GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.86

Large-scale portfolios using realized covariance matrix:
evidence from the Japanese stock market

Masato Ubukata

September 2009GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Large-scale portfolios using realized covariance matrix:
evidence from the Japanese stock market∗
Masato Ubukata †Graduate School of Economics, Osaka University

Abstract
The objective of this paper is to examine effects of realized covariance matrix estimators
based on intraday returns on large-scale minimum-variance equity portfolio optimization. We
empirically assess out-of-sample performance of portfolios with different covariance matrix
estimators: the realized covariance matrix estimators and Bayesian shrinkage estimators basedon the past monthly and daily returns. The main results are: (1) the realized covariance matrix
estimators using the past intraday returns yield a lower standard deviation of the large-scale
portfolio returns than the Bayesian shrinkage estimators based on the monthly and daily historical returns; (2) gains to switching to strategies using the realized covariance matrix estimators
are higher for an investor with higher relative risk aversion; and (3) the better portfolio performance of the realized covariance approach implied by ex-post returns in excess of the risk-freerate, the standard deviations of the excess returns, the return per unit of risk (Sharpe ratio) andthe switching fees seems to be robust to the level of transaction costs.

Keywords: Large-scale portfolio selection; Realized covariance matrix; Intraday data
JEL classification: G11∗

Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Young Scientists (Start-up), 20830047,
2009.
†
Graduate School of Economics, Osaka University, 560-0043, Machikaneyama, Toyonaka, Osaka, Japan. E-Mail:
ubukata@econ.osaka-u.ac.jp

1 Introduction
Measuring and controlling the risk of assets are important factors in portfolio management, together with option pricing and value-at-risk analysis. In the finance literature, risk is usually indicated by the variability of asset returns. Thus, estimating their covariance matrix plays a key rolefor determining the portfolio allocation, for example, using mean-variance portfolio optimization
based on the investment theory of Markowitz (1952). High-frequency financial data, which havebecome widely available by the development of computers and data recording systems, potentially
include much information about the dynamics of asset prices. In addition, nonparametric varianceand covariance measures have attracted the attention of financial econometricians. The measures
are called realized variance and covariance, which are constructed by summing outer-products of
intraday return data. They potentially provide very accurate estimates of the underlying quadratic
variation and covariation. Also, the covariance matrix of asset returns becomes observable by therealized covariance approach.
There are some earlier studies for evaluating the economic benefit of the realized covarianceapproach in the context of investment decisions. Recent work is used to construct estimates of thedaily covariance matrix using the intraday historical returns and to examine the performance ofsmall- or medium-scale portfolios. Fleming et al. (2003) consider that a risk-averse investor uses
conditional mean-variance analysis to allocate funds composed of three assets: S&P 500 futures,
Treasury bond futures, and gold futures. They empirically show that the economic gains yielded byrolling covariance matrix estimators, based on intraday returns, are economically large. It is wellknown that the realized variance and covariance measures deteriorate with market microstructure
effects that are induced by various market frictions such as the discreteness of price changes, bidask bounces, and asymmetric information across traders, inter alia. Bandi et al. (2008) and Pooter
et al. (2008) examine the effect of the realized covariance estimators that are constructed under
optimal sampling frequency on optimized portfolios using three stocks and 78 stocks in S&P 100
index constituents, respectively.
This paper examines the economic benefit of large-scale minimum variance equity portfolio

1

optimized using the realized covariance approach. For the analysis of a large-scale portfolio, Chanet al. (1999) construct a portfolio made up of 250 stocks randomly selected from domestic common
stock issues on the New York Stock Exchange and the American Stock Exchange. They show thatthe large-scale portfolio optimization using a factor model with the monthly historical returns is
helpful for risk control. Clarke et al. (2006) construct a minimum-variance portfolio consisting of
the 1,000 largest market capitalization U.S. stocks using Bayesian shrinkage estimators proposedin Ledoit and Wolf (2003) with the past monthly and daily returns, and conduct an empiricalresearch that is independent of any particular assumption about target expected returns.
Overall, there has been a shortage of empirical evidence evaluating portfolio performance using
the intraday returns and different risk optimization methods. To our knowledge, there has been noresearch that applies the realized covariance approach to the large-scale portfolio optimization. It is
important to compare the realized covariance approach with current monthly or daily return-based
covariance matrix structuring methodology suited for the large-scale portfolio optimization.
We construct large-scale realized covariance matrix estimators based on the past intraday returns for 500 of the largest market capitalization JPN stocks and assess the out-of-sample performance of the minimum-variance portfolios by comparing with portfolios based on Bayesianshrinkage estimators using the past monthly and daily returns. The empirical results report thestatistics, including annualized ex-post returns in excess of the risk-free rate, a standard deviationof the portfolio excess returns, the return per unit of risk (Sharpe ratio), and the net excess returnwith transaction costs. In addition, we estimate how much a risk-averse investor would be willingto pay to use the realized covariance estimators based on the intraday returns.
Our main findings are: (1) the realized covariance matrix estimators using the past intraday
returns yield a lower standard deviation of the large-scale portfolio than the Bayesian shrinkage
estimators based on the monthly and daily historical returns; (2) gains to switching to strategies
using realized covariance matrix estimators are higher for an investor with higher relative risk
aversion; and (3) the better performance using the realized covariance matrix estimators seems to
be robust to the level of transaction costs.
The remainder of the paper is organized as follows. Section 2 describes the data and our2methodology, including the construction of the large-scale covariance matrix estimators based on
the past monthly, daily, and intraday returns, the minimum-variance methodology, and the performance measurement criteria. Empirical results are discussed in Section 3. Section 4 concludes.

