Guideline on Actuarial Calculations (Appendix No. 17)

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Republic of Azerbaijan
Collegium of the Ministry of Finance
Approved by Resolution No. Q-08 dated March 27, 2014.
Appendix No. 17
Guideline on Actuarial Calculations

1. General Provisions
   1.1. The Guideline on Actuarial Calculations (hereinafter "the Guideline") was prepared to implement paragraph 4.1.6 of the Presidential Decree No. 49 dated December 16, 2013, amending the Presidential Decree No. 735 dated March 13, 2008, on the implementation of the Law No. 519-IIIQ dated December 25, 2007, and the Law No. 806-IVQD dated October 29, 2013, amending the Law on Insurance Activity of the Republic of Azerbaijan. It aligns with Article 81-2.2 of the Law on Insurance Activity and regulates requirements for methods and principles of calculations stipulated in Article 81-1.
   1.2. For the purposes of this Guideline, the following basic concepts are used:
   1.2.1. insurance unit – the measurement unit used for calculating the insurance tariff;
   1.2.2. insurance tariff – the value of the insurance unit, consisting of the net insurance premium and loading;
   1.2.3. net insurance premium – the portion ensuring the formation of insurance reserves for insurance payments, consisting of the base part and risk loading;
   1.2.4. base part – the expected value (mathematical expectation) of insurance payments;
   1.2.5. risk loading – the portion reflecting random deviations from the expected value of insurance payments;
   1.2.6. loading – the portion ensuring costs for conducting insurance operations, financing preventive measures, and tariff profit. A detailed financial analysis of the composition components of the loading must be conducted and included in calculations;
   1.2.7. risk classification – a list of criteria characterizing the degree of the insured or risk, with corresponding differentiation coefficients;
   1.2.8. differentiation coefficients – coefficients applied to the insurance tariff to account for risk factors of the insured subject matter.

2. General Requirements for Non-Life Insurance Tariff Calculation
   2.1. Derivations of formulas used in calculations must be provided fully and clearly explained. Distribution functions for insurance events and losses must be justified using probability theory and mathematical statistics methods. Numerical values of all necessary parameters for calculations, along with detailed information on the sources and time periods of initial statistical indicators used, must be provided.
   2.2. Tariff calculations under this Guideline must be conducted for all insurance classes and types of insurance that combine certain classes. Insurers must review calculations at least once every three years.
   2.3. Statistical data characterizing insurance risks are used in tariff calculations.
   2.4. For calculating tariffs by insurance class (type), statistical data available for the last three or more complete financial years prior to the calculation date must be used.
   2.5. When applying initial statistical data for tariff calculations, one of the following must be followed:
   2.5.1. Tariff calculation by insurance class (type) is based on initial statistical data for one or several main insurance risks that most significantly affect the tariff price;
   2.5.2. Tariff calculation by insurance class (type) is based on aggregate initial statistical data for all insured risks providing coverage.
   2.6. Macroeconomic indicators used (inflation indicator, minimum calculation indicator) must correspond to official data available at the calculation date.
   2.7. The following formula is used for calculating insurance tariffs for non-life insurance classes:
   B = N / (1 - f)
   Where:
   B – insurance tariff;
   N – net insurance premium; f – loading (expressed as a percentage of the insurance premium).
   If statistics for an insurance class are available, actuaries may use the following formula to calculate the net premium:
   N = (M[X] + √D[X]) / n × 1/S
   Where:
   n – number of insured (insurance) subject matters; S – insurance amount for each homogeneous insurance portfolio subject matter; X – random variable representing total loss for a homogeneous insurance portfolio; M[X] – mathematical expectation of X; D[X] – variance of X; γ – probability that net insurance premiums are sufficient for insurance payments (guarantee probability); γ̃ – coefficient corresponding to the guarantee probability. Insurers may use the following table for values of γ and γ̃:
   γ: 0.84, 0.90, 0.95, 0.98, 0.9986
   γ̃: 1.0, 1.3, 1.645, 2.0, 3.0
   If statistics are available for an insurance class, the mathematical expectation and variance of X can be estimated using mathematical statistics methods.
   2.8. If statistics are not available for a specific insurance class, insurers must use statistical indicators published by the insurance regulatory authority for each insurance class in the market.

