AAA_Equity_Generator.jl

### A Pluto.jl notebook ###
# v0.19.18

using Markdown
using InteractiveUtils

# ╔═╡ 0d04723f-2b88-4467-9b7b-fbbf1a29f88c
using Distributions

# ╔═╡ 96c6047b-4cf1-4be0-a803-2bf57368cf6c
using Random

# ╔═╡ 05ba01cc-8b80-4d36-b478-74457c014476
using LabelledArrays

# ╔═╡ 5cf21c5b-44e8-4319-9a0c-5770d823b5f7
md" Initial attempt to replicate AAA equtity generator. 



Todo for prototype:
- The summary statistics don't match the paper for the US diversified equity
  - potentially caused by incorrect correlation matrix? what should be the 1st entry?

Reference: [https://www.actuary.org/sites/default/files/pdf/life/c3supp_march05.pdf](https://www.actuary.org/sites/default/files/pdf/life/c3supp_march05.pdf)"

# ╔═╡ 067a7524-ef60-4ebd-be38-54459c2ea7d6
md"## Model Parameters"

# ╔═╡ 008699e5-03de-431c-8c0a-f36825627f3b
# use a labelled array for easy reference of the parameters 
params = @LArray [
    0.12515 0.14506 0.16341 0.20201 # τ
    0.35229 0.41676 0.3632 0.35277 # ϕ
    0.32645 0.32634 0.35789 0.34302 # σ_v
    -0.2488 -0.1572 -0.2756 -0.2843 #ρ
    0.055 0.055 0.055 0.055 # A
    0.56 0.466 0.67 0.715 # B
    -0.9 -0.9 -0.95 -1.0 #C
    0.1476 0.1688 0.2049 0.2496 # σ_0
    0.0305 0.0354 0.0403 0.0492 # σ_m
    0.3 0.3 0.4 0.55 # σ_p
    0.7988 0.4519 0.9463 1.1387 # σ⃰
] ( # define the regions each label refers to
	τ = (1,:),
	ϕ = (2,:),
	σ_v = (3,:),
	ρ = (4,:),
	A = (5,:),
	B = (6,:),
	C = (7,:),
	σ_0 = (8,:),
	σ_m = (9,:),
	σ_p = (10,:),
	σ⃰ = (11,:)
)

# ╔═╡ d04a7924-39c7-42a6-8068-02b9987dce54
md"## Stochastic Log Volatility Model
Note that the `@.` and other broadcasting (`.` symbol) allows us to operate on multiple funds at once.
"

# ╔═╡ a2d95991-9f15-46b2-9d01-84bc71551bb9
function v(v_prior,params,Zₜ) 
	(;σ_v, σ_m,σ_p,σ⃰,ϕ,τ) = params
	
	v_m = log.(σ_m)
	v_p = log.(σ_p)
	v⃰ = log.(σ⃰)

	# vol are the odd values in the random array
	ṽ =  @. min(v_p, (1 - ϕ) * v_prior + ϕ * log(τ) ) + σ_v * Zₜ[[1,3,5,7]]
	
	v = @. max(v_m, min(v⃰,ṽ))

	return v
end

# ╔═╡ b089355b-852b-4649-bb0b-b3ac2454f00e
function scenario(params,Z;months=1200)
	(;σ_v,σ_0, ρ,A,B,C) = params

	n_funds = size(params,2)
	
	#initilize/pre-allocate
	Zₜ = rand(Z)
	v_t = log.(σ_0)
	σ_t = zeros(n_funds)
	μ_t = zeros(n_funds)
	
	log_returns = map(1:months) do t
		Zₜ = rand!(Z,Zₜ)
		v_t .= v(v_t,params,Zₜ)

		σ_t .= exp.(v_t)

		@. μ_t =  A + B * σ_t + C * (σ_t)^2

		# return are the even values in the random array
		log_return = @. μ_t / 12 + σ_t / sqrt(12) * Zₜ[[2,4,6,8]]
	end

	# convert vector of vector to matrix
	reduce(hcat,log_returns)
		
		
		
end

# ╔═╡ 256e10bf-89e1-4f8e-aacb-ec161447cff3
md" ## Scenarios and validation

### A single scenario"

# ╔═╡ 38d7927f-aac9-466c-ba4b-be2efe39d883
md"### Validation of summary statistics"

# ╔═╡ 81405282-74f7-497f-9b56-59a03fe3c407
md"The summary statistics expected (per paper Table 8):

- `μ ≈ [0.0060, 0.0062, 0.0063, 0.0065]`
- `σ ≈ [0.0436, 0.0492, 0.0590, 0.0724]`
"

# ╔═╡ bc0dde6d-8df9-4d1d-bec5-1f668f173b5b
md"## The Multivariate normal and covariance matrix"

# ╔═╡ 997a67f2-bafe-4618-9abc-5b9b87e607f7
# 11 columns because it's got the bond returns in it
cov_matrix = [
	1.000	-0.249	0.318	-0.082	0.625	-0.169	0.309	-0.183	0.023	0.075	0.080;
	-0.249	1.000	-0.046	0.630	-0.123	0.829	-0.136	0.665	-0.120	0.192	0.393;
	0.318	-0.046	1.000	-0.157	0.259	-0.050	0.236	-0.074	-0.066	0.034	0.044;
	-0.082	0.630	-0.157	1.000	-0.063	0.515	-0.098	0.558	-0.105	0.130	0.234;
	0.625	-0.123	0.259	-0.063	1.000	-0.276	0.377	-0.180	0.034	0.028	0.054;
	-0.169	0.829	-0.050	0.515	-0.276	1.000	-0.142	0.649	-0.106	0.067	0.267;
	0.309	-0.136	0.236	-0.098	0.377	-0.142	1.000	-0.284	0.026	0.006	0.045;
	-0.183	0.665	-0.074	0.558	-0.180	0.649	-0.284	1.000	0.034	-0.091	-0.002;
	0.023	-0.120	-0.066	-0.105	0.034	-0.106	0.026	0.034	1.000	0.047	-0.028;
	0.075	0.192	0.034	0.130	0.028	0.067	0.006	-0.091	0.047	1.000	0.697;
	0.080	0.393	0.044	0.234	0.054	0.267	0.045	-0.002	-0.028	0.697	1.000;
]

# ╔═╡ 0c1c7cd8-b154-48da-931f-44e0f0b0c840
	Z = MvNormal(
		zeros(11), #means for return and volatility
		cov_matrix # covariance matrix
		# full covariance matrix in AAA Excel workook on Parameters tab
	)


# ╔═╡ 0b85a138-827e-11ed-05ad-5daf08af6bee
x =scenario(params,Z;months=10)

# ╔═╡ 4e1a31ce-8164-4c00-bc68-ceefc7ccff3a
let
	# generate 1000 scenarios and calculate summary statisics
	scens = [scenario(params,Z) for _ in 1:1000]
	
	μ = mean(mean(x,dims=2) for x in scens)
	σ = mean(std(x,dims=2) for x in scens)
	(;μ,σ)
end

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[[deps.Test]]
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[[deps.UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[deps.Unicode]]
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version = "17.4.0+0"
"""

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