Robust formulation

The robust version of the mean-variance problem can be expressed as
max⁡w∈Cmin⁡(μ^,Σ^)∈U(μ^,Σ^)μ^′w−βw′Σ^w\max_{w\in C}\min_{(\hat{\mu}, \hat{\Sigma})\in U_(\hat{\mu}, \hat{\Sigma})}\hat{\mu}'w-\beta w'\hat{\Sigma}w w∈Cmax(μ^,Σ^)∈U(μ^,Σ^)minμ^′w−βw′Σ^w
where U(μ^,Σ^)U_{(\hat{\mu}, \hat{\Sigma})}U(μ^,Σ^) is a joint uncertainty set of (μ^,Σ^)(\hat{\mu}, \hat{\Sigma})(μ^,Σ^).
U(μ^,Σ^)={(μ^1,Σ^1),…,(μ^I,Σ^I)}μ^i=1J∑jri,jΣ^j=1J−1∑j(ri,j−μ^i)(ri,j−μ^i)′U_{(\hat{\mu}, \hat{\Sigma})}=\{(\hat{\mu}_1, \hat{\Sigma}_1),\dots, (\hat{\mu}_I, \hat{\Sigma}_I) \}\\ \hat{\mu}_i=\frac{1}{J}\sum_j r_{i, j}\\ \hat{\Sigma}_j=\frac{1}{J-1}\sum_j(r_{i, j}-\hat{\mu}_i)(r_{i,j}-\hat{\mu}_i)' U(μ^,Σ^)={(μ^1,Σ^1),…,(μ^I,Σ^I)}μ^i=J1j∑ri,jΣ^j=J−11j∑(ri,j−μ^i)(ri,j−μ^i)′

Skew normal distribution

A random variable XXX follows a skew nromal distribution with a shape parameter α,X∼SN(α)\alpha, X\sim SN(\alpha)α,X∼SN(α), if its probability density function is
fX(x∣α)=2ϕ(x)Φ(αx)f_X(x\mid\alpha)=2\phi(x)\Phi(\alpha x) fX(x∣α)=2ϕ(x)Φ(αx)
where x∈Rx\in\mathbb{R}x∈R, ϕ(x)\phi(x)ϕ(x) and Φ(x)\Phi(x)Φ(x) are the probability density function and the cumulative distribution function of a standard normal random variable, respectively, and α∈R\alpha\in\mathbb{R}α∈R. Furthermore, a skew normal distribution has the following main properties:

The skewness of distribution increases as the shape parameter α\alphaα increases. When α=0\alpha=0α=0, the distribution becomes standard normal.
E(X)=2πd,V(x)=1−2d2π\mathbb{E}(X)=\sqrt{\frac{2}{\pi}}d, \mathbb{V}(x)=1-\frac{2d^2}{\pi}E(X)=π2d,V(x)=1−π2d2 and γ1=(4−π)E(X)32V(X)32\gamma_1=\frac{(4-\pi)\mathbb{E}(X)^3}{2\mathbb{V}(X)^{\frac{3}{2}}}γ1=2V(X)23(4−π)E(X)3 and d=α1+α2d=\frac{\alpha}{\sqrt{1+\alpha^2}}d=1+α2α.
The linear transform Y=m+wXY=m+wXY=m+wX is skew normal random variable, and denoted by Y∼SN(m,w2,α)Y\sim SN(m, w^2, \alpha)Y∼SN(m,w2,α).
E(Y)=m+wX,V=w2V(X)\mathbb{E}(Y)=m+wX, \mathbb{V}=w^2\mathbb{V}(X)E(Y)=m+wX,V=w2V(X), and skewness of YYY is γ1\gamma_1γ1.
The sum of skew normal random variables is skew normal.
γ1∈(−0.995,0.995)\gamma_1\in (-0.995, 0.995)γ1∈(−0.995,0.995).

Restate this observation from the perspective of portfolio management. Suppose portfolio has a return distribution that is negatively skewed. Then, for the period that its realized return is relatively low, its realized volatility is likely to be high, and vice versa. When the portfolio return is positively skewed, the realized volatility tends to increase as realized return increases.
In fact, one can show that covariance between sample mean and sample variance is proportionate to the skewness
Cov(E(⋅),V(⋅))=μ3JCov(\mathbb{E}(\cdot), \mathbb{V}(\cdot))=\frac{\mu_3}{J} Cov(E(⋅),V(⋅))=Jμ3

The same intuition holds for the kurtosis. The worst case solution tends to get worse as the variance of fi(w)f_i(w)fi(w) increases, and because V[Si2]=μ4−σ4\mathbb{V}[S_i^2]=\mu_4-\sigma^4V[Si2]=μ4−σ4, a portfolio with a heavy tailed return distribution is penalized by problem. Findings show that robust portfolios tend to produce return distributions with thinner tails than their nonrobust counterparts and are thus less likely to be affected by outliers.