2 Data and Methodology
The data set was obtained from the Nikkei NEEDS-TICK data and the Nikkei NEEDS Financial
Quest. We analyze large-scale minimum variance equity portfolios optimized from covariancematrix estimators that are constructed by different frequency data such as intraday, daily, andmonthly data. On the assessment of the three different strategies, the sample consists of intraday
and daily returns from November 1999 to February 2007 and monthly returns from November
1995 to February 2007. The out-of-sample period for an examination of the portfolios is every
month (76 months) from November 2000 to February 2007. Among the Tokyo Stock Exchange(TSE) 1st section-listed stocks, we pick the 500 largest market capitalization JPN stocks with nostock splits during the sample period.

2.1

Large-scale covariance matrix estimators

The objective of our analysis is to investigate whether or not the monthly covariance matrix estimators based on intraday data improves large-scale equity portfolio performance. First, we definea monthly risk measure for period t computed with intraday data that has higher frequency than theforecasting horizon. The high-frequency data were obtained from the Nikkei NEEDS-TICK data.
The dataset includes all transactions and quotes for the 500 largest market capitalization stocks
traded at the TSE. On the estimation of the large-scale covariance matrix, we use the transaction
price series that is time stamped to the nearest minute. For each day τ , the total trading time on
the TSE is 270 minutes, from 9:00 to 11:00 in the morning session and from 12:30 to 15:00 in theafternoon session. On the other hand, we cannot obtain the high-frequency data from 15:00 to 9:00
(overnight) and from 11:00 to 12:30 (lunchtime).
Let p, h(ON ) , and h(L) denote the (N × 1) vector logarithmic prices and time intervals for3overnight and lunchtime, where N = 500 is the number of stocks. Trading sessions on the
TSE are divided into the time intervals of equal length h. Suppose that on day τ , we havethe vector of overnight returns, r τ −1+h(ON ) := pτ −1+h(ON ) − pτ −1 , returns in the morning session, r τ −1+h(ON ) +ih := pτ −1+h(ON ) +ih − pτ −1+h(ON ) +(i−1)h for i = 1, . . . , I, lunchtime returnsr τ −1+h(ON ) +Ih+h(L) := pτ −1+h(ON ) +Ih+h(L) − pτ −1+h(ON ) +Ih , and returns in the afternoon session,r τ −1+h(ON ) +Ih+h(L) +jh := pτ −1+h(ON ) +Ih+h(L) +jh − pτ −1+h(ON ) +Ih+h(L) +(j−1)h for j = 1, . . . , J,
where h(ON ) + h(L) + (I + J)h = 1. Then, the realized covariance matrix Vt,h in month t isdefined as:

Vt,h =X0r τ −1+h(ON ) r τ −1+h(ON ) +

τ ∈t

+

X

I
XX

r τ −1+h(ON ) +ih r 0 τ −1+h(ON ) +ih(1)τ ∈t i=1

r τ −1+h(ON ) +Ih+h(L) r 0 τ −1+h(ON ) +Ih+h(L)

τ ∈t

+

J
XX

r τ −1+h(ON ) +Ih+h(L) +jh r 0 τ −1+h(ON ) +Ih+h(L) +jh .

