{ "_id": "6a1ee864b401979e73411a6b", "Package": "mqriskR", "Type": "Package", "Title": "Actuarial Risk Modeling and Life Contingencies", "Version": "0.1.0", "Authors@R": "c(\nperson(\n\"Nii\", \"Okine\",\nemail = \"okinean@appstate.edu\",\nrole = c(\"aut\", \"cre\")\n)\n)", "Description": "Provides functions for actuarial risk modeling, including\nsurvival models, life annuities, multiple-decrement models, and\nmortality improvement projections. The package is designed to\nalign with standard actuarial notation and supports teaching,\nexam preparation, and reproducible actuarial analysis. The\nmethods are based on standard actuarial references including\nCamilli, Duncan and London (2014, ISBN:9781625423474) \"Models\nfor Quantifying Risk\" and Dickson, Hardy and Waters (2020,\nISBN:9781108478083) \"Actuarial Mathematics for Life Contingent\nRisks\".", "License": "MIT + file LICENSE", "Encoding": "UTF-8", "Config/testthat/edition": "3", "RoxygenNote": "7.3.3", "VignetteBuilder": "knitr", "Repository": "https://aokine.r-universe.dev", "Date/Publication": "2026-04-30 16:48:08 UTC", "RemoteUrl": "https://github.com/aokine/mqriskr", "RemoteRef": "HEAD", "RemoteSha": "34ca1815335b2c508c56745605194b7121f4fa7c", "NeedsCompilation": "no", "Packaged": { "Date": "2026-05-19 10:38:17 UTC", "User": "root" }, "Author": "Nii Okine [aut, cre]", "Maintainer": "Nii Okine ", "MD5sum": "98758c7e53e7bb5da0d2e92ca697521a", "_user": "aokine", "_type": "src", "_file": "mqriskR_0.1.0.tar.gz", "_fileid": "cf752329e675a48db65d6d30dacbdbc0061a55cb2a2b3dbb6bb908bd002767bd", "_filesize": 994384, 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"A2x_m", "title": "Second moment of m-thly whole life insurance PV", "topics": [ "A2x_m" ] }, { "page": "A2xn", "title": "Second moment of endowment insurance PV", "topics": [ "A2xn" ] }, { "page": "A2xn_m", "title": "Second moment of m-thly endowment insurance PV", "topics": [ "A2xn_m" ] }, { "page": "A2xn1", "title": "Second moment of term insurance PV", "topics": [ "A2xn1" ] }, { "page": "A2xn1_m", "title": "Second moment of m-thly term insurance PV", "topics": [ "A2xn1_m" ] }, { "page": "AAL_PUC_db", "title": "Projected Unit Credit accrued liability for a DB plan", "topics": [ "AAL_PUC_db" ] }, { "page": "AAL_TUC_db", "title": "Traditional Unit Credit accrued liability for a DB plan", "topics": [ "AAL_TUC_db" ] }, { "page": "AB_cae", "title": "Accrued benefit for a career average earnings plan", "topics": [ "AB_cae" ] }, { "page": "AB_fas", "title": "Accrued benefit for a final average salary plan", "topics": [ "AB_fas" ] }, { "page": "Abarx", "title": "Continuous whole life insurance APV", "topics": [ "Abarx" ] }, { "page": "Abarx_udd", "title": "UDD approximation of continuous whole life insurance", "topics": [ "Abarx_udd" ] }, { "page": "abarx_y", "title": "Continuous reversionary annuity to (y) after death of (x)", "topics": [ "abarx_y" ] }, { "page": "Abarxj_md", "title": "Continuous multiple-decrement insurance APV \\overline{A}_{x}^{(j)}", "topics": [ "Abarxj_md" ] }, { "page": "Abarxn", "title": "Continuous endowment insurance APV", "topics": [ "Abarxn" ] }, { "page": "Abarxn_udd", "title": "UDD approximation of continuous endowment insurance", "topics": [ "Abarxn_udd" ] }, { "page": "Abarxn1", "title": "Continuous term insurance APV", "topics": [ "Abarxn1" ] }, { "page": "Abarxn1_udd", "title": "UDD approximation of continuous term insurance", "topics": [ "Abarxn1_udd" ] }, { "page": "Abarxy", "title": "Continuous joint-life whole life insurance", "topics": [ "Abarxy" ] }, { "page": "abarxy_ch12", "title": "Continuous joint-life whole life