Replicating the AAA equtity generator

using Distributions

using Random

using LabelledArrays

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

v (generic function with 1 method)

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

scenario (generic function with 1 method)

x = scenario(params,Z;months=1200)

4×1200 Matrix{Float64}:
 -0.0328665  -0.0126225  -0.0205485   …   0.0266938   0.00851183  0.0831278
  0.0134244   0.0179725  -0.0314215       0.0576454   0.0174838   0.0307883
 -0.10297     0.0207088  -0.0391713       0.0231305   0.00612645  0.0968145
 -0.08964     0.0202982   0.00053222     -0.00105492  0.0287865   0.0774308

# generate 1000 scenarios 
scens = [scenario(params,Z) for _ in 1:1000];

let
    # compute summary statistics
    μ = vec(mean(mean(x,dims=2) for x in scens))
    σ = vec(mean(std(x,dims=2) for x in scens))
    (;μ,σ)
end

(μ = [0.0060431989496125835, 0.006185811839010164, 0.006289265872131668, 0.006462537162096842], σ = [0.043540417636768906, 0.04908847920116854, 0.059096677232561706, 0.07240146588182594])

using CairoMakie, ColorSchemes

let 
    f = Figure()
    n = 25
    colors = ColorSchemes.Johnson
    ax = Axis(f[1,1],yscale=log10,ylabel="index value",
        title="$n realizations of 4 correlated equity funds per AAA ESG")
    for s in scens[1:n]
        for i in 1:4
        lines!(ax,cumprod(exp.(s[i,:])), color=(colors[i],0.3),label="fund $i")
        end
    end
    axislegend(ax,unique=true,position=:lt)
    f
end

# 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,:)
)

11×4 LabelledArrays.LArray{Float64, 2, Matrix{Float64}, (τ = (1, Colon()), ϕ = (2, Colon()), σ_v = (3, Colon()), ρ = (4, Colon()), A = (5, Colon()), B = (6, Colon()), C = (7, Colon()), σ_0 = (8, Colon()), σ_m = (9, Colon()), σ_p = (10, Colon()), σ⃰ = (11, Colon()))}:
   :τ => 0.12515    :τ => 0.14506  …    :τ => 0.20201
   :ϕ => 0.35229    :ϕ => 0.41676       :ϕ => 0.35277
 :σ_v => 0.32645  :σ_v => 0.32634     :σ_v => 0.34302
   :ρ => -0.2488    :ρ => -0.1572       :ρ => -0.2843
      ⋮                            ⋱  
 :σ_m => 0.0305   :σ_m => 0.0354      :σ_m => 0.0492
 :σ_p => 0.3      :σ_p => 0.3         :σ_p => 0.55
   :σ⃰ => 0.7988     :σ⃰ => 0.4519   …    :σ⃰ => 1.1387

    Z = MvNormal(
        zeros(11), #means for return and volatility
        cov_matrix # covariance matrix
        # full covariance matrix in AAA Excel workook on Parameters tab
    )

FullNormal(
dim: 11
μ: [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
Σ: [1.0 -0.249 … 0.075 0.08; -0.249 1.0 … 0.192 0.393; … ; 0.075 0.192 … 1.0 0.697; 0.08 0.393 … 0.697 1.0]
)

# 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;
]

11×11 Matrix{Float64}:
  1.0    -0.249   0.318  -0.082   0.625  …   0.309  -0.183   0.023   0.075   0.08
 -0.249   1.0    -0.046   0.63   -0.123     -0.136   0.665  -0.12    0.192   0.393
  0.318  -0.046   1.0    -0.157   0.259      0.236  -0.074  -0.066   0.034   0.044
 -0.082   0.63   -0.157   1.0    -0.063     -0.098   0.558  -0.105   0.13    0.234
  0.625  -0.123   0.259  -0.063   1.0        0.377  -0.18    0.034   0.028   0.054
 -0.169   0.829  -0.05    0.515  -0.276  …  -0.142   0.649  -0.106   0.067   0.267
  0.309  -0.136   0.236  -0.098   0.377      1.0    -0.284   0.026   0.006   0.045
 -0.183   0.665  -0.074   0.558  -0.18      -0.284   1.0     0.034  -0.091  -0.002
  0.023  -0.12   -0.066  -0.105   0.034      0.026   0.034   1.0     0.047  -0.028
  0.075   0.192   0.034   0.13    0.028      0.006  -0.091   0.047   1.0     0.697
  0.08    0.393   0.044   0.234   0.054  …   0.045  -0.002  -0.028   0.697   1.0

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