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The paper proposes a new methodological approach for the product performance analysis into the actuarial context. Two indexes are proposed as restyled versions of the corresponding most popular ones: They have been adapted into the actuarial assessment preserving the plainness in the interpretation of the numerical results. The paper offers a practical implementation of the new approach in the case of a specific contract, containing itself innovative profiles: It concerns a life annuity in which the installments are scaled by a demographic index and contains an embedded option linked to the financial profit participating quota. It is a new life product linked at the same time to the financial and demographic volatility. The product project is studied in its profitability performance assuming stochastic hypotheses for the financial and demographic systematic risks. The indexes are implemented in a conditional quantile simulated framework and tables and graphs illustrate their trends as function of time. The results give an example of the usefulness of the proposed indexes in the phase of decisions about the product design feasibility. Moreover some suggestions concerning the consumer’s perception of the contract profitability are obtained by means of a utility-equivalent fixed annuity.

The new regulations currently in progress in the insurance sector make the estimation procedures involving the capital amount to be allocated, according to the requirements of Solvency II, a compelling issue; a primary constraint for an insurance company is basically the entrepreneurial capability for applying the correct strategies, as regards solvency and related issues, which also accord with these guidelines. This issue is of course linked to strategic investment decisions concerning business policies, such as defining and allocating new products. The current macroeconomic environment also requires careful consideration of several variables, which impact on the ability of a company to confront adverse scenarios (see for example the stress test implemented for insurance companies in [

The paper is organized as follows: in Section 2 the profit and risk parameters are presented and discussed; in Section 3, aiming at an applicative procedure of the scheme proposed in the previous section, the new contract architecture is defined and the two indexes are specified in their expression in the considered contractual case; in Section 4 the model is implemented in a stochastic context for both the financial and demographic risk drivers. The performance analysis is developed by means of simulating techniques within a conditional quantile framework and numerical evidences are collected in tables, illustrating the profitability index behavior as a function of the contract duration. In Section 5 an analysis of the insured’s point of view is considered. In Section 6 some final conclusions are given.

The company’s performance and its overall efficiency are well described by the profitability ratios in [

In light of these considerations we propose a new version of two traditional and popular profitability indexes [

As with the ROE index, the AROE, referring to one balance sheet year, is given by the ratio of profit to equity. When AROE is employed and its results are studied as performance measures, essential to the aim of an efficient financial interpretation is how the profit and the equity are valued. We will construct the ratio setting as numerator the profit the company achieves through its business and as denominator the surplus the business provides, in the context referred to above (the difference between prospective loss and retrospective gain). The valuations at time t are relative to the beginning of the t-th year in a forward perspective, in the sense that the information flow the insurer is given is at hand at time 0. The numerical results for AROE express how much the total amount collected by the insurer to manage the product yields in a certain time interval. This is a very common profitability measure, quick to communicate and easily interpreted, for which good performances will be represented by high percentages.

The second index we will deal with is based on the Return on Asset concept. This ratio seems to be very informative in the case of annuities as they are contracts producing large assets, especially if compared with other kinds of life insurance contracts. As suggested in [

The results derived for this index quantify the efficiency the company achieves in using the assets arising from the product under consideration. The ARG index measures the efficiency the company shows in managing its investment in assets for generating profit and of course in this case too, the higher the percentage, the better. The actuarial calculations will be performed in a forward perspective. In this case, as with the AROE, useful indications can be deduced from the description of the evolution over time of the stochastic process ARG. These include the measure of how profit is yielded from the financial resources invested in the product itself.

We develop the application of the proposed performance analysis to a specific contract, falling under the category of the participating policies. The contract contains innovative characteristics, interesting within the financial and demographic scenarios in which the insurance activity is performed.