τ ∈t j=1

The realized covariance matrix estimator in (1) is based on the calendar time sampling, which is awell-used scheme sampled at equidistantly spaced intervals with the length of h over the trading
session. However, the raw intraday data are usually unevenly spaced because the transaction timeis randomly determined. In this paper, the previous-tick interpolation method is applied to obtain ahomogeneous time series, which is an artifact constructed from the raw intraday data. Let k denotethe irregularly spaced sequence of the raw intraday data, for example, the time τ − 1 + h(ON ) + ih
in the morning session is bracketed by:k ∗ = max(k | τk ≤ τ − 1 + h(ON ) + ih), τk∗ ≤ τ − 1 + h(ON ) + ih < τk∗ +1 .

(2)Then, the previous-tick interpolation is defined as the manipulation of taking the most recent priceas follows:pτ −1+h(ON ) +ih = pk∗ .

4(3)

Although a linear interpolation exists (weighted average of the most recent price and the immediate price) as the alternative interpolation method, Dacorogna et al. (2001) show that the realized
variance and covariance with the linear interpolation do not converge in probability to the quadratic
variance and covariance, and Barucci and Renó (2002) also show the bias by the linear interpolation using Monte Carlo simulation.
The realized variance and covariance measures constructed in (1) have the potential to provide
very accurate estimates of the underlying quadratic variations and covariations. However, thesemeasures have been shown to be sensitive to market microstructure effects that can be induced byvarious market frictions such as the discreteness of price changes, bid-ask bounces, and asymmetric
information across traders, inter alia.1 When we estimate the underlying quadratic variations andcovariations of the stocks from high-frequency observations, it is necessary to use a sufficiently
large sample size while we avoid the market microstructure effects because the realized covarianceestimator with finer high-frequency data may have larger bias and variance. In our analysis, we
calculate the realized covariance matrix based on five-minute returns (I +J = 54) because realized
variance and covariance plots for every sampling frequency showed the estimates from under fiveminutes to 20 minutes were comparatively stable.2 In addition, the average of realized covariancematrices calculated by using intraday returns from the preceding six and 12 months are constructed
for the portfolio optimization as follows:1(6)
Vt,h =

5
X

6 k=0

11

Vt−k,h ,(12)Vt,h =

1 X
Vt−k,h .
12 k=0
(6)

(4)

(12)

The simple average realized covariance matrix estimators Vt,h and Vt,h are input to the optimization routine.
1

For example, the growing literature on market microstructure provides important insights from early studies including Roll (1984), who derives a simple estimator of the bid-ask spread based on the negative autocovariance ofreturns. Harris (1990) examines the rounding effects emanating from the discreteness of transaction prices. Ubukata
and Oya (2009) analysis a dependence of the market microstructure effects.
2
Recent literature on integrated variance and covariance estimation with market microstructure effects have beendeveloped by Zhou (1996), Zhang et al. (2005), Zhang (2006), Hansen and Lunde (2005, 2006), Voev and Lunde(2007), and Barndorff-Nielsen et al. (2008), inter alia. Also, the optimal frequency based on the minimization of themean squared errors has been proposed by Bandi and Russell (2008). We do not apply the bias corrected or optimal
frequency-based estimators as proposed above.

5

Second, we construct the covariance matrix using monthly and daily closing prices obtained
from the Nikkei NEEDS Financial Quest. The standard approach is, for example, to use the sample
covariance matrix estimators with the past 60 months of monthly returns or the past year of daily
returns. However, it is widely known that the approach for the large-scale optimized portfolio
creates some problems in which the sample covariance matrix may be singular (noninvertible) andestimation outliers can dominate the optimized portfolio. Ledoit and Wolf (2003, 2004) show that
their shrinkage estimators of the covariance matrix improve the portfolio performance over the
sample covariance matrix estimator. Therefore, we compute the Bayesian shrinkage covariancematrix proposed in Ledoit and Wolf (2003). Let r = (r 1 , r 2 , . . . , r N )0 denote the (N × K) matrixof the historical returns. K represents 60 months or about 240 days for the use of the corresponding
monthly and daily historical data, respectively. Then the sample covariance matrix, which is notdivided by K, is calculated by V = rr 0 . The Bayesian shrinkage covariance matrix (denote VBS )is defined as the weighted average of the two-parameter prior covariance matrix Vprior and thesample covariance matrix V, as follows:

VBS = λVprior + (1 − λ)V,(5)

where a scalar shrinkage parameter λ bounded between zero and one is given by:λ=

SUM[r.2 r 0 .2 ] − SUM[V.2 ]/K,
SUM[(V − Vprior ).2 ](6)

where .2 is the element-by-element squaring and SUM[ ] is the sum of the matrix elements. The
diagonal and off-diagonal elements in the prior covariance matrix Vprior are the average value ofthe diagonal elements and the N (N − 1)/2 off-diagonal elements in the sample covariance matrix,
respectively. Finally, we divide the Bayesian shrinkage covariance matrix VBS by K and, if thecalculation is based on daily historical returns, the daily Bayesian shrinkage covariance matrix
is multiplied by the number of trading days per month. Thus, we input the Bayesian shrinkage
covariance matrices based on monthly and daily historical returns to the optimization routine.

