| Second moment of continuous whole life insurance PV | A2barx |
| Second moment of continuous endowment insurance PV | A2barxn |
| Second moment of continuous term insurance PV | A2barxn1 |
| Second moment of continuous deferred insurance PV | A2nAbarx |
| Second moment of deferred insurance PV | A2nAx |
| Second moment of m-thly deferred insurance PV | A2nAx_m |
| Second moment of pure endowment PV | A2nEx |
| Second moment of whole life insurance PV | A2x |
| Second moment of m-thly whole life insurance PV | A2x_m |
| Second moment of endowment insurance PV | A2xn |
| Second moment of m-thly endowment insurance PV | A2xn_m |
| Second moment of term insurance PV | A2xn1 |
| Second moment of m-thly term insurance PV | A2xn1_m |
| Projected Unit Credit accrued liability | AAL_PUC_db |
| Traditional Unit Credit accrued liability | AAL_TUC_db |
| Accrued benefit under a career-average-earnings plan | AB_cae |
| Accrued benefit under a final-average-salary plan | AB_fas |
| Continuous whole life insurance APV | Abarx |
| UDD approximation of continuous whole life insurance | Abarx_udd |
| Continuous multiple-decrement insurance present value | Abarxj_md |
| Continuous endowment insurance APV | Abarxn |
| UDD approximation of continuous endowment insurance | Abarxn_udd |
| Continuous term insurance APV | Abarxn1 |
| UDD approximation of continuous term insurance | Abarxn1_udd |
| AG 38 prefunding ratio | ag38_prefunding_ratio |
| AG 38 reserve calculation | ag38_reserve_ul |
| Annual annuity functions | abarx abarxn adotx adotxn annuity_annual ax axn nabarx nadotx nax sbarxn sdotxn sxn |
| UDD annuity approximations | abarxn_udd abarx_udd adotxn_m_udd adotx_m_udd annuity_approximations_udd axn_m_udd ax_m_udd nabarx_udd nadotx_m_udd nax_m_udd sdotxn_m_udd sxn_m_udd |
| Woolhouse 2-term annuity approximations | abarx_woolhouse2 adotxn_m_woolhouse2 adotx_m_woolhouse2 annuity_approximations_woolhouse2 axn_m_woolhouse2 ax_m_woolhouse2 nadotx_m_woolhouse2 nax_m_woolhouse2 sdotxn_m_woolhouse2 sxn_m_woolhouse2 |
| Woolhouse 3-term annuity approximations | abarx_woolhouse3 adotxn_m_woolhouse3 adotx_m_woolhouse3 annuity_approximations_woolhouse3 axn_m_woolhouse3 ax_m_woolhouse3 nadotx_m_woolhouse3 nax_m_woolhouse3 |
| Present value of a level annuity-certain | annuity_certain |
| m-thly contingent annuity functions | adotxn_m adotx_m annuity_mthly axn_m ax_m nadotx_m nax_m sdotxn_m sxn_m |
| Annuity-insurance relationships | annuity_identity_abarx annuity_identity_abarxn annuity_identity_adotx annuity_identity_adotxn annuity_identity_ax annuity_identity_axn annuity_identity_nabarx annuity_identity_nadotx annuity_identity_nax annuity_relationships |
| Varying-payment annuity functions | annuity_varying_payments Dabarxn Dadotxn Daxn Iabarx Iabarxn Iadotx Iadotxn Iax Iaxn |
| Actuarial present value of gross premiums | APV_gross_premiums |
| Actuarial present value of a normal retirement benefit | APV_NR_db |
| Projected asset-share path for two decrement causes | AS_path |
| General projected asset-share path | AS_path_md |
| Type A universal life account-value path | AV_path_ul_typeA |
| Type B universal life account-value path | AV_path_ul_typeB |
| Accumulated value of defined contribution plan contributions | AVz_dc |
| Whole life insurance APV | Ax |
| m-thly whole life insurance APV | Ax_m |
| UDD approximation of m-thly whole life insurance | Ax_m_udd |
| Discrete multiple-decrement insurance present value | Axj_md |
| Endowment insurance APV | Axn |
| m-thly endowment insurance APV | Axn_m |
| UDD approximation of m-thly endowment insurance | Axn_m_udd |
| Term insurance APV | Axn1 |
| m-thly term insurance APV | Axn1_m |
| UDD approximation of m-thly term insurance | Axn1_m_udd |
