These packages are available for use in your project. Scroll down for more information on each one.
Easily work with standard tables and parametric models with common survival calculations.  Insurance, annuity, premium, and reserve maths. 
Robust and fast calculations for  Meeting your exposure calculation needs. 
Composable contracts, models, and functions that allow for modeling of both simple and complex financial instruments.  Easytouse scenario generation that's FinanceModels.jl compatible. 
For consistency, you can lock any package in its current state and not worry about breaking changes to any code that you write. Julia's package manager lets you exactly recreate a set of code and its dependencies. (More).
There are two ways to add packages:
In the code itself: using Pkg; Pkg.add("MortalityTables")
In the REPL, hit ]
to enter Pkg mode and type add MortalityTables
More info can be found at the Pkg manager documentation.
To use packages in your code:
using PackageName
Hasslefree mortality and other rate tables.
Full set of SOA mort.soa.org tables included
survival
and decrement
functions to calculate decrements over period of time
Partial year mortality calculations (Uniform, Constant, Balducci)
Friendly syntax and flexible usage
Extensive set of parametric mortality models.
Load and see information about a particular table:
julia> vbt2001 = MortalityTables.table("2001 VBT Residual Standard Select and Ultimate  Male Nonsmoker, ANB")
MortalityTable (Insured Lives Mortality):
Name:
2001 VBT Residual Standard Select and Ultimate  Male Nonsmoker, ANB
Fields:
(:select, :ultimate, :metadata)
Provider:
Society of Actuaries
mort.SOA.org ID:
1118
mort.SOA.org link:
https://mort.soa.org/ViewTable.aspx?&TableIdentity=1118
Description:
2001 Valuation Basic Table (VBT) Residual Standard Select and Ultimate Table  Male Nonsmoker.
Basis: Age Nearest Birthday.
Minimum Select Age: 0.
Maximum Select Age: 99.
Minimum Ultimate Age: 25.
Maximum Ultimate Age: 120
The package revolves around easytoaccess vectors which are indexed by attained age:
julia> vbt2001.select[35] # vector of rates for issue age 35
0.00036
0.00048
โฎ
0.94729
1.0
julia> vbt2001.select[35][35] # issue age 35, attained age 35
0.00036
julia> vbt2001.select[35][50:end] # issue age 35, attained age 50 through end of table
0.00316
0.00345
โฎ
0.94729
1.0
julia> vbt2001.ultimate[95] # ultimate vectors only need to be called with the attained age
0.24298
Calculate the force of mortality or survival over a range of time:
julia> survival(vbt2001.ultimate,30,40) # the survival between ages 30 and 40
0.9894404665434904
julia> decrement(vbt2001.ultimate,30,40) # the decrement between ages 30 and 40
0.010559533456509618
Nonwhole periods of time are supported when you specify the assumption (Constant()
, Uniform()
, or Balducci()
) for fractional periods:
julia> survival(vbt2001.ultimate,30,40.5,Uniform()) # the survival between ages 30 and 40.5
0.9887676470262408
Over 20 different models included. Example with the Gompertz
model
m = MortalityTables.Gompertz(a=0.01,b=0.2)
m[20] # the mortality rate at age 20
decrement(m,20,25) # the five year cumulative mortality rate
survival(m,20,25) # the five year survival rate
MortalityTables package on Github ๐กญ
A collection of common functions/manipulations used in Actuarial Calculations.
A collection of common functions/manipulations used in Actuarial Calculations.
cfs = [5, 5, 105]
times = [1, 2, 3]
discount_rate = 0.03
present_value(discount_rate, cfs, times) # 105.65
duration(Macaulay(), discount_rate, cfs, times) # 2.86
duration(discount_rate, cfs, times) # 2.78
convexity(discount_rate, cfs, times) # 10.62
duration
:
Calculate the Macaulay
, Modified
, or DV01
durations for a set of cashflows
convexity
for price sensitivity
Flexible interest rate options via the FinanceModels.jl
package.
internal_rate_of_return
or irr
to calculate the IRR given cashflows (including at timepoints like Excel's XIRR
)
breakeven
to calculate the breakeven time for a set of cashflows
accum_offset
to calculate accumulations like survivorship from a mortality vector
eurocall
and europut
for BlackScholes option prices
Calculate risk measures for a given vector of risks:
CTE
for the Conditional Tail Expectation, or
VaR
for the percentile/Value at Risk.
duration
:
Calculate the duration given an issue date and date (a.k.a. policy duration)
ActuaryUtilities package on GitHub ๐กญ
Common life contingent calculations with a convenient interface.
