Características de las UFs

This section will provide empirical evidence that estimates of graphical models are useful in the estimation of the inverse covariance matrix, and they produce consistent and robust portfolio performance for minimum-variance portfolios. We perform the back- testing using historical returns of all component stocks of Dow Jones Industrial Aver- age (DJIA) and of S&P 500 Index, respectively, from 1996-Jan-01 to 2018-Feb-28. We compare the performance of the minimum-variance portfolio under different estimation methods for the (inverse) covariance matrix. All data were downloaded from Yahoo! Finance (https://finance.yahoo.com).

Generally, the true covariance matrix of asset returns is not stationary over time. A standard approach to dealing with the non-stationary of the financial time series is to use a periodic rebalancing strategy. That is, Σ (or Ω) will be re-calibrated every fixed amount of time. The periodic rebalancing strategy is defined as follows:

Definition 1 (Periodic rebalancing strategy) It is a portfolio rebalancing strategy where a portfolio will be rebalanced at time points T1, T2, · · · with one of the following

cases:

2. {Tj}j=1,2,··· are the calendar time points on a periodic basis, for example, monthly,

quarterly, semiannually, or annually.

For the sake of illustration, we focus on the first case of periodic rebalancing strategy, that is, Tj+1 − Tj is fixed for all j. In particular, we estimate the (inverse) covariance

matrices from daily arithmetic returns of DJIA (or S&P 500 Index) component stocks for various choice of the lengths of two consecutive rebalancing time points n = 35, 40, 45, 50, 75, 150, 225. For each case of n, we divide the entire time horizon into a sequence of consecutive non-overlapping time periods whose starting points are T1, T2, · · · , TT (n),

where T (n) is the maximal total number of periods having full length of sample size n such that TT (n)+ n is no later than 2018-Feb-28 (i.e. T1 =1996-Jan-02, and Tj+1− Tj = n

for j = 1, 2, · · · , T (n) − 1). We call these periods as investment periods to which the estimated portfolio allocation w would be applied. For each investment period with starting point Tj, we construct the corresponding estimation period ranging from Tj−n to

Tj− 1 as a collection of sample data to estimate S or the inverse covariance matrix ˆΩReg.

We then calculate the minimum-variance portfolio allocation ˆw∗ = (e>ΩˆRege)−1ΩˆRege where e = (1, 1, · · · , 1)> for the investment period from Tj to Tj + n. We repeat the

process from the first up to the last period and summarize the portfolio performance in Table 3.1 with different performance metrics, including the condition-number regularized estimation (CondReg) [33] and Ledoit-Wolf estimation (Ledoit) [85]. We assume that the initial investment of portfolio is $1 and the annualized risk-free interest rate is 5%. We use the definitions of various portfolio performance metrics from [40]. Please see the Supplementary materials B.1 for metrics definitions.

Estimates of regularized methods are generally computed numerically via software packages. In this empirical study, we use R packages, including the package gconcord package for both CONCORD and PseudoNet methods, the package glasso1 _{for Glasso,}

1

CondReg2 for condition-number-regularized estimator, and corpcor3 _{for Ledoit-Wolf es-}

timator.

The results in Table 3.1 and 3.2 show that, as the sample size becomes smaller, portfolios using biased estimates of graphical models, as well as other regularized models, achieve higher Sharpe ratio, higher realized return, higher terminal wealth, lower realized risk, lower turnover rate, and lower short size, compared with using unbiased estimates of the sample covariance matrix. The performance of portfolios that use these regularized estimations is also relatively consistent and stable, regardless of the choices of n. This indicates that the benefit of using biased estimates outweighs the effect of bias.

We repeat the empirical study by using all component stocks of S&P 500 Index and present the corresponding results in Table 3.3 and Table 3.4. We observe that under our dataset, results from S&P 500 have generally lower realized risks, higher realized returns, higher Sharpe ratios, higher terminal wealth, higher turnover rates, and higher short sides. This phenomenon can be explained by the fact that S&P 500 covers a larger set of assets thus a minimum-variance portfolio constructed from Dow Jones Industrial Average component stocks is relatively sub-optimal compared with that constructed from S&P 500. Higher short sides and turnover rates can be explained as Dow Jones stocks are biased towards large-cap stocks while companies of S&P 500 stocks are more diverse. In addition, we observe that within each performance metric, the order of the relative performance for each estimation method is almost the same between Dow Jones and S&P 500. A detailed result is presented in the Table 3.3 and 3.4.