3. Requirements for Non-Life Insurance Tariff Valuation and Calculation Methods
   3.1. Tariff calculations by insurance class (type) are conducted according to the selected method.
   3.2. The calculation method must reflect:
   3.2.1. detailed information on the source of data used for tariff calculation;
   3.2.2. a full explanation of the method used.
   3.3. The explanation of the tariff calculation method is provided separately, independent of the calculations themselves.
   3.4. The tariff calculation method determines the following components:
   3.4.1. base part of the net insurance premium;
   3.4.2. risk loading;
   3.4.3. net insurance premium;
   3.4.4. loading.
   3.5. Adequacy and compatibility of calculation methods with initial statistical data are determined by the following criteria:
   3.5.1. consistency between initial statistical data and calculation parameters participating in tariff calculations, according to the calculation method;
   3.5.2. the selected method accounts for trends of increase or decrease in calculation indicators affecting tariff prices (probability of insurance events, loss severity, loss ratio, etc.);
   3.5.3. intermediate and final results obtained using the selected method must have economic meaning and purposefulness.

4. General Requirements for Life Insurance Premium Calculation
   4.1. For life insurance classes (types), detailed information must be provided on the sources of indicators in mortality and disability tables used, methods for constructing these tables (mathematically fully justified), and the mentioned tables must be attached to calculations.
   4.2. Quantities used for calculating premiums for life insurance against death, life savings insurance, and annuity insurance are calculated using the following formulas:
   The expected present value of insurance payments for life insurance against survival risk for a person aged x over n years is calculated as:
   E_x^n = Σ (from t=0 to n-1) p_x^t * v^t
   Where p_x^n is the probability of a person aged x surviving n years, and i is the annual interest rate. Their values are noted in Appendix 1 and Appendix 2. v is the discount factor, calculated as: v = (1+i)^-1 The formula for E_x^n expressed through commutation functions is:
   E_x^n = D_{x+n} / D_x
   The expected present value of insurance payments for life insurance against death risk for a person aged x over n years is calculated as:
   A_x:n|1 = Σ (from t=0 to n-1) v^{t+1} * p_x^t * q_{x+t} Where q_x is the probability of a person aged x dying in the following year, with values noted in Appendix 1. The formula for A_x:n|1 expressed through commutation functions is:
   A_x:n|1 = (N_x - N_{x+n}) / D_x
   Considering that death does not occur only at the discrete end of time, the expected present value is calculated as:
   A_x:n|1 = Σ (from t=0 to n-1) v^{t+1} * p_x^t * q_{x+t} Where δ is the intensity of the annual interest rate, calculated as: δ = ln(1+i) The expected present value of unit amounts paid at the beginning of each insurance year for a person aged x over n years is calculated as:
   ä_x:n = Σ (from t=0 to n-1) v^t * p_x^t
   The formula expressed through commutation functions is:
   ä_x:n = (M_x - M_{x+n}) / D_x
   For annuity insurance with term n years, the expected present value of amounts paid m times per year (1/m amount at the beginning of each 1/m fraction) is calculated as:
   a_x:n^(m) = a_x:n - (m-1)/(2m)
   The formula is:
   a_x:n^(m) = a_x:n - (m-1)/(2m)
   4.3. When the insurance amount (S) for life insurance against death is known, the equal periodic premium (P_m) paid m times per year for k years (k≤n) within an n-year coverage term is calculated as:
   P_m = [S * A_x:k|1 + S * (a_x:n - a_x:k) * ρ] / [a_x:k^(m) * (1+β) + a_x:n * γ] When the premium is known, the insurance amount is calculated as:
   S = P_m * a_x:k^(m) / [A_x:k|1 + (a_x:n - a_x:k) * ρ] The single premium (P) is calculated as:
   P = S * A_x:n|1
   When the single premium is known, the insurance amount is calculated as:
   S = P / A_x:n|1
   4.4. For life savings insurance, when the insurance amount is known, the equal periodic premium (P_m) paid m times per year for k years within an n-year term is calculated as:
   P_m = [S * (a_x:n - a_x:k) + S * E_x^n] / [a_x:k^(m) * (1+β) + a_x:n * γ] When the premium is known, the insurance amount is calculated as:
   S = P_m * a_x:k^(m) / [(a_x:n - a_x:k) + E_x^n]
   4.5. For annuity insurance, if premiums are paid m times per year from age x to (x+k), and payments of equal amount (S_q) are made q times per year from age (x+k) to (x+n), the premium is calculated as:
   P_m = [S_q * a_x:n^(q) - S_q * (a_x:k^(m)) * ρ] / [a_x:k^(m) * (1+β) + a_x:n * γ] When the premium is known, the insurance payment is calculated as:
   S_q = P_m * a_x:k^(m) / [a_x:n^(q) - (a_x:k^(m)) * ρ]
   4.6. Mathematical reserves at the end of the t-th year for life insurance against death, savings, and annuity insurance are calculated as follows:
   For t < k: V_x^t = [P_m * a_x:t^(m) - S_q * (a_x:n^(q) - a_x:k^(m)) * ρ] / [1 + β * (a_x:t^(m)/a_x:k^(m))] For t ≥ k: V_x^t = [P_m * a_x:t^(m) - S_q * (a_x:n^(q) - a_x:k^(m)) * ρ] / [1 + β * (a_x:t^(m)/a_x:k^(m))] For fractional years between insurance periods, mathematical reserves are estimated using linear interpolation:
   V_x^{t+s} = (1-s)V_x^t + s V_x^{t+1}, where 0 < s < 1.
   4.7. If a life insurance contract is terminated early at the end of the t-th year, the refund amount (tSV_x) is calculated as:
   tSV_x = tV_x - (S - tV_x) * 2%
   4.8. Coefficients α, β, γ, ρ_1, and ρ_2 in formulas 4.3–4.6 have the following meanings:
   α – coefficient (expressed as a percentage) representing costs incurred when concluding the insurance contract (excluding commissions paid to brokers); β – coefficient (expressed as a percentage) representing costs associated with premium payments during the payment period; γ – coefficient (expressed as a percentage) representing annual administrative expenses incurred during the contract validity period; ρ_1 – coefficient (expressed as a percentage) representing claim settlement costs for death events; ρ_2 – coefficient (expressed as a percentage) representing claim settlement costs for survival events. Values of these coefficients are shown in Appendix 2.
   4.9. Insurers may calculate premiums for life insurance classes based on their own mortality and disability tables and selected method, provided that premiums calculated without commissions are not lower than those calculated per sections 4.3–4.5 and 4.8.
   Note: Coefficients in formulas of Part 4 represent:
   x – person's age; n – term (years); i – interest rate; v – discount factor; p_x^n – probability of x surviving n years; E_x^n – expected present value of savings insurance for x over n years; S – insurance amount; a_x:n^(m) – present value of annuity payments (m times per year); q_{x+t} – probability of x dying within t years; q_x – probability of x dying in the following year; A_x:n|1 – expected present value for death risk (discrete case); D_x, M_x, N_x – commutation functions; δ – intensity of annual interest rate.

