCR^(d)_i;t and theM V M_i;t^(d) are now given by the two simultaneous equations

SCR^(d)_i;t = q_99:5% 1

1 +rM V M_i;t+1^(d) M V M_i;t^(d)+L^(1;d)_i;t Tt (5.14) M V M_i;t = c

i+JX1

j=t

E

"

SCR^(d)_i;j (1 +r)^{j t} Tt

#

: (5.15)

Note thatSCR^(d)_i;i+J is still null.

The SCR^(d)_t and theM V M_t^(d) for all accident years are de…ned by the two simultaneous equa- tions

SCR^(d)_t = q_99:5% 1

1 +rM V M_t+1^(d) M V M_t^(d)+L^(1;d)_t Tt (5.16) M V M_t = c

I+JX1

j=t

E

"

SCR^(d)_j (1 +r)^{j t} Tt

#

(5.17) where

L^(1;d)_t =

XI

i=t J+1

L^(1;d)_i;t = XI

i=t J+1

X_{i;t+1 i} 1 +r + 1

1 +rBEL^(d)_i;t+1 BEL^(d)_i;t

=

XI

i=t J+1

0

@Xi;t+1 i

1 +r + 1 1 +rE

2 4

Xi+J

j=t+2

Xi;j i

(1 +r)^{j t 1} T_t+1 3 5 E

2 4

i+JX

j=t+1

Xi;j i

(1 +r)^{j t} T_t 3 5

1 A: Note thatSCR^(d)_I+J = 0.

Throughout this section, we will use the following notation: for I t < l i+j, log( _i;j;l;t) = 1

2

l iX1

k=t i

(s^(t)_k )²+ ²_k +1 2

Xj

k=l i

(s^(t)_k )²+ ²_k Xl t

q=1

(t+q+1:l) k

(t+q) k

2

+ Xj

k=l i

(s^(t)_k )² X

1 p<q l t

(t+p+1:l) k

(t+q+1:l) k

(t+p) k

(t+q) k

and 0otherwise.

5.1 MVM for one accident year By using(5:14)and (5:15), and since

L^(1;d)_i;t = X_{i;t+1 i} 1 +r + 1

1 +rE 2 4

Xi+J

j=t+2

X_{i;j i}

(1 +r)^{j t 1} Tt+1

3 5 E

2 4

Xi+J

j=t+1

X_{i;j i} (1 +r)^{j t} Tt

3 5

the SCR expression for accident yearican be developed as follows SCR^(d)_i;t = q_99:5%

0

@c

i+JX1

j=t+1

E

"

SCR_i;j^(d) (1 +r)^{j t} T_t+1

#

+ Xi;t+1 i

1 +r + 1 1 +rE

2 4

Xi+J

j=t+2

Xi;j i

(1 +r)^{j t 1} T_t+1 3 5 T_t

1 A

c

i+JX1

j=t

E

"

SCR^(d)_i;j (1 +r)^{j t} Tt

# E

2 4

Xi+J

j=t+1

Xi;j i

(1 +r)^{j t} Tt

3 5:

Since Eh

SCR^(d)_i;t jT_ti

=SCR^(d)_i;t , we derive the recursive equations for the SCRs: fort i+J 1

SCR^(d)_i;t = 1 1 +c

0

@q_99:5%

0

@c

i+JX1

j=t+1

E

"

SCR^(d)_i;j (1 +r)^{j t} Tt+1

#

+ Xi;t+1 i

1 +r +E 2 4

Xi+J

j=t+2

Xi;j i

(1 +r)^{j t} Tt+1

3 5 Tt

1 A

1 A (5.18) 1

1 +c 0

@c

i+JX1

j=t+1

E

"

SCR^(d)_i;j (1 +r)^{j t} T_t

# +E

2 4

Xi+J

j=t+1

Xi;j i

(1 +r)^{j t} T_t 3 5

1 A

These equations have to be solved by starting at the …nal time period (i+J 1) and working backwards recursively. But it is necessary to approximate the quantiles of the sum of the SCRs and the current an future discounted cash ‡ows in such a way that we are able to construct analytical recursive formulas for the SCRs.

Let us introduce some notation:

r^(t)_i;j = 1

&^(t)_i;j&^(t)_S;i 0

@

j 1

X

l=t i (t)

i;l(&^(t)_i;l)²+ (&^(t)_i;j)²

JX1

l=j (t) i;l

1 A

where

(&^(t)_i;j)² = ((s^(t)_{t i})²+ ²_{t i}) + Xj

l=t+1 i

( ^(t+1)_l )²((s^(t)_l )²+ ²_l)

(&^(t)_S;i)² =

JX1

j=t i (t) i;j

2(&^(t)_i;j)²+ 2 X

t i j<l J 1 (t) i;j

(t) i;l(&^(t)_i;j)²

and

log( ^(t)_i;j) = log(A^(2);d_i;j;t ) + log(C_{i;I i}) +M_{I i:t i}^(I) ₁+ 0:5S_{I i:t i}^(I) ₁ +

Xj

l=t+1 i (t+1) l

2

4 ^(I)_l + 0:5 ^(t+1)_l ((s^(I)_l )²+ ²_l) Xt I

q=1

( ^(I+q+1:t)_l ^(I+q)_l )² 3 5

+ Xj

l=t+1 i

( ^(t+1)_l )²(s^(I)_l )² X

1 p<q t I

(I+q+1:t) l

(I+q) l

(I+p+1:t) l

(I+p) l

+ ^(I)_{t i}+ Xj

l=t+1 i (t+1) j

(I) l

!

