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    },
    {
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    },
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      "topics": [
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    },
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      "title": "Compute the aggregate loss distribution",
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      "title": "Calculate moments of a collective aggregate loss",
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    },
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    },
    {
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    },
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    {
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    {
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    },
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      "concept": [
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      "topics": [
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    },
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      "topics": [
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    },
    {
      "page": "bootstrap_spot_curve",
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      "concept": [
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      "topics": [
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    {
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      "topics": [
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      "title": "Specify a collective risk model",
      "topics": [
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    },
    {
      "page": "commutation_table",
      "title": "Build an annual commutation table (discrete ages)",
      "topics": [
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      ]
    },
    {
      "page": "convexity_cash_flow",
      "title": "Convexity of a general cash-flow stream",
      "topics": [
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    },
    {
      "page": "coverage_spec",
      "title": "Specify an insurance coverage contract",
      "topics": [
        "coverage_spec"
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    },
    {
      "page": "discount_factor",
      "title": "Discount factor under conventional or time-varying interest",
      "concept": [
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        "time-value"
      ],
      "topics": [
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    },
    {
      "page": "discount_factor_spot",
      "title": "Spot discount factor",
      "concept": [
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      "topics": [
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    },
    {
      "page": "discretize_severity",
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      "topics": [
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    },
    {
      "page": "distribution_convolution",
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      "topics": [
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    },
    {
      "page": "distribution_moments",
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    },
    {
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    {
      "page": "e_x",
      "title": "Expected future lifetime from an annual life table",
      "topics": [
        "e_x"
      ]
    },
    {
      "page": "e_xy",
      "title": "Expected future lifetime for two independent lives",
      "topics": [
        "e_xy"
      ]
    },
    {
      "page": "empirical_frequency",
      "title": "Specify an empirical claim frequency distribution",
      "topics": [
        "empirical_frequency"
      ]
    },
    {
      "page": "empirical_severity",
      "title": "Specify an empirical claim severity distribution",
      "topics": [
        "empirical_severity"
      ]
    },
    {
      "page": "forward_curve",
      "title": "Construct a forward-rate curve from spot rates",
      "concept": [
        "interest"
      ],
      "topics": [
        "forward_curve"
      ]
    },
    {
      "page": "forward_rate",
      "title": "Compute an implied forward rate from a discrete spot curve",
      "concept": [
        "interest"
      ],
      "topics": [
        "forward_rate"
      ]
    },
    {
      "page": "frequency_spec",
      "title": "Specify a claim frequency distribution",
      "topics": [
        "frequency_spec"
      ]
    },
    {
      "page": "future_value",
      "title": "Future value of a single payment",
      "concept": [
        "time-value"
      ],
      "topics": [
        "future_value"
      ]
    },
    {
      "page": "fv_flow",
      "title": "Future value of a general cash flow",
      "concept": [
        "time-value"
      ],
      "topics": [
        "fv_flow"
      ]
    },
    {
      "page": "immunize_duration",
      "title": "Duration-based immunization with multiple assets",
      "concept": [
        "immunization"
      ],
      "topics": [
        "immunize_duration"