5. Requirements for Results of Tariff/Premium Calculation and Formalization
   5.1. The following indicators must be determined and justified for tariff calculations by insurance class (type):
   5.1.1. base, minimum base, and maximum base values of the net insurance premium;
   5.1.2. risk factors conditioning the application of differentiation coefficients to the base net premium value (with justification for coefficient values);
   5.1.3. detailed classification of risks by insurance classes (types);
   5.1.4. distribution tables of tariffs (premium amounts) by insured categories and accepted classifications.
   5.2. The requirement of paragraph 5.1.4 does not apply to insurers operating in life insurance.
   5.3. Documents for tariff/premium calculations must be certified by the actuary with their signature and seal (if any), indicating the certificate number, issue date, actuary's full name.

6. Requirements for Required Capital Calculation
   6.1. Required capital is determined separately for life and non-life insurance.
   6.2. Actuarial calculations for determining the required solvency level of non-life insurers are based on "premiums" and "claims" methods.
   6.3. Required capital calculation is regulated by the regulator's adopted "Rules for Determining Required Capital of Insurers."

7. Requirements for Total Capital Calculation
   7.1. According to Article 79.5 of the Law on Insurance Activity, total capital cannot be less than required capital.
   7.2. Total capital calculation considers the insurer's own funds directed to investments, as indicated in Article 63.1 of the Law on Insurance Activity.
   7.3. Liquidity and diversification coefficients are applied to asset groups used in total capital calculation.
   7.4. Total capital calculation is regulated by the regulator's adopted "Rules for Investment Operations of Insurers."