Integration with other JuliaActuary packages such as MortalityTables.jl
Fast calculations, with some parts utilizing parallel processing power automatically
Use functions that look more like the math you are used to (e.g. A
, aฬ
) with Unicode support
All of the power, speed, convenience, tooling, and ecosystem of Julia
Flexible and modular modeling approach
Leverages MortalityTables.jl for
the mortality calculations
Contains common insurance calculations such as:
Insurance(life,yield)
: Whole life
Insurance(life,yield,n)
: Term life for n
years
aฬ(life,yield)
: present_value
of lifecontingent annuity
aฬ(life,yield,n)
: present_value
of lifecontingent annuity due for n
years
Contains various commutation functions such as D(x)
,M(x)
,C(x)
, etc.
SingleLife
and JointLife
capable
Interest rate mechanics via FinanceModels.jl
More documentation available by clicking the DOCS badges at the top of this README
Calculate various items for a 30yearold male nonsmoker using 2015 VBT base table and a 5% interest rate
using LifeContingencies
using MortalityTables
using FinanceModels
import LifeContingencies: V, aฬ # pull the shortform notation into scope
# load mortality rates from MortalityTables.jl
vbt2001 = MortalityTables.table("2001 VBT Residual Standard Select and Ultimate  Male Nonsmoker, ANB")
issue_age = 30
life = SingleLife( # The life underlying the risk
mortality = vbt2001.select[issue_age], #  Mortality rates
)
yield = FinanceModels.Yield.Constant(0.05)
lc = LifeContingency(life, yield) # LifeContingency joins the risk with interest
ins = Insurance(lc) # Whole Life insurance
ins = Insurance(life, yield) # alternate way to construct
With the above life contingent data, we can calculate vectors of relevant information:
cashflows(ins) # A vector of the unit cashflows
timepoints(ins) # The timepoints associated with the cashflows
survival(ins) # The survival vector
benefit(ins) # The unit benefit vector
probability(ins) # The probability of benefit payment
Some of the above will return lazy results. For example, cashflows(ins)
will return a Generator
which can be efficiently used in most places you'd use a vector of cashflows (e.g. pv(...)
or sum(...)
) but has the advantage of being nonallocating (less memory used, faster computations). To get a computed vector instead of the generator, simply call collect(...)
on the result: collect(cashflows(ins))
.
Or calculate summary scalars:
present_value(ins) # The actuarial present value
premium_net(lc) # Net whole life premium
V(lc,5) # Net premium reserve for whole life insurance at time 5
Other types of life contingent benefits:
Insurance(lc,10) # 10 year term insurance
AnnuityImmediate(lc) # Whole life annuity due
AnnuityDue(lc) # Whole life annuity due
aฬ(lc) # Shortform notation
aฬ(lc, 5) # 5 year annuity due
aฬ(lc, 5, certain=5,frequency=4) # 5 year annuity due, with 5 year certain payable 4x per year
... # and more!
SingleLife(vbt2001.select[50]) # no keywords, just a mortality vector
SingleLife(vbt2001.select[50],issue_age = 60) # select at 50, but now 60
SingleLife(vbt2001.select,issue_age = 50) # use issue_age to pick the right select vector
SingleLife(mortality=vbt2001.select,issue_age = 50) # mort can also be a keyword
LifeContingencies package on GitHub ๐กญ
Flexible and composable yield curves and interest functions.
FinanceModels.jl provides a set of composable contracts, models, and functions that allow for modeling of both simple and complex financial instruments. The resulting models, such as discount rates or term structures, can then be used across the JuliaActuary ecosystem to perform actuarial and financial analysis.