      ]
    },
    {
      "page": "immunize_duration_convexity",
      "title": "Duration and convexity immunization with multiple assets",
      "concept": [
        "immunization"
      ],
      "topics": [
        "immunize_duration_convexity"
      ]
    },
    {
      "page": "insurance_variable_k",
      "title": "Actuarial present value of a life insurance with variable k-thly benefits",
      "concept": [
        "life-contingencies"
      ],
      "topics": [
        "insurance_variable_k"
      ]
    },
    {
      "page": "insurance_x",
      "title": "Actuarial present value of a life insurance",
      "concept": [
        "life-contingencies"
      ],
      "topics": [
        "insurance_x"
      ]
    },
    {
      "page": "insurance_xj",
      "title": "Cause-specific term/whole-life insurance APV under multiple decrements",
      "concept": [
        "multiple-decrements"
      ],
      "topics": [
        "A_xj",
        "insurance_xj"
      ]
    },
    {
      "page": "insurance_xy",
      "title": "Actuarial present value of a two-life insurance",
      "concept": [
        "life-contingencies"
      ],
      "topics": [
        "insurance_xy"
      ]
    },
    {
      "page": "interest_equivalents",
      "title": "Equivalent interest rates in FM actuarial notation",
      "concept": [
        "interest"
      ],
      "topics": [
        "interest_equivalents"
      ]
    },
    {
      "page": "irr_flow",
      "title": "Internal rate of return for a cash flow",
      "concept": [
        "time-value"
      ],
      "topics": [
        "irr_flow"
      ]
    },
    {
      "page": "irr_flow_multi",
      "title": "Multiple internal rates of return for a cash flow",
      "concept": [
        "time-value"
      ],
      "topics": [
        "irr_flow_multi"
      ]
    },
    {
      "page": "km_lifetable",
      "title": "Kaplan-Meier survival curve and a lifetable-style life table",
      "topics": [
        "km_lifetable"
      ]
    },
    {
      "page": "life_contract",
      "title": "Create a life-contingency contract specification",
      "concept": [
        "life-contingencies"
      ],
      "topics": [
        "life_contract"
      ]
    },
    {
      "page": "lifetable",
      "title": "Build an annual life table (tidy tibble) from lx, qx, px, or mx",
      "topics": [
        "lifetable"
      ]
    },
    {
      "page": "loans_sample",
      "title": "Sample loan contracts for amortization examples",
      "topics": [
        "loans_sample"
      ]
    },
    {
      "page": "lt_tau",
      "title": "Total-decrement lifetable from a multiple decrement table: lt_tau",
      "topics": [
        "lt_tau"
      ]
    },
    {
      "page": "mc_annuity",
      "title": "Compute simulated present values for life annuities",
      "concept": [
        "monte-carlo"
      ],
      "topics": [
        "mc_annuity"
      ]
    },
    {
      "page": "mc_insurance",
      "title": "Compute simulated present values for life insurance benefits",
      "concept": [
        "monte-carlo"
      ],
      "topics": [
        "mc_insurance"
      ]
    },
    {
      "page": "mc_loss",
      "title": "Compute Monte Carlo loss random variables for life contingencies",
      "concept": [
        "monte-carlo"
      ],
      "topics": [
        "mc_loss"
      ]
    },
    {
      "page": "mc_multilife_status",
      "title": "Compute multiple-life simulated status variables",
      "concept": [
        "monte-carlo"
      ],
      "topics": [
        "mc_multilife_status"
      ]
    },
    {
      "page": "mc_premium",
      "title": "Compute Monte Carlo net premiums for life contingencies",
      "concept": [
        "monte-carlo"
      ],
      "topics": [
        "mc_premium"
      ]
    },
    {
      "page": "mc_reserve",
      "title": "Compute Monte Carlo prospective reserves for life contingencies",
      "concept": [
        "monte-carlo"
      ],
      "topics": [
        "mc_reserve"
      ]
    },
    {
      "page": "md_table",
      "title": "Multiple decrement table (annual, discrete ages)",
      "topics": [
        "md_table"
      ]
    },
    {
      "page": "mortality_colombia_tables",
      "title": "Colombian mortality tables",
      "topics": [
        "mortality_colombia_tables"
      ]
    },
    {
      "page": "mortality_law_table",
      "title": "Generate a tidy life table from a theoretical mortality law",
      "concept": [
        "life-tables"
      ],
      "topics": [
        "mortality_law_table"
      ]
    },
    {
      "page": "mortality_world_sample_2015_2023",
      "title": "World mortality sample panel, 2015-2023",
      "topics": [
        "mortality_world_sample_2015_2023"
      ]
    },
    {
      "page": "mortality_world_sample_2023",
      "title": "World mortality sample, 2023",
      "topics": [
        "mortality_world_sample_2023"
      ]
    },
    {
      "page": "multiple_decrement_sample",
      "title": "Sample multiple decrement probabilities",
      "topics": [
        "multiple_decrement_sample"
      ]
    },
