Nested_Stochastic.jl

### A Pluto.jl notebook ###
# v0.19.5

using Markdown
using InteractiveUtils

# ╔═╡ fbce1e27-45db-465d-83f0-a0a7935abf14
begin
	using ActuaryUtilities
	using DataFrames
	using Setfield
end

# ╔═╡ ba85bf56-5401-4430-a0e2-eef7e06347a5
md""" # Nested Projection Mechanics

A simple example of how one could define a nested projection system. Includes the following examples:
- Outer loop policy projections only
- Outer + Inner loop policy projections with padded cashflows determining reserves and capital
- Outer + Inner loop with a stochastic interest rate for the reserves

In this notebook, we define a term life policy, implement the mechanics for "outer" projected values, as well as "inner" projections so that we can determine a projection-based reserve. This is done with both a deterministic and stochastic "inner" loop.
"""

# ╔═╡ 1f4e28e0-c2f0-4c06-a9f8-d0d71999114a
md" ## Policy Data & Methods

We will use a simple Term policy as an example for the mechanics. We don't need to, but to illustrate that we could easily implement different product types, we first define an `abstract type` for `Policy`s, and then the specific `Term` implementation.

The type annotations aren't strictly necessary, but knowing the types in advance helps Julia specialize the code for it."

# ╔═╡ 001d049c-efd2-11ec-0083-cff70f84b072
abstract type Policy end

# ╔═╡ c7dbfd5b-a97e-4628-a2f8-8651f12c60f6
struct Term <: Policy 
	inforce::Float64 # count of inforce
	term::Int        # length of benefit period (in months?)
	premium::Float64
	face::Float64
end

# ╔═╡ 50569a1e-5deb-48b6-be72-5be41dcb5b43
md"""
### Policy mechanics

For consistency across different calculated amounts, we will keep the function signature consistent, even if not all of the arguments are used:

```
(policy,assumptions,time) -> result
```
"""

# ╔═╡ 280c7ef0-61d5-4d28-87b2-d26855c76b6b
premiums(pol::Term,assumptions,time) = pol.inforce * pol.premium / 12

# ╔═╡ 741777ff-2d40-4393-a0d4-669f3460cc64
qx(pol::Term,assumptions,time)= assumptions.q / 12

# ╔═╡ b8fe0af1-29b7-419a-ad3e-7b120eb5cbe1
deaths(pol::Term,assumptions,time) = pol.inforce * qx(pol,assumptions,time)

# ╔═╡ 35bed858-6b76-4dbb-b49a-505f36b13434
claims(pol::Term,assumptions,time) = deaths(pol,assumptions,time) * pol.face

# ╔═╡ 1f59e516-4dc1-450e-87a4-9ce6ac64029d
md" ## Projection Assumptions

We define some global assumptions that get passed around. It's good practice (and performant) to pass variables into functions instead of just referring to global variable values."

# ╔═╡ dde19062-8a22-4caa-ae68-23895628dc3e
assumptions = (
	q = 0.012,
	int_reserve = 0.02,
	capital_factor = 0.1, # rate * reserves
)

# ╔═╡ 19e3a49e-ccb3-45a6-9bbd-7ac1bf3b93c9
md"### Innerloop assumption

In this example, we're assuming just a PADed mortality rate for the inner loop. We take the assumption set and use `Setfield.@set` to return a new immutable named tuple with just that value modified:
"

# ╔═╡ 18e32b95-2d13-4502-8500-48b0c380291c
innerloop_assumption(outer_assump) = @set outer_assump.q *= 1.2

# ╔═╡ 10c9e253-2913-4f85-9989-8b3fffba423e
md"""## Projection Logic

The architecture takes inspiration from web server architecture where data is passed through multiple processing steps before being returned. The logic is contained within a function called `project`, which:

1. Project a single timestep and create a tuple of values: `(policy, assumptions, result)`
2. Apply the function `additional_processing` which takes as an argument `(policy, assumptions, result)`.
3. `additional_processing` can then define an "inner" loop, which could just be to apply the `project` with a modified set of assumptions. In this way, one or more "inner" loops can be defined.
4. The final `additional_processing` function should return whatever you want to return as a result.

By default, the `additional_processing` will simply return the last argument, `result` and therefore will not have any inner loops.

"""

# ╔═╡ ac656984-3bdd-4a33-90be-45042e0080a5
"""
	project(policy,assumptions;start_time=1,additional_processing=res)

The kwarg `additional_processing` defines an intermediate processing step where one can take the current model state and perform additional work, including nested projections. If left out of the arguments, the default for `additional_processing` is `res`, where `res` is (pol,assumptions, result)->result (ie will just return the model point's results with no additional work being done).