Empirical Tests

To find the robust optimal portfolio from the problem, we solve the following equivalent convex quadratically constrained quadratic program (QCQP) formulation
max⁡w∈Czs.t.z≤μ^iw−βw′Σ^iw(i=1,2,…,I)\max_{w\in C} z\\ s.t.\quad z\leq \hat{\mu}_iw-\beta w'\hat{\Sigma}_iw\quad (i=1,2, \dots, I) w∈Cmaxzs.t.z≤μ^iw−βw′Σ^iw(i=1,2,…,I)

Matlab Code

main function

%% generate simulated data
rng(729);
n = 10;
nDays = 500;
I = 100;
muA = zeros(n, I);
sigmaA = zeros(n, n, I);
for i = 1:Ireturns = randn(nDays, n);muA(:, i)= mean(returns)';sigmaA(:,:,i)=sqrt(cov(returns));
endopt_robust_cvx = func_qcqp_cvx(muA, sigmaA);
opt_robust_YALMIP = func_qcqp_YALMIP(muA, sigmaA);

cvx verision

function w = func_qcqp_cvx(muA, sigmaA)% input:% muA: n x I matrix where each column is a sample mean% sigmaA: n x n x I a three dimensional matrix where each page (n x n)% is a sample covariance matrix% output:% w: portfolio weights[n, I] = size(muA);beta = 1;cvx_beginvariables z w(n);maximize z;subject toones(1, n)*w==1;% I scenariosfor i=1:Iz<=muA(:, i)'*w-beta*quad_form(w, sigmaA(:, :, i));end  cvx_end
end

YALMIP version

function w = func_qcqp_YALMIP(muA, sigmaA)[n, I] = size(muA);beta = 1;w=sdpvar(n, 1);z=sdpvar;F = [sum(w)==1];for i=1:IS = sigmaA(:, :, i)'*sigmaA(:, :, i); % nxnF = [F, z<=muA(:, i)'*w-beta*w'*S*w];endobj = -z;optimize(F, obj);w = value(w);
end

【RO】Robust formulation controls higher moments相关推荐

【AP】Robust multi-period portfolio selection(3)
Navigator Pre Blog Link Robust conterparts of multi-period portfolio problems The relationship betwe ...
【AP_EJOR】Robust solutions to multi-objective linear programs with uncertain data(1)
Navigator Paper Link Introduction main contributions Preliminaries Paper Link Robust solutions to mu ...
【AP】Robust multi-period portfolio selection(2)
Navigator Pre Blog Link Robust multi-period portfolio selection(1) Robust optimization and asymmetri ...
【翻译】Robust High-Resolution Video Matting with Temporal Guidance
Robust High-Resolution Video Matting with Temporal Guidance 论文地址 RobustVideoMatting 代码地址论文阅读笔记版权声明 ...
【论文】——Robust High-Resolution Video Matting with Temporal Guidance浅读
视频matting 时序监督摘要我们介绍了一种稳健.实时.高分辨率的人类视频抠图方法,该方法取得了新的最先进性能.我们的方法比以前的方法轻得多,可以在Nvidia GTX 1080Ti GPU上以 ...
VINS-初始化:【翻译】Robust Initialization of Monocular Visual-Inertial Estimation on Aerial Robots
目录用于空中机器人的单目视觉惯性估计的鲁棒初始化摘要简介综述方法纯视觉结构 IMU预积分视觉-惯性对齐非线性VINS估计讨论实验结果在公共数据集上的性能真实世界的实验结论用 ...
【SIGGRAPH】用【有说服力的照片真实】技术实现最终幻想15的视觉特效
原文:西川善司 http://www.4gamer.net/games/075/G007535/20160726064/ 最终幻想15的演讲会场.相当大,听众非常多. 在本次计算机图形和交互技术大会[ ...
【论文】解读Robust bike-sharing stations allocation and path network design: a two-stage stochastic...
解读Robust bike-sharing stations allocation and path network design: a two-stage stochastic programmin ...
【Verilog】跨时钟域设计Clock Domain Crossing Design（Multi cycle path formulation with feedback acknowledge）
上次写了跨时钟域设计MCP公式不带反馈的实现[Verilog]跨时钟域设计Clock Domain Crossing (CDC) Design(MCP formulation without feed ...

【RO】Robust formulation controls higher moments

Navigator

Robust formulation

Skew normal distribution

Empirical Tests

Matlab Code

【RO】Robust formulation controls higher moments相关推荐

最新文章

热门文章