Bases-Stein Efficient Frontier can be considered as a better replacement in this case. Furthermore, we interestingly found out that besides widely-used diversification effect, we can also use Monte-Carlo simulation method to build up a better efficient frontier. It should also be noted that the diversification effect we experimented in this study failed to meet the requirement in portfolio performance improvement. Part l: Introduction I. Background and motivation This study aims to help investors and portfolio managers to make the “best” hooch of portfolios via using NEFF.
Furthermore, the paper will also provide some more advanced techniques to get better portfolio performance as well as tools to test it, namely Jobs-Kookier test and Eldest-Wolf test. The remainder of the paper is structured as follows. Part II provides the comparison of Margarita and Bases-Stein as well as briefing through the methods used to test the in-sample and out-of-sample performance of the optimal portfolios. In Part Ill some further studies on improving the portfolio performance are derived. And our conclusions are included in Part IV. II.
Data & Methodology Data The data used in this study comprises of 8 US industry portfolios from January 1981 to December 2014. Industry portfolios used in this study are from Food, Oil, Clothes, Chemicals, Steel, Cars, Utilities and Finance sectors. For further analysis we have selected one individual stock from each above industry sector. Those stocks are PepsiCo, Exxon Oil, PH, DOD Chemicals, Uncut Steel, Ford, Duke Energy Corporation and Joanna Chase. The selecting criteria are: (1) all companies are enlisted in NYSE, (2) companies have historical data room January, 1981 and (3) these companies are among top 10 in their industry sectors.
Regarding the history of the chosen companies, we tried to select companies with longer histories. However, 1981 is the furthest point we can reach. In the scope of our study, we also construct the efficient frontier using Fame-French formed portfolios. The efficient frontiers built from these portfolios is somewhat similar to a benchmark for other efficient frontiers in this study. Methodology Margarita efficient frontier is the widely-known method to be used in choosing the optimal portfolios. However, this method is criticized against its poor performance on out-of-sample basis due to its estimation error.
In short, the Margarita frontier use the mean and covariance matrix estimated from the data sample, therefore it naturally leads to: (1) Extreme weights for “lucky” asset, (2) Under-diversification due to the over-weighting of those “lucky’ asset, (3) Unstable optimal portfolio: slight changes in the input mean parameter may lead to substantial changes in the estimated allocations especially when asset correlations are low. On the other hand, Bases-Stein s considered a more accurate estimator by using a coefficient to “shrink” the means of the assets toward a global mean according to Jargon (1986) and Stevenson (2001).
This effectively reduces the difference between extreme observations. Thus this method can reduce the degree of estimation error. Particularly, the general form for the estimators can be defined as: Do)RI Where RI is adjusted mean, rug is the global mean, is the original asset mean and w is the shrinkage factor which can be estimated from a suitable prior: Where S is the sample covariance metric and T is the sample size. For testing the performance of our constructed portfolios, we use Jobs-Kookier test (Jobs and Kookier 1981 ) to compare the performance of the first and second sub-periods.
In particular, for the same portfolio in two different periods i and n, we test the null hypothesis HO: Shih – Sin = Or Shin = O The transformed difference for the Sharp measure has variance as below: 1022 clinically kill 0 20 20 ii no in on O i O TO 20 n Thus, the asymptotic normality of suggests the following test statistics. T] Shin However, the standard tests are not valid when returns have tails heavier than he normal distribution or for time series and panel datasets (see Eldest and Wolf, 2008).
In other words, the Jobs-Kookier test K test) is no longer reliable when it comes to testing non-did time series data. In order to solve this problem, Eldest and Wolf (2008) (LAW test) suggested two procedures. To put it simple, if SSH = O is not contained in the confidence interval, we can reject the null hypothesis that the performance of the portfolio in the first sub-period is equal to that in the second sub-period. 3 Part II: Portfolio Optimization – Margarita vs.. Bases-Stein l.
Optimization for individual stocks Graph 1: Margarita efficient frontier Graph 2: Bases – Stein efficient frontier constructed from stock portfolios We observe some strange patterns of Efficient Frontier constructed from stock portfolio for both Margarita and Bases-Stein methods. The slope of the frontier should be upward, however in this case all of them are downward which means the Sharpe ratios are negative. Often, when tangency portfolio returns are smaller than risk-free rates, leads to this situation. We will observe more “natural” versions of Efficient Frontier in the Optimization for Industry Portfolios section.