When a country’s financial systems provide strong social security, the demand for voluntary life annuity is rather soft as a consequence. On the contrary, because of the contemporary financial crisis affecting the majority of Western economies, the present scenario is characterized by a weak social security system. In such situations, an increasing demand for voluntary life annuity as supplementary pension or pension fund arises [

From the actuarial evaluation viewpoint, the participating business, when connected to a pension annuity policy, triggers a complex contractual structure. Correct contract management requires a fine awareness, applied at the time of issue, of the evolution in time of both the insurer’s obligations and the income capabilities or, more synthetically, a competent awareness of the design of the payment phase, as is pointed out in [

margin ( t + 1 ) = [ Premium ( t ) + V ( t ) ] ( 1 + i ) − p [ Benefit ( t + 1 ) + V ( t + 1 ) ] − ExpensesandTaxes ( t + 1 ) (1)

where p is the probability that the annuitant is living at time t + 1 and i is a suitable interest rate. The formula clearly shows that the margin in t + 1 comes from the insurer’s obligations―in case of life―to pay the Benefit and constitute the reserve V ( t + 1 ) at time t + 1 , constituted by the Premium eventually collected together with the reserve at time t, both capitaòized. Expenses and taxes have to be subtracted.

In the paper we will assume that the annuitants participate in the investment profit and in the mortality experience, thus transferring both the financial and the demographic volatility to capital markets, in a profit sharing mechanism. The benefit amount, although influenced by the volatility in financial and demographic scenarios, will be constrained within appropriate limits, as guarantees for both insured and insurer.

As long ago as 2007, [

From the perspective of prudential fund management, within the context of fair safeguarding of the insureds’ rights, the kind of margin the insureds participate in is specified for each life annuity participating product. From these general observations, participating life annuity structures are different from country to country and they can even differ among insurance companies of the same country. Recently [

The contract we are going to propose is a life annuity characterized by installments contextually containing an imbedded option linked to the financial margin participating quota and an indexation by a demographic ratio. The monetary amount we call margin has to be accurately handled. Insurers are institutionally required to maintain minimum solvency margins for facing future risk, thus guaranteeing their solvency. Therefore, the margin has to be considered not only as an amount to distribute but, first of all, as the amount to reserve and to accrue for adverse evolution of the business due to its randomness. From these considerations, a very careful quantification of the margin participating quota has to be realized. The participating quota has a very meaningful strategic role: the insurer has to attribute to it the right value, realizing a form of equilibrium between, on one hand, its solvency obligations and the advantages of keeping more money free to invest in several industrial activities (new product perspectives, new strategic possibilities) and, on the other hand, the advantages of distributing more money to the insureds (serving the aims of competitiveness and product attractiveness).

The careful choice of the participating quota impacts on the premium calculation and on the profitability results. This choice can be performed by means of scenario analysis based on variable values of the participating quota, as shown in the numerical example in Section 4. Moreover, from the insureds point of view, the attractiveness of the product is based on the clear knowledge of the minim value of the installments guaranteed by the threshold inserted in the contract. There is little doubt that the non-systematic demographic risk component, also called micro-longevity risk [

According with the aforementioned overview, we now consider a contractual scheme in which the installments contain the embedded option linked to the financial margin participating quota and are scaled by a demographic index based on the ratio between the forecast projected survival trend and the observed one.

Following the basic line in [

S I x , t = p t x proj p t x obs (2)

where p t x proj is the survival probability for an annuitant aged x being alive at age x + t , inferable from a proper projected demographic model, and p t x obs is the analogous survival probability deduced from the observed data. In particular the insurer, referring to the cohort of annuitants aged x at contract inception, at each payment time t , compares the probability p t x proj (previously estimated by means of a projected model) to p t x obs , being referred to the fraction of the population which is still living at time t . Moreover we assume:

S t = max { min { S I x , t , S I max } , 1 } (3)

where S I max represents the maximum level obtained by an appropriate contractual equilibrium between marketing appeal and capability to meet the demographic risk.