Additionally, the models can be used to project contracts through time: most basically as a series of cashflows but more complex output can be defined for contracts.
using FinanceModels
# a set of marketobserved prices we wish to calibrate the model to
# annual effective unless otherwise specified
q_rate = ZCBYield([0.01,0.02,0.03]);
q_spread = ZCBYield([0.01,0.01,0.01]);
# bootstrap a linear spline yield model
model_rate = fit(Spline.Linear(),q_rate,Fit.Bootstrap());โ
model_spread = fit(Spline.Linear(),q_spread,Fit.Bootstrap());
# the zero rate is the combination of the two underlying rates
zero(m_spread + m_rate,1) # 0.02 annual effective rate
# the discount is the same as if we added the underlying zero rates
discount(m_spread + m_rate,0,3) โ discount(0.01 + 0.03,3) # true
# compute the present value of a contract (a cashflow of 10 at time 3)
present_value(m_rate,Cashflow(10,3)) # 9.15...
Often we start with observed or assumed values for existing contracts. We want to then use those assumed values to extend the valuation logic to new contracts. For example, we may have a set of bond yields which we then want to discount a series of insurance obligations.
In the language of FinanceModels, we would have a set of Quote
s which are used to fit a Model
. That model is then used to discount
a new series of cashflows.
That's just an example, and we can use the various components in different ways depending on the objective of the analysis.
Contracts are a way to represent financial obligations. These can be valued using a model, projected into a future steam of values, or combined with assumed prices as a Quote
.
Included are a number of primitives and convenience methods for contracts:
Existing struct
s:
Cashflow
Bond.Fixed
Bond.Floating
Forward
(an obligation with a forward start time)
Composite
(combine two other contracts, e.g. into a swap)
EuroCall
CommonEquity
Commonly, we deal with conventions that imply a contract and an observed price. For example, we may talk about a treasury yield of 0.03
. This is a description that implies a Quote
ed price for an underling fixed bond. In FinanceModels, we could use CMTYield(rate,tenor)
which would create a Quote(price,Bond.Fixed(...))
. In this way, we can conveniently create a number of Quote
s which can be used to fit models. Such convenience methods include:
ZCBYield
ZCBPrice
CMTYield
ParYield
ParSwapYield
ForwardYield
FinanceModels offers a way to define new contracts as well.
A Cashflow
s obligation are themselves a contract, but other contracts can be considered as essentially anything that can be combined with assumptions (a model) to derive a collection of cashflows.
For example, a obligation that pays 1.75 at time 2 could be represented as: Cashflow(1.75,2)
.
Models are objects that can be fit to observed prices and then subsequently used to make valuations of other cashflows/contracts.
Yield models include:
Yield.Constant
Bootstrapped Spline
s
Yield.SmithWilson
Yield.NelsonSiegel
Yield.NelsonSiegelSvensson
The models can be used to compute various rates of interest:
discount(curve,from,to)
or discount(curve,to)
gives the discount factor
accumulation(curve,from,to)
or accumulation(curve,to)
gives the accumulation factor
zero(curve,time)
or zero(curve,time,Frequency)
gives the zerocoupon spot rate for the given time.
forward(curve,from,to)
gives the zero rate between the two given times
par(curve,time;frequency=2)
gives the couponpaying par equivalent rate for the given time.
Other models include:
BlackScholesMerton
derivative valuation
Most basically, we can project a contract into a series of Cashflow
s:
julia> b = Bond.Fixed(0.04,Periodic(2),3)
FinanceModels.Bond.Fixed{Periodic, Float64, Int64}(0.04, Periodic(2), 3)
julia> collect(b)
6element Vector{Cashflow{Float64, Float64}}:
Cashflow{Float64, Float64}(0.02, 0.5)
Cashflow{Float64, Float64}(0.02, 1.0)
Cashflow{Float64, Float64}(0.02, 1.5)
Cashflow{Float64, Float64}(0.02, 2.0)
Cashflow{Float64, Float64}(0.02, 2.5)
Cashflow{Float64, Float64}(1.02, 3.0)
However, Projection
s allow one to combine three elements which can be extended to define any desired output (such as amortization schedules, financial statement projections, or account value rollforwards). The three elements are:
the underlying contract of interest
the model which includes assumptions of how the contract will behave
a ProjectionKind
which indicates the kind of output desired (cashflow stream, amortization schedule, etc...)
Model Method
 
 
fit(Spline.Cubic(), CMTYield.([0.04,0.05,0.055,0.06,0055],[1,2,3,4,5]), Fit.Bootstrap())


Quotes
Model could be Spline.Linear()
, Yield.NelsonSiegelSvensson()
, Equity.BlackScholesMerton(...)
