JuliaActuary

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 mort.SOA.org 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.

Simple and composable yield curves and calculations.

Meeting your exposure calculation needs.

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`

Hassle-free mortality and other rate tables.

Full set of SOA mort.soa.org tables included

`survival`

and`decrement`

functions to calculate decrements over period of timePartial 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 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
```

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 🡭

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 sensitivityFlexible 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

`eurocall`

and`europut`

for Black-Scholes 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 🡭

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`

,`ä`

) with Unicode supportAll 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`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`

capableInterest rate mechanics via

`Yields.jl`

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

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!
```

```
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 🡭

Flexible and composable yield curves and interest functions.

**Yields** 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).

```
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
```

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.

```
Yields.Continuous(0.05) # 5% continuously compounded
Yields.Periodic(0.05,2) # 5% compounded twice per period
```

These are both subtypes of the parent `Rate`

type and are instantiated as:

```
Yields.Rate(0.05,Continuous()) # 5% continuously compounded
Yields.Rate(0.05,Periodic(2)) # 5% compounded twice per period
```

Broadcast over a vector to create `Rates`

with the given compounding:

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

There are a several ways to construct a yield curve object.

`rates`

can be a vector of `Rate`

s described above, or will assume `Yields.Periodic(1)`

if the functions are given `Real`

number values

`Yields.Zero(rates,maturities)`

using a vector of zero, or spot, rates`Yields.Forward(rates,maturities)`

using a vector of one-period (or`periods`

-long) forward rates`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`

`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.`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.

`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.ZeroCouponQuote`

s,`Yields.SwapQuote`

s, or`Yields.BulletBondQuote`

s.

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`forward(curve,from,to)`

gives the average rate between the two given times`zero(curve,time)`

or`zero(curve,time,CompoundingFrequency)`

gives the zero-coupon spot rate for the given time.

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
```

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
```

Meeting your exposure calculation needs.

```
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"))
```

`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 🡭

Resources to help get started.

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.

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`

tableUniversal 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 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
```

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

LifeTable.jl will caculate life tables from the Human Mortality Database.

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.

© JuliaActuary Contributors. Last modified: May 17, 2022. Website built with Franklin.jl and Julia.