6

2.2

Minimum-variance portfolio

The portfolio optimization reduces to finding the asset weights that minimize the portfolio covolatility while aiming for a target expected return or maximize the portfolio return while targeting a certain covolatility. We conduct the different covolatility optimizations by constructing
two scenarios of minimum-variance portfolios. Each month, we solve the following minimization
problem:³´
min w0t+1 Σt+1 wt+1
w(7)

subject to :Scenario 1 :wt+1 > 0 and w0t+1 1 = 1,

Scenario 2 : w0t+1 µt+1 = µp , wt+1 > 0 and w0t+1 1 = 1,
where wt+1 = (w1 , w2 , . . . , wN )0 is the (N × 1) vector of portfolio weights, 1 is an (N × 1) vectorof ones, and Σt+1 is the (N × N ) conditional covariance matrix. We use the estimator of Σt+1 as
(6)

(12)

Vt,h , Vt,h , and the Bayesian shrinkage covariance estimators based on monthly and daily historical returns. For the constraints on the portfolio weights, the portfolio weights are required to benonnegative because short selling is not generally a common practice for most investors. The total
portfolio weight is assumed to be equal to one. We do not impose an upper bound on the portfolio
weights. In scenario 1, we construct the minimum-variance portfolio that minimizes risk without
an expected return because we consider purely different volatility-timing strategies that the portfolio weights are determined by the estimators of the conditional covariance matrix. The scenario
is associated with the previous study of Clark (2006), which also examined large-scale minimumvariance equity portfolios that do not rely on any specific expected return. They show that theminimum-variance portfolios based on the past monthly and daily returns give higher realized returns and lower realized standard deviations than a market-capitalization weighted portfolio. We
also conduct scenario 2 where the minimum-variance portfolio is determined by minimizing covolatility given a target expected return. µt+1 is set as the average monthly returns in the complete
out-of-sample period from November 2000 to February 2007 and the target expected return µp is
7set to 10% (annualized). All the inputs are fed into a quadratic optimization software that computes
the optimal weights wt+1 . The minimum-variance optimization is conducted using the numericalroutines in the MATLAB procedure. Given the optimized weights, we calculate buy-and-hold returns on the portfolio for the one month, at the end of which estimation of the covariance matrix
and the optimization procedures are repeated. The resulting time series of monthly realized returns
permit us to characterize the performance and other properties of the optimized portfolios that are
based on the estimators with different sampling intervals.

2.3

Assessing portfolio performance

We assess the empirical out-of-sample performance of portfolios based on the different covariancematrix estimators on several grounds. The performance of the portfolios is evaluated using the
ex-post realized portfolio returns over 76 months from November 2000 to February 2007. First,
we compare the return per unit of risk, that is, measure the corresponding Sharpe ratios given by:

SR =r̄P − rf
,
σ̂rP

(8)

where rf is a risk-free rate and we use the unsecured one-month call rate to proxy the risk-free rate.
r̄P − rf represents the mean of the ex-post realized portfolio returns in excess of the risk-free rate.
σ̂rP is the standard deviation of the realized excess returns. The Sharpe ratio is used to characterize
how well the realized return of the portfolio compensates the investor for the risk. A higher Sharpe
ratio implies that a portfolio’s risk-adjusted performance is better.
Second, we evaluate the economic benefit of the different covariance matrix estimators following Fleming et al. (2001, 2003). On a utility-based approach to measure the value of the portfolio’s
performance gains, we assume a risk-averse investor with the following quadratic utility:³
U (rP t+1 ) = W0 (1 + rf + rP t+1 ) −´γ(1 + rf + rP t+1 )2 ,2(1 + γ)(9)where rP t+1 is the portfolio return and γ is the investor’s relative risk aversion. W0 is initial wealth8and is set equal to one for simplicity. Let rP1 t+1 and rP2 t+1 be the portfolio returns on the strategies
using the two different covariance matrix estimators. The maximum amount ∆γ that the investor
would be willing to pay to switch from the first strategy to the second is then determined by:TX

