導航
- CDF函數(normal distribution)
- CDF函數(t-location-scale distribution)
- CAPM模型 β \beta β計算
- 誤差項 ε i \varepsilon_i εi計算
-
- correlation and covariance
- t t t分布拟合序列
- Generalized Hyperbolic Distribution
-
- approximation method 1
- approximation method 2
- Goodness of Fit
-
- Anderson & Darling Test
- Kolmogorv Distance
- L1 Distance
- L2 Distance
- References
CDF函數(normal distribution)
計算
normal distribution
拟合下累積分布函數
function [Fncap] = minux_CDF_normal(returns)
% Normal cumulative distribution function, the prob. the return
% is lower than or equal to the return in cell(i, j) of returns where
% assuming returns follow a normal distribution.
% input:
% returns: matrix of stock returns
% output:
% Fncap: matrix of cumulative distribution function
[n, m]=size(returns);
Fncap = zeros(n, m);
for j=1:m
A = fitdist(returns(:, j), 'Normal'); % 使用正态分布拟合資産收益率
for i = 1:n
Fncap(i, j)=normcdf(returns(i, j), A.mu, A.sigma);
end
end
plot(sort(returns), sort(Fncap));
end
CDF函數(t-location-scale distribution)
Theis preferred nub cases oft-location-scale distribution
, as expressed by the distribution1 of returns(使用t分布可以更好地刻畫收益率序列中的尖峰厚尾現象)leptokurtosis
給出 T ( x , μ , σ , ν ) T(x, \mu, \sigma, \nu) T(x,μ,σ,ν)分布的pdf如下
T ( x , μ , σ , ν ) = Γ ( ν + 1 2 ) σ ν π Γ ( ν 2 ) [ ν + ( x − μ σ ) 2 ν ] − ν + 1 2 T(x, \mu, \sigma, \nu)=\frac{\Gamma(\frac{\nu+1}{2})}{\sigma\sqrt{\nu\pi}\Gamma(\frac{\nu}{2})}\bigg[\frac{\nu+(\frac{x-\mu}{\sigma})^2}{\nu}\bigg]^{-\frac{\nu+1}{2}} T(x,μ,σ,ν)=σνπ
Γ(2ν)Γ(2ν+1)[νν+(σx−μ)2]−2ν+1
由于 Γ \Gamma Γ結構的存在，是以無法直接計算
t-location-scale
的
CDF
，可以使用标準化的方法，轉為
student t-distribution
求解，
pdf
的轉化關系如下
T ( y , μ , σ , ν ) = 1 σ PDF tStudent ( y − μ σ ) T(y, \mu, \sigma, \nu)=\frac{1}{\sigma}\text{PDF}_{\text{tStudent}}(\frac{y-\mu}{\sigma}) T(y,μ,σ,ν)=σ1PDFtStudent(σy−μ)
得到
CDF
之間的轉化關系
F t S t u ( y − μ σ ) = F t L o c ( y ) F_{tStu}(\frac{y-\mu}{\sigma})=F_{tLoc}(y) FtStu(σy−μ)=FtLoc(y)
function [Ftcap] = minux_CDF_t(returns)
% t-location scale cumulative distribution function
% in cell(i,j) there is the prob. that the return is lower than or
% equal to the return in cell of returns, if returns follow a
% t-location scale distribution
[n, m] = size(returns);
Ftcap = zeros(n, m);
for j = 1:m
A = fitdist(returns(:, j), 'tLocationScale');
for i = 1:n
Ftcap(i, j)=tcdf((returns(i, j)-A.mu)/A.sigma, A.nu);
end
end
plot(sort(returns), sort(Ftcap));
end
CAPM模型 β \beta β計算
CAPM模型
E ( R i ) = R f + β i [ E ( R m ) − R f ] + ε i E(R_i)=R_f+\beta_i[E(R_m)-R_f]+\varepsilon_i E(Ri)=Rf+βi[E(Rm)−Rf]+εi
可以使用簡單線性回歸的方法或者公式法計算 β i \beta_i βi的值
function [beta] = minux_beta(returns, retmkt)
%% generate a column vector that contains the betas of all stocks
% input:
% returns: matrix of stocks returns
% retmkt: column vector of returns of the market
% output:
% beta: column vector of stocks betas
% 回歸方法計算
m = size(returns, 2);
beta = zeros(m, 1);
for i=1:m
beta(i, 1)=regress(returns(:, i), retmkt);
end
% 矩陣方法計算
%{
cov_mat = cov(returns, retmkt);
beta = cov_mat(1, 2)/cov_mat(2, 2);
%}
end
誤差項 ε i \varepsilon_i εi計算
設定誤差項 ε i \varepsilon_i εi為從如下機率空間抽取的獨立随機變量(
independent random variables
).
