feature request: Relaxed Risk Parity

[ghost]

Hi, this is an amazing library! I have a small feature request: Risk Return Trade-Off Relaxed Risk Parity Portfolio Optimization, which might be an interesting addition to your library. Please let me know what you think, and if its possible to be implemented in your library. I have written some simple code for this although its not in your library's style, but I am happy to put it here to help in any way if you want to implement this paper :)

[dcajasn]

Hi @blenderben2

Thanks for your support. First let me read the paper and related papers to know if this model can be posed as a DCP problem, after a fast read, I think it can be posed using SOCP or POWER CONE, but It could take me some time to check the maths (with the maths make the implementations is very fast). Now I'm working on adding a feature to limit the number of assets and also on a pro version of this library that include some backtest capabilities and a GUI for non programer users to finance the developed of this open source library.

Best,
Dany

[ghost]

Glad to hear that it is possible to be implemented. Here is my source code that I done awhile back, it was part of an implementation with cvxpylayers, but I hope it can be useful in a way for you. Oh and all the best with your pro version! :)

Σ = cp.Parameter((n_assets, n_assets))
Σ_sqrt = cp.Parameter((n_assets, n_assets))
μ = cp.Parameter(n_assets)
R = cp.Parameter(n_assets, nonneg=True)
Θ = cp.diag(cp.diag(Σ_sqrt))
        
assert penalty >= 0, 'Penalty, λ, term must be greater or equal than 0, ideally between 0 to (not far after) 1'
λ = cp.Constant(penalty) # penalty, λ
# λ = cp.Parameter(1, nonneg=True)

x = cp.Variable(n_assets)
ζ = cp.Variable(n_assets)
ψ = cp.Variable()
γ = cp.Variable()
ρ = cp.Variable(nonneg=True) # regulating-term, ρ

objective = cp.Minimize(ψ - γ)

constraint_soc1 = [ζ[i] == (Σ @ x)[i] for i in range(n_assets)] # ζ == Σ @ x
constraint_soc2 = [cp.quad_over_lin(γ, x[i]) <= ζ[i] for i in range(n_assets)] # x * ζ >= γ ** 2
constraint_soc3 = [ζ[i] >= 0 for i in range(n_assets)] # ζ >= 0 # redundant
constraint_soc4 = [x[i] >= 0 for i in range(n_assets)] # x >= 0 # redundant

constraint_soc5 = [x[i] <= max_weight for i in range(n_assets)] # personally added constraint

constraint_soc = constraint_soc1 + constraint_soc2 + constraint_soc3 + constraint_soc4 + constraint_soc5

constraint_other = [cp.quad_over_lin(Σ_sqrt @ x, (ψ + ρ)) <= n_assets * (ψ - ρ),
                    λ * cp.quad_over_lin(Θ @ x, ρ) <= ρ, 
                    # use λ.value when parameter: λ.value * cp.quad_over_lin(Θ @ x, ρ) <= ρ,
                    μ.T @ x >= R,
                    cp.sum(x) == 1, ψ >= 0, γ >= 0] # redundant (ψ >= 0)

constraint_all = constraint_other + constraint_soc

prob = cp.Problem(objective, constraint_all)

assert prob.is_dcp()
assert prob.is_dpp()

cvxpylayer = CvxpyLayer(prob, parameters=[Σ, Σ_sqrt, μ, R], variables=[x, ζ, ψ, γ, ρ])

[uhasan1]

Hi, this is an amazing library! I have a small feature request: Risk Return Trade-Off Relaxed Risk Parity Portfolio Optimization, which might be an interesting addition to your library. Please let me know what you think, and if its possible to be implemented in your library. I have written some simple code for this although its not in your library's style, but I am happy to put it here to help in any way if you want to implement this paper :)

Great read!!

[uhasan1]

Glad to hear that it is possible to be implemented. Here is my source code that I done awhile back, it was part of an implementation with cvxpylayers, but I hope it can be useful in a way for you. Oh and all the best with your pro version! :)

Σ = cp.Parameter((n_assets, n_assets))
Σ_sqrt = cp.Parameter((n_assets, n_assets))
μ = cp.Parameter(n_assets)
R = cp.Parameter(n_assets, nonneg=True)
Θ = cp.diag(cp.diag(Σ_sqrt))
        
assert penalty >= 0, 'Penalty, λ, term must be greater or equal than 0, ideally between 0 to (not far after) 1'
λ = cp.Constant(penalty) # penalty, λ
# λ = cp.Parameter(1, nonneg=True)

x = cp.Variable(n_assets)
ζ = cp.Variable(n_assets)
ψ = cp.Variable()
γ = cp.Variable()
ρ = cp.Variable(nonneg=True) # regulating-term, ρ

objective = cp.Minimize(ψ - γ)

constraint_soc1 = [ζ[i] == (Σ @ x)[i] for i in range(n_assets)] # ζ == Σ @ x
constraint_soc2 = [cp.quad_over_lin(γ, x[i]) <= ζ[i] for i in range(n_assets)] # x * ζ >= γ ** 2
constraint_soc3 = [ζ[i] >= 0 for i in range(n_assets)] # ζ >= 0 # redundant
constraint_soc4 = [x[i] >= 0 for i in range(n_assets)] # x >= 0 # redundant

constraint_soc5 = [x[i] <= max_weight for i in range(n_assets)] # personally added constraint

constraint_soc = constraint_soc1 + constraint_soc2 + constraint_soc3 + constraint_soc4 + constraint_soc5

constraint_other = [cp.quad_over_lin(Σ_sqrt @ x, (ψ + ρ)) <= n_assets * (ψ - ρ),
                    λ * cp.quad_over_lin(Θ @ x, ρ) <= ρ, 
                    # use λ.value when parameter: λ.value * cp.quad_over_lin(Θ @ x, ρ) <= ρ,
                    μ.T @ x >= R,
                    cp.sum(x) == 1, ψ >= 0, γ >= 0] # redundant (ψ >= 0)

constraint_all = constraint_other + constraint_soc

prob = cp.Problem(objective, constraint_all)

assert prob.is_dcp()
assert prob.is_dpp()

cvxpylayer = CvxpyLayer(prob, parameters=[Σ, Σ_sqrt, μ, R], variables=[x, ζ, ψ, γ, ρ])

Blenderben2 -- thanks for posting. Would you consider contributing some time to turn this into a quick Jupyter NB with comments? Would really appreciate that very much.

[dcajasn]

Hi @blenderben2,

I've finally had time to try the relaxed risk parity. It works like in the paper, however, the model concentrate portfolio weights in assets with the higher expected (mean) returns and keep other weights equal. Due this behavior, I try another approach and I find a generalized form to include returns constraints in risk parity models. Comparing both models, I find that the new approach get lower risk than the relaxed risk parity with the same return constraint. I'm going to write a small working paper about the new approach (that works for all risk measures) before I implement both models in Riskfolio-Lib. I hope for the end of september or first week of octuber, I will add this feature to Riskfolio-Lib. As an example of both models, here are some plots:

Original Relaxed Risk Parity
Expected return: 0.18505469
Expected risk (std. dev.): 0.12813528781895778

New General Approach:
Expected return: 0.18505469
Expected risk (std. dev.): 0.12433758127346449

Best,
Dany

[ghost]

Thanks @dcajasn , looking forward to see the results.

Sorry @uhasan1 , but the code is quite self-explanatory if you understand the paper, and its quite similar to the equations listed in the paper

[dcajasn]

Hi @blenderben2, I implement models A, B and C of relaxed risk parity model in version 2.0.0. Try to update Riskfolio-Lib to last version.