annuity", "topics": [ "abarxy", "abarxy_ch12" ] }, { "page": "Abarxy1", "title": "Continuous contingent insurance: benefit on death of (x) if before (y)", "topics": [ "Abarxy1" ] }, { "page": "Abarxy2", "title": "Continuous contingent insurance: benefit on death of (x) if after (y)", "topics": [ "Abarxy2" ] }, { "page": "Abarxybar", "title": "Continuous last-survivor whole life insurance", "topics": [ "Abarxybar" ] }, { "page": "abarxybar_ch12", "title": "Continuous last-survivor whole life annuity", "topics": [ "abarxybar", "abarxybar_ch12" ] }, { "page": "abary_x", "title": "Continuous reversionary annuity to (x) after death of (y)", "topics": [ "abary_x" ] }, { "page": "Abaryx1", "title": "Continuous contingent insurance: benefit on death of (y) if before (x)", "topics": [ "Abaryx1" ] }, { "page": "Abaryx2", "title": "Continuous contingent insurance: benefit on death of (y) if after (x)", "topics": [ "Abaryx2" ] }, { "page": "adotxy", "title": "Joint-life whole life annuity-due", "topics": [ "adotxy" ] }, { "page": "adotxybar", "title": "Last-survivor whole life annuity-due", "topics": [ "adotxybar" ] }, { "page": "adotxybarn", "title": "Last-survivor temporary annuity-due", "topics": [ "adotxybarn" ] }, { "page": "adotxyn", "title": "Joint-life temporary annuity-due", "topics": [ "adotxyn" ] }, { "page": "ag38_prefunding_ratio", "title": "AG 38 prefunding ratio", "topics": [ "ag38_prefunding_ratio" ] }, { "page": "ag38_reserve_ul", "title": "AG 38 reserve calculation", "topics": [ "ag38_reserve_ul" ] }, { "page": "alphaF", "title": "Full preliminary term first-year modified premium", "topics": [ "alphaF" ] }, { "page": "annuity_annual", "title": "Annual annuity functions (Chapter 8)", "topics": [ "abarx", "abarxn", "adotx", "adotxn", "annuity_annual", "ax", "axn", "nabarx", "nadotx", "nax", "sbarxn", "sdotxn", "sxn" ] }, { "page": "annuity_approximations_udd", "title": "UDD annuity approximations", "topics": [ "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" ] }, { "page": "annuity_approximations_woolhouse2", "title": "Woolhouse 2-term annuity approximations", "topics": [ "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" ] }, { "page": "annuity_approximations_woolhouse3", "title": "Woolhouse 3-term annuity approximations", "topics": [ "abarx_woolhouse3", "adotxn_m_woolhouse3", "adotx_m_woolhouse3", "annuity_approximations_woolhouse3", "axn_m_woolhouse3", "ax_m_woolhouse3", "nadotx_m_woolhouse3", "nax_m_woolhouse3" ] }, { "page": "annuity_certain", "title": "Present value of a level annuity-certain", "topics": [ "annuity_certain" ] }, { "page": "annuity_mthly_accum_due", "title": "Temporary m-thly annuity-due actuarial accumulated value", "topics": [ "annuity_mthly_accum_due", "sdotxn_m" ] }, { "page": "annuity_mthly_accum_immediate", "title": "Temporary m-thly annuity-immediate actuarial accumulated value", "topics": [ "annuity_mthly_accum_immediate", "sxn_m" ] }, { "page": "annuity_mthly_deferred_due", "title": "Deferred whole life m-thly annuity-due", "topics": [ "annuity_mthly_deferred_due", "nadotx_m" ] }, { "page": "annuity_mthly_deferred_immediate", "title": "Deferred whole life m-thly annuity-immediate", "topics": [ "annuity_mthly_deferred_immediate", "nax_m" ] }, { "page": "annuity_mthly_temp_due", "title": "Temporary m-thly annuity-due", "topics": [ "adotxn_m", "annuity_mthly_temp_due" ] }, { "page": "annuity_mthly_temp_immediate", "title": "Temporary m-thly annuity-immediate", "topics": [ "annuity_mthly_temp_immediate", "axn_m" ] }, { "page": "annuity_mthly_whole_due", "title": "Whole life m-thly annuity-due", "topics": [ "adotx_m", "annuity_mthly_whole_due" ] }, { "page": "annuity_mthly_whole_immediate", "title": "Whole life m-thly annuity-immediate", "topics": [ "annuity_mthly_whole_immediate", "ax_m" ] }, { "page": "annuity_relationships", "title": "Annuity-insurance relationships (Chapter 8)", "topics": [ "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" ] }, { "page": "annuity_varying_payments", "title": "Varying-payment annuity functions (Chapter 8)", "topics": [ "annuity_varying_payments", "Dabarxn", "Dadotxn", "Daxn", "Iabarx", "Iabarxn", "Iadotx", "Iadotxn", "Iax", "Iaxn" ] }, { "page": "APV_gross_premiums", "title": "APV of gross premiums under a risk discount rate", "topics": [ "APV_gross_premiums" ] }, { "page": "APV_NR_db", "title": "APV of normal retirement benefit for a DB plan", "topics": [ "APV_NR_db" ] }, { "page": "AS_path", "title": "Projected asset share path {}_{k}AS", "topics": [ "AS_path" ] }, { "page": "AS_path_md", "title": "General projected asset share path (multiple decrements)", "topics": [ "AS_path_md" ] }, { "page": "AV_path_ul_typeA", "title": "Account-value path for Type A universal life", "topics": [ "AV_path_ul_typeA" ] }, { "page": "AV_path_ul_typeB", "title": "Account-value path for Type B universal life", "topics": [ "AV_path_ul_typeB" ] }, { "page": "AVz_dc", "title": "Accumulated value of defined contribution plan contributions", "topics": [ "AVz_dc" ] }, { "page": "Ax", "title": "Whole life insurance APV", "topics": [ "Ax" ] }, { "page": "ax_improved", "title": "Whole life annuity-immediate under mortality improvement", "topics": [ "ax_improved" ] }, { "page": "Ax_m", "title": "m-thly whole life insurance APV", "topics": [ "Ax_m" ] }, { "page": "Ax_m_udd", "title": "UDD approximation of m-thly whole life insurance", "topics": [ "Ax_m_udd" ] }, { "page": "ax_y", "title": "Reversionary annuity to (y) after death of (x)", "topics": [ "ax_y" ] }, { "page": "Axj_md", "title": "Discrete multiple-decrement insurance APV A_{x}^{(j)}", "topics": [ "Axj_md" ] }, { "page": "Axn", "title": "Endowment insurance APV", "topics": [ "Axn" ] }, { "page": "axn_improved", "title": "Temporary annuity-immediate under mortality improvement", "topics": [ "axn_improved" ] }, { "page": "Axn_m", "title": "m-thly endowment insurance APV", "topics": [ "Axn_m" ] }, { "page": "Axn_m_udd", "title": "UDD approximation of m-thly endowment insurance", "topics": [ "Axn_m_udd" ] }, { "page": "Axn1", "title": "Term insurance APV", "topics": [ "Axn1" ] }, { "page": "Axn1_m", "title": "m-thly term insurance APV", "topics": [ "Axn1_m" ] }, { "page": "Axn1_m_udd", "title": "UDD approximation of m-thly term insurance", "topics": [ "Axn1_m_udd" ] }, { "page": "Axy", "title": "Joint-life whole life insurance", "topics": [ "Axy" ] }, { "page": "axy_ch12", "title": "Joint-life whole life annuity-immediate", "topics": [ "axy", "axy_ch12" ] }, { "page": "Axybar", "title": "Last-survivor whole life insurance", "topics": [ "Axybar" ] }, { "page": "axybar_ch12", "title": "Last-survivor whole life annuity-immediate", "topics": [ "axybar", "axybar_ch12" ] }, { "page": "Axybarn", "title": "Last-survivor endowment insurance", "topics": [ "Axybarn" ] }, { "page": "axybarn_ch12", "title": "Last-survivor temporary annuity-immediate", "topics": [ "axybarn", "axybarn_ch12" ] }, { "page": "Axybarn1", "title": "Last-survivor term insurance", "topics": [ "Axybarn1" ] }, { "page": "Axyn", "title": "Joint-life endowment insurance", "topics": [ "Axyn" ] }, { "page": "axyn_ch12", "title": "Joint-life temporary annuity-immediate", "topics": [ "axyn", "axyn_ch12" ] }, { "page": "Axyn1", "title": "Joint-life term insurance", "topics": [ "Axyn1" ] }, { "page": "ay_x", "title": "Reversionary annuity to (x) after death of (y)", "topics": [ "ay_x" ] }, { "page": "betaF", "title": "Full preliminary term renewal modified premium", "topics": [ "betaF" ] }, { "page": "chapter15_spot_interest_apv", "title": "Spot-rate actuarial present value functions", "topics": [ "Axn1_spot", "Axn_spot", "axn_spot", "chapter15_spot_interest_apv", "nEx_spot" ] }, { "page": "chapter15_variable_interest_apv", "title": "Variable-interest actuarial present value functions", "topics": [ "Axn1_var", "Axn_var", "axn_var", "chapter15_variable_interest_apv", "nEx_var" ] }, { "page": "coi_ul_typeB", "title": "Cost of insurance for Type B universal life", "topics": [ "coi_ul_typeB" ] }, { "page": "contribution_rate_target", "title": "Target contribution rate for a defined contribution plan", "topics": [ "contribution_rate_target" ] }, { "page": "cov_term_deferred", "title": "Covariance of term and deferred insurance PVs", "topics": [ "cov_term_deferred" ] }, { "page": "cov_term_endow", "title": "Covariance of term insurance and pure endowment PVs", "topics": [ "cov_term_endow" ] }, { "page": "cumhaz0", "title": "Cumulative hazard for age-at-failure T0", "topics": [ "cumhaz0" ] }, { "page": "DAbarxn1", "title": "Piecewise-continuous decreasing n-year term insurance", "topics": [ "DAbarxn1" ] }, { "page": "DAxn1", "title": "Decreasing n-year term insurance", "topics": [ "DAxn1" ] }, { "page": "DbarAbarxn1", "title": "Fully continuous decreasing n-year term insurance", "topics": [ "DbarAbarxn1" ] }, { "page": "decompGg_disc", "title": "Ordered gross gain decomposition", "topics": [ "decompGg_disc" ] }, { "page": "discount", "title": "Discount factor for compound interest", "topics": [ "discount" ] }, { "page": "discounted_payback_period", "title": "Discounted payback period", "topics": [ "discounted_payback_period" ] }, { "page": "dist0", "title": "Distribution functions for age-at-failure T0", "topics": [ "dist0", "F0", "f0" ] }, { "page": "double_force_delta", "title": "Doubled force of interest", "topics": [ "double_force_delta" ] }, { "page": "double_force_i", "title": "Effective annual interest at doubled force", "topics": [ "double_force_i" ] }, { "page": "dx", "title": "Compute deaths between ages x and x+1", "topics": [ "dx" ] }, { "page": "dxj", "title": "Cause-specific decrements d_x^{(j)}", "topics": [ "dxj" ] }, { "page": "dxtau", "title": "Total decrements d_x^{(tau)}", "topics": [ "dxtau" ] }, { "page": "ELtx", "title": "Mean present value of loss at duration t for whole life insurance", "topics": [ "ELtx" ] }, { "page": "ex_complete", "title": "Complete expectation of life at age x", "topics": [ "ex_complete" ] }, { "page": "ex_complete_tab", "title": "Complete expectation of life from a life table", "topics": [ "ex_complete_tab" ] }, { "page": "ex_curtate", "title": "Curtate expectation of life at age x", "topics": [ "ex_curtate" ] }, { "page": "ex_curtate_tab", "title": "Curtate expectation of life from a life table", "topics": [ "ex_curtate_tab" ] }, { "page": "ex_temp_complete_tab", "title": "Temporary complete expectation of life from a life table", "topics": [ "ex_temp_complete_tab" ] }, { "page": "ex_temp_curtate_tab", "title": "Temporary curtate expectation of life from a life table", "topics": [ "ex_temp_curtate_tab" ] }, { "page": "fnk_from_z", "title": "Forward rate f_{n,k} from spot rates", "topics": [ "fnk_from_z" ] }, { "page": "forward_matrix_from_z", "title": "Matrix of all determinable forward rates from spot rates", "topics": [ "forward_matrix_from_z" ] }, { "page": "fx", "title": "Conditional density