ε i = σ i T t S t u ( ν i ) \varepsilon_i=\sigma_iT_{tStu}(\nu_i) εi=σiTtStu(νi)
function [e] = minux_err(returns, retmkt, beta, rf)
% generate the error values for the CAPM formula from a process
% following an fitted t-Location Scale Distribution
% 根據CAPM模型計算誤差項
% input:
% returns: matrix of stocks returns， 資産收益率矩陣
% retmkt: matrix of market returns，市場收益率向量
% beta，beta列向量
% rf，無風險利率
% output:
% e, matrix of error terms for each stock
[n, m] = size(returns);
e = zeros(n, m);
for i=1:m
e(:, i) = returns(:, i)-rf-beta(i, 1)*(retmkt-rf);
end
end
The expected value of the returns of the market is calculated with the same method as the error term. With all the variables needed, we can simply follow the CAPM formula and calculate the expected returns of security i i i following a t t t-location scale distribution.
根據 t t t分布的性質模拟出場景代碼如下
function [rbarStocks] = minux_SCENARIOS(retmkt, beta, rf, e)
% this function creates the matrix of predicted returns following the
% CAPM
% input:
% retmkt: matrix of market returns
% beta: risk free rate of returns
% e: error matrix
% output:
% rbarStocks: matrix of predicted stocks returns following a t-location
% scale
[n, m] = size(e);
rbarStocks = zeros(n, m);
dMarket = fitdist(retmkt, 'tLocationScale'); % 使用t分布拟合市場收益率序列
for i = 1:m
dErr = fitdist(e(:, i), 'tLocationScale'); % t分布拟合誤差序列
for j=1:n
rbarStocks(j, i) = rf(j, 1)+beta(i, 1)* (random(dMarket)-rf(j, 1))+random(dErr); % 根據拟合市場序列，beta值，誤差項生成stock scenarios
end
end
end
correlation and covariance
One of the major issues related to the use of the CAPM model is the correlation between the predicted returns and the error term. In fact it would mean that the model, with the current structure and variables is not explaining the behavior of returns in a complete way.