| Cost of insurance for Type B universal life | coi_ul_typeB |
| Continuous multi-life annuities | abarxy abarxybar abarx_y abary_x continuous_multilife_annuities |
| Continuous multi-life insurance | Abarxy Abarxy1 Abarxy2 Abarxybar Abaryx1 Abaryx2 continuous_multilife_insurance |
| Target contribution rate for a defined contribution plan | contribution_rate_target |
| Covariance of term and deferred insurance PVs | cov_term_deferred |
| Covariance of term insurance and pure endowment PVs | cov_term_endow |
| Cumulative hazard for age-at-failure | cumhaz0 |
| Piecewise-continuous decreasing n-year term insurance | DAbarxn1 |
| Decreasing n-year term insurance | DAxn1 |
| Fully continuous decreasing n-year term insurance | DbarAbarxn1 |
| Ordered decomposition of gross gain | decompGg_disc |
| Deferred insurance reserves | deferred_insurance_reserves htVnAx tVnAx |
| Discount factor for compound interest | discount |
| Discounted payback period | discounted_payback_period |
| Distribution functions for age-at-failure | dist0 F0 f0 |
| Doubled force of interest | double_force_delta |
| Effective annual interest at doubled force | double_force_i |
| Compute deaths between ages x and x+1 | dx |
| Cause-specific numbers of decrements | dxj |
| Total number of decrements | dxtau |
| Mean present value of loss at duration t for whole life insurance | ELtx |
| Complete expectation of life | ex_complete |
| Complete expectation of life from a life table | ex_complete_tab |
| Curtate expectation of life | ex_curtate |
| Curtate expectation of life from a life table | ex_curtate_tab |
| Temporary complete expectation of life from a life table | ex_temp_complete_tab |
| Temporary curtate expectation of life from a life table | ex_temp_curtate_tab |
| Forward rate implied by spot rates | fnk_from_z |
| Matrix of forward rates implied by spot rates | forward_matrix_from_z |
| Fractional-duration whole life reserves | fractional_duration_reserves meanVx tsVx |
| Fractional-duration term and endowment reserves | fractional_duration_term_endowment_reserves tsVxn tsVxn1 |
| Full preliminary term modified premiums and reserves | alphaF betaF full_preliminary_term tVFx |
| Conditional density | fx |
| Fractional conditional density from a life table | fx_tab |
| Gain or loss in a two-cause multiple-decrement model | gain_loss_md |
| Interest gain for a continuous-style recursion | GI_cont |
| Interest gain for a discrete insurance contract | GI_disc |
| Mortality gain for a continuous-style recursion | GM_cont |
| Mortality gain for a discrete insurance contract | GM_disc |
| Guaranteed maturity fund roll-forward | GMF_rollforward_ul |
| Whole life gross premium and expense reserves | gross_premium_expense_reserves tVEx tVGx |
| Total gain for a continuous-style one-step recursion | GT_cont |
| Total gain for a discrete insurance contract | GT_disc |
| Total gross gain for a discrete insurance contract | GTg_disc |
| Hazard or force of mortality for age-at-failure | hazard0 |
| h-pay whole life net level premium reserve | htVx |
| Credited rates from index growth rates | i_credit_eiul |
| Piecewise-continuous increasing whole life insurance | IAbarx |
| Increasing whole life insurance | IAx |
| Increasing n-year term insurance | IAxn1 |
| Fully continuous increasing whole life insurance | IbarAbarx |
| Fully continuous increasing n-year term insurance | IbarAbarxn1 |
| Monthly-average index growth rate | iMA_eiul |
| Retirement income from a defined contribution accumulation | Income_dc |
| Convert between compound-interest quantities | interest_convert |
| Point-to-point index growth rates | iP_eiul |
| Internal rate of return | IRR_profit |
| Joint-life annuities | adotxy adotxyn axy axyn joint_life_annuities |