U (rP1 t+1 ) =

t=1

T
X

U (rP2 t+1 − ∆γ ),

(10)

t=1

where T = 76 is the out-of-sample period for the portfolio performance. Comparing the realized
(6)

(12)

performance fees for Vt,h and Vt,h over Bayesian shrinkage covariance estimators measures the
improvement due to the use of intraday data. We report the value of the switching fee ∆γ as theannualized percentage at the relative risk aversion parameters of γ = 1 and 10.
Third, we assess the different portfolio performances that incorporate transaction costs. The
transaction costs play a nontrivial role for portfolio selections because the higher turnover implies
that the investor has to pay a higher cost by more active trading and, then, the net returns of
the portfolio decrease. However, it is not generally easy to compute the total transaction cost,
including stock trading commissions, the bid–ask spread, and the account management fee, interalia. Following Pooter et al. (2008), we assume that transaction costs amount to the sum of absolutechanges in the portfolio weights multiplied by a fixed percentage cost c as follows:

ct+1 = cNX| wi,t+1 − wi,t |,

(11)

i=1

where ct+1 , intuitively, represents a cost to reallocate the portfolio at the point of rebalancing. c is
set to 2% and 4%, expressed in annualized percentage.3 Then, the net portfolio return is given by
rP t+1 − rf − ct+1 . We also report a portfolio turnover as the total amount of purchases and sales
over the same month divided by the total net asset value of the portfolio.

3

In the fixed-amount system that trading fees are determined by the total amount of purchases and sales per day,stock trading commissions to Japanese brokerage roughly range from 1% to 4% (annualized) in case of monthly
rebalancing.

9

3 Empirical Results
Table 1 summarizes statistics of minimum-variance portfolios optimized under the conditions in
scenario 1 (without expected return input). The panels A, B, and C show the statistics where thelevel of annualized transaction costs c take 0%, 2%, and 4%, respectively. For each panel, the firstrow contains the performance for a simple diversification strategy that involves no optimization;
namely, the 500 stocks value-weighted portfolio composed of the same stocks as the others. The
500 stocks value-weighted portfolio might be regarded as the market portfolio because its realized
returns are highly correlated with those of the Tokyo stock price index (TOPIX), which is a stock
market index based on the total number of shares, tracking the TSE 1st section-listed stocks. The
Sharpe ratio of the market portfolio is the lowest of all due to the lowest mean return in excess of therisk-free rate and the highest standard deviation of the portfolio excess returns. This result implies
that to conduct optimized procedures is very helpful to the improvement of portfolio performance
although the value-weighted portfolio may be often considered as a passive benchmark portfolio.
(6)

For panel A in Table 1, we find that using two realized covariance matrix estimators Vt,h(12)

and Vt,h yields higher Sharpe ratios of 0.270 and 0.259 than the Bayesian shrinkage covarianceestimators based on the past monthly and daily returns. This is because of the high reduction of
(6)

(12)

the standard deviation for the estimators Vt,h and Vt,h . Figure 1 plots the cumulative portfolio
returns over 76 months by the estimators with monthly, daily, and intraday returns. Although
the three cumulative portfolio returns tend to move similarly through the whole period, we can(6)

see that the starting cumulative returns for Vt,h have less volatility and their large rise and drop
are relatively not seen so much. Going back to Table 1, the returns of the realized covariancematrix estimators has a lower correlation with TOPIX and the tracking error takes a higher value
than the Bayesian shrinkage covariance estimators. This means that the intraday returns-based
strategy takes a different investment style from the value-weighted strategy than the monthly or
daily returns-based strategies.
(6)

(12)

In addition, the turnover of Vt,h and Vt,h takes annualized 27.699% and 19.578% values
that are higher than those of the 500 stocks value-weighted portfolio and the portfolio using the10Bayesian shrinkage estimator based on the monthly returns. In order to evaluate the effect of thetransaction costs, on panels B and C in Table 1, we describe the statistics of minimum-variance
portfolios when the two levels of transaction cost (c = 2%, 4%) defined as (11) are imposed on
(6)

(12)

every monthly rebalancing. Even in the cases of c = 2% and 4%, the Vt,h and Vt,h still yieldhigher Sharpe ratios and lower standard deviations than the portfolio using the Bayesian shrinkage(12)estimator based on the monthly returns. The Sharpe ratios for Vt,h are 0.238 and 0.218, which
(6)

(6)

exceed those for Vt,h although Vt,h earns the highest Sharpe ratio in the case without transaction(12)

cost, as in panel A. It is noted that Vt,h is the realized covariance estimator using more historical
(6)

