2.1. IFRS 17: concepts and principles

2.1.1. Initial recognition: CSM and LC

Consider a group of contracts acknowledged during the reporting period t ₀. The CSM constitutes a tack of the asset or liability associated with the group of insurance contracts, representing the unearned profit that the entity expecting recognizing in future service provisions. The CSM cannot take a negative value. Hence, if the insurance liability value exceeds any cash, liabilities from contracts on that date are considered as an LC incorporated into the liability value rather than contributing to the CSM (International Accounting Standards Board, 2017a).

Upon recognition of the group of contracts, either the contractual service margin $\hbox{CSM}^{( t_0) }$ or the loss component $\hbox{LC}^{( t_0) }$ will be determined. Let $P^{( t_0) }$ denote the premium amount at t ₀. The following equations define the (initial) recognition at t ₀ (Palmborg et al., Reference Palmborg, Lindholm and Lindskog2021a):

(3)$$\hbox{CSM}^{( t_0) } = \left(P^{( t_0) } - L_{RC}^{( t_0) } \right)^ + $$

(4)$$\hbox{LC}^{( t_0) } = \left(L_{RC}^{( t_0) } - P^{( t_0) } \right)^ + $$

It is crucial that the initial recognition does not incorporate the influence of discount rates and the time value of money. Additionally, during the initial recognition, any collective unearned profits are not obligatory to be allocated to profit or loss (International Accounting Standards Board, 2017a). In accordance with the standard, the CSM remaining at the end of the reporting period is evenly distributed among the units of coverage provided and the anticipated remaining coverage units throughout the period (International Accounting Standards Board, 2017b).

2.1.2. The weighting of the contractual service margin (W _t)

The CSM encompasses weights W _t, reflecting the rate of unearned profit for the period post-t at time t. W _t is a random variable within the range [0,  1], known at time t. W _t acts as a weighting factor that ensures the profit recognition process is proportional to the remaining coverage obligations, allowing a more accurate reflection of unearned profits as they are earned over time. This weighting captures the time-based progression of profit realization in accordance with the remaining service provided by the contract. 1 − W _t signifies the proportion of unearned profit at period t allocated to profit or loss at time t.

The quantity of coverage units within an insurance group is determined by the benefits conferred under a contract and its anticipated coverage period (International Accounting Standards Board, 2017a). Let $B_s^{( k) }$, $s = t + 1,\; \, \dots ,\; \, t_0 + \tau$, represent the coverage provided under the k-th contract in period s. Additionally, $T_t^{( k) }$ denote the time remaining from time t to the contract end date. Equation (5) defines the coverage unit $\hbox{CU}_s^{( t, k) }$ in period s for T ≤ s:

(5)$$\hbox{CU}_s^{( t, k) } {\, \colon = }B_s^{( k) } {\opf I}_{\{ T_t^{( k) } > s - t - 1\} }$$

Here, ${\opf I}$ is an indicator function, and $\hbox{CU}_s^{( t, k) }$ is known at time t. When $T_t^{( k) } = 0$, $\hbox{CU}_t^{( t, k) } = B_t^{( k) }$ represents the last unit of coverage provided under the contract, and $\hbox{CU}_t^{( t, k) }$ is defined for (t − 1,  t]. The coverage unit $\hbox{CU}_s^{( t) }$ is the total of the coverage units $\hbox{CU}_s^{( t, k) }$ summed over the index k (Palmborg et al., Reference Palmborg, Lindholm and Lindskog2021a).