Graph 3: Margarita and Bases – Stein mean-variance efficient frontier Plotting Margarita and Bases-Stein mean-variance portfolios in the same figure gives us a more thorough understanding of the Bases-Stein method. These two efficient frontiers are quite close around global minimum portfolios, but spread out as the expected return increases, because global minimum portfolio does not depend on the average return. In other word, the global minimum portfolio only depends on the covariance matrix which is unknown, but more stable over time, according to Morton, therefore the shrinkage factor has no strong effect on lobar minimum 4 portfolio.
Apparently, the Bases-Stein line lies “inside” the Margarita line due to the shrinkage factor (1 -w) of Bases-Stein method. This also means the returns associate with Bases-Stein efficient frontier is closer to the true value than Margarita’s. Observing this figure leads to an interesting implication. One can logically think that because Bases-Stein BEEF is closer to true value than Margarita’s, Margarita’s would probably lie inside Bases-Stein’s in out-of- sample tests. However, in the scope of this study, we shall not try to prove that implication. II.
Optimization for Fame-French factor-mimicking portfolios Graph 4: Margarita and Bases – Stein mean-variance efficient frontier constructed from Fame – French factor-mimicking portfolios For all three Fame-French datasets, we observe similar patterns as in the previous section. The Bases-Stein BEEF lies inside the Margarita BEEF. We will see the same pattern in the next section of Optimization for Industry Portfolios. Therefore, it is now safe to conclude that NEFF indeed overestimates the expected returns of the portfolios, where as Bases-Stein provides us a closer approximate. Ill. Optimization for Industry Portfolios
As mentioned in section one, here we observe more “natural” Efficient Frontier for both methods, which is upward instead of downward. This means we can earn higher returns than risk-free rates from the portfolios constructed. From two figures, it is safe to conclude that the sign of Sharp ratio should not be taken lightly in portfolio construction. Particularly, when the Sharp ratio is negative (downward DEED, it is better if we consider replacing or adding one or more items in the portfolio to see if we can build up an upward efficient frontier (positive Sharp ratio), which means we can earn excess returns over risked dates. Graph 5: Margarita efficient frontier Graph 6: Bases – Stein efficient frontier constructed from industry portfolios Graph 7: Margarita and Bases – Stein mean-variance efficient frontier Graph 8: Margarita and Bases – Stein mean-variance efficient frontier 6 Above is the plot of all previous Efficient Frontier for both methods. In general, individual stock portfolio frontier lies at the bottom of the figure. This can be interpreted that portfolios in the black line yield the least returns for a given level of standard deviation or vice versa, highest standard deviation for a given level of expected returns.
Meanwhile, the industry portfolio frontier lies at the very top of the figure. Moreover, in the previous part, it can be seen that while efficient frontier constructed from individual stock has negative Sharp ratio, one constructed from industry portfolio has positive Sharp ratio. They strongly illustrate the benefits of diversification which we will talk in details in section 3 & 4 of Part Ill. IV. Performance Test Jobs-Kookier test We use Jobs and Kookier test to investigate whether the Sharp ratio in the first sample is equal to that in the second sample. Null hypothesis : SSL =so Alternatives : so 1: Sharp ratio in the 1st period. 2: Sharp ratio in the 2nd period. Here, we perform the test at 5% significance level. The Jobs and Kookier test for Margarita Tangency Portfolio is shown as below: Table 1 : Jobs – Kookier test for Margarita tangency portfolios Tangency portfolio Test stats Low-critical value Up-critical value Result Industry portfolio 3. 021 1 -1. 9600 1 . 9600 Stock portfolio -4. 0317 IF 3 factors -1 . 5422 -1 . 9600 1. 9600 SSL = so IF 5 factors (ex.) -1. 9807 IF 5 factors (2x2x2x2) -1. 4539 SSL =so We fail to reject the null hypothesis for Fame – French 3 factor and Fame – French factor (2x2x2x2) portfolios.
It seems like the NV BEEF did a poor job since only two out of five portfolios that have matched Sharpe ratios. However, the conclusion is not actually robust because although using the same weight as the first period, the return movements of each items in the tangency portfolio in the second period is quite different with the first one. Particularly, as the result of stable of average return and covariance matrix of risk factors over two periods, the test statistic of Fame – French risk factors is smaller than one of industry portfolios and stock portfolios.