for Tx", "topics": [ "fx" ] }, { "page": "fx_tab", "title": "Fractional conditional density from a life table", "topics": [ "fx_tab" ] }, { "page": "gain_loss_md", "title": "Gain or loss in a multiple-decrement model", "topics": [ "gain_loss_md" ] }, { "page": "GI_cont", "title": "Interest gain helper for continuous-style recursion", "topics": [ "GI_cont" ] }, { "page": "GI_disc", "title": "Interest gain for a discrete insurance contract", "topics": [ "GI_disc" ] }, { "page": "GM_cont", "title": "Mortality gain helper for continuous-style recursion", "topics": [ "GM_cont" ] }, { "page": "GM_disc", "title": "Mortality gain for a discrete insurance contract", "topics": [ "GM_disc" ] }, { "page": "GMF_rollforward_ul", "title": "Guaranteed maturity fund roll-forward", "topics": [ "GMF_rollforward_ul" ] }, { "page": "GT_cont", "title": "Total gain for a continuous-style one-step recursion", "topics": [ "GT_cont" ] }, { "page": "GT_disc", "title": "Total gain for a discrete insurance contract", "topics": [ "GT_disc" ] }, { "page": "GTg_disc", "title": "Total gross gain for a discrete insurance contract", "topics": [ "GTg_disc" ] }, { "page": "hazard0", "title": "Hazard / force for age-at-failure T0", "topics": [ "hazard0" ] }, { "page": "htVx", "title": "h-pay whole life net level premium reserve", "topics": [ "htVx" ] }, { "page": "i_credit_eiul", "title": "Credited rates from raw index growth rates", "topics": [ "i_credit_eiul" ] }, { "page": "IAbarx", "title": "Piecewise-continuous increasing whole life insurance", "topics": [ "IAbarx" ] }, { "page": "IAx", "title": "Increasing whole life insurance", "topics": [ "IAx" ] }, { "page": "IAxn1", "title": "Increasing n-year term insurance", "topics": [ "IAxn1" ] }, { "page": "IbarAbarx", "title": "Fully continuous increasing whole life insurance", "topics": [ "IbarAbarx" ] }, { "page": "IbarAbarxn1", "title": "Fully continuous increasing n-year term insurance", "topics": [ "IbarAbarxn1" ] }, { "page": "iMA_eiul", "title": "Monthly-average index growth rate", "topics": [ "iMA_eiul" ] }, { "page": "Income_dc", "title": "Retirement income from a defined contribution accumulation", "topics": [ "Income_dc" ] }, { "page": "interest_convert", "title": "Convert between compound-interest quantities", "topics": [ "interest_convert" ] }, { "page": "iP_eiul", "title": "Point-to-point index growth rates", "topics": [ "iP_eiul" ] }, { "page": "IRR_profit", "title": "Internal rate of return of a profit signature", "topics": [ "IRR_profit" ] }, { "page": "life_table", "title": "Construct a life table", "topics": [ "life_table" ] }, { "page": "lx", "title": "Extract life-table survivor values", "topics": [ "lx" ] }, { "page": "lx_select", "title": "Extract select-table survivor value", "topics": [ "lx_select" ] }, { "page": "lx_to_S0", "title": "Convert life-table values to survival probabilities", "topics": [ "lx_to_S0" ] }, { "page": "markov_nstep_prob", "title": "n-step transition probability for a discrete-time Markov chain", "topics": [ "markov_nstep_prob" ] }, { "page": "md_table", "title": "Build a multiple-decrement table", "topics": [ "md_table" ] }, { "page": "meanVx", "title": "Mean reserve for whole life insurance", "topics": [ "meanVx" ] }, { "page": "mux_tab", "title": "Fractional force of mortality from a life table", "topics": [ "mux_tab" ] }, { "page": "nAbarx", "title": "Continuous deferred insurance APV", "topics": [ "nAbarx" ] }, { "page": "nAbarx_udd", "title": "UDD approximation of continuous deferred insurance", "topics": [ "nAbarx_udd" ] }, { "page": "nAx", "title": "Deferred insurance APV", "topics": [ "nAx" ] }, { "page": "nAx_m", "title": "m-thly deferred insurance APV", "topics": [ "nAx_m" ] }, { "page": "nAx_m_udd", "title": "UDD approximation of m-thly deferred insurance", "topics": [ "nAx_m_udd" ] }, { "page": "naxn_improved", "title": "Deferred temporary annuity-immediate under mortality improvement", "topics": [ "naxn_improved" ] }, { "page": "NC_EAN_db", "title": "Entry Age Normal normal cost for a DB plan", "topics": [ "NC_EAN_db" ] }, { "page": "NC_PUC_db", "title": "Projected Unit Credit normal cost for a DB plan", "topics": [ "NC_PUC_db" ] }, { "page": "NC_TUC_db", "title": "Traditional Unit Credit normal cost for a DB plan", "topics": [ "NC_TUC_db" ] }, { "page": "ndx", "title": "Compute deaths over an n-year interval from a life table", "topics": [ "ndx" ] }, { "page": "nEx", "title": "Pure endowment APV", "topics": [ "nEx" ] }, { "page": "nExy", "title": "Joint-life pure endowment", "topics": [ "nExy" ] }, { "page": "nExybar", "title": "Last-survivor pure endowment", "topics": [ "nExybar" ] }, { "page": "nkqx", "title": "Curtate death probability from a life table", "topics": [ "nkqx" ] }, { "page": "nmxq", "title": "Deferred death probability from a life table", "topics": [ "nmxq" ] }, { "page": "nmxq_select", "title": "Deferred select-life death probability", "topics": [ "nmxq_select" ] }, { "page": "NPV_partial", "title": "Partial net present values", "topics": [ "NPV_partial" ] }, { "page": "NPV_profit", "title": "Net present value of a profit signature", "topics": [ "NPV_profit" ] }, { "page": "npx", "title": "Compute n-year survival probability from a life table", "topics": [ "npx" ] }, { "page": "npx_select", "title": "Select-life survival probability", "topics": [ "npx_select" ] }, { "page": "npxtau_md", "title": "{}_n p_x^{(tau)} from a multiple-decrement table", "topics": [ "npxtau_md" ] }, { "page": "nqx", "title": "Compute n-year death probability from a life table", "topics": [ "nqx" ] }, { "page": "nqx_select", "title": "Select-life death probability", "topics": [ "nqx_select" ] }, { "page": "nqxj_md", "title": "{}_n q_x^{(j)} from a multiple-decrement table", "topics": [ "nqxj_md" ] }, { "page": "nqxtau_md", "title": "{}_n q_x^{(tau)} from a multiple-decrement table", "topics": [ "nqxtau_md" ] }, { "page": "PAB_cae", "title": "Projected annual benefit under a career average earnings DB plan", "topics": [ "PAB_cae" ] }, { "page": "PAB_fas", "title": "Projected annual benefit under a final average salary DB plan", "topics": [ "PAB_fas" ] }, { "page": "Pbar_trapz_ms", "title": "Continuous premium approximation \\overline{P} by trapezoidal rule", "topics": [ "Pbar_trapz_ms" ] }, { "page": "Pi_signature", "title": "Profit signature from a profit vector", "topics": [ "Pi_signature" ] }, { "page": "PnAdotx", "title": "Deferred annuity-due premium", "topics": [ "PnAdotx" ] }, { "page": "Pnax", "title": "Deferred annuity-immediate premium", "topics": [ "Pnax" ] }, { "page": "Pr_vector_disc", "title": "Profit vector for a discrete profit-analysis model", "topics": [ "Pr_vector_disc" ] }, { "page": "premium_ch9", "title": "Chapter 9 premium, loss, and expense functions", "topics": [ "EL0barAbarx", "EL0x", "EL0xn", "EL0xn1", "Gx", "PbarAbarx", "PbarAbarxn", "PbarAbarxn1", "Pbarx", "Pbarxn", "Pbarxn1", "PnAx", "PnAx_m", "PnEx", "premium_ch9", "Px", "Pxn", "Pxn1", "Pxn1_m", "Pxn_m", "Px_m", "tPnAx", "tPnEx", "tPx", "tPxn", "tPxn1", "varL0barAbarx", "varL0x", "varL0xn", "varL0xn1" ] }, { "page": "profit_margin", "title": "Profit margin", "topics": [ "profit_margin" ] }, { "page": "pv_cashflows", "title": "Present value of cash flows at time 0", "topics": [ "pv_cashflows" ] }, { "page": "pv_spot_cashflows", "title": "Present value of cash flows using spot rates", "topics": [ "pv_spot_cashflows" ] }, { "page": "px_proj", "title": "Project one-year survival probability under mortality improvement", "topics": [ "px_proj" ] }, { "page": "px_to_lx", "title": "Construct life-table values from p_x values", "topics": [ "px_to_lx" ] }, { "page": "pxtau", "title": "Survival probability p_x^{(tau)}", "topics": [ "pxtau" ] }, { "page": "pxtau_ul", "title": "One-year persistency rates for universal life", "topics": [ "pxtau_ul" ] }, { "page": "qx_dep_cf", "title": "Dependent probabilities q_x^{(j)} from independent probabilities q_x^{\\prime(j)} under constant force", "topics": [ "qx_dep_cf" ] }, { "page": "qx_dep_sudd", "title": "Dependent probabilities q_x^{(j)} from independent probabilities q_x^{\\prime(j)} under SUDD", "topics": [ "qx_dep_sudd" ] }, { "page": "qx_proj", "title": "Project one-year death probability under mortality improvement", "topics": [ "qx_proj" ] }, { "page": "qx_tab", "title": "Compute one-year death probability from a life table", "topics": [ "qx_tab" ] }, { "page": "qx_to_lx", "title": "Construct life-table values from q_x values", "topics": [ "qx_to_lx" ] }, { "page": "qxprime_mudd", "title": "Independent probabilities q_x^{\\prime(j)} from dependent probabilities q_x^{(j)} under MUDD", "topics": [ "qxprime_mudd" ] }, { "page": "qxprime_sudd", "title": "Independent probabilities q_x^{\\prime(j)} from dependent probabilities q_x^{(j)} under SUDD", "topics": [ "qxprime_sudd" ] }, { "page": "qxtau", "title": "Total probability of decrement q_x^{(tau)}", "topics": [ "qxtau" ] }, { "page": "replacement_ratio_db", "title": "Replacement ratio for a defined benefit plan", "topics": [ "replacement_ratio_db" ] }, { "page": "replacement_ratio_dc", "title": "Replacement ratio for a defined contribution plan", "topics": [ "replacement_ratio_dc" ] }, { "page": "reserve_deferred_insurance_ch10", "title": "Deferred insurance reserve functions", "topics": [ "htVnAx", "reserve_deferred_insurance_ch10", "tVnAx" ] }, { "page": "rt_ul", "title": "Ratio r_t = AV_t / GMF_t capped at 1", "topics": [ "rt_ul" ] }, { "page": "S0", "title": "Survival function for age-at-failure T0", "topics": [ "S0" ] }, { "page": "S0_to_lx", "title": "Convert survival probabilities to life-table values", "topics": [ "S0_to_lx" ] }, { "page": "salary_scale", "title": "Salary scale under constant annual growth", "topics": [ "salary_scale" ] }, { "page": "select_life_table", "title": "Construct a select life table", "topics": [ "select_life_table" ] }, { "page": "solve_yield", "title": "Solve the yield rate by the equation of value", "topics": [ "solve_yield" ] }, { "page": "thiele_backward_path", "title": "Backward Euler reserve path from maturity", "topics": [ "thiele_backward_path" ] }, { "page": "thiele_backward_step", "title": "One backward Euler-style Thiele step", "topics": [ "thiele_backward_step" ] }, { "page": "thiele_dVdt", "title": "Reserve derivative from Thiele's equation", "topics": [ "thiele_dVdt" ] }, { "page": "thiele_dVdt_01", "title": "Reserve derivatives for the disability model with recovery", "topics": [ "thiele_dVdt_01" ] }, { "page": "thiele_path_01", "title": "Backward reserve path for the disability model with recovery", "topics": [ "thiele_path_01" ] }, { "page": "tp00_tp01_euler", "title": "Euler approximation for {}_{t}p_{x}^{00} and {}_{t}p_{x}^{01}", "topics": [ "tp00_tp01_euler" ] }, { "page": "tpx", "title": "Conditional survival probability for Tx", "topics": [ "tpx" ] }, { "page": "tpx_improved", "title": "Multi-year