W _t is defined as the ratio of the expected number of remaining coverage units at time t to the total number of coverage units for the group. The coverage unit is described by Palmborg et al. (Reference Palmborg, Lindholm and Lindskog2021a):

(6)$$\eqalign{& \hbox{ERCU}^{( t) }{\, \colon = } {\opf E}_t \left[\sum_{s = t + 1}^{t_0 + \tau} \hbox{CU}_s^{( t) } \right]\cr & W_t = {\hbox{ERCU}^{( t) }\over \hbox{CU}_t^{( t) } + \hbox{ERCU}^{( t) }}}$$

2.1.4. Contractual service margin and loss component

At the conclusion of the reporting period, the carrying amount of the CSM for a group of contracts encompasses the impact of several factors. These factors include the effects from newly initiated contracts, the interest incorporated into the CSM throughout the period, alterations in fulfillment cash flows, and the influence of currency exchange differences on the CSM (International Accounting Standards Board, 2017a). Let's consider (W _t)_t∈T and (U _t)_t∈T.

If CSM^(t−1) ≥ 0 and LC^(t−1) : = 0, are,

(7)$$\Delta_1 = {d_{t_0, t-1}\over d_{t_0, t}} \hbox{CSM}^{( t-1) } + {d_{t_0, t-1}\over d_{t_0, t}} L_{FS, t_0}^{( t-1) } - L_{RC, t_0}^{( t) } + P^{( t) }$$

becomes $\hbox{CSM}^{( t) } {\, \colon = }\ W_t \Delta _1^ +$ and $\hbox{LC}^{( t) } {\, \colon = }\ \Delta _1^-$ for the next period.

If CSM^(t−1) = 0 and LC^(t−1) > 0, are,

(8)$$\Delta_2 = -\hbox{LC}^{( t-1) } - U_t \left(L_{RC}^{( t) } - L_{RC}^{( t-1) } + {d_{t_0, t-1}\over d_{t_0, t}} L_{FS, t_0}^{( t-1) } - L_{RC, t_0}^{( t) } \right)$$

(9)$$\Delta_3 = -\Delta_2^- + {d_{t_0, t-1}\over d_{t_0, t}} L_{FS, t_0}^{( t-1) } - L_{RC, t_0}^{( t) } + P^{( t) }$$

calculate with $\hbox{CSM}^{( t) } {\, \colon = } \ W_t \Delta _3^ +$ and $\hbox{LC}^{( t) }{\, \colon = } \ \Delta _3^-$ for the next period (Palmborg et al., Reference Palmborg, Lindholm and Lindskog2021a).

$L_{FS, t_0}^{( t-1) }$ represents the liability value at t − 1 allocated to cash flows occurring after time t measured with initial discount rates at t ₀. ($L_{RC, t_0}^{( t_0) } = L_{RC}^{( t_0) }$, $L_{FS, t_0}^{( t_0) } = L_{FS}^{( t_0) }$). (${d_{t_0, t-1}}/{d_{t_0, t}}) L_{FS, t_0}^{( t-1) }$ is the adjustment for the liability measured with the initial discount rate (at time t ₀). $d_{t_0, t}$ is the discount factor between t ₀ and t, known at time t ₀ ($d_{t_0, t_0} = 1$). (${d_{t_0, t-1}}/{d_{t_0, t}}) L_{FS, t_0}^{( t-1) } - L_{RC, t_0}^{( t) }$ is the change in the value of the future service liability during the period, measured at the discount rates determined at the initial recognition of contracts. P ^(t) and L ^(t−1) − L ^(t) denote premiums in period t and the change in total liability value, respectively. It is needed to assess whether an economically disadvantaged group results in a CSM at the end of the period. At the end of the contract, both the CSM and the LC equal zero ($\hbox{LC}^{( t_0 + \tau ) } = 0$, $\hbox{CSM}^{( t_0 + \tau ) } = 0$).

2.1.5. Profit or loss (P&L)

Let I _t represent the net cash flow in (t − 1,  t]. The following equation shows profit or loss for reporting time t (Palmborg et al., Reference Palmborg, Lindholm and Lindskog2021a):

(10)$$\hbox{P\&L}_t = L^{( t-1) } + \hbox{CSM}^{( t-1) } + P^{( t) } - \left(L^{( t) } + \hbox{CSM}^{( t) } \right)- I_t$$

More comprehensive information regarding the principles of IFRS 17 can be found in Palmborg et al. (Reference Palmborg, Lindholm and Lindskog2021a, Reference Palmborg, Lindholm and Lindskog2021b).

