Forecast of realized covariance matrix based on asymptotic distribution of the LU decomposition with an application for balancing minimum variance portfolio

ABSTRACT

We derive the asymptotic distribution for the LU decomposition, that is, the Cholesky decomposition, of realized covariance matrix. Distributional properties are combined with an existing generalized heterogeneous autoregressive (GHAR) method for forecasting realized covariance matrix, which will be referred to as a generalized HARQ (GHARQ) method. An out-of-sample forecast comparison of a real data set shows that the proposed GHARQ method outperforms other existing methods in terms of optimizing the variances of portfolios.

KEYWORDS:

• Cholesky decomposition
• realized covariance
• LU decomposition
• GHAR
• minimum variance portfolio
• portfolio optimization

JEL CLASSIFICATION:

• C22
• C32

Acknowledgments

The authors are very thankful for the valuable comments of a referee which improve the article considerably. This research is supported by a grant from the National Research Foundation of Korea (2016R1A2B4008780)

Disclosure statement

No potential conflict of interest was reported by the authors.

Appendix – Asymptotic normality of ${\mathit{U}}_t$ and consistent estimators of variances of elements of $\sqrt{M}{U}_t$

The analysis of this Appendix provides materials for the GHARQ method proposed for two-asset portfolio in Section 2 to be extended to a general multiple-asset setup. Let us consider a portfolio of q assets, $q\ge 2$. Let $t$ be given. Let a realized covariance matrix $S_t=(S_{ijt}{)}_{q\times q}$ and the corresponding intraday log return $\left\{r_{ijt},j=1,\dots ,M,i=1,2\right\}$ be given. Let $V_t=vech(S_t)=(S_{11t},\dots ,S_{q1t},S_{22t},\dots ,S_{q2t},\dots ,S_{qqt}{)}^{\prime }$. Recall that the LU decomposition of ${\mathrm{S}}_{\mathrm{t}}$ is $S_t=U_t^{\prime }{U}_t$ in which $U_t$ is upper triangular. The elements of $U_t=(U_{ijt})$ are computed by the following algorithm:

for $i=1,\dots ,q,$

(7) $U_{ijt}=0,j=1,\dots ,i-1;$(7)

(8) $U_{iit}=\sqrt{S_{iit}-\sum _{k^{\prime }=1}^{i-1}{U}_{k^{\prime }it}^2};$(8)

(9) $U_{ijt}=\frac{1}{U_{iit}}\left[S_{ijt}-\sum _{k\prime =1}^{i-1}{U}_{k\prime it}{U}_{k\prime jt}\right],j=i+1,\dots ,q.$(9)

Forecasts of $S_{t+1}$ is obtained by fitting q-variate version of (4) for which all the elements other than $\sqrt{\widehat{Var}(U_{k,t-1})}$ can be computed from (7)–(9). Estimators of the other elements are obtained from the asymptotic normality of $P_t=vech(U_t)=(U_{11t},U_{12t},U_{22t},U_{13t},\dots ,U_{1qt},\dots ,U_{qqt}{)}^{\prime }$ which is, as $M\to \mathrm{\infty }$,

$\sqrt{M}(P_t-I{P}_t)\stackrel{d}{⟶}N\left(0,\frac{\mathrm{\partial }I{P}_t}{\mathrm{\partial }I{V}_t^{\prime }}{\mathrm{\Gamma }}_t\frac{\mathrm{\partial }I{P}_t^{\prime }}{\mathrm{\partial }I{V}_t}\right),I{P}_t=\mathrm{p}\mathrm{l}\mathrm{i}{\mathrm{m}}_{M\to \mathrm{\infty }}{P}_t,I{V}_t=\mathrm{p}\mathrm{l}\mathrm{i}{\mathrm{m}}_{M\to \mathrm{\infty }}{V}_t,$