Then, within the context of a market, where insurance policies with profit currently represent an interesting potential in the insurance sector, we add in the contract the participation of the insured to the financial profits eventually arising from the investments. In this framework, we assume that the profit sharing is based on the differences among income, capital gains and losses [

On the basis of the aforementioned considerations, we propose a new model, where, together with the longevity risk sharing structure, we introduce a participation rate applied to financial result for the period, following the lines in [

So, the benefit flow b ˜ t payable to the insureds can be represented as follows:

b ˜ t = { b t S t + α ( R t − γ ) if ( R t − γ ) > 0 b t S t if ( R t − γ ) ≤ 0 (4)

where b t is the basic installment. Equation (4) guarantees at least the basic installment to the annuitant. The architecture of the annuity described above achieves a balance market appeal on the part of annuitants, who focus on the financial competitiveness of their investments, and the point of view of the insurers, who pay attention to profitability combined with decreasing levels of risks.

According to the scheme described and recalling [

V t = ∑ i = t ∞ ( b ˜ i 1 ( T ≤ i ≤ K ( x ) ) − P i 1 ( i < τ / K ( x ) > i ) ) v ( t , i )

The indicator function 1 ( T ≤ i ≤ K ( x ) ) takes the value 1 if T ≤ i ≤ K ( x ) ( K ( x ) being the random curtate lifetime of an annuitant aged x at the issue time of the contract), 0 otherwise, whilst the indicator function 1 ( i < τ / K ( x ) > i ) takes the value 1 if i < τ if the insured is alive, 0 otherwise.

We pose the financial result R t + 1 of the (t + 1)-th accounting period as follows:

R t + 1 = ( V t + P t 1 ( i < τ / K ( x ) > i ) ) v ( t + 1 , t ) − ( b ˜ t + 1 + V t + 1 ) 1 ( T ≤ t + 1 ≤ K ( x ) ) .

If R t + 1 is greater than γ , the insurer pays the additional bonus α ( R t + 1 − γ ) to the insured ( 0 < α < 1 ) .

Hence we can write the total (t + 1)-th financial result T R t + 1 as:

T R t + 1 = R t + 1 − α ( R t + 1 − γ ) + = min ( R t + 1 , ( 1 − α ) R t + 1 + α γ )

and the surplus at time t + 1 is:

S t + 1 = ∑ j = 0 ∞ X j v ( t + 1 , j ) (5)

where

X j = ( P j 1 ( j < τ / K ( x ) > J ) − b ˜ j 1 T ≤ j ≤ K ( x ) ) sign ( t − j ) (6)

The two indexes presented in Section 2 are specified in the contractual case described in Section 3.1. We can express the AROE index as the ratio between profit to surplus:

AROE ( t + 1 ) = R t + 1 − α ( R t + 1 − γ ) + ∑ j = 0 ∞ X j v ( t + 1 , j ) (7)

with R t + 1 the financial result of the (t + 1)-th accounting period.

Within a managerial context, information about the trend of the stochastic process AROE can be particularly useful if applied to asset management policy, for example in choosing the profit-participation level or setting the cap and floor boundaries.

The ARG index is

ARG ( t + 1 ) = S t + 1 R G t + 1 (8)

with S t + 1 given by formula (7) and R G t + 1 the retrospective gain produced by the life annuity contract at time t + 1, that is:

R G t + 1 = ∑ s = 0 K ( x ) ∧ t X s v − 1 ( s , t + 1 )

where x ∧ y = min ( x , y ) and K ( x ) the random life-time of an individual aged x .

In this section we want to show how the profitability indexes presented in Section 2 can describe the performance of a homogeneous portfolio of policies structured as proposed in Section 3. We refer to a portfolio of 1000 immediate annuities, with deferred installments, issued to lives aged 65. In this application each insured pays a single premium at the issue time (the example can be easily extended to the case of a deferred life annuity with periodic premiums). The pure premium paid by each insured will be calculated at a fixed technical rate and according to a suitable projected probability law, taking into account the variable installments of the described contractual architecture, the assumption is that the benefits due to each annuitant, if alive at the payment time, are adjusted by the scale factor defined in (2) and increased by α ( R t + 1 − γ ) , if positive. In order to investigate the values of the survival ratio in formula (2), based on the comparison between the observed survival probabilities and the forecast ones, we consider the period from 1970 to 2009 in our assumptions. We use Italian male survival annual rates: the observed probabilities at the denominator are drawn by the Human Mortality Database website and we choose to describe the probabilities at the numerator by the Lee Carter model^{1} with parameters estimated as in 1970, on the basis of the dataset available at that date.

