Packages

These packages are available for use in your project. Scroll down for more information on each one.

Adding and Using Packages

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

MortalityTables.jl

Hassle-free mortality and other rate tables.

Features

• 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.

Quickstart

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 easy-to-access vectors which are indexed by attained age:

julia> vbt2001.select[35]          # vector of rates for issue age 35
 0.00036
 0.00048
 ⋮
 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
 ⋮
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

Non-whole 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

Parametric Models

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 🡭

ActuaryUtilities.jl

A collection of common functions/manipulations used in Actuarial Calculations.

A collection of common functions/manipulations used in Actuarial Calculations.

Quickstart

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

Features

Financial Maths

• duration:

  • Calculate the Macaulay, Modified, or DV01 durations for a set of cashflows

• convexity for price sensitivity

• Flexible interest rate options via the Yields.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

Options Pricing

• eurocall and europut for Black-Scholes option prices

Risk Measures

• Calculate risk measures for a given vector of risks:

  • CTE for the Conditional Tail Expectation, or

  • VaR for the percentile/Value at Risk.

Insurance mechanics

• duration:

  • Calculate the duration given an issue date and date (a.k.a. policy duration)

ActuaryUtilities package on GitHub 🡭

LifeContingencies.jl

Common life contingent calculations with a convenient interface.

Features

• 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, ä) with Unicode support

• All of the power, speed, convenience, tooling, and ecosystem of Julia

• Flexible and modular modeling approach

Package Overview

• 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

  • ä(life,yield): present_value of life-contingent annuity

  • ä(life,yield,n): present_value of life-contingent 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 Yields.jl

• More documentation available by clicking the DOCS badges at the top of this README

Examples

Basic Functions

Calculate various items for a 30-year-old male nonsmoker using 2015 VBT base table and a 5% interest rate

using LifeContingencies
using MortalityTables
using Yields
import LifeContingencies: V, ä     # 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
    mort = vbt2001.select[issue_age],    # -- Mortality rates
)

yield = Yields.Constant(0.05)      # Using a flat 5% interest rate

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 non-allocating (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
ä(lc)                              # Shortform notation
ä(lc, 5)                           # 5 year annuity due
ä(lc, 5, certain=5,frequency=4)    # 5 year annuity due, with 5 year certain payable 4x per year
...                                # and more!

Constructing Lives

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 🡭

Yields.jl

Flexible and composable yield curves and interest functions.

Yields.jl provides a simple interface for constructing, manipulating, and using yield curves for modeling purposes.

It's intended to provide common functionality around modeling interest rates, spreads, and miscellaneous yields across the JuliaActuary ecosystem (though not limited to use in JuliaActuary packages).

Quickstart

using Yields

riskfree_maturities = [0.5, 1.0, 1.5, 2.0]
riskfree    = [5.0, 5.8, 6.4, 6.8] ./ 100     #spot rates, annual effective if unspecified

spread_maturities = [0.5, 1.0, 1.5, 3.0]      # different maturities
spread    = [1.0, 1.8, 1.4, 1.8] ./ 100       # spot spreads

rf_curve = Yields.Zero(riskfree,riskfree_maturities)
spread_curve = Yields.Zero(spread,spread_maturities)


yield = rf_curve + spread_curve               # additive combination of the two curves

discount(yield,1.5)                           # 1 / (1 + 0.064 + 0.014) ^ 1.5

Usage

Rates

Rates are types that wrap scalar values to provide information about how to determine discount and accumulation factors.

There are two CompoundingFrequency types:

• Yields.Periodic(m) for rates that compound m times per period (e.g. m times per year if working with annual rates).

• Yields.Continuous() for continuously compounding rates.

Examples

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

Broadcast over a vector to create Rates with the given compounding:

Periodic.([0.02,0.03,0.04],2) 
Continuous.([0.02,0.03,0.04])

Rates can also be constructed by specifying the CompoundingFrequency and then passing a scalar rate:

Periodic(1)(0.05)
Continuous()(0.05)

Conversion

Convert rates between different types with convert. E.g.:

r = Rate(Yields.Periodic(12),0.01)             # rate that compounds 12 times per rate period (ie monthly)

convert(Yields.Periodic(1),r)                  # convert monthly rate to annual effective
convert(Yields.Continuous(),r)          # convert monthly rate to continuous

Arithmetic

Adding, subtracting, and comparing rates is supported.

Curves

There are a several ways to construct a yield curve object. If maturities is omitted, the method will assume that the timepoints corresponding to each rate are the indices of the rates (e.g. generally one to the length of the array for standard, non-offset arrays).

Fitting Curves to Rates

There is a set of constructor methods which will return a yield curve calibrated to the given inputs.

• Yields.Zero(rates,maturities) using a vector of zero rates (sometimes referred to as "spot" rates)

• Yields.Forward(rates,maturities) using a vector of forward rates

• Yields.Par(rates,maturities) takes a series of yields for securities priced at par. Assumes that maturities <= 1 year do not pay coupons and that after one year, pays coupons with frequency equal to the CompoundingFrequency of the corresponding rate (2 by default).

• Yields.CMT(rates,maturities) takes the most commonly presented rate data (e.g. Treasury.gov) and bootstraps the curve given the combination of bills and bonds.

• Yields.OIS(rates,maturities) takes the most commonly presented rate data for overnight swaps and bootstraps the curve. Rates assume a single settlement for <1 year and quarterly settlements for 1 year and above.

