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DiffEq.jl
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DiffEq.jl
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### A Pluto.jl notebook ###
# v0.19.9
using Markdown
using InteractiveUtils
# ╔═╡ ceff6126-51fc-11eb-3cf5-dd30d637e267
begin
using Dates
using MortalityTables
using DifferentialEquations
using Plots
using ActuaryUtilities
end
# ╔═╡ cd5e7080-5348-11eb-2c22-455a3556cab0
md"
# Universal Life Policy Account Mechanics as a Differential Equation
"
# ╔═╡ de0ace1a-5348-11eb-35c9-5f0ff4009d97
md"""
## Introduction
This demonstrates an example of defining an universal life policy value roll like a discrete differential equation.
It then uses the [SciML DifferentialEquations](https://sciml.ai/) package to "solve" the policy projection given a single point, but also to see how the policy projection behaves under different premium and interest rate conditions.
The first time this is run, it will download and precompile some large packages.
"""
# ╔═╡ ef26ffb6-5348-11eb-04a8-c5cbfdcab72a
md"""
Let's use the 2001 CSO table as the basis for cost of insurance charges:
"""
# ╔═╡ 45dab59c-528d-11eb-357d-af3664466910
cso = MortalityTables.table("2001 CSO Super Preferred Select and Ultimate - Male Nonsmoker, ANB")
# ╔═╡ ffe9b6e0-5348-11eb-2d38-a5613eb24383
md"""
Next, policy mechanics are coded. It's essentially a discrete differential equation, so it leverages [`DifferentialEquations.jl`](https://diffeq.sciml.ai/dev/)
The projection is coded in the [Discrete DifferentialEquation format](https://diffeq.sciml.ai/dev/types/discrete_types/):
``u_{n+1} = f(u,p,t_{n+1})``
In the code below, this translates to:
- `u` is the *state* of the system. We will track three variables to represent the `state`:
- `state[1]` is the account value
- `state[2]` is the premium paid
- `state[3]` is the policy duration
- `p` are the parameters of the system.
- `t` is the time, which will represent days since policy issuance
"""
# ╔═╡ da199644-51fc-11eb-0aab-dd68b1fcddc6
function policy_projection(state,p,t)
# grab the state from the inputs
av = state[1]
# calculated variables
cur_date = p.issue_date + Day(t)
dur = duration(p.issue_date,cur_date)
att_age = p.issue_age + dur - 1
# lapse if AV <= 0
lapsed = (av <= 0.0 ) & (t > 1)
if !lapsed
monthly_coi_rate = (1 - (1-p.mort_assump[att_age]) ^ (1/12))
## Periodic Policy elements
# annual events
if Dates.monthday(cur_date) == Dates.monthday(p.issue_date) ||
cur_date ==p.issue_date + Day(1) # OR first issue date
premium = p.annual_prem
else
premium = 0.0
end
# monthly_events
if Dates.day(cur_date) == Dates.day(p.issue_date)
coi = max((p.face - av) * monthly_coi_rate,0.0)
else
coi = 0.0
end
# daily events
int(av) = av * ((1 + p.int_rate) ^ (1 / 360) - 1.0)
# av
new_av = max(0.0,av - coi + premium + int(av-coi))
# new state
return [new_av, premium, dur] # AV, Prem, Dur
else
# new state
return [0.0, 0.0, dur] # AV, Prem, Dur
end
end
# ╔═╡ 8a227236-5349-11eb-353d-1d3f94a968d0
md"The following function will create a named tuple of parameters given a varying `prem` (premium) and `int` (credit rate)."
# ╔═╡ 956811e4-5349-11eb-2b34-b5441def32a1
params(prem,int) = (
int_rate = int,
issue_date = Date(2010,1,1),
face = 1e6,
issue_age = 25,
mort_assump = MortalityTables.table("2001 CSO Super Preferred Select and Ultimate - Male Nonsmoker, ANB").ultimate,
projection_years = 75,
annual_prem = prem,
)
# ╔═╡ 9ee35710-5349-11eb-0d45-23bc292d8a3d
md"""
## Runing the system
This results in the following plot. The tracked output variables u1 and u2 represent the two vars that we tracked above: account value and cumulative premium.