+ 0:5 ((s^(I)_{t i})²+ ²_{t i}) + Xj

l=t+1 i

( ^(t+1)_l )²((s^(I)_l )²+ ²_l)

!

withA^(2);d_i;j;t de…ned in the following proposition.

Proposition 6 For i=t J+ 1; : : : ; I

SCR^(d)_i;t 'C_{i;t i}

JX1

j=t i

e^Mt(t)i:ja_i;j;t where a_i;j;t is de…ned recursively by

- for j=I i; : : : ; J 1 and t i+j+ 1, ai;j;t = 0 - for j=t i; : : : ; J 1, we have

a_i;j;t= 1

1 +c A^(2);d_i;j;t ^(d)_i;j;t A^(1);d_i;j;t where

A^(1);d_{i;t i;t} = r

(1 +r)²e^{0:5St(t)i:t i} A^(1);d_i;j;t =

Xi+j

l=t+1

c

(1 +r)^{l t}a_{i;j;l i;j;l;t}+ r

(1 +r)^{j t+i+2}e^0:5St(t)i:j; j=t+ 1 i; : : : ; J 2 A^(1);d_{i;J 1;t} =

i+JX1

l=t+1

c

(1 +r)^{l t}a_{i;J 1;l i;j;l;t}+ 1

(1 +r)^{J t+1+i}e^{0:5S(t)t i:J 1} and

A^(2);d_{i;t i;t} = r (1 +r)² A^(2);d_i;j;t =

Xi+j

l=t+1

c

(1 +r)^{l t}a_{i;j;l i;j;l;t+1}+ r

(1 +r)^{j t+i+2}e^{0:5St+1(t+1)i:j}

!

; j=t+ 1 i; : : : ; J 2

A^(2);d_{i;J 1;t} =

"_{i+J 1} X

l=t+1

c

(1 +r)^{l t}a_{i;J 1;l i;J 1;l;t+1}

#

+ 1

(1 +r)^{J t+1+i}e^{0:5S(t+1)t+1 i:J 1}

! : and

(d)

i;j;t=e^{0:5(1 (r(t)i;j)2)(&(t)i;j)2+r(t)i;j&(t)i;j'}: Moreover, at the current date t=I, we have

M V M_i;I^(d)'cCi;I i J 1

X

j=I i

e^MI(I)i:j

"_{i+j 1} X

l=I

a_{i;j;l i;j;l;I}

# :

5.2 MVM for aggregated accident years

For completeness sake, we give the MVM for all accident years. By using the same arguments as previously, we derive the recursive equations for the SCRs for all accident years: for t=I; : : : ; I+

J 1

SCR^(d)_t (5.19)

= 1

1 +c 0

@q_99:5%

0

@c

I+JX1

j=t+1

E

"

SCR^(d)_j (1 +r)^{j t} Tt+1

# +

XI

i=t J+1

0

@X_{i;t+1 i} 1 +r +E

2 4

Xi+J

j=t+2

X_{i;j i}

(1 +r)^{j t} Tt+1

3 5

1 A Tt

1 A

1 A

1 1 +c

0

@c

I+JX1

j=t+1

E

"

SCR^(d)_j (1 +r)^{j t} Tt

# +

XI

i=t J+1

E 2 4

Xi+J

j=t+1

X_{i;j i} (1 +r)^{j t} Tt

3 5

1 A: Let

r_i;j^(t) = 1

&^(t)_i;j&^(t)_S XI

k=t J+1 J 1

X

l=t k (t)

k;l(&^(t)_i^k;j^l)² where

(&^(t)_i;j)² = ((s^(t)_{t i})²+ ²_{t i}) + Xj

l=t+1 i

( ^(t+1)_l )²((s^(t)_l )²+ ²_l)

(&^(t)_S )² = X

t J+1 i;k I J 1

X

j=t i JX1

l=t k (t) i;j

(t)

k;l(&^(t)_i^k;j^l)² and

log( ^(t)_i;j) = log(B_i;j;t^(2);d) + log(C_{i;I i}) +M_{I i:t i}^(I) ₁+ 0:5S_{I i:t i}^(I) ₁ +

Xj

l=t+1 i (t+1) l

2

4 ^(I)_l + 0:5 ^(t+1)_l ((s^(I)_l )²+ ²_l) Xt I

q=1

( ^(I+q+1:t)_l ^(I+q)_l )² 3 5

+ Xj

l=t+1 i

( ^(t+1)_l )²(s^(I)_l )² X

1 p<q t I

(I+q+1:t) l

(I+q) l

(I+p+1:t) l

(I+p) l

+ ^(I)_{t i}+ Xj

l=t+1 i (t+1) j

(I) l

!

+ 0:5 ((s^(I)_{t i})²+ ²_{t i}) + Xj

l=t+1 i

( ^(t+1)_l )²((s^(I)_l )²+ ²_l)

!

withB^(2);d_i;j;t de…ned in the following proposition.

Proposition 7 For i=t J+ 1; : : : ; I SCR^(d)_i;t '

XI

i=t J+1

Ci;t i J 1

X

j=t i

e^Mt(t)i:jbi;j;t

where b_i;j;t is de…ned recursively by

- for j=I i; : : : ; J 1 and t i+j+ 1, b_i;j;t= 0