    {
      "page": "par_rate",
      "title": "Par coupon rate implied by a spot yield curve",
      "concept": [
        "interest"
      ],
      "topics": [
        "par_rate"
      ]
    },
    {
      "page": "plot_cash_flow",
      "title": "Plot a cash-flow diagram",
      "concept": [
        "time-value"
      ],
      "topics": [
        "plot_cash_flow"
      ]
    },
    {
      "page": "plot_contingent_cash_flow",
      "title": "Plot discrete life-contingent cash flows",
      "topics": [
        "plot_contingent_cash_flow"
      ]
    },
    {
      "page": "plot_immunization_gap",
      "title": "Plot immunization performance under interest-rate shifts",
      "concept": [
        "immunization"
      ],
      "topics": [
        "plot_immunization_gap"
      ]
    },
    {
      "page": "plot_km",
      "title": "Plot a Kaplan-Meier survival curve",
      "topics": [
        "plot_km"
      ]
    },
    {
      "page": "portfolio_convexity",
      "title": "Compute portfolio convexity as a market-value-weighted average",
      "concept": [
        "bonds"
      ],
      "topics": [
        "portfolio_convexity"
      ]
    },
    {
      "page": "portfolio_duration",
      "title": "Compute portfolio duration as a market-value-weighted average",
      "concept": [
        "bonds"
      ],
      "topics": [
        "portfolio_duration"
      ]
    },
    {
      "page": "premium_gross",
      "title": "Gross premium under a simple expense-loaded equivalence principle",
      "concept": [
        "life-contingencies"
      ],
      "topics": [
        "premium_gross"
      ]
    },
    {
      "page": "premium_x",
      "title": "Net premium for single-life insurance by the equivalence principle",
      "concept": [
        "life-contingencies"
      ],
      "topics": [
        "premium_x"
      ]
    },
    {
      "page": "premium_xy",
      "title": "Net premium for two-life insurance by the equivalence principle",
      "concept": [
        "life-contingencies"
      ],
      "topics": [
        "premium_xy"
      ]
    },
    {
      "page": "present_value",
      "title": "Present value of a single payment",
      "concept": [
        "time-value"
      ],
      "topics": [
        "present_value"
      ]
    },
    {
      "page": "print.tidyact_aggregate_distribution",
      "title": "Print an aggregate loss distribution",
      "topics": [
        "print.tidyact_aggregate_distribution"
      ]
    },
    {
      "page": "print.tidyact_collective_risk_spec",
      "title": "Print a collective risk model specification",
      "topics": [
        "print.tidyact_collective_risk_spec"
      ]
    },
    {
      "page": "print.tidyact_convolution_spec",
      "title": "Print a convolution specification",
      "topics": [
        "print.tidyact_convolution_spec"
      ]
    },
    {
      "page": "print.tidyact_frequency_spec",
      "title": "Print a frequency distribution specification",
      "topics": [
        "print.tidyact_frequency_spec"
      ]
    },
    {
      "page": "print.tidyact_payment_spec",
      "title": "Print a payment severity specification",
      "topics": [
        "print.tidyact_payment_spec"
      ]
    },
    {
      "page": "print.tidyact_severity_spec",
      "title": "Print a severity distribution specification",
      "topics": [
        "print.tidyact_severity_spec"
      ]
    },
    {
      "page": "pv_flow",
      "title": "Present value of a general cash flow",
      "concept": [
        "time-value"
      ],
      "topics": [
        "pv_flow"
      ]
    },
    {
      "page": "reserve_x",
      "title": "Benefit reserves for single-life insurance",
      "concept": [
        "life-contingencies"
      ],
      "topics": [
        "reserve_x"
      ]
    },
    {
      "page": "reserve_xy",
      "title": "Benefit reserves for two-life insurance",
      "concept": [
        "life-contingencies"
      ],
      "topics": [
        "reserve_xy"
      ]
    },
    {
      "page": "s_angle",
      "title": "Level annuity accumulation factor s-angle-n",
      "concept": [
        "annuities"
      ],
      "topics": [
        "s_angle"
      ]
    },
    {
      "page": "severity_spec",
      "title": "Specify an individual claim severity distribution",
      "topics": [
        "severity_spec"
      ]
    },
    {
      "page": "simulate_aggregate",
      "title": "Simulate a collective risk model",
      "topics": [
        "simulate_aggregate"
      ]
    },
    {
      "page": "simulate_annuity_x",
      "title": "Monte Carlo simulation of a life annuity",
      "concept": [
        "life-contingencies",
        "simulation"
      ],
      "topics": [
        "simulate_annuity_x"
      ]
    },
    {
      "page": "simulate_frequency",
      "title": "Simulate claim counts from a frequency model",
      "topics": [
        "simulate_frequency"
      ]
    },
    {
      "page": "simulate_insurance_x",
      "title": "Monte Carlo simulation of a life insurance",
      "concept": [
        "life-contingencies",
        "simulation"
      ],