3.1. Mortality model

Life insurance companies commonly utilize fixed mortality rates in calculating present values. However, this deterministic approach fails to capture the current mortality rates accurately. In contrast, introducing randomness through stochastic mortality models can address this limitation. However, stochastic models do not reflect the effects of shocks on mortality rates, as observed during the recent COVID-19 pandemic. The present study integrates a mortality model with a jump component. First, the Poisson log-bilinear model, which we consider as the permanent jump model, is addressed in this section. Later, the temporary mortality jump model is defined to explain unexpected changes in mortality.

3.1.1. Poisson log-bilinear Lee–Carter model

The paper is based on calculating the parameters of the Poisson log-bilinear model for death rates, which was put out by Brouhns et al. (Reference Brouhns, Denuit and Vermunt2002). In this model, a Poisson regression model is incorporated into the Lee–Carter model for mortality rates. Herein, let D _x,t represent the number of deaths in the population at the age of x in the period t. The probability distribution of D _x,t conditioned on E _x,t and μ _x,t is expressed as a Poisson distribution:

(11)$${\cal L}( D_{x, t} \mid E_{x, t},\; \mu_{x, t}) = \hbox{Pois}( E_{x, t} \mu_{x, t}) $$

At age x, in period t, E _x,t shows risk exposure, and μ _x,t denotes the mortality rate. The formulation of the mortality rate model is presented by

(12)$$\log( \mu_{x, t}) = a_x + \beta_x \kappa_t,\; \quad \kappa_t = \delta + \kappa_{t-1} + \xi_t$$

Here, a _x represents the age effect, κ _t expresses the mortality trend, and β _x is the age sensitivity of the change in κ _t. Notably, κ _t is a Gaussian random walk, where δ is the drift parameter and ξ _t is an independent and identically distributed (i.i.d.) variable with $\xi _t \sim N( 0,\; \, \sigma _k^2)$. The constraints $\sum _t \kappa _t = 0$ and $\sum _x \beta _x = 1$ are maintained. It is assumed that the mortality rates remain constant after the age of 90. Furthermore, the mortality trend is characterized as a non-stationary series. The derivation of survival probabilities using this model is presented in detail in Appendix A.

3.1.2. Mortality risk including temporary jump

The mortality model, incorporating the temporary jump, is expressed as

(13)$$\mu_{x, t}^j = \mu_{x, t} e^{H_{x, t}}$$

Here, μ _x,t represents the force of mortality without the jump, as defined in equation (12). The transient mortality jump process H _x,t is defined by

(14) $$H_{x, t} = \sum_{\,j = 1}^{\infty} b_j B_j( x) \exp( -k_j ( t-\tau_j) ) {\opf I}_{( t \geq \tau_j) }$$

Here, τ _j represents the time of the j-th adverse mortality event, and b _j is the maximum severity of the j-th jump event. The time τ _j can be modeled as the arrival times of a Poisson process with a jump size b _j > 0. B _j(x) distributes the jumping effect of the mortality rate among different ages (0 ≤ B _j(x) ≤ 1). For instance, in the flu epidemic of 1918, people younger than 50 were more likely to be affected, while the situation was vice versa for COVID-19. As demonstrated by these examples, the impact of the jump in the mortality rate may vary according to age. The term $\exp ( -k_j ( t-\tau _j) )$ is used to model the temporal nature of the shock, k _j can be estimated using historical data or set to 1, as commonly done in the literature. The higher k _j implies a faster cessation of the jump's effect. ${\opf I}$ is the indicator function, equal to 1 if a jump event occurs and 0 otherwise.