8. Requirements for Calculation of Insurer's Own Funds
   8.1. Calculation considers the total balance sheet value of assets, insurance-related balance liabilities (excluding insurance reserves), delayed premiums (>90 days, excluding state mandatory personal insurance premiums due), insurance reserves, total balance values of assets backing reserves, non-insurance balance liabilities, off-balance sheet liabilities (total guaranteed amounts), and total balance values of assets outside groups specified in Articles 63.1.1–63.1.6 and 64.1.1–64.1.7 of the Law on Insurance Activity.
   8.2. Calculation is regulated by the regulator's adopted "Rules for Investment Operations of Insurers."

9. Requirements for Insurance Reserves Calculation
   9.1. Insurers form insurance reserves adequate to their liabilities under life and reinsurance contracts.
   9.2. Insurers may use an alternative method for calculating the Incurred But Not Reported (IBNR) reserve, provided it is not lower than IBNR calculated according to the regulator's "Rules for Forming Insurance Reserves for Life and Non-Life Insurance."
   9.3. Reserve calculation is regulated by the regulator's adopted "Rules for Forming Insurance Reserves for Life and Non-Life Insurance."

10. Requirements for Loss Ratio Calculation
    10.1. For this Guideline, loss ratio is determined only for non-life insurance classes.
    10.2. Calculations use indicators related to the insurance year of events.
    10.3. Calculation is regulated by the regulator's adopted "Rules for Forming Insurance Reserves for Life and Non-Life Insurance."

11. Requirements for Individual Exposure Calculation
    11.1. Insurers' individual exposure is determined separately for life and reinsurance contracts.
    11.2. Individual exposure is determined based on the subject matter and insurance (reinsurance) contract.
    11.3. According to Article 79.6 of the Law on Insurance Activity, individual exposure volume per subject matter must not exceed 10% of total capital.
    11.4. According to Article 79.7, considering Article 79.6, individual exposure volume for property insurance contracts must not exceed 30% of total capital.

Appendix 1 to the "Guideline on Actuarial Calculations"
Mortality Table x | l_x | d_x | q_x | x | l_x | d_x | q_x | x | L_x | d_x | Q_x 0 | 1,000,000 | 12,887 | 0.0129 | 36 | 958,740 | 1,859 | 0.0019 | 72 | 533,874 | 32,267 | 0.0604 1 | 987,113 | 1,906 | 0.0019 | 37 | 956,881 | 2,010 | 0.0021 | 73 | 501,607 | 33,200 | 0.0662 2 | 985,207 | 1,068 | 0.0011 | 38 | 954,871 | 2,276 | 0.0024 | 74 | 468,407 | 33,449 | 0.0714 3 | 984,139 | 701 | 0.0007 | 39 | 952,595 | 2,300 | 0.0024 | 75 | 434,958 | 34,489 | 0.0793 4 | 983,438 | 536 | 0.0005 | 40 | 950,294 | 2,547 | 0.0027 | 76 | 400,470 | 34,058 | 0.0850 5 | 982,902 | 487 | 0.0005 | 41 | 947,747 | 2,797 | 0.0030 | 77 | 366,412 | 33,118 | 0.0904 6 | 982,415 | 464 | 0.0005 | 42 | 944,950 | 3,016 | 0.0032 | 78 | 333,294 | 32,793 | 0.0984 7 | 981,951 | 429 | 0.0004 | 43 | 941,934 | 3,317 | 0.0035 | 79 | 300,501 | 31,694 | 0.1055 8 | 981,522 | 411 | 0.0004 | 44 | 938,616 | 3,561 | 0.0038 | 80 | 268,807 | 30,578 | 0.1138 9 | 981,111 | 403 | 0.0004 | 45 | 935,055 | 3,986 | 0.0043 | 81 | 238,229 | 27,247 | 0.1144 10| 980,708 | 392 | 0.0004 | 46 | 931,069 | 4,296 | 0.0046 | 82 | 210,982 | 25,502 | 0.1209 11| 980,316 | 375 | 0.0004 | 47 | 926,773 | 4,863 | 0.0052 | 83 | 185,480 | 24,176 | 0.1303 12| 979,941 | 342 | 0.0003 | 48 | 921,910 | 5,371 | 0.0058 | 84 | 161,305 | 22,252 | 0.1380 13| 979,599 | 343 | 0.0004 | 49 | 916,539 | 5,882 | 0.0064 | 85 | 139,052 | 19,549 | 0.1406 14| 979,256 | 376 | 0.0004 | 50 | 910,658 | 6,653 | 0.0073 | 8+ | ... | ... | ...

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