"""
# ╔═╡ b65ef28c-5349-11eb-1baa-6575ad7fb946
begin
p = params(
8000.0, # 8,000 annual premium
0.08 # 8% interest
)
# calculate the number of days to project
projection_end_date = p.issue_date + Year(p.projection_years)
days_to_project = Dates.value(projection_end_date - p.issue_date)
# the [0.0,..] are the initial conditions for the tracked variables
prob = DiscreteProblem(policy_projection,[0.0,0.0,0],(0,days_to_project),p)
proj = solve(prob,FunctionMap())
plot(proj)
end
# ╔═╡ c4637664-5349-11eb-0e81-59e39974f02e
md"""
## Moving up the ladder of abstraction
An excellent way to understand the behavior of a model is to [move up the ladder of abstraction](http://worrydream.com/LadderOfAbstraction/). Below, we will see what happens to the projection at varying levels of credit rates and annual premiums.
"""
# ╔═╡ a0f5551d-9e1b-4d2a-85f0-7ebb14b6bd88
md"The full range of simulated outcomes can take a couple of minutes to run:"
# ╔═╡ 02cc352a-5409-11eb-0fda-336d5fa42fa9
begin
prem_range = 1000.0:100.0:9000.0
int_range = 0.02:0.0025:0.08
function ending_av(ann_prem,int,days_to_project)
p = params(ann_prem,int)
prob = DiscreteProblem(policy_projection,[0.0,0.0,0],(0,days_to_project),p)
proj = solve(prob,FunctionMap())
end_av = proj[end][1]
if end_av == 0.0
lapse_time = findfirst(isequal(0.0),proj[1,2:end])
else
lapse_time = length(proj)
end
duration = proj[3,lapse_time]
end_age = p.issue_age + duration - 1.0
return end_av,end_age
end
end_age = zeros(length(prem_range),length(int_range))
end_av = zeros(length(prem_range),length(int_range))
# loop through each projection we did and fill our ranges with the ending AV and ending age
for (i,vp) in enumerate(prem_range)
for (j,vi) in enumerate(int_range)
end_av[i,j],end_age[i,j] = ending_av(vp,vi,days_to_project)
end
end
end
# ╔═╡ 280d25a6-5409-11eb-17d9-674f218f4351
begin
using ColorSchemes # for Turbo colors, which emphasize readability
viz = plot(layout=2) # side by side plot
#
contour!(viz[1],int_range,
prem_range,
end_av ./ 1e6, # scale to millions for readability
contour_labels=true,
c=cgrad(ColorSchemes.turbo.colors),
fill=true,
title="AV at age 100 (\$M)",
ylabel="Annual Premium (\$)",
xlabel="credit rate"
)
contour!(viz[2],int_range,
prem_range,
end_age,
contour_labels=true,
c=cgrad(ColorSchemes.turbo.colors),
fill=true,
yaxis=false,
title="Age at Lapse",
xlabel="credit rate"
)
annotate!(viz[2],[0.055,7000,Plots.text("Doesn't lapse \nbefore age 100", 8, :white, :center)])
end
# ╔═╡ 21e53f60-5409-11eb-291d-d903da8e16b4
md"
Now let's plot the result. Not surprising, interest has a huge effect on the policy projection. Premium is also a major influence.
One thing that's remarkable is how going from 2000 premium to just ~2200 of premium results in about a $5m difference at 8% interest. The power of compound interest!
"
# ╔═╡ 684f7efc-5409-11eb-0757-19b40c5d418e
md"
## Conclusion
This shows how universal life mechanics are a dynamic system. the growth/decay is governed by two competing feedback loops:
- Growth: the force of interest lets the balance grow exponentially over long periods of time
- Decay: low balances increase the net amount at risk and the resulting COI charges.
"
# ╔═╡ 7e80faf0-5409-11eb-33f6-4f6be6fea835
md"
## Endnotes
This is not meant to represent any particular insurance product, nor fully replicate typical account mechanics.
"
# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
ActuaryUtilities = "bdd23359-8b1c-4f88-b89b-d11982a786f4"
ColorSchemes = "35d6a980-a343-548e-a6ea-1d62b119f2f4"
Dates = "ade2ca70-3891-5945-98fb-dc099432e06a"
DifferentialEquations = "0c46a032-eb83-5123-abaf-570d42b7fbaa"
MortalityTables = "4780e19d-04b9-53dc-86c2-9e9aa59b5a12"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
[compat]
ActuaryUtilities = "~3.4.2"
ColorSchemes = "~3.19.0"
DifferentialEquations = "~7.2.0"
MortalityTables = "~2.3.0"
Plots = "~1.31.2"
"""
# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
# This file is machine-generated - editing it directly is not advised
[[ActuaryUtilities]]
deps = ["Dates", "Distributions", "ForwardDiff", "QuadGK", "Roots", "StatsBase", "Yields"]
git-tree-sha1 = "6006b05894a4d1d10b2f61b94105918ee0fa86fe"
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[[ArnoldiMethod]]
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[[Artifacts]]
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