3.2. Modeling short rate

Interest rates play a significant role in determining the present value of future cash flows and assessing risks from a financial perspective. The insurance industry often relies on a fixed-rate assumption. However, this approach may struggle to adapt to markets with high financial volatility. In contrast, stochastic interest rates offer a more realistic approach to modeling uncertainties and fluctuations in the real world. Acknowledging this, the Vasicek model, well-known for its convenience and widespread use, was selected as the preferred framework for modeling stochastic interest rates in this study.

Vasicek model was introduced by Vasicek (Reference Vasicek1977) to model short-term interest rates under the assumption that of interest rates can based on a single stochastic factor. Vasicek model is described by

(15)$$dr_t = \gamma( \theta - r_t) dt + \sigma dW_t$$

γ, θ, and σ are non-negative constants. W _t represents the random market risk defined as the Wiener process, θ represents the long-term average interest rate, γ indicates the rate of return to θ, and σ is the volatility of the short-term interest rate. The short-term rate is pushed up and pulled down, respectively, when it tends below and above the long-term rate (Burgess, Reference Burgess2014).

The Vasicek model is the Ornstein–Uhlenbeck process. The interest rate for a future time t, where 0 ≤ s ≤ t, is defined by (Burgess, Reference Burgess2014)

$$r_t = e^{-\gamma( t-s) } r_s + \theta( 1 - e^{-\gamma( t-s) }) + \sigma \int_{u = s}^{t} e^{-\gamma( t-u) } dW_u$$

The parameters of the Vasicek model, where r _t is the current interest rate, can be estimated by using the maximum-likelihood method. The estimation of parameters is

$$\eqalign{& \hat{\gamma} = -{1\over dt} \log\left({n\sum_{i = 1}^n r_{t_i}r_{t_{i-1}} - \sum_{i = 1}^n r_{t_i} \sum_{i = 1}^n r_{t_{i-1}}\over n\sum_{i = 1}^n r_{t_i}^2 - \left(\sum_{i = 1}^n r_{t_{i-1}}\right)^2}\right)\cr & \hat{\theta} = {1\over n( 1 - e^{-\hat{\gamma} dt}) } \left(\sum_{i = 1}^n r_{t_i} - e^{-\hat{\gamma} dt} \sum_{i = 1}^n r_{t_{i-1}} \right)\cr & \hat{\sigma}^2 = {2\hat{\gamma}\over n( 1 - e^{-2\hat{\gamma} dt}) } \sum_{i = 1}^n \left(r_{t_i} - r_{t_{i-1}} e^{-\hat{\gamma} dt} - \hat{\theta}( 1 - e^{-\hat{\gamma} dt}) \right)^2}$$

where $dt = t_i - t_{i-1},\; \, i = 1,\; \, 2,\; \, \dots ,\; \, n$ (Tanaka et al., Reference Tanaka, Xiao and Yu2020).

The discount rate is used to calculate the present value of cash flows expected in a certain period. The discount rate is shown with the interest rate as

(16)$$d_{t_0, t} = e^{-\int_{t_0}^{t} r_s ds}$$

4.1. Mortality

We use mortality table for both sexes from age of 35 to 90 in the United States from 1960 to 2019 that is taken from the Human Mortality Database (The Human Mortality Database, 2023), Mortality rates and parameter estimations are derived using Poisson log-bilinear mortality model and the R package “StMoMo” is employed for this purpose (Lee Carter, Reference Lee and Carter1992; Brouhns et al., Reference Brouhns, Denuit and Vermunt2002; Villegas et al., Reference Villegas, Millossovich and Kaishev2018). The most suitable model for the mortality trend is the ARIMA(0,1,0), with parameters δ = −0.6291, $\sigma _k^2 = 0.5961$. Figure 1 presents the results of the parameter estimation for the mortality model.

Figure 1.

Parameters estimation of mortality model for the United States.

The parameters shown in Figure 1, α _x and β _x vary with age, while κ _t varies with time. κ _t represents a time series modeled with ARIMA(0,1,0), and κ _t for the next decade is predicted using this time series. To estimate the probabilities of permanent jumps, 10,000 simulations are generated using a 10-year time series forecast of κ _t.