| Joint-life insurance functions | Axy Axyn Axyn1 joint_life_insurance |
| Last-survivor annuity functions | adotxybar adotxybarn axybar axybarn last_survivor_annuities |
| Last-survivor insurance functions | Axybar Axybarn Axybarn1 last_survivor_insurance |
| Construct a life table | life_table |
| Extract life-table survivor values | lx |
| Extract select-table survivor value | lx_select |
| Convert life-table values to survival probabilities | lx_to_S0 |
| Multi-step transition probability | markov_nstep_prob |
| Construct a multiple-decrement table | md_table |
| Mortality improvement projection functions | axn_improved ax_improved mortality_improvement_projection naxn_improved px_proj qx_proj tpx_improved |
| Contingent multi-life probabilities | multilife_contingent_probabilities tqxy1 tqxy2 tqyx1 tqyx2 |
| Multi-life pure endowments | multilife_pure_endowments nExy nExybar |
| Multi-life survival and failure probabilities | multilife_survival_probabilities tpxy tpxybar tqxy tqxybar |
| Fractional force of mortality from a life table | mux_tab |
| Continuous deferred insurance APV | nAbarx |
| UDD approximation of continuous deferred insurance | nAbarx_udd |
| Deferred insurance APV | nAx |
| m-thly deferred insurance APV | nAx_m |
| UDD approximation of m-thly deferred insurance | nAx_m_udd |
| Entry Age Normal normal cost | NC_EAN_db |
| Projected Unit Credit normal cost | NC_PUC_db |
| Traditional Unit Credit normal cost | NC_TUC_db |
| Compute deaths over an n-year interval from a life table | ndx |
| Pure endowment APV | nEx |
| Curtate death probability from a life table | nkqx |
| Deferred death probability from a life table | nmxq |
| Deferred select-life death probability | nmxq_select |
| Partial net present values | NPV_partial |
| Net present value of a profit signature | NPV_profit |
| Compute n-year survival probability from a life table | npx |
| Select-life survival probability | npx_select |
| Multiple-decrement survival probability from a table | npxtau_md |
| Compute n-year death probability from a life table | nqx |
| Select-life death probability | nqx_select |
| Cause-specific multiple-decrement probability from a table | nqxj_md |
| Total multiple-decrement probability from a table | nqxtau_md |
| Projected annual benefit under a career-average-earnings plan | PAB_cae |
| Projected annual benefit under a final-average-salary plan | PAB_fas |
| Continuous premium approximation in a disability model | Pbar_trapz_ms |
| Profit signature | Pi_signature |
| Net premium for a deferred annuity-due | PnAdotx |
| Net premium for a deferred annuity-immediate | Pnax |
| Profit vector for a discrete profit-analysis model | Pr_vector_disc |
| Premium, loss, and expense functions | EL0barAbarx EL0x EL0xn EL0xn1 Gx PbarAbarx PbarAbarxn PbarAbarxn1 Pbarx Pbarxn Pbarxn1 PnAx PnAx_m PnEx premium_functions Px Pxn Pxn1 Pxn1_m Pxn_m Px_m tPnAx tPnEx tPx tPxn tPxn1 varL0barAbarx varL0x varL0xn varL0xn1 |
| Profit margin | profit_margin |
| Present value of cash flows at time 0 | pv_cashflows |
| Present value of deterministic cash flows using spot rates | pv_spot_cashflows |
| Construct life-table values from p_x values | px_to_lx |
| Total one-year survival probability | pxtau |
| Universal life persistency probabilities | pxtau_ul tpxtau_ul |
| Multiple-decrement probabilities under constant forces | qx_dep_cf |
| Multiple-decrement probabilities under SUDD | qx_dep_sudd |
| Compute one-year death probability from a life table | qx_tab |
| Construct life-table values from q_x values | qx_to_lx |
| Associated single-decrement probabilities under MUDD | qxprime_mudd |
| Associated single-decrement probabilities under SUDD | qxprime_sudd |
| Total one-year decrement probability | qxtau |