, etc.
Quote could be CMTYield
s, ParYield
s, Option.Eurocall
, etc.
Method could be Fit.Loss(x>x^2)
, Fit.Loss(x>abs(x))
, Fit.Bootstrap()
, etc.
This unified way to fit models offers a much simpler way to extend functionality to new models or contract types.
After being fit, models can be used to value contracts:
present_value(model,cashflows)
Additionally, ActuaryUtilities.jl offers a number of other methods that can be used, such as duration
, convexity
, price
which can be used for analysis with the fitted models.
Rates are types that wrap scalar values to provide information about how to determine discount
and accumulation
factors.
There are two Frequency
types:
Periodic(m)
for rates that compound m
times per period (e.g. m
times per year if working with annual rates).
Continuous()
for continuously compounding rates.
Continuous(0.05) # 5% continuously compounded
Periodic(0.05,2) # 5% compounded twice per period
These are both subtypes of the parent Rate
type and are instantiated as:
Rate(0.05,Continuous()) # 5% continuously compounded
Rate(0.05,Periodic(2)) # 5% compounded twice per period
Rates can also be constructed by specifying the Frequency
and then passing a scalar rate:
Periodic(1)(0.05)
Continuous()(0.05)
Convert rates between different types with convert
. E.g.:
r = Rate(FinanceModels.Periodic(12),0.01) # rate that compounds 12 times per rate period (ie monthly)
convert(FinanceModels.Periodic(1),r) # convert monthly rate to annual effective
convert(FinanceModels.Continuous(),r) # convert monthly rate to continuous
Adding, substracting, multiplying, dividing, and comparing rates is supported.
FinanceModels package on GitHub ๐กญ
Meeting your exposure calculation needs.
df = DataFrame(
policy_id = 1:3,
issue_date = [Date(2020,5,10), Date(2020,4,5), Date(2019, 3, 10)],
end_date = [Date(2022, 6, 10), Date(2022, 8, 10), Date(2022,12,31)],
status = ["claim", "lapse", "inforce"]
)
df.policy_year = exposure.(
ExperienceAnalysis.Anniversary(Year(1)),
df.issue_date,
df.end_date,
df.status .== "claim"; # continued exposure
study_start = Date(2020, 1, 1),
study_end = Date(2022, 12, 31)
)
df = flatten(df, :policy_year)
df.exposure_fraction =
map(e > yearfrac(e.from, e.to + Day(1), DayCounts.Thirty360()), df.policy_year)
# + Day(1) above because DayCounts has Date(2020, 1, 1) to Date(2021, 1, 1) as an exposure of 1.0
# here we end the interval at Date(2020, 12, 31), so we need to add a day to get the correct exposure fraction.
policy_id<br>Int64  issue_date<br>Date  end_date<br>Date  status<br>String  policy_year<br>@NamedTuple{from::Date, to::Date, policy\_timestep::Int64}  exposure_fraction<br>Float64 

1  20200510  20220610  claim  (from = Date("20200510"), to = Date("20210509"), policy_timestep = 1)  1.0 
1  20200510  20220610  claim  (from = Date("20210510"), to = Date("20220509"), policy_timestep = 2)  1.0 
1  20200510  20220610  claim  (from = Date("20220510"), to = Date("20230509"), policy_timestep = 3)  1.0 
2  20200405  20220810  lapse  (from = Date("20200405"), to = Date("20210404"), policy_timestep = 1)  1.0 
2  20200405  20220810  lapse  (from = Date("20210405"), to = Date("20220404"), policy_timestep = 2)  1.0 
2  20200405  20220810  lapse  (from = Date("20220405"), to = Date("20220810"), policy_timestep = 3)  0.35 
3  20190310  20221231  inforce  (from = Date("20200101"), to = Date("20200309"), policy_timestep = 1)  0.191667 
3  20190310  20221231  inforce  (from = Date("20200310"), to = Date("20210309"), policy_timestep = 2)  1.0 
3  20190310  20221231  inforce  (from = Date("20210310"), to = Date("20220309"), policy_timestep = 3)  1.0 
3  20190310  20221231  inforce  (from = Date("20220310"), to = Date("20221231"), policy_timestep = 4)  0.808333 
ExperienceAnalysis.Anniversary(period)
will give exposures periods based on the first date
ExperienceAnalysis.Calendar(period)
will follow calendar periods (e.g. month or year)
ExperienceAnalysis.AnniversaryCalendar(period,period)
will split into the smaller of the calendar or policy period.