COVID-19 data are utilized to estimate the parameters for the transient mortality model (H _x,t) as defined in equations (13) and (14). For the COVID-19 data, excess deaths in the United State between 2020 and 2022, as reported by the Center for Disease Control and Prevention (CDC) (Center for Disease Control and Prevention, 2023), are taken into account. Figure 2 shows the distribution of excess deaths by age group from the CDC. Notably, the distribution of excess deaths by age groups has changed over the years. As demonstrated in Figure 2, COVID-19 has a notable impact on people in middle age and older. The year 2020 observed the highest deaths attributed to COVID-19 among those 85 years and older. However, following years, notably 2022, observed a decline in fatalities owing to pandemic control measures and widespread vaccination efforts. Additionally, the demographic groups most severely affected by COVID-19 have shifted over time. For instance, while the number of deaths among people aged 85 and older was higher in 2020, the number of deaths in the 65–74 and 75–84 age groups was higher in 2021. It cannot be determined in which age group COVID-19 is more effective with the excess death numbers, but this conclusion can be reached by examining the excess death rates.

Figure 2.

Excess death numbers from 2020 to 2022.

The excess death rate for each age group is calculated as (Zhang et al., Reference Zhang, Liao and Chen2021)

(17)$$\hbox{excess death rate}_t = {\hbox{excess deaths}_t\over \hbox{the average deaths}_{2015-2019}},\; \quad t = 2020,\; 2021,\; 2022$$

Table 1 shows the excess death rate computed for each age group.

Table 1.

Excess death rate for each age group from 2020 to 2022

The linear interpolation method is used to estimate excess mortality for each age. This method depends on the assumption that the probabilities of deaths presented in Table 1 represent the median value for each age group. Additionally, it is assumed that those under 24, and over 85 years old remained constant. In other words, the excess mortality rate for people within this age range is determined by the median value, and no interpolation is applied.

The rates estimated for each age are utilized to construct a distribution the impact of mortality rate. This distribution denotes the adverse mortality effect and is denoted by B _j(x) for x age. Since pandemic data from 2020 to 2022 are utilized in the paper, the parameter B _j can be estimated for these years. In years beyond 2022, it is assumed that the distribution observed in 2022, denoted as B _j, continues to be utilized. Figure 3 shows the predicted B _j values. While the age distributions for 2020 and 2022 exhibit similarities in adverse mortality effects, the influence of COVID-19 is more pronounced among younger age groups in 2021. This trend could be attributed to the increased transmission rate of the epidemic during that period.

Figure 3.

Adverse mortality effect across ages.

b _j signifies the maximum size of the jump event. The maximum value in Table 1 for excess death rates is considered as the value of b _j for the respective year.

Considering the Spanish flu that occurred between 1918 and 1920 and the ongoing COVID-19 pandemic that began in 2019, it can be inferred that pandemic events occur approximately once every century (Zhang et al., Reference Zhang, Liao and Chen2021). In addition, according to studies in the literature, the probability of the event occurring is accepted as τ _j = 0.01. This assumption is also accepted in the paper.

The k _j value shown in equation (14) reflects information about the duration of the pandemic. If k _j, greater than 1, the jump effect ends quickly; otherwise, it persists for a long time. The k _j can be estimated using mortality data for 2020 and 2021. However, due to not announced of mortality data for the years after 2021, the assumption k _j = 1, which is frequently used in the literature, is accepted. The mean square error is utilized to estimate k _j for 2020 and 2021. These estimates are evaluated in 0.001 increments for 0 < k _j ≤ 2. k _j is estimated as 1.27 for 2020. Assuming that the k _j in 2020 is 1.27, k _j for 2021 is estimated as 0.71.