^{1}The Lee-Carter (LC) model [

ln m x , t = α x + β x k t + ε x , t

with

− m x , t = death rate at the agex in the year t

− α x = age-specific component (not dependent on time) at age x

− β x = age-specific component (not dependent on time), which incorporates mortality variations linked with variations of the general mortality level (sensitivity parameter)

− k t = time component which expresses the general mortality level in year t

− ε x , t = error term with zero mean and finite variance.

In particular, α x is the mean of ln m x , t throughout the observation period. In order to estimate the parameters, Lee and Carter used the following normalizing positions: ∑ t k t = 0

so:

α ¯ x = ∑ t ln m x , t n = ln ( ∏ t m x , t ) 1 n

and

∑ x β x = 1

hence

k ¯ t = ∑ t ln m x , t − ∑ x α x

β ¯ x can be obtained by a linear regression and:

ln m ¯ x , t = α ¯ x + β ¯ x

Lee and Carter modeled k t by means of an ARIMA ( 0 , 1 , 0 ) process.

In

In order to limit the impact of this index on the annuity payments, we assume S I max = 1.2 , then the benefit at time t is:

b ˜ t = b t max { min { S I x , t , 1.2 } , 1 } and for simplicity purposes we set b t = 1 . The AROE and ARG functions are based on investment and discounting operations. Within the performance analysis from the insurer’s point of view, for describing both the evolution in time of the rate of return on investment of collected premiums and the dynamics of the discounting rate required for valuing the reserves, we choose the Vasicek mean reversion model [

d r t = β ( μ − r t ) d t + σ d W t

with β , μ and σ positive constants and W t a standard Wiener process. Of course we calibrated the model coherently with the demographic hypotheses and we refer to the monthly yield over the period January 1960-January 1970, on a basket of Treasury Italian bonds. Such parameters are reflecting the results of the insurer’s actual investment strategy. So we obtained μ = 0.0389 , β = 0. 3263 , σ = 0.054 . We pose the annual expenses of the portfolio equal to 60. The single premium is calculated at the technical rate of 2% and on the basis of the aforementioned Lee-Carter model, considering two specific values for α , i.e. α = 2 0 % and α = 8 0 % .

The numerical analysis was performed using the conditional quantile technique carried out by a simulation procedure applied to the empirical distributions of the stochastic variables AROE and ARG as described in section 3.2 and studied in function of the time. Fixing a threshold value depending on the confidence interval, the results were interpreted as the worst cases scenarios with an occurrence probability lower than the threshold and their average can be regarded as the expected worst occurrence corresponding to that level of confidence. Moreover the values for expected ARG and AROE was interpreted as measures of mean profitability per policy.

Beginning with the AROE index, the simulation procedure was implemented with 10,000 AROE values for each t and the expected values and the C-quantiles with confidence level 0.95 and 0.99 were obtained. The results, calculated in t = 0 , 5 , 10 , 15 , 20 , 25 , 30 , 35 , 40 are shown in

Likewise for the AROE, the ARG analysis was performed by a simulation procedure for getting the empirical distribution of ARG values for each t, implemented on 10,000 ARG outputs for each value of t. Again in the case of an annuitant aged 65 and the participating quota of 20%,

t | E[AROE]% | Minimum | Maximum | cq (0.95) | cq (0.99) |
---|---|---|---|---|---|

0 | 2.34542 | 1.87298 | 3.34245 | 2.19645 | 2.00290 |

5 | 2.00863 | 1.57663 | 3.51297 | 1.95231 | 1.93460 |

10 | 1.64217 | 1.36129 | 2.57190 | 1.62701 | 1.60120 |

15 | 1.58753 | 1.24358 | 2.15253 | 1.57734 | 1.52347 |

20 | 1.50907 | 1.24286 | 2.04465 | 1.48923 | 1.45209 |

25 | 1.45953 | 1.22108 | 1.86286 | 1.44100 | 1.42147 |

30 | 1.43741 | 1.21368 | 1.82375 | 1.42253 | 1.40054 |

35 | 1.41934 | 1.20237 | 1.79042 | 1.40866 | 1.38234 |

40 | 1.39442 | 1.20857 | 1.78789 | 1.38342 | 1.36479 |

t | E[AROE]% | Minimum | Maximum | cq (0.95) | cq (0.99) |
---|---|---|---|---|---|