survival probability under mortality improvement", "topics": [ "tpx_improved" ] }, { "page": "tpx_tab", "title": "Fractional survival probability from a life table", "topics": [ "tpx_tab" ] }, { "page": "tpx_tau_cf", "title": "Total survival probability {}_t p_x^{(tau)} under constant forces", "topics": [ "tpx_tau_cf" ] }, { "page": "tpxprimej_cf", "title": "Single-decrement survival probability {}_t p_x^{\\prime(j)} under constant force", "topics": [ "tpxprimej_cf" ] }, { "page": "tpxtau_ul", "title": "Cumulative persistency to the end of each policy year", "topics": [ "tpxtau_ul" ] }, { "page": "tpxy", "title": "Joint-life survival probability", "topics": [ "tpxy" ] }, { "page": "tpxybar", "title": "Last-survivor survival probability", "topics": [ "tpxybar" ] }, { "page": "tqx", "title": "Conditional failure probability for Tx", "topics": [ "tqx" ] }, { "page": "tqx_tab", "title": "Fractional failure probability from a life table", "topics": [ "tqx_tab" ] }, { "page": "tqxj_cf", "title": "Cause-specific probability {}_t q_x^{(j)} under constant forces", "topics": [ "tqxj_cf" ] }, { "page": "tqxprime_mudd", "title": "Independent probabilities {}_t q_x^{\\prime(j)} under MUDD", "topics": [ "tqxprime_mudd" ] }, { "page": "tqxprimej_cf", "title": "Single-decrement failure probability {}_t q_x^{\\prime(j)} under constant force", "topics": [ "tqxprimej_cf" ] }, { "page": "tqxy", "title": "Joint-life failure probability", "topics": [ "tqxy" ] }, { "page": "tqxy1", "title": "Probability that (x) fails before (y) within n years", "topics": [ "tqxy1" ] }, { "page": "tqxy2", "title": "Probability that (x) fails after (y) within n years", "topics": [ "tqxy2" ] }, { "page": "tqxybar", "title": "Last-survivor failure probability", "topics": [ "tqxybar" ] }, { "page": "tqyx1", "title": "Probability that (y) fails before (x) within n years", "topics": [ "tqyx1" ] }, { "page": "tqyx2", "title": "Probability that (y) fails after (x) within n years", "topics": [ "tqyx2" ] }, { "page": "tsVx", "title": "Fractional-duration whole life reserve", "topics": [ "tsVx" ] }, { "page": "tsVxn", "title": "Fractional-duration endowment reserve", "topics": [ "tsVxn" ] }, { "page": "tsVxn1", "title": "Fractional-duration term reserve", "topics": [ "tsVxn1" ] }, { "page": "tVbarAbarx", "title": "Fully continuous whole life reserve", "topics": [ "tVbarAbarx" ] }, { "page": "tVbarx", "title": "Whole life reserve with continuous premiums", "topics": [ "tVbarx" ] }, { "page": "tVEx", "title": "Whole life expense reserve", "topics": [ "tVEx" ] }, { "page": "tVFx", "title": "Full preliminary term reserve for whole life insurance", "topics": [ "tVFx" ] }, { "page": "tVGx", "title": "Whole life gross premium reserve", "topics": [ "tVGx" ] }, { "page": "tVnAdotx", "title": "Deferred annuity-due reserve", "topics": [ "tVnAdotx" ] }, { "page": "tVnax", "title": "Deferred annuity-immediate reserve", "topics": [ "tVnax" ] }, { "page": "tVnEx", "title": "Pure endowment net level premium reserve", "topics": [ "tVnEx" ] }, { "page": "tVx", "title": "Whole life net level premium reserve", "topics": [ "tVx" ] }, { "page": "tVx_m", "title": "Whole life reserve with m-thly premiums", "topics": [ "tVx_m" ] }, { "page": "tVx_ret", "title": "Whole life net level premium reserve by retrospective method", "topics": [ "tVx_ret" ] }, { "page": "tVxn", "title": "Endowment insurance net level premium reserve", "topics": [ "tVxn" ] }, { "page": "tVxn_ret", "title": "Endowment insurance reserve by retrospective method", "topics": [ "tVxn_ret" ] }, { "page": "tVxn1", "title": "Term 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