Figure 4 shows the transient death jump process H _x,t with parameters based on some assumptions and estimates. Figure 4 reflects the changes in the temporal jump process over the years for ages 40, 50, and 60. The impact of shocks on mortality increases in the first 2 years and decreases in the following years. After 2024, although the impact of the pandemic has eased, it still has an impact on mortality. As shown in Figure 3, the impact of the temporary shock on mortality across ages varies by year.

Figure 4.

Temporary mortality jump for simulation in the next decade for ages 40, 50, and 60, starting from 2018. The point values in the chart represent the value at the end of the year.

The coherence of the simulations created using the estimated parameters with the observed values is shown in Table 2 and Figure 5.

Table 2.

Estimation of k _j for 2020 and 2021 with mean square error

Figure 5.

Estimation of mortality rate and observed mortality rate for 2020 and 2021.

Note: The observed mortality rate is estimated using q _x = m _x/(1 + 0.5m _x). The solid black dashed line indicates the difference between the observed mortality probabilities and the predicted temporary mortality jump model. The solid gray line represents the difference between the observed mortality probabilities and the permanent mortality jump model.

Table 2 contains the mean square errors of the temporary and permanent model for 2020 and 2021. The persistent jump model contains probabilities from simulations created with Poisson-log bilinear Lee–Carter model estimated parameters. The temporary jump model contains the probabilities obtained from the jump model added to the mortality model. As can be seen from Table 2, the temporary mortality jump model is a more appropriate model to explain the unexpected shock in mortality.

Figure 5 presents a detailed analysis of simulation results for 2020 and 2021, focusing on age-specific discrepancies between observed mortality rates and modeled outcomes. Interestingly, both years show similar results for people aged 35–45 across both models, suggesting a minimal pandemic impact on this group. In particular, the difference between the temporary mortality jump model and observations is less for the middle and elderly population, which is significantly affected by the pandemic. In other words, the temporary mortality jump model provides a more realistic simulation for all ages.

4.2. Interest rate

We utilize daily data of 3-month bond yields of the United States from January 2, 2008 to December 29, 2017. Data are sourced from the Federal Reserve Bank of St. Louis and are shown in Figure 6.

Figure 6.

3-Month bond yields: actual data and simulated paths.

Note: The solid black lines represent the actual daily data of 3-month bond yields for the United States, while the dashed lines illustrate a few examples of simulation paths based on parameters from Table 3.

Table 3.

Parameter estimation for Vasicek model

Maximum-likelihood calibration is applied to estimate the parameters of the Vasicek model. Table 3 involves parameter estimates. Moreover, Figure 6 presents several simulated paths generated using the estimated parameters from Table 3. The simulated paths closely match the actual data trends, demonstrating a strong alignment between them. Showing only a few selected simulation paths is sufficient to illustrate this consistency, as they effectively represent the overall behavior of bond yields during this period.

A value greater than one for γ indicates that interest rates return to their average level more quickly and fluctuations reach equilibrium more quickly. The parameters also show that the interest pattern moves steadily within a certain range of variability in the long run. With estimates of parameters, 10,000 simulations are created, starting in 2018 and continuing until 2028.

4.3. Insurance coverage

Financial performance of the insurance company is analyzed using estimated interest rates with both zero interest rates and the Vasicek model. It is assumed that the starting point is t ₀ = 2018. Moreover, no new survival benefit policies are introduced. Three distinct cohorts were considered for people aged 40, 50, and 60 years, each comprising 10,000 people. The survival benefit policy is considered to yield benefits (B = 1) if the policyholder lives for 10 years or more; if not, the policyholder is not eligible for benefits. Considering these assumptions, the insurer's liability over the years is estimated for a portfolios of 10,000 people and the average of the resulting simulated outcomes is calculated.