(12)

intraday data than Vt,h . The less effect of the portfolio for Vt,h on the transaction costs implies
that the portfolio determined by the realized covariance estimator with the longer past intraday
returns would be characterized as the active portfolio with a lower turnover. We think that, forthe construction of the portfolio based on intraday data, the investor can use the length of the past
intraday returns according to their preference to degrees of the activity of buying and selling.
Table 2 shows the annualized fees ∆γ that the investor with relative risk aversion parameterγ = 1 and 10 switches from the strategies using the Bayesian shrinkage estimators based onmonthly and daily returns to using the realized covariance matrix estimators based on the intraday
returns. A risk-averse investor pays the positive switching fees ∆γ with γ = 1 and 10. We alsofind that an investor with high relative risk aversion γ = 10 would be willing to pay larger fees
than an investor with low relative risk aversion γ = 1. The increase of the fees from γ = 1
to γ = 10 consistent with the investor with high relative risk aversion is preferable to switching
to the portfolios using realized covariance matrix estimators that yield lower standard deviations.
Therefore, the strategies of using intraday returns also make more economic gains than those basedon the monthly and daily returns.
So far, we have discussed the performance of the minimum-variance portfolios optimized under no constraint for the target expected return in scenario 1. It is also important to consider thecase where the portfolio weight is determined by minimizing a variance under a given target expected return. Table 3 reports the statistics of minimum-variance portfolios with annualized target
expected return equal to 10% in scenario 2. For panel A, all the means of the annualized excess
11returns naturally are around the target expected return of 10%, but the standard deviations of theportfolios based on the intraday returns are lower than those using the monthly and daily returns.
(6)

(12)

In consequence, we obtain the higher Sharpe ratios of 0.371 and 0.331 for Vt,h and Vt,h . For
panels B and C where the transaction costs of c = 2% and 4% are imposed, the portfolio for the(6)

realized covariance estimator Vt,h is the most efficient in the sense of the trade-off return and risk,
although the Sharp ratio for the Bayesian shrinkage estimator based on the monthly returns exceeds(12)that for Vt,h because of the lowest turnover of the portfolio using the monthly returns. Table 4
also represents annualized performance fees ∆γ to switch from the monthly or daily returns for theBayesian shrinkage covariance estimator to the average realized covariance matrix with six and 12
(6)

(12)

months, Vt,h and Vt,h . In cases without transaction costs and with c = 2%, all of the performance fees take positive values. For the transaction cost of c equal to 4%, the fees from using the(6)monthly or daily returns to using Vt,h are unalterably positive in cases with γ = 1 and 10. By(12)contrast, the performance fee for Vt,h versus the use of monthly returns falls slightly below zero,−0.065, at the low relative risk aversion γ = 1, but the fee at the high relative risk aversion γ = 10
considerably exceeds zero, 13.052. The empirical results show that the large-scale portfolio optimization based on the realized covariance matrix estimators using the past intraday returns canyield substantial benefits in terms of risk reduction. Overall, the results are also suggestive of thebetter performance of the large-scale portfolio constructed by the realized covariance approach.

4

Conclusion

The objective of this paper is to examine effects of the realized covariance matrix estimators basedon intraday returns on large-scale minimum-variance equity portfolio optimization. We empirically assess out-of-sample performance of portfolios with different covariance matrix estimators:
the realized covariance matrix estimators and the Bayesian shrinkage estimators based on the pastmonthly and daily returns. The main results are: (1) the realized covariance matrix estimators
using the past intraday returns yield a lower standard deviation of the large-scale portfolio returns
than the Bayesian shrinkage estimators based on the monthly and daily historical returns; (2) gains

12

to switching to strategies using the realized covariance matrix estimators are higher for an investor
with higher relative risk aversion; and (3) the better portfolio performance of the realized covariance approach implied by ex-post returns in excess of the risk-free rate, the standard deviationsof the excess returns, the return per unit of risk (Sharpe ratio) and the switching fees seems to berobust to the level of transaction costs.