Fitting techniques

There are multiple curve fitting methods available:

• Boostrap(interpolation_method) (the default method)

  • where interpolation can be one of the built-in QuadraticSpline() (the default) or LinearSpline(), or a user-supplied function.

• Two methods from the Nelson-Siegel-Svensson family, where τ_initial is the starting τ point for the fitting optimization routine:

  • NelsonSiegel(τ_initial=1.0)

  • NelsonSiegelSvensson(τ_initial=[1.0,1.0])

To specify which fitting method to use, pass the object to as the first parameter to the above set of constructors, for example: Yields.Par(NelsonSiegel(),rates,maturities).

Kernel Methods

• Yields.SmithWilson curve (used for discounting in the EU Solvency II framework) can be constructed either directly by specifying its inner representation or by calibrating to a set of cashflows with known prices.

  • These cashflows can conveniently be constructed with a Vector of Yields.ZeroCouponQuotes, Yields.SwapQuotes, or Yields.BulletBondQuotes.

Other Curves

• Yields.Constant(rate) takes a single constant rate for all times

• Yields.Step(rates,maturities) doesn't interpolate - the rate is flat up to the corresponding time in times

Functions

Most of the above yields have the following defined (goal is to have them all):

• 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,CompoundingFrequency) gives the zero-coupon spot rate for the given time.

• forward(curve,from,to) gives the zero rate between the two given times

• par(curve,time) gives the coupon-paying par equivalent rate for the given time.

Combinations

Different yield objects can be combined with addition or subtraction. See the Quickstart for an example.

When adding a Yields.AbstractYield with a scalar or vector, that scalar or vector will be promoted to a yield type via Yield(). For example:

y1 = Yields.Constant(0.05)
y2 = y1 + 0.01                # y2 is a yield of 0.06

Forward Starting Curves

Constructed curves can be shifted so that a future timepoint becomes the effective time-zero for a said curve.

julia> zero = [5.0, 5.8, 6.4, 6.8] ./ 100
julia> maturity = [0.5, 1.0, 1.5, 2.0]
julia> curve = Yields.Zero(zero, maturity)
julia> fwd = Yields.ForwardStarting(curve, 1.0)

julia> discount(curve,1,2)
0.9275624570410582

julia> discount(fwd,1) # `curve` has effectively been reindexed to `1.0`
0.9275624570410582

Yields package on GitHub 🡭

ExperienceAnalysis.jl

Meeting your exposure calculation needs.

Quickstart

using ExperienceAnalysis
using Dates

issue = Date(2016, 7, 4)
termination = Date(2020, 1, 17)
basis = ExperienceAnalysis.Anniversary(Year(1))
exposure(basis, issue, termination)

This will return an array of tuples with a from and to date:

4-element Array{NamedTuple{(:from, :to),Tuple{Date,Date}},1}:
 (from = Date("2016-07-04"), to = Date("2017-07-04"))
 (from = Date("2017-07-04"), to = Date("2018-07-04"))
 (from = Date("2018-07-04"), to = Date("2019-07-04"))
 (from = Date("2019-07-04"), to = Date("2020-01-17"))

Available Exposure Basis

• 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 🡭

EconomicScenarioGenerators.jl

Easy-to-use scenario generation that's Yields.jl compatible.

Models

Interest Rate Models

• Vasicek

• CoxIngersolRoss

• HullWhite

EquityModels

• BlackScholesMerton

Interest Rate Model Examples

Vasicek

m = Vasicek(0.136,0.0168,0.0119,Continuous(0.01)) # a, b, σ, 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 Yields.jl:

YieldCurve(s)

will produce a yield curve object:

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀Yield Curve (Yields.BootstrapCurve)⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀           
              ┌────────────────────────────────────────────────────────────┐           
         0.03 │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠔⠒⠉⠉⠒⠒⠒⠒⠒⠤⣄⣀│ Zero rates
              │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│           
              │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│           
              │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│           
              │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀⡤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│           
              │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠋⠁⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│           
              │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│           
   Continuous │⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠓⠦⠤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│           
              │⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│           
              │⠀⠀⣀⠖⠢⡀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│           
              │⠉⠉⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│           
              │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│           
              │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│           
              │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│           
            0 │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│           
              └────────────────────────────────────────────────────────────┘           
              ⠀0⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀time⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀30⠀

EconomicScenarioGenerators package on GitHub 🡭

Community

Learn

Resources to help get started.

Programming and Julia

• JuliaLang.org, the home site with the downloads to get started, and links to learning resources.

• JuliaHub indexes open-source 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 quick-start if you are already comfortable with coding.

• Think Julia, a free e-book (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, one-off scripts to designing larger packages and projects.

Actuarial Usage and Examples

Documentation

Each package includes examples on the Github site and in the documentation.

Walkthroughs and tutorials

• Getting MortalityTables.jl data into a dataframes

Benchmarks

Benchmarks of Actuarial workflows can be found on the Benchmarks page.

Miscellaneous

• Interactive exploration of the AAA's Economic Scenario Generator

• Interactive mortality table comparison tool for any mort.soa.org table

• Interactive cashflow analysis

• Universal Life Policy Account Mechanics as a Differential Equation

Help mode

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 non-whole 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
  ≡≡≡≡≡≡≡≡≡≡

  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

Other Repositories of Interest for Actuaries

• RMInsurance is the code and examples for the second edition of the book "Value-Oriented Risk Management of Insurance Companies"

• LifeTable.jl will calculate life tables from the Human Mortality Database.