Figure 7 shows the insurer's remaining liability over time for the ages of 40, 50, and 60 for both temporary and permanent mortality jump models. For the two-mortality model, the values of the liability over time are depicted, both with and without consideration of the interest rate. Additionally, it is important to note that the disparity in the value of the remaining liability at the initial time (t ₀ = 2018) stems from interest rate. The temporary jump model, unlike the permanent jump model, leads to a decrease in liabilities with an increase in mortality. Every year, a consistent decline in the insurer's liability is observed, albeit at varying rates. It is a well-known fact that mortality rates increase with age. This trend is particularly pronounced for older ages, especially during the COVID-19 pandemic. As reflected in the graphs, the difference between the two mortality models increases proportionally with age. This phenomenon remains unchanged when the interest rate is taken as zero or estimated using the Vasicek model. Furthermore, the fluctuation in liabilities depending on the applied interest rates highlights the impact of interest rate adjustments on the insurer's financial position.

Figure 7.

Average remaining liability of the two models by years.

Note: The probabilities and the methodology for calculating liabilities are elucidated in Appendices A and B.

The premium amount for the insurance portfolio is estimated from the actuarial equilibrium equation. Pure premiums that equalize the liability for each age are obtained. A loading factor is added to the pure premium (P _pure) to cover the insurer's operational expenses, administrative and risk costs, and profit margin. Thus, the premium (P) for each age is estimated as P = (1 + m)P _pure. Premiums for each age are shown in Table 4, assuming that the premium (P) is received only at time t ₀ = 2018 and m = 0.1.

Table 4.

Premiums

In the paper, benefits are committed to those who are survive at the end of 10 years. Since the 40-year-old is considered more risky by the insurance company compared to other ages, higher premiums are requested. Also, based on the data within the scope of the paper, it can be said that interest rates play an important role in determining insurance premiums.

The premium at time t ₀ is an crucial factor affecting the CSM and LC. Higher premium can increase insurance companies’ future profit and increase CSM while at the same time providing greater financial assurance against future benefits and reducing LC.

The results derived from the CSM and LC defined by equations (7), (8), and (9) through simulation are presented in Figure 8 over the years. CSM and LC must equal zero at the end of the contract. The initial values of CSM and LC resulting from premiums and liabilities at time t ₀ are differ for each age. The impact of the mortality increase in 2020 is reflected differently in CSM and LC; in this case, it acts more on the CSM and less on the LC. The effect of evaluating interest rates with the Vasicek model on CSM and LC can be observed in the figure. The existence of the interest rate for ages 40 and 50 causes a change in the CSM value, but the result is still as CSM. When the interest rate is zero for the 60 age, it is recognized as LC; however, when assumed with the Vasicek model, the result is recognized as CSM.

Figure 8.

Contractual service margin (CSM) and loss component (LC).

CSM and LC are financial estimates and cannot be directly recognized as profit or loss. CSM and LC are used to help evaluate the financial performance, while indicators such as profit or loss are often used to assess financial position and performance. The profit or loss assessment of the insurance portfolio within the scope of the paper is shown in Figure 9.

Figure 9.

Distribution of profit or loss for different margins by year.

Figure 9 shows profits or losses of the insurance portfolios over years. Specifically, for adults aged 40 and 50, Figure 8 clearly indicates that the contract service margin will translate into profit. However, the situation is different for those over 60. Although there is a loss at the beginning, there is a trend in which profits increase over time. It is noted that among 60-year-olds, who are most impacted by the jump in mortality, this effect is more distinctly reflected in profit or loss compared to other ages.

When evaluating the implementation of the Vasicek model, it is noticeable that the profit or loss is more variable over the years. Especially in some years, significant losses or profits become evident. It can be expressed that interest rates cause fluctuations in profit and loss, and financial performance is more variable.

According to the results, the financial position of the portfolios varies over the years. However, it has been observed that all portfolios, including those that started with a loss, were in a profit position, albeit with different amounts, at the time the contract ended. In addition, the volatility of financial performance and position increases over the years, especially when the interest rate model is taken into account. This volatility may require portfolio managers to pursue a more careful risk management strategy. Additionally, financial positions in the portfolio may need to be managed more dynamically due to the impact of interest rates.