0 | 0.67479 | 0.34287 | 1.68634 | 0.66452 | 0.65280 |

5 | 0.53243 | 0.25872 | 1.55239 | 0.51239 | 0.50342 |

10 | 0.47354 | 0.20432 | 1.49752 | 0.46231 | 0.45265 |

15 | 0.42496 | 0.15321 | 1.45321 | 0.40856 | 0.39312 |

20 | 0.37553 | 0.12865 | 1.40729 | 0.34569 | 0.33264 |

25 | 0.36333 | 0.11867 | 1.39634 | 0.35001 | 0.33142 |

30 | 0.35233 | 0.11745 | 1.38634 | 0.31987 | 0.30963 |

35 | 0.35005 | 0.10998 | 1.37523 | 0.31867 | 0.30856 |

40 | 0.34758 | 0.09723 | 1.37231 | 0.30645 | 0.30021 |

t | E[ARG]% | Minimum | Maximum | cq (0.95) | cq (0.99) |
---|---|---|---|---|---|

0 | 0.89367 | 0.87739 | 0.91924 | 0.89418 | 0.87922 |

5 | 0.97520 | 0.96826 | 1.00013 | 0.96401 | 0.96031 |

10 | 1.15109 | 1.15108 | 1.19647 | 1.02297 | 0.98735 |

15 | 1.27279 | 1.19356 | 1.39885 | 1.20453 | 1.19428 |

20 | 1.36341 | 1.24928 | 1.57392 | 1.28398 | 1.25465 |

25 | 1.42728 | 1.28308 | 1.74562 | 1.32298 | 1.29120 |

30 | 1.48836 | 1.29481 | 1.81914 | 1.34175 | 1.32631 |

35 | 1.50740 | 1.31042 | 1.87954 | 1.36789 | 1.32962 |

40 | 1.52289 | 1.31896 | 1.90108 | 1.37072 | 1.34907 |

t | E[ARG]% | Minimum | Maximum | cq (0.95) | cq (0.99) |
---|---|---|---|---|---|

0 | 0.25386 | 0.24317 | 0.27967 | 0.24985 | 0.243775 |

5 | 0.28530 | 0.26984 | 0.29764 | 0.27452 | 0.26987 |

10 | 0.31086 | 0.27862 | 0.33275 | 0.30845 | 0.28653 |

15 | 0.33818 | 0.27923 | 0.40243 | 0.32579 | 0.31289 |

20 | 0.40325 | 0.29456 | 0.44276 | 0.33478 | 0.32674 |

25 | 0.45926 | 0.31645 | 0.63407 | 0.34679 | 0.33376 |

30 | 0.51820 | 0.32032 | 0.71324 | 0.36417 | 0.35786 |

35 | 0.52083 | 0.33631 | 0.76497 | 0.38785 | 0.36897 |

40 | 0.52331 | 0.33953 | 0.77428 | 0.39002 | 0.37634 |

C-quantiles.

In this section we provide some brief guidelines about the consumer’s point of view.

Consumer protection is one of the main concerns within the current activities of the Committee on Consumer Protection and Financial Innovation (CCPFI) of the European Insurance and Occupational Pension Authority (see for instance [

This problem invests into areas concerning vigilance over products and governance systems. The current financial literature is paying increasing attention to the varied world of decision making under uncertainty from the insured’s perspective. The many aspects of the question are treated by means of different approaches, which range from the expected utility theory to methodologies of behavioral finance as in [

[

U = E [ ∑ i = 1 ω − x − 1 ϕ i i p x b ˜ i 1 − γ ] (9)

with γ the relative risk aversion, ϕ the discount factor arising from the insureds subjective preferences. With reference to the numerical example presented in the previous section, we consider three different risk aversion levels, which represent, respectively, low, medium and high risk aversion ( γ = 2 , 5 , 10 , respectively) as in [

In both the two tables it is evident that the installments of all the UEAs decrease with an increasing risk aversion, whilst they increase with a decreasing patience level. The constant installments obtained for the UEAs are greater than 1, as a consequence of the peculiarities of the equivalent insurance contract.

