# LifeContingencies.jl

LifeContingencies.InsuranceMethod
Insurance(lc::LifeContingency; n=nothing)
Insurance(life,interest; n=nothing)

Life insurance with a term period of n. If n is nothing, then whole life insurance.

Issue age is based on the issue_age in the LifeContingency lc.

Examples

ins = Insurance(
SingleLife(mort = UltimateMortality([0.5,0.5]),issue_age = 0),
Yields.Constant(0.05),
n = 1
) 
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LifeContingencies.JointAssumptionType
JointAssumption()

An abstract type representing the different assumed relationship between the survival of the lives on a JointLife. Available options to use include:

• Frasier()
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LifeContingencies.JointLifeType
struct JointLife
lives
contingency
joint_assumption
end

A Life object containing the necessary assumptions for contingent maths related to a joint life insurance. Use with a LifeContingency to do many actuarial present value calculations.

Keyword arguments:

• lives is a tuple of two SingleLifes
• contingency default is LastSurvivor(). It is the trigger for contingent benefits. See ?Contingency.
• joint_assumption Default value is Frasier(). It is the assumed relationship between the mortality of the two lives. See ?JointAssumption.

Examples

using MortalityTables
mort = MortalityTables.table("2001 VBT Residual Standard Select and Ultimate - Male Nonsmoker, ANB")

l1 = SingleLife(
mort       = mort.select[30],
issue_age  = 30
)
l2 = SingleLife(
mort       = mort.select[30],
issue_age  = 30
)

jl = JointLife(
lives = (l1,l2),
contingency = LastSurvivor(),
joint_assumption = Frasier()
)
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LifeContingencies.SingleLifeType
struct SingleLife
mort
issue_age::Int
alive::Bool
fractional_assump::MortalityTables.DeathDistribution
end

A Life object containing the necessary assumptions for contingent maths related to a single life. Use with a LifeContingency to do many actuarial present value calculations.

Keyword arguments:

• mort pass a mortality vector, which is an array of applicable mortality rates indexed by attained age
• issue_age is the assumed issue age for the SingleLife and is the basis of many contingency calculations.
• alive Default value is true. Useful for joint insurances with different status on the lives insured.
• fractional_assump. Default value is Uniform(). This is a DeathDistribution from the MortalityTables.jl package and is the assumption to use for non-integer ages/times.

Examples

using MortalityTables
mort = MortalityTables.table("2001 VBT Residual Standard Select and Ultimate - Male Nonsmoker, ANB")

SingleLife(
mort       = mort.select[30],
issue_age  = 30
)
source
LifeContingencies.APVMethod
APV(lc::LifeContingency,to_time)

The actuarial present value which is the survival times the discount factor for the life contingency.

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LifeContingencies.AnnuityDueMethod
AnnuityDue(lc::LifeContingency; n=nothing, start_time=0; certain=nothing,frequency=1)
AnnuityDue(life, interest; n=nothing, start_time=0; certain=nothing,frequency=1)

Annuity due with the benefit period starting at start_time and ending after n periods with frequency payments per year of 1/frequency amount and a certain period with non-contingent payments.

Examples

ins = AnnuityDue(
SingleLife(mort = UltimateMortality([0.5,0.5]),issue_age = 0),
Yields.Constant(0.05),
n = 1
) 
source
LifeContingencies.AnnuityImmediateMethod
AnnuityImmediate(lc::LifeContingency; n=nothing, start_time=0; certain=nothing,frequency=1)
AnnuityImmediate(life, interest; n=nothing, start_time=0; certain=nothing,frequency=1)

Annuity immediate with the benefit period starting at start_time and ending after n periods with frequency payments per year of 1/frequency amount and a certain period with non-contingent payments.

Examples

ins = AnnuityImmediate(
SingleLife(mort = UltimateMortality([0.5,0.5]),issue_age = 0),
Yields.Constant(0.05),
n = 1
) 
source
LifeContingencies.CMethod
C(lc::LifeContingency, to_time)

$C_x$ is a retrospective actuarial commutation function which is the product of the discount factor and the difference in l ($l_x$).

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LifeContingencies.DMethod
D(lc::LifeContingency, to_time)

$D_x$ is a retrospective actuarial commutation function which is the product of the survival and discount factor.

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LifeContingencies.MMethod
M(lc::LifeContingency, from_time)

The $M_x$ actuarial commutation function where the from_time argument is x. Issue age is based on the issue_age in the LifeContingency lc.

source
LifeContingencies.NMethod
N(lc::LifeContingency, from_time)

$N_x$ is a prospective actuarial commutation function which is the sum of the D ($D_x$) values from the given time to the end of the mortality table.

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LifeContingencies.lMethod
l(lc::LifeContingency, to_time)

$l_x$ is a retrospective actuarial commutation function which is the survival up to a certain point in time. By default, will have a unitary basis (ie 1.0), but you can specify basis keyword argument to use something different (e.g. 1000 is common in the literature.)

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LifeContingencies.premium_netMethod
premium_net(lc::LifeContingency)
premium_net(lc::LifeContingency,to_time)

The net premium for a whole life insurance (without second argument) or a term life insurance through to_time.

The net premium is based on 1 unit of insurance with the death benfit payable at the end of the year and assuming annual net premiums.

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MortalityTables.decrementMethod
decrement(lc::LifeContingency,to_time)
decrement(lc::LifeContingency,from_time,to_time)

Return the probablity of death for the given LifeContingency.

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MortalityTables.omegaMethod
omega(lc::LifeContingency)
omega(l::Life)
omega(i::InterestRate)

Lifes and LifeContingencys

Returns the last defined timeperiod for both the interest rate and mortality table. Note that this is different than calling omega on a MortalityTable, which will give you the last attainedage.

Example: if the LifeContingency has issue age 60, and the last defined attained age for the MortalityTable is 100, then omega of the MortalityTable will be 100 and omega of the LifeContingency will be 40.

InterestRates

The last period that the interest rate is defined for. Assumed to be infinite (Inf) for functional and constant interest rate types. Returns the lastindex of the vector if a vector type.

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MortalityTables.survivalMethod
survival(lc::LifeContingency,from_time,to_time)
survival(lc::LifeContingency,to_time)

Return the probablity of survival for the given LifeContingency.

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