References
Bandi, F. M., J. R. Russell and Y. Zhu (2008), ”Using High-Frequency Data In Dynamic Portfolio
Choice” Econometric Reviews, 27, 163-198.
Barndorff-Nielsen, O. E., P. R. Hansen, A. Lunde and N. Shephard. (2008). ”Designing Realised
Kernels to Measure the Ex-Post Variation of Equity Prices in the Presence of Noise.” Econometrica, forthcoming.
Barucci, E. and R. Renó. (2002). ”On measuring volatility of diffusion processes with high frequency data,” Economics Letters, 74, 371-378.
Clarke, R., H. D. Silva and S. Thorley (2006) ”Minimum-Variance Portfolios in the U.S Equity
Market” Journal of Portfolio Management, 33, 10-24.
Dacorogna, M. M., R. Gencay, U. Müller, R. B. Olsen and O. V. Pictet. (2001). An Introduction to
High-Frequency Finance, Academic Press, London.
Fleming, J., C. Kirby and B. Ostdiek (2001) ”The economic value of volatility timing” Journal of
Finance, 56, 329-352.
Fleming, J., C. Kirby and B. Ostdiek (2003) ”The economic value of volatility timing using ”realized” volatility” Journal of Financial Economics, 67, 473-509.
Gosier, K., A. Madhavan, V. Serbin and J. Yang (2005) ”Toward Better Risk Forecasts” Journal ofPortfolio Management, 31, 82-91.
Hansen, P. R., and A. Lunde. (2005). ”A realized variance for the whole day based on intermittent
high-frequency data.” Journal of Financial Econometrics, 20, 525-554.
Hansen, P. R., and A. Lunde. (2006). ”Realized Variance and Market Microstructure Noise.”13

Journal of Business and Economic Statistics, 24, 127-161.
Harris, L. (1990). ”Estimation of Stock Price Variances and Serial Covariances from Discrete
Observations.” The Journal of Financial and Quantitative Analysis, 25, 291-306.
Ledoit, O., and M. Wolf (2003). ”Improved Estimation of the Covariance Matrix of Stock Returns
with an Application to Portfolio Selection” Journal of Empirical Finance, 10, 603-621.
Ledoit, O., and M. Wolf (2004). ”Honey, I Shrunk the Sample Covariance Matrix” Journal ofPortfolio Management, Summer, 110-119.
Markowitz, H. (1952). ”Portfolio Selection” Journal of Finance, 7, 77-91.
Pooter, M. D., M. Martens and D. V. Dijk (2008). ”Predicting The Daily Covariance Matrix For
S&P 100 Stocks Using Intraday Data-But Which Frequency To Use” Econometric Reviews, 27,
199-229.
Roll, R. (1984). ”A Simple Implicit Measure of the Effective Bid-Ask Spread in an Efficient
Market.” The Journal of Finance, 39, 1127-1139.
Ubukata, M., and K. Oya (2009). ”Estimation and Testing for Dependence in Market Microstructure Noise,” Journal of Financial Econometrics, 7, 106-151.
Voev, V, and A. Lunde. (2007). ”Integrated Covariance Estimation using High-frequency Data in
the Presence of Noise.” Journal of Financial Econometrics, 5, 68-104.
Zhang, L. (2006). ”Efficient estimation of stochastic volatility using noisy observations: a multiscale approach.” Bernoulli, 12, 1019-1043.
Zhang, L, P. A. Mykland and Y. Aı̈t-Sahalia. (2005). ”A tale of two time scales: Determining integrated volatility with noisy high frequency data.” Journal of the American Statistical Association,
100, 1394-1411.
Zhou, B. (1996). ”High-Frequency Data and Volatility in Foreign-Exchange Rates.” Journal ofBusiness and Economic Statistics, 14, 45-52.

14

Monthly
Intraday: 6 month

500

Daily

400

300

200

100

0

−100

−200
2001

2002

2003

2004

2005

Figure 1: Cumulative portfolio returns

15

2006

2007

Table 1: Performance of minimum-variance portfolios (scenario 1)
Annualized Std Dev Sharpe Correlation Turnover Tracking
Mean
Ratio with TOPIX
Error
Panel A: c = 0
500 stocks
value-weighted
Monthly
Daily
(6)

Vt,h

(12)

Vt,h

Panel B: c = 2
500 stocks
value-weighted
Monthly
Daily
(6)

Vt,h

(12)

Vt,h

Panel C: c = 4
500 stocks
value-weighted
Monthly
Daily
(6)

Vt,h

(12)