2_{Download from https://cran.r-project.org/web/packages/CondReg/index.html} 3_{Download from https://cran.r-project.org/web/packages/corpcor/index.html}

Table 3.1: A comparison of the performance of portfolios over Dow Jones component stocks from Jan-02-1996 to Feb-28-2018 using different methods to estimate the (inverse) covariance matrix of asset returns. Portfolios using regularized inverse covariance esti- mation achieves higher Sharpe ratio, higher realized return, higher terminal wealth, and lower realized risk, lower turnover rate, lower short size. Performance of portfolios that use regularized estimations is also relatively consistent and stable.

Realized Risk (%)

n PseudoNet CONCORD Glasso CondReg LW Sample 35 14.73 14.88 14.44 15.61 14.64 29.41 40 14.47 14.66 14.12 15.51 14.36 23.32 45 14.50 14.64 14.21 15.53 14.34 21.30 50 14.58 14.65 14.28 15.64 14.46 20.00 75 14.58 14.72 14.21 15.65 14.32 16.71 150 14.52 14.50 14.15 15.54 14.35 15.12 225 14.80 14.78 14.56 15.60 14.68 15.24 Realized Return (%)

n PseudoNet CONCORD Glasso CondReg LW Sample 35 11.38 11.67 10.17 12.90 11.18 3.03 40 12.94 13.25 11.91 13.90 12.75 7.22 45 12.72 13.05 11.64 13.70 12.37 11.96 50 11.71 12.24 10.33 13.30 11.33 7.58 75 12.30 12.34 11.30 13.55 12.30 10.97 150 13.39 13.31 12.36 14.38 12.96 11.16 225 13.49 13.36 12.93 14.18 13.02 12.29 Sharpe Ratio

n PseudoNet CONCORD Glasso CondReg LW Sample 35 0.43 0.45 0.36 0.51 0.42 -0.07 40 0.55 0.56 0.49 0.57 0.54 0.10 45 0.53 0.55 0.47 0.56 0.51 0.33 50 0.46 0.49 0.37 0.53 0.44 0.13 75 0.50 0.50 0.44 0.55 0.49 0.36 150 0.58 0.57 0.52 0.60 0.55 0.41 225 0.57 0.57 0.54 0.59 0.55 0.48

Table 3.2: Continuing to Table 3.1. Turnover Rate

n PseudoNet CONCORD Glasso CondReg LW Sample 35 0.93 0.92 1.54 0.68 0.98 8.07 40 0.89 0.89 1.47 0.65 0.95 6.07 45 0.88 0.89 1.45 0.63 0.92 5.14 50 0.85 0.88 1.41 0.61 0.93 4.73 75 0.83 0.87 1.37 0.62 0.91 3.17 150 0.84 0.90 1.29 0.64 0.91 2.19 225 0.87 0.94 1.30 0.72 0.98 2.04 Short Side (%)

n PseudoNet CONCORD Glasso CondReg LW Sample 35 5.01 4.99 15.59 1.20 6.03 39.93 40 4.84 4.72 14.91 0.95 5.94 37.76 45 5.21 5.68 15.57 1.08 6.23 36.03 50 4.99 5.62 15.24 0.87 6.50 34.77 75 5.03 5.67 15.06 1.00 6.73 29.77 150 5.55 6.72 15.62 1.29 7.14 24.49 225 5.78 7.43 15.42 1.80 8.34 22.85 Terminal Wealth ($)

n PseudoNet CONCORD Glasso CondReg LW Sample 35 9.76 10.35 7.55 13.23 9.36 0.75 40 13.84 14.73 11.17 16.52 13.33 2.73 45 13.01 13.92 10.38 15.56 12.11 8.44 50 10.47 11.71 7.81 14.30 9.66 3.44 75 11.92 11.97 9.68 15.13 11.31 8.29 150 15.18 14.90 12.24 18.23 13.87 9.12 225 14.23 13.86 12.73 16.07 12.93 10.85