      "topics": [
        "simulate_insurance_x"
      ]
    },
    {
      "page": "simulate_lifetime",
      "title": "Simulate future lifetimes from a life table",
      "concept": [
        "monte-carlo"
      ],
      "topics": [
        "simulate_lifetime"
      ]
    },
    {
      "page": "simulate_lifetimes",
      "title": "Simulate future lifetimes for multiple lives",
      "concept": [
        "monte-carlo"
      ],
      "topics": [
        "simulate_lifetimes"
      ]
    },
    {
      "page": "simulate_severity",
      "title": "Simulate individual claim severities",
      "topics": [
        "simulate_severity"
      ]
    },
    {
      "page": "sinking_fund_schedule",
      "title": "Sinking fund amortization schedule for a loan",
      "concept": [
        "amortization"
      ],
      "topics": [
        "sinking_fund_schedule"
      ]
    },
    {
      "page": "soa08lt",
      "title": "SOA Illustrative Life Table",
      "topics": [
        "soa08lt"
      ]
    },
    {
      "page": "solve_value",
      "title": "Solve a scalar equation of value",
      "concept": [
        "time-value"
      ],
      "topics": [
        "solve_value"
      ]
    },
    {
      "page": "standardize_interest",
      "title": "Standardize an interest rate to the annual effective rate i",
      "concept": [
        "interest"
      ],
      "topics": [
        "standardize_interest"
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    },
    {
      "page": "stop_loss_premium",
      "title": "Net stop-loss premium",
      "topics": [
        "stop_loss_premium"
      ]
    },
    {
      "page": "summary_mc",
      "title": "Summarise Monte Carlo simulation output",
      "concept": [
        "monte-carlo"
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      "topics": [
        "summary_mc"
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    {
      "page": "t_Ex",
      "title": "Pure endowment (discounted survival): {}_tE_x",
      "topics": [
        "t_Ex"
      ]
    },
    {
      "page": "t_px",
      "title": "t-year survival probability from a life table",
      "topics": [
        "t_px"
      ]
    },
    {
      "page": "t_pxy",
      "title": "Two-life survival probability for independent lives",
      "topics": [
        "t_pxy"
      ]
    },
    {
      "page": "t_qx",
      "title": "t-year death probability from a life table",
      "topics": [
        "t_qx"
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    },
    {
      "page": "t_qxj",
      "title": "t-year probability of decrement by cause j: t_qxj",
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        "t_qxj"
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    {
      "page": "tail_value_at_risk",
      "title": "Tail Value at Risk (Expected Shortfall)",
      "topics": [
        "tail_value_at_risk"
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    },
    {
      "page": "to_payment_basis",
      "title": "Specify a change from per-loss to per-payment basis",
      "topics": [
        "to_payment_basis"
      ]
    },
    {
      "page": "value_at_risk",
      "title": "Value at Risk for actuarial losses",
      "topics": [
        "value_at_risk"
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    },
    {
      "page": "yield_curve",
      "title": "Validate a yield curve and compute discount factors",
      "concept": [
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        "yield_curve"
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      "title": "Bonds, Duration, Convexity, and Immunization",
      "author": "Julian Mauricio Fajardo",
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      "headings": [
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        "$$935{,}000",
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        "$$\\frac",
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      "modified": "2026-09-14 21:33:11",
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      "filename": "financial-mathematics.html",
      "title": "Financial Mathematics with tidyactuarial",
      "author": "Julian Mauricio Fajardo",
      "engine": "knitr::rmarkdown",
      "headings": [
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        "$$\\boxed{A(s,t)",
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        "\\frac",
        "$$A(s,t)",
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        "$$\\boxed{\\sum_{j=1}^{n}C_j A(t_j,f)",
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      "modified": "2026-09-14 21:33:11",
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      "title": "Loans, Amortization, and General Payment Schedules",
      "author": "Julian Mauricio Fajardo",
      "engine": "knitr::rmarkdown",
      "headings": [
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        "Final actuarial reading"
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      "created": "2026-09-14 21:33:11",
      "modified": "2026-10-25 11:21:04",
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