Vt,h

2.502

49.315

0.051

0.984

2.851

6.670

31.045

0.215

0.734

9.107

34.051

7.220

29.575

0.244

0.695

27.38635.0137.340

27.147

0.270

0.675

27.69936.5437.273

28.086

0.259

0.688

19.57835.5352.386

49.318

0.048

6.274

31.038

0.202

6.485

29.636

0.219

6.427

27.089

0.237

6.692

28.067

0.238

2.270

49.322

0.046

5.879

31.032

0.189

5.751

29.703

0.194

5.513

27.034

0.204

6.110

28.0500.218

Note: The table compares the performance of minimum-variance portfolios using the optimization procedure in scenario 1. The sample comprises the 500 largest market capitalization JPN stocks on the TSE. The
out-of-sample period is 76 months from November 2000 to February 2007. We compute the realized returns
yielded by the different strategies using the 500 stocks value-weighted rate, using the Bayesian shrinkage
estimators based on monthly and daily returns “Monthly” and “Daily” and realized covariance matrix es(6)(12)
timators based on the prior six- and 12-month intraday returns “Vt,h ” and “Vt,h ”. The table reports theaverage annualized return in excess of the risk-free rate, annualized standard deviation of the excess returns,
Sharpe ratio, correlation with TOPIX, turnover for each strategy, and tracking error from the 500 stocks
value-weighted portfolio. Panels A, B, and C show the statistics where the level of transaction costs c are
0%, 2%, and 4%, respectively.

16

Table 2: The economic gains of strategies using intraday returns (scenario 1)

(6)

Vt,h

(12)

Vt,h

vs monthly
vs dailyvs monthly
vs dailyc=0
c=2∆γ = 1 ∆γ = 10 ∆γ = 1 ∆γ = 101.863
12.7341.355
11.998
0.849
9.380
0.701
8.927
1.526
10.8981.340
10.439
0.509
6.943
0.683
6.826

c=4
∆γ = 1 ∆γ = 100.846
11.264
0.553
8.506
1.153
9.988
0.858
6.752

Note: The table represents the annualized fees ∆γ that the investor with relative risk aversion parameterγ = 1 and 10 switches from the strategies using the Bayesian shrinkage estimators based on monthly anddaily returns “Monthly” and “Daily” to using the realized covariance matrix estimators based on the prior(6)(12)
six- and 12-month intraday returns “Vt,h ” and “Vt,h ”.

17Table 3: Performance of minimum-variance portfolios (scenario 2)
Annualized Std Dev Sharpe
Mean
Ratio

TurnoverPanel A: c = 0
Monthly

10.70132.791

0.326

10.043

Daily

9.512

31.2260.305

20.837

10.57728.4950.371

25.053

Vt,h

9.766

29.4920.331

18.324

Panel B: c = 2
Monthly

10.21733.000

0.310

Daily

8.798

31.447

0.280

9.741

28.606

0.341

Vt,h

9.190

29.6650.310

Panel C: c = 4
Monthly

9.796

32.993

0.297

Daily

8.089

31.460

0.257

8.813

28.539

0.309

8.589

29.642

0.290

(6)

Vt,h

(6)

(6)

Vt,h

(12)

(6)

Vt,h

(12)

Vt,h

Note: The table compares the performance of minimum-variance portfolios using the optimization procedure in scenario 2. The target expected return is set to annualized 10%. The sample comprises the 500
largest market capitalization JPN stocks on the TSE. The out-of-sample period is 76 months from November 2000 to February 2007. We compute the realized returns yielded by the different strategies using theBayesian shrinkage estimators based on monthly and daily returns “Monthly” and “Daily” and using the(6)realized covariance matrix estimators based on the prior six- and 12-month intraday returns “Vt,h ” and(12)“Vt,h ”. The table reports the average annualized return in excess of the risk-free rate, annualized standard
deviation of the excess returns, Sharpe ratio, correlation with TOPIX, turnover for each strategy, and tracking error from the 500 stocks value-weighted portfolio. Panels A, B, and C show the statistics where thelevel of transaction costs c are 0%, 2%, and 4%, respectively.

18

Table 4: The economic gains of strategies using intraday returns (scenario 2)

(6)

Vt,h

(12)

Vt,h

vs monthly
vs dailyvs monthly
vs dailyc=0
c=2∆γ = 1 ∆γ = 10 ∆γ = 1 ∆γ = 10
1.322
16.8751.000
16.152
1.952
13.426
1.862
12.833
0.196
14.1940.115
13.6230.829
10.132
0.981
9.693

c=4
∆γ = 1 ∆γ = 100.505
15.3241.660
12.160-0.06513.052
1.095
9.295

Note: The table represents the annualized fees ∆γ that the investor with relative risk aversion parameterγ = 1 and 10 switches from the strategies using the Bayesian shrinkage estimators based on monthly anddaily returns “Monthly” and “Daily” to using the realized covariance matrix estimators based on the prior(6)(12)
six- and 12-month intraday returns “Vt,h ” and “Vt,h ”.

19