For every fixed impatience level, when the risk aversion increases, the constant installments of the UEAs decrease; this is congruent with a high risk aversion, which implies low installments, while avoiding randomness.

For every fixed aversion level, on the contrary, lower impatience is satisfied by lower installments.

All the values in

The overall trend of the expected utility is shown in

It is also evident that the expected utility strictly decreases when the subjective discount factor increases.

A further discerning aspect could depend on the premium amount in the cases of utility-equivalent certain annuities, consistently with their risk aversion and impatience profile. In

Subjective discount factor | Low risk adverse | Medium risk adverse | High risk adverse |
---|---|---|---|

1.214637 | 1.188294 | 1.140241 | |

1.236498 | 1.192035 | 1.167642 | |

1.253928 | 1.203672 | 1.173338 |

Subjective discount factor | Low risk adverse | Medium risk adverse | High risk adverse |
---|---|---|---|

1.225520 | 1.194290 | 1.157643 | |

1.240045 | 1.203625 | 1.160199 | |

1.264984 | 1.217635 | 1.187646 |

Annuity | Premium |
---|---|

Participating life annuity ? Financial and demographic indexing | 8.214 |

Participating life annuity ? Financial indexing | 8.055 |

Constant life annuity | 7.985 |

Utility-equivalent annuity ? normal individuals with medium risk aversion | 9.582 |

cases as follows:

1) the ten-years annuity as considered in the paper (financial and demographic indexing, ( α = 0.40 ) ;

2) the ten-years annuity with only the financial indexing ( α = 0.40 ) ;

3) the ten-years life annuity with constant unitary installments;

4) the ten-years utility-equivalent annuity. As an example considered for an individual normally patient and with medium risk aversion, with constant installments equal to 1.203625 (case of Vasicek interest rates).

We sum up the results outlined throughout the paper. Referring to the values of AROE and ARG in figures and tables, and pointing up their informative potential, we can argue that the participating annuity we have depicted is characterized by a profitability for the insurer that, even when strictly decreasing as in the AROE case, moves within a narrow interval and tends to steady right from the second half of the contract duration. The trend of the AROE index represented in its worst cases by means of the C-quantiles points out these considerations. The ARG increasing trend shows the improving performance of the business in the profitability of its assets. The range of ARG and AROE values is narrower when α is higher. The negative impact of higher values of the participating quota is clear and easily conceivable.

The aforesaid analysis mainly focuses on the insurer’s point of view. Nevertheless, it is important to point out conceivable motives from the insureds’ perspective. In our numerical example we can observe that the projected survival probabilities fit well the observed ones, so the survival index is always greater than 1. Nevertheless, the paper does not neglect the point of view of the insured, an aspect more and more relevant in the product efficiency analysis. The topic is deepened in the Utility Equivalent Annuity framework, providing tables quantifying the perception of the new product in terms of the traditional one. In our example, on the base of different degree of risk aversion and of subjective discount factor, the analysis points out that in any case the insured considers equivalent to the new variable product, a traditional annuity with benefits larger than 1. A table reports the single premiums of comparable annuities and a graph designs the trend of the expected life-time utility.

We think that this research line could be certainly interesting in a context paying more and more attention to the insurance product profitability combined with the unavoidable consumer care. In the paper we outline the scheme for a clear analysis of a new life insurance product, consisting in three basic steps: the product analytical description, the study of the its profitability from the point of view of the insurer and the exam of the product perception on the part of the insured, in order to have a measure of their potential interest in the product itself.

Di Lorenzo, E., Orlando, A. and Sibillo, M. (2017) Measuring Risk-Adjusted Performance and Product Attractiveness of a Life Annuity Portfolio. Journal of Mathematical Finance, 7, 83-101. https://doi.org/10.4236/jmf.2017.71005