Where period
is a Period Type from the Dates standard library.
Calculate exposures with exposures(basis,from,to,continue_exposure)
.
continue_exposures
indicates whether the exposure should be extended through the full exposure period rather than terminate at the to
date.
ExperienceAnalysis package on GitHub ๐กญ
Easytouse scenario generation that's FinanceModels.jl compatible.
Vasicek
CoxIngersolRoss
HullWhite
BlackScholesMerton
m = Vasicek(0.136,0.0168,0.0119,Continuous(0.01)) # a, b, ฯ, initial Rate
s = ScenarioGenerator(
1, # timestep
30, # projection horizon
m, # model
)
This can be iterated over, or you can collect all of the rates like:
rates = collect(s)
or
for r in s
# do something with r
end
And the package integrates with FinanceModels.jl:
YieldCurve(s)
will produce a yield curve object:
โ โ โ โ โ โ โ โ โ โ โ โ โ โ Yield Curve (FinanceModels.BootstrapCurve)โ โ โ โ โ โ โ โ โ โ โ โ โ
โโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโ
0.03 โโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โฃโ คโ คโ โ โ โ โ โ โ โ โ โ คโฃโฃโ Zero rates
โโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โฃโ คโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
โโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โฃโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
โโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โฃโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
โโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โขโฃโฃโ โ โฃโกคโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
โโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โฃโ คโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
โโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โฃโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
Continuous โโ โ โ โ โ โ โ โ โ โฃโกคโ โ โ ฆโ คโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
โโ โ โ โ โ โ โ โขฐโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
โโ โ โฃโ โ ขโกโกฐโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
โโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
โโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
โโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
โโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
0 โโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
โโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโ
โ 0โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ timeโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ 30โ
EconomicScenarioGenerators package on GitHub ๐กญ
Resources to help get started.
JuliaLang.org, the home site with the downloads to get started, and links to learning resources.
JuliaHub indexes opensource Julia packages and makes the entire ecosystem and documentation searchable from one place.
JuliaAcademy, which has free short courses in Data Science, Introduction to Julia, DataFrames.jl, Machine Learning, and more.
Data Science Tutorials from the Alan Turing Institute.
Learn Julia in Y minutes, a great quickstart if you are already comfortable with coding.
Think Julia, a free ebook (or paid print edition) book which introduces programming from the start and teaches you valuable ways of thinking.
Design Patterns and Best Practices, a book that will help you as you transition from smaller, oneoff scripts to designing larger packages and projects.
Each package includes examples on the Github site and in the documentation.
Benchmarks of Actuarial workflows can be found on the Benchmarks page.
Interactive exploration of the AAA's Economic Scenario Generator
Interactive mortality table comparison tool for any mort.soa.org
table
Universal Life Policy Account Mechanics as a Differential Equation
You can also access help text when using the packages in the REPL by activating help mode, e.g.:
julia> ? survival
survival(mortality_vector,to_age)
survival(mortality_vector,from_age,to_age)
Returns the survival through attained age to_age. The start of the
calculation is either the start of the vector, or attained age `from_age`
and `to_age` need to be Integers.
Add a DeathDistribution as the last argument to handle floating point
and nonwhole ages:
survival(mortality_vector,to_age,::DeathDistribution)
survival(mortality_vector,from_age,to_age,::DeathDistribution)
If given a negative to_age, it will return 1.0. Aside from simplifying the code,
this makes sense as for something to exist in order to decrement in the first place,
it must have existed and survived to the point of being able to be decremented.
Examples
โกโกโกโกโกโกโกโกโกโก
julia> qs = UltimateMortality([0.1,0.3,0.6,1]);
julia> survival(qs,0)
1.0
julia> survival(qs,1)
0.9
julia> survival(qs,1,1)
1.0
julia> survival(qs,1,2)
0.7
julia> survival(qs,0.5,Uniform())
0.95
RMInsurance is the code and examples for the second edition of the book "ValueOriented Risk Management of Insurance Companies"
LifeTable.jl will calculate life tables from the Human Mortality Database.