"""
function project(
		pol::Term,
		assumptions;
		start_time=1,
		additional_processing=(pol,assumptions, result)->result
	)

	# alias the assumptions to A for brevity
	A = assumptions

	# iterate over the policy from the start time to the end of the policy's term
	map(start_time:pol.term) do t
		
		# calculate components of the projection
		timestep = t
		premium = premiums(pol,A,t)
		q = qx(pol,A,t)
		death = deaths(pol,A,t)
		claim = claims(pol,A,t)
		net_cf = premium-claim
		inforce = pol.inforce - death
		pol = @set pol.inforce = inforce

		# return a vector of name tuples with the results
		result = (;
			timestep,
			premium,
			death,
			claim,
			net_cf,
			inforce,
			q,
		)

		# apply additional processing function 
		additional_processing(pol,A,result)
	end
end

# ╔═╡ 1a903be4-57fb-466c-96e3-f2ad36acc77c
# Function signature: (policy, assumptions, result) -> updated result
function run_inner(policy,assumptions,result)
	A = innerloop_assumption(assumptions)
	p = project(policy,A;start_time=result.timestep+1)

	# calculate the reserves as the present value of the 
	# cashflows within the inner loop projections
	# discounted at the reserve interest rate
	reserves = -pv(A.int_reserve,[modelpoint.net_cf for modelpoint in p])
	capital = reserves * A.capital_factor
	additional_results = (; reserves, capital)
	return merge(result,additional_results)
	
end


# ╔═╡ 295f57d0-59ae-4f50-9b5d-fce8df76be5c
md" And a stochastic version:"

# ╔═╡ 324be56b-512b-4664-9f5b-943b6d9aeade
# Function signature: (policy, assumptions, result) -> updated result
function run_inner_stochastic(policy,assumptions,result)
	A = innerloop_assumption(assumptions)
	p = project(policy,A;start_time=result.timestep+1)

	# simple stochastic interest rate 
	n = 100

	reserves = let 
		i = A.int_reserve
		f = pv(i + 0.005 * randn(),[modelpoint.net_cf for modelpoint in p])

		-sum(f for _ in 1:n) / n
		
	end
	capital = reserves * A.capital_factor
	additional_results = (; reserves, capital)
	return merge(result,additional_results)
	
end


# ╔═╡ 5c13430a-96c4-447f-89a7-1fe99078feee
md" ## Projections

First, define a sample policy:"

# ╔═╡ be7bbb76-fc0f-40b5-982b-7e46126f9c28
p = Term(1.,120,1300.,100_000.)

# ╔═╡ b5013701-848a-4920-b4f5-360815a9f2e2
md" A projection without any additional processing:"

# ╔═╡ 7dd4015f-6b58-4cf2-af3f-30bf3d43d468
md" And an example which uses a PADed inner loop to determine the resserves and capital:"

# ╔═╡ 2e7a438e-435f-4dc1-86c2-947ea8ceb8d4
project(p,assumptions;additional_processing = run_inner) |> DataFrame

# ╔═╡ 9e824c6d-a303-427f-973c-bc1b9658709d
project(p,assumptions;additional_processing = run_inner) |> DataFrame

# ╔═╡ 973a5a58-9b42-4532-be80-21f18fb6791e
project(p,assumptions;additional_processing = run_inner_stochastic) |> DataFrame

# ╔═╡ 98720096-e605-40e9-a32f-d1d5ade04be6
md"## Endnotes

### Further Work

This example is simple, but could be greatly optimized to reduce intermediate variable allocations, refine the timing of cashflows, add additional decrements, handle different types of `Policy`s, abstract some of the projection mechanics into an `Iterable` object, etc.

### Disclaimer
Created as a proof of concept and not indended to be interpreted as a meaningful projection."

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"""

# ╔═╡ Cell order:
# ╟─ba85bf56-5401-4430-a0e2-eef7e06347a5
# ╠═fbce1e27-45db-465d-83f0-a0a7935abf14
# ╟─1f4e28e0-c2f0-4c06-a9f8-d0d71999114a
# ╠═001d049c-efd2-11ec-0083-cff70f84b072
# ╠═c7dbfd5b-a97e-4628-a2f8-8651f12c60f6
# ╟─50569a1e-5deb-48b6-be72-5be41dcb5b43
# ╠═280c7ef0-61d5-4d28-87b2-d26855c76b6b
# ╠═b8fe0af1-29b7-419a-ad3e-7b120eb5cbe1
# ╠═35bed858-6b76-4dbb-b49a-505f36b13434
# ╠═741777ff-2d40-4393-a0d4-669f3460cc64
# ╟─1f59e516-4dc1-450e-87a4-9ce6ac64029d
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