Practical, extensible, and open-source actuarial modeling and analysis.


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 internal_rate_of_return, duration, convexity, present_value, breakeven, and more.


Meeting your exposure calculation needs.


Composable contracts, models, and functions that allow for modeling of both simple and complex financial instruments.


Easy-to-use 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).

Adding and Using Packages

There are two ways to add packages:

More info can be found at the Pkg manager documentation.

To use packages in your code:

using PackageName


Hassle-free mortality and other rate tables.



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):
       2001 VBT Residual Standard Select and Ultimate - Male Nonsmoker, ANB
       (:select, :ultimate, :metadata)
       Society of Actuaries
   mort.SOA.org ID:
   mort.SOA.org link:
       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
julia> vbt2001.select[35][35]      # issue age 35, attained age 35
julia> vbt2001.select[35][50:end] # issue age 35, attained age 50 through end of table

julia> vbt2001.ultimate[95]        # ultimate vectors only need to be called with the attained age

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

julia> decrement(vbt2001.ultimate,30,40) # the decrement between ages 30 and 40

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

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 ๐Ÿกญ


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


Financial Maths

Options Pricing

Risk Measures

Insurance mechanics

ActuaryUtilities package on GitHub ๐Ÿกญ


Common life contingent calculations with a convenient interface.


Package Overview

the mortality calculations


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 FinanceModels
import LifeContingencies: V, aฬˆ     # 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 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
aฬˆ(lc)                              # Shortform notation
aฬˆ(lc, 5)                           # 5 year annuity due
aฬˆ(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 ๐Ÿกญ


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 market-observed 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‰ˆ 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...

Overview of FinanceModels

A conceptual sketch of FinanceModels.jl

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 Quotes 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 and Quotes

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 structs:

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 Quoteed 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 Quotes which can be used to fit models. Such convenience methods include:

FinanceModels offers a way to define new contracts as well.


A Cashflows 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:

The models can be used to compute various rates of interest:

Other models include:


Most basically, we can project a contract into a series of Cashflows:

julia> b = Bond.Fixed(0.04,Periodic(2),3)
FinanceModels.Bond.Fixed{Periodic, Float64, Int64}(0.04, Periodic(2), 3)

julia> collect(b)
6-element 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, Projections 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:

Fitting Models

Model                                                               Method
          |                                                                   |
  	|------------|                                                     |---------------|
fit(Spline.Cubic(), CMTYield.([0.04,0.05,0.055,0.06,0055],[1,2,3,4,5]), Fit.Bootstrap())

This unified way to fit models offers a much simpler way to extend functionality to new models or contract types.

Using Models

After being fit, models can be used to value contracts:


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:


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:



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 ๐Ÿกญ


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.(
    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>Int64issue_date<br>Dateend_date<br>Datestatus<br>Stringpolicy_year<br>@NamedTuple{from::Date, to::Date, policy\_timestep::Int64}exposure_fraction<br>Float64
12020-05-102022-06-10claim(from = Date("2020-05-10"), to = Date("2021-05-09"), policy_timestep = 1)1.0
12020-05-102022-06-10claim(from = Date("2021-05-10"), to = Date("2022-05-09"), policy_timestep = 2)1.0
12020-05-102022-06-10claim(from = Date("2022-05-10"), to = Date("2023-05-09"), policy_timestep = 3)1.0
22020-04-052022-08-10lapse(from = Date("2020-04-05"), to = Date("2021-04-04"), policy_timestep = 1)1.0
22020-04-052022-08-10lapse(from = Date("2021-04-05"), to = Date("2022-04-04"), policy_timestep = 2)1.0
22020-04-052022-08-10lapse(from = Date("2022-04-05"), to = Date("2022-08-10"), policy_timestep = 3)0.35
32019-03-102022-12-31inforce(from = Date("2020-01-01"), to = Date("2020-03-09"), policy_timestep = 1)0.191667
32019-03-102022-12-31inforce(from = Date("2020-03-10"), to = Date("2021-03-09"), policy_timestep = 2)1.0
32019-03-102022-12-31inforce(from = Date("2021-03-10"), to = Date("2022-03-09"), policy_timestep = 3)1.0
32019-03-102022-12-31inforce(from = Date("2022-03-10"), to = Date("2022-12-31"), policy_timestep = 4)0.808333

Available Exposure Basis

Where period is a Period Type from the Dates standard library.

Calculate exposures with exposures(basis,from,to,continue_exposure).

ExperienceAnalysis package on GitHub ๐Ÿกญ


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


Interest Rate Models


Interest Rate Model Examples


m = Vasicek(0.136,0.0168,0.0119,Continuous(0.01)) # a, b, ฯƒ, 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)


for r in s
    # do something with r

And the package integrates with FinanceModels.jl:


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 €           
         0.03 โ”‚โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โฃ€โ คโ คโ ”โ ’โ ‰โ ‰โ ’โ ’โ ’โ ’โ ’โ คโฃ„โฃ€โ”‚ Zero rates
              โ”‚โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โฃ€โ 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 €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โฃ€โ ”โ Šโ โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ”‚           
              โ”‚โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โฃ€โ ”โ ‹โ โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ”‚           
              โ”‚โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โข€โฃ€โฃ€โ €โ €โฃ€โกคโ –โ Šโ ‰โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ”‚           
              โ”‚โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โฃ€โ 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 €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โฃ€โ ”โ ‹โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €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 โ”‚โ €โ €โ €โ €โ €โ €โ €โ €โ €โฃ€โกคโ ’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”‚           
              โ”‚โ €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”‚           
              โ”‚โ €โ €โฃ€โ –โ ขโก€โกฐโ ƒโ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ”‚           
              โ”‚โ ‰โ ‰โ โ €โ €โ ‰โ โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ”‚           
              โ”‚โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ”‚           
              โ”‚โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ”‚           
              โ”‚โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €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 โ”‚โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ €โ”‚           
              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 ๐Ÿกญ



Resources to help get started.

Programming and Julia

Actuarial Usage and Examples


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

Walkthroughs and tutorials


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


Help mode

You can also access help text when using the packages in the REPL by activating help mode, e.g.:

julia> ? survival

  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:


  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.


  julia> qs = UltimateMortality([0.1,0.3,0.6,1]);

  julia> survival(qs,0)
  julia> survival(qs,1)

  julia> survival(qs,1,1)
  julia> survival(qs,1,2)

  julia> survival(qs,0.5,Uniform())

Other Repositories of Interest for Actuaries

The packages in JuliaActuary are open-source and liberally licensed (MIT License) to allow wide private and commercial usage of the packages, like the base Julia language and many other packages in the ecosystem. See terms of this site.