| Replacement ratio for a defined benefit plan | replacement_ratio_db |
| Replacement ratio for a defined contribution plan | replacement_ratio_dc |
| Reversionary annuity functions | ax_y ay_x reversionary_annuities |
| Account-value to guaranteed-fund ratio | rt_ul |
| Survival function for age-at-failure | S0 |
| Convert survival probabilities to life-table values | S0_to_lx |
| Salary scale under constant annual growth | salary_scale |
| Construct a select life table | select_life_table |
| Solve the yield rate by the equation of value | solve_yield |
| Actuarial present values under spot rates | Axn1_spot Axn_spot axn_spot nEx_spot spot_interest_apvs |
| Backward reserve path from a terminal value | thiele_backward_path |
| One backward numerical step for Thiele's equation | thiele_backward_step |
| Reserve derivative from Thiele's equation | thiele_dVdt |
| Reserve derivatives for a disability model with recovery | thiele_dVdt_01 |
| Backward reserve path for a disability model with recovery | thiele_path_01 |
| Euler approximation of disability-state probabilities | tp00_tp01_euler |
| Conditional survival probability | tpx |
| Fractional survival probability from a life table | tpx_tab |
| Total survival under constant cause-specific forces | tpx_tau_cf |
| Single-decrement survival under a constant force | tpxprimej_cf |
| Conditional failure probability | tqx |
| Fractional failure probability from a life table | tqx_tab |
| Cause-specific decrement probability under constant forces | tqxj_cf |
| Fractional-year associated single-decrement probabilities under MUDD | tqxprime_mudd |
| Single-decrement failure under a constant force | tqxprimej_cf |
| Fully continuous whole life reserve | tVbarAbarx |
| Whole life reserve with continuous premiums | tVbarx |
| Reserve for a deferred annuity-due | tVnAdotx |
| Reserve for a deferred annuity-immediate | tVnax |
| Pure endowment net level premium reserve | tVnEx |
| Whole life net level premium reserve | tVx |
| Whole life reserve with m-thly premiums | tVx_m |
| Retrospective whole life reserve | tVx_ret |
| Endowment insurance net level premium reserve | tVxn |
| Retrospective endowment insurance reserve | tVxn_ret |
| Term insurance net level premium reserve | tVxn1 |
| Retrospective term insurance reserve | tVxn1_ret |
| UDD multiplier for continuous insurance approximations | udd_continuous_multiplier |
| UDD multiplier for m-thly insurance approximations | udd_mthly_multiplier |
| Zeroized reserves for a discrete death-benefit contract | V_zeroized |
| Variance of continuous whole life insurance PV | var_Abarx |
| Variance of continuous endowment insurance PV | var_Abarxn |
| Variance of continuous term insurance PV | var_Abarxn1 |
| Variance of whole life insurance PV | var_Ax |
| Variance of m-thly whole life insurance PV | var_Ax_m |
| Variance of endowment insurance PV | var_Axn |
| Variance of m-thly endowment insurance PV | var_Axn_m |
| Variance of term insurance PV | var_Axn1 |
| Variance of m-thly term insurance PV | var_Axn1_m |
| Variance of continuous deferred insurance PV | var_nAbarx |
| Variance of deferred insurance PV | var_nAx |
| Variance of m-thly deferred insurance PV | var_nAx_m |
| Variance of pure endowment PV | var_nEx |
| Actuarial present values under variable annual interest rates | Axn1_var Axn_var axn_var nEx_var variable_interest_apvs |
| Variance of present value of loss at duration t for whole life insurance | varLtx |
| Pre-floor CRVM reserve | Vprefloor_crvm_ul |
| Discount factors under variable annual interest rates | vt_var |
| Bootstrap annual effective spot rates | z_from_coupon_annual |
| Bootstrap semiannual nominal spot rates | z_from_coupon_semi |
| Spot rates from one-year forward rates | z_from_fn1 |
