Using the CNSF $2013$ table,With an annual interest rate of $5\%,$ only the reserve corresponding to the risk should be calculated; no expenses are indicated.

$1.$ Present the general calculation formulas in detail for determining probabilities $($for a person of age x$,$ with coverage period n and premium payment period k years$),$ and the specific formulas $($age and numerical temporalities, both for coverage and premium payment$).$ Provide numerical values for:

a$)$ Tariff premiums

b$)$ Terminal reserves for years $2$ and $4$ using:

i$)$ Prospective method

ii$)$ Retrospective method

iii$)$ Flackler method

For the following insurances:

I$)$ Lifelong insurance that pays $$1,500,000$ upon the insured$$s death $($i$.$e$.,$ lifelong insurance for a person aged $45$ with a premium payment period of $3$ years$).$

III$)10-$year mixed endowment insurance with a death benefit of $$1,500,000$ and a survival benefit of $$2,000,000$; with a $3-$year premium payment period.

In this section, $12$ cases must be presented:

i$.$ For Lifelong Insurance I, prospective method, reserve at the end of year $2.$

ii$.$ For Lifelong Insurance I, prospective method, reserve at the end of year $4.$

iii. For Lifelong Insurance I, retrospective method, reserve at the end of year $2.$

iv$.$ For Lifelong Insurance I, retrospective method, reserve at the end of year $4.$

v$.$ For Lifelong Insurance I, Flackler method, reserve at the end of year $2.$

vi$.$ For Lifelong Insurance I, Flackler method, reserve at the end of year $4.$

vii. For Endowment Insurance II$,$ prospective method, reserve at the end of year $2.$

viii. For Endowment Insurance II$,$ prospective method, reserve at the end of year $4.$

ix$.$ For Endowment Insurance II$,$ retrospective method, reserve at the end of year $2.$

x$.$ For Endowment Insurance II$,$ retrospective method, reserve at the end of year $4.$

xi$.$ For Endowment Insurance II$,$ Flackler method, reserve at the end of year $2.$

xii. For Endowment Insurance II$,$ Flackler method, reserve at the end of year $4.$