where ${\mathrm{\Gamma }}_t$ is the asymptotic variance matrix of $\sqrt{M}{V}_t$ which is given in Barndorff-Nielsen and Shephard (2004). Then, $\widehat{Var}(U_{k,t-1})$ is the estimated value of the corresponding element of $diag\left(\frac{\mathrm{\partial }I{P}_t}{\mathrm{\partial }I{V}_t^{\prime }}{M}^{-1}{\mathrm{\Gamma }}_t\frac{\mathrm{\partial }I{P}_t^{\prime }}{\mathrm{\partial }I{V}_t}\right)$. We need estimators of $\frac{\mathrm{\partial }I{P}_t}{\mathrm{\partial }I{V}_t^{\prime }}$ and $M^{-1}{\mathrm{\Gamma }}_t$. Obvious estimator of $\frac{\mathrm{\partial }I{P}_t}{\mathrm{\partial }I{V}_t^{\prime }}$ is $\frac{\mathrm{\partial }{P}_t}{\mathrm{\partial }{V}_t^{\prime }}$. Let $U_{ij,lkt}=\frac{\mathrm{\partial }{U}_{ijt}}{\mathrm{\partial }{S}_{lkt}},$ for $i,j,k,l=1,\dots ,q$, the elements of $\frac{\mathrm{\partial }{P}_t}{\mathrm{\partial }{V}_t^{\prime }}$. Differentiating (7)–(9), we get,

for $i=1,\dots ,q,$

$\left\{\begin{array}{c}{U}_{ij,lkt}=0,l,k=1,\dots ,q,j=1,\dots ,i-1;\end{array}\right.$

$\left\{\begin{array}{c}{U}_{ii,lkt}=-\frac{1}{U_{iit}}{\sum }_{k{}^{\prime }=1}^{i-1}{U}_{k{}^{\prime }it}{U}_{k{}^{\prime }i,lkt},l=1,\dots ,i-1,k=1,\dots ,i;\\ U_{ii,lkt}=\frac{1}{2U_{iit}},l=k=i;\\ U_{ii,lkt}=0,\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}(l,k);\end{array}\right.$

$\left\{\begin{array}{c}{U}_{ij,lkt}=-\frac{U_{ii,lkt}}{U_{iit}^2}[S_{ijt}-{\sum }_{k^{\prime }=1}^{i-1}{U}_{k^{\prime }it}{U}_{k^{\prime }jt}]-\frac{1}{U_{iit}}\sum _{k^{\prime }=1}^{i-1}(U_{k^{\prime }i,lkt}{U}_{k^{\prime }jt}+U_{k^{\prime }it}{U}_{k^{\prime }j,lkt}),\\ l=1,\dots ,i,k=1,\dots ,j,(l,k)\ne (i,j),j=i+1,\dots ,q;\\ U_{ij,lkt}=\frac{1}{U_{iit}},l=i,k=j,j=i+1,\dots ,q;\\ U_{ij,lkt}=0,\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}(l,k),j=i+1,\dots ,q.\end{array}\right.$

As discussed in Barndorff-Nielsen and Shephard (2004), a consistent estimator of the element ${\gamma }_{kl,k{}^{\prime }l{}^{\prime },t}$ of ${\mathrm{\Gamma }}_t$ corresponding to the asymptotic covariance of $(\sqrt{M}{S}_{klt},\sqrt{M}{S}_{k^{\prime }{l}^{\prime }t})$ is

${\hat{\gamma }}_{kl,k{}^{\prime }l{}^{\prime },t}=M\left[\sum _{j=1}^M{r}_{kjt}{r}_{ljt}{r}_{k{}^{\prime }jt}{r}_{l{}^{\prime }jt}\right.-\frac{1}{2}\left(\sum _{j=1}^{M-1}{r}_{kjt}{r}_{ljt}{r}_{k{}^{\prime },j+1,t}{r}_{l{}^{\prime },j+1,t}\right.+\left.\left.\sum _{j=1}^{M-1}{r}_{k{}^{\prime }jt}{r}_{l{}^{\prime }jt}{r}_{k,j+1,t}{r}_{l,j+1,t}\right)\right],$

for $k,l,k{}^{\prime },l{}^{\prime }=1,\dots ,q$. Since ${\hat{\gamma }}_{kl,k{}^{\prime }l{}^{\prime },t}$ is consistent, so is the proposed estimator of $\mathrm{M}\mathrm{V}\mathrm{a}\mathrm{r}({\mathrm{U}}_{\mathrm{k},\mathrm{t}-1})$.

Additional information

Funding

This work was supported by the National Research Foundation of Korea [2016R1A2B4008780]