計算誤差與模拟收益率序列之間的協方差代碼如下
function [CovER, Cor] = minux_cov(rbarStocks, e)
% input:
% rbarStocks: matrix of predicted stock returns
% e: matrix of error returns
% CovER: 模拟序列和誤差項之間的協方差
% Cor: 模拟序列和誤差項之間的相關系數
[n, m] = size(rbarStocks);
Cor = zeros(m, 1);
for j=1:m
A = cov(rbarStocks(:, j)', e(:, j));
if j>=2
CovER = [CovER; A(1, 2)] ;
else
CovER = A(1, 2);
end
end
for j=1:m
Cor(j, 1) = corr(rbarStocks(:, j), e(:, j));
end
end
t t t分布拟合序列
The prediction of scenarios coherent with the t − t- t−location scale distribution, without using CAPM
%% t-分布拟合
rbar_tloc = minux_t_fit(returns);
sample_ret = returns(:, 1);
A = fitdist(sample_ret, 'tLocationScale');
xmin = min(sample_ret)-0.01;
xmax = max(sample_ret)+0.01;
xp = xmin: 0.01:xmax;
hold on;
yyaxis left; % 激活左軸
yp = pdf(A, xp);
plot(xp, yp, 'b--');
yyaxis right; % 激活右軸
histogram(sample_ret, 'Normalization', 'probability', 'FaceAlpha', 0.4);
legend('t-fitted', 'returns');
hold off;
拟合函數為
function [rbarStocks_tloc] = minux_t_fit(rStocks)
[n, m]=size(rStocks);
rbarStocks_tloc = zeros(n, m);
for j = 1:m
A = fitdist(rStocks(:, j), 'tLocationScale'); % 将收益率序列使用t-dist拟合
for i=1:n
rbarStocks_tloc(i, j)=A.mu+A.sigma*trnd(A.nu);
end
end
end
t t t拟合後在雙坐标軸上繪制pdf圖像，可以發現重合度較高，可以描述收益率尖峰厚尾的現象
對比使用正态分布的拟合圖像如下
Generalized Hyperbolic Distribution
廣義雙曲線分布(
GH
)的機率密度函數為
d G H ( λ , α , β , δ , μ ) ( x ) = a ( λ , α , β , δ , μ ) ( δ 2 + ( x − μ ) 2 ) ( 2 λ − 1 4 ) e β ( x − μ ) × K λ − 1 2 ( α δ 2 + ( x − μ ) 2 ) d_{{GH}(\lambda, \alpha, \beta, \delta, \mu)}(x)=a(\lambda, \alpha, \beta, \delta, \mu)(\delta^2+(x-\mu)^2)^{(\frac{2\lambda-1}{4})}e^{\beta(x-\mu)}\times K_{\lambda-\frac{1}{2}}(\alpha\sqrt{\delta^2+(x-\mu)^2}) dGH(λ,α,β,δ,μ)(x)=a(λ,α,β,δ,μ)(δ2+(x−μ)2)(42λ−1)eβ(x−μ)×Kλ−21(αδ2+(x−μ)2
)
其中， α > 0 \alpha>0 α>0為形狀參數，偏度由 β \beta β的絕對值确定，且 0 ≤ ∣ β ∣ < α 0\leq |\beta|<\alpha 0≤∣β∣<α， μ ∈ R \mu\in R μ∈R為位置參數， λ ∈ R \lambda\in R λ∈R刻畫尾部形狀， δ > 0 \delta>0 δ>0描述尺度參數，正則化函數 a ( λ , α , β , δ , μ ) a(\lambda, \alpha, \beta, \delta, \mu) a(λ,α,β,δ,μ)
a ( λ , α , β , δ , μ ) = ( α 2 − β 2 ) λ 2 2 π α ( λ − 1 2 ) δ λ K λ ( δ α 2 − β 2 ) a(\lambda, \alpha, \beta, \delta, \mu)=\frac{(\alpha^2-\beta^2)^{\frac{\lambda}{2}}}{\sqrt{2\pi}\alpha^{(\lambda-\frac{1}{2})}\delta^\lambda K_\lambda(\delta\sqrt{\alpha^2-\beta^2})} a(λ,α,β,δ,μ)=2π
α(λ−21)δλKλ(δα2−β2
)(α2−β2)2λ
其中函數 K λ K_\lambda Kλ表示第三種
modified Bessel function with index
λ \lambda λ.
GH
分布依賴于
Generalized Inverse Gaussian Distribution (GIG)
，設 δ α 2 + β 2 = ζ \delta\sqrt{\alpha^2+\beta^2}=\zeta δα2+β2
=ζ，
GH
的期望值為
E ( G H ) = μ + β δ 2 ζ K λ + 1 ( ζ ) K λ ( ζ ) = μ + β E ( G I G ) \mathbb{E}(GH)=\mu+\frac{\beta \delta^2}{\zeta}\frac{K_{\lambda+1}(\zeta)}{K_\lambda(\zeta)}=\mu+\beta\mathbb{E}(GIG) E(GH)=μ+ζβδ2Kλ(ζ)Kλ+1(ζ)=μ+βE(GIG)
方差為
V ( G H ) = δ 2 ζ K λ + 1 ( ζ ) K λ ( ζ ) + β 2 δ 4 ζ 2 ( K λ + 2 ( ζ ) K λ ( ζ ) − K λ + 1 2 ( ζ ) K λ 2 ( ζ ) ) = E ( G I G ) + β 2 V ( G I G ) \mathbb{V}(GH)=\frac{\delta^2}{\zeta}\frac{K_{\lambda+1}(\zeta)}{K_\lambda(\zeta)}+\beta^2\frac{\delta^4}{\zeta^2}(\frac{K_{\lambda+2}(\zeta)}{K_\lambda(\zeta)}-\frac{K_{\lambda+1}^2(\zeta)}{K_\lambda^2(\zeta)})=\mathbb{E}(GIG)+\beta^2\mathbb{V}(GIG) V(GH)=ζδ2Kλ(ζ)Kλ+1(ζ)+β2ζ2δ4(Kλ(ζ)Kλ+2(ζ)−Kλ2(ζ)Kλ+12(ζ))=E(GIG)+β2V(GIG)
approximation method 1
構造一個嚴格遞增的函數 h ( x ; μ ) = F G H ( x ) − μ h(x; \mu)=F_{GH}(x)-\mu h(x;μ)=FGH(x)−μ，其中 u u u是從 ( 0 , 1 ) (0, 1) (0,1)均勻分布中的随機抽樣，可以得到數值解
h ( x ; u ) = ∫ ∞ x d G H ( λ , α , β , δ , μ ) ( x ) d x − μ h(x; u)=\int_\infty^xd_{GH(\lambda, \alpha, \beta, \delta, \mu)}(x)dx-\mu h(x;u)=∫∞xdGH(λ,α,β,δ,μ)(x)dx−μ
approximation method 2
根據
Newton-Raphson
算法近似
{ x 1 start point x k + 1 = x k − F G H ( x k ) − μ d G H ( λ , α , β , δ , μ ) \begin{cases} x_1\quad\text{start point}\\ x_{k+1}=x_k-\frac{F_{GH}(x_k)-\mu}{d_{GH(\lambda, \alpha, \beta, \delta, \mu)}} \end{cases} {x1start pointxk+1=xk−dGH(λ,α,β,δ,μ)FGH(xk)−μ
Goodness of Fit
Anderson & Darling Test
Anderson & Darling test
檢測任意分布的公式為
A D = max x ∈ R ∣ F e m p ( x ) − F e s t ( x ) ∣ F e s t ( x ) ( 1 − F e s t ( x ) ) AD = \max\limits_{x\in R}\frac{|F_{emp}(x)-F_{est}(x)|}{\sqrt{F_{est}(x)(1-F_{est}(x))}} AD=x∈RmaxFest(x)(1−Fest(x))
∣Femp(x)−Fest(x)∣
其中 F e m p ( x ) F_{emp}(x) Femp(x)表示
empirical CDF
， F e s t ( x ) F_{est}(x) Fest(x)表示
estimated CDF
Kolmogorv Distance
CDF
函數內插補點的絕對值的上界
K D = sup x ∈ R ∣ F e m p ( x ) − F e s t ( x ) ∣ KD=\sup_{x\in R}|F_{emp}(x)-F_{est}(x)| KD=x∈Rsup∣Femp(x)−Fest(x)∣
L1 Distance
L 1 = ∑ i ∣ F e m p ( x i ) − F e s t ( x i ) ∣ L_1=\sum_i|F_{emp}(x_i)-F_{est}(x_i)| L1=i∑∣Femp(xi)−Fest(xi)∣
L2 Distance
L 2 = ∑ i ∣ F e m p ( x i ) − F e s t ( x i ) ∣ 2 L_2=\sqrt{\sum_i|F_{emp}(x_i)-F_{est}(x_i)|^2} L2=i∑∣Femp(xi)−Fest(xi)∣2
References
optimization of Conditional Value-at-Risk
- Location-Scale Distributions ↩︎