6.2 The BAB portfolio 33
The BAB-MR portfolio has signicant lower drawdown-risk that the average feature selection portfolio and the maximum drawdown for the two portfolios are respectively 6.5% and 41.2% The correlation coecient is rho = 0.42, indicating a very low correlation.
A t-value for BAB-MR log returns ist = 2.73, i.e. the null-hypothesis of the true mean equal to zero can be rejected on a 95% condence level. This conrms the ndings inFrazzini and Pedersen[2014], which indicates that the security market line is not linear, i.e. the gradient is lower for high beta stocks than for low beta stocks. Constrained investors prefer high-beta instruments, hence the expected return per unit risk (beta) are lower than for low-beta instruments.
CHAPTER 7
Performance evaluation
In this chapter an ex-post analysis is performed. The results of the back-test of the portfolios are analyzed and the performance is evaluated using a various selection of plots and ratios. The portfolios are compared with the purpose to evaluate the strategies and each step in the portfolio management framework presented.
7.1 Back-testing and performance evaluation
A set of the strategies are selected for further analysis and comparison.N1 , zn1 and median 1nsample portfolios are all, as mentioned benchmark portfolios, but zn1 best reect a market portfolio for the portfolios produced by the feature selection algorithm. The F-STARR-AR portfolio is selected from the group of feature selection portfolios for three reasons. First, it was one of the best performing portfolios, hence the potential for improvement are less likely, Secondly, 7 instruments are representing the full universe throughout the period and is therefore the fairest benchmark to compare with for the CVaR optimal portfolios which can allocate weights within the same universe. All CVaR optimal portfolios are included as they represent investors with dierent risk-preferences.
F-beta-AR are included because it was the portfolio with the highest cumulative return despite the relative low volatility. Finally, both BAB portfolios are included in the analysis.
In gure 7.1 the distributional properties of the daily log returns are compared with box plots. The hinges are indicating the 1. and 3. quantile and the whiskers are showing the variation og the log-returns. All the distributions are rather bell shaped and symmetric around zero. As expected, the portfolio targeting the risk averse investor has a small variance compared to the other portfolios. The BAB portfolios exhibit low volatility as they are supposed to be indierent regarding market risk (ex-ante). The OPT-M portfolios has almost the same 1. and 3. quantiles as OPT-L, but the trade-o between accepting more risk and seeking return results in more extreme variations. The portfolio with the highest volatility seems to be OPT-H, which is also expected as risk is not considered when the optimization algorithm allocates weights. The two market portfolios median
1
n and N1 exhibits relatively high volatility, probably due the high weighting of equity-based ETFs discussed earlier. This conrms, that the average feature selection portfolio zn1
is the best benchmark for the portfolios constructed from second step in the framework and onwards. The average feature selection portfolio has a slightly higher variance than the other feature selection portfolios, but this was also expected as they had a
7.1 Back-testing and performance evaluation 35
● ●● ●
● ●● ●● ●●● ●
● ●● ● ●
● ●●● ●● ● ● ●
●● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●● ●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●
●●● ●
● ●● ●●● ●●● ●
● ●● ● ●
● ●● ● ●● ● ● ●
● ●● ●● ●●●●●● ●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●● ● ●
● ● ●
● ● ●●●●● ●● ●
●
● ●
● ●
● ● ●
● ●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●● ●●●●●●●● ●●●●●●●●●●●●●●●●●●● ● ●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●
● ●●●●●●●● ●●●● ●●●●●● ●● ●●●
●
● ● ● ●
●● ● ●● ●
● ● ●● ● ●
● ● ● ●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ●●●●●● ●●●●● ●●● ●●●●●●●●●●●●●●●●●●● ●●●● ●● ●●●●●●●●●● ●●●●●●● ●●●●●● ●●●●●●●●● ●●
● ● ●●●●●● ●●●● ●●●●●● ●● ●
● ●●
● ●●● ●● ● ●
●
●●●●●●●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●● ●●●
●
● ●
●● ●●● ●●●●●●●●●● ●● ●●●●●●●●●●●●●
●●●●●●●● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●● ● ●●● ●●●●● ●● ●●●●● ●●● ●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●● ●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●
● ●
● ●●● ●●●● ●●● ●
●●● ●●●●
●●●● ●● ●●●● ● ●
●●●●●●●●●●●●●●●●●●●● ● ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●● ●●●●●●●●●●●●●● ●●●●●●● ●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●●●●●● ●● ●●●●● ●●●●● ●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●
● ● ● ● ●●●●● ●●● ●●●●●●●● ●●●●● ●
● ●●● ●● ●
● ●
● ●● ● ●
● ● ●●●● ●●●●●●●●
● ●●●●●●● ●●●●●● ● ●●● ●●
●●● ● ●●●●●●●●●●● ●●●●●● ●●●● ●
● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●● ● ●
● ●● ● ●●● ●● ●
● ● ●●●●● ● ●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●● ●●● ●●● ●
● ●
●●● ●
● ●
●● ●
● ●
● ●●●●●●● ●●●●●●●●● ●●● ●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●● ●
● ●
●●● ●
● ●
●● ●
● ●
●●●●●●● ●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●● ●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●● ● ●
Portfolio Return Distribution
Log return
●
●
●
●
●
●
●
●
●
●
1/N median 1/n 1/zn F−STARR−AR F−beta−AR OPT−L OPT−M OPT−H BAB−MR BAB−ACMR
−0.10 −0.05 0.00 0.05 0.10
Figure 7.1: Distributions of portfolio log-return.
selection criterion specically focusing on lowering risk. The box-plot is illustrating that feature selection are reducing risk, but how much it does aect the risk adjusted return, must be analyzed using other measures.
Summary statistics, risk and performance measurements of the portfolios are listed in table 7.1.T-test and performance ratios are based on log-return as they are closest to being normal and considers the time weighting of return (geometric mean of regular return), and risk and return measurement are calculated using regular returns to interpret the results directly. The zn1 portfolio is dened as the market portfolio when calculating the alpha and beta in the one-factor CAPM with the 1Y Euro-bond as the risk-free rate. The target return in the Sortino ratio is conveniently set to zero, as is does not aect the internal ranking.
All portfolios yield a positive annual return on average (geometric), where the high risk optimal portfolio yields the highest return on 7.2% annually and 91.7% in total and the BAB-ACMR has the lowest return of 3.4%
annually 36% in total. The alphas produced are near zero, and only signicant dierent from zero for OPT-L and BAB-MR portfolios. The zero-beta portfolios do not have zero beta, still low value though, which indicates that the beta value of an instrument changes over time despite eort of rebalancing and re-clustering trying to cover the latest behavior. The BAB portfolio that are re-clustered annually has a beta of 0.1 which is lower than the beta of the BAB portfolio with one-time clustering with a value of 0.189, however high turnover, hence high trading cost especially when the market changes might aect the volatility on the returns and therefore also estimates of beta. Therefore BAB-MR is accepted at the true ex-ante zero-beta portfolio in the rest of this study.
The risk measurements show thatN1 , zn1 and median n1 portfolios are all riskier, both on average, but also considering tail risk. They experience massive losses during the nancial crisis, way more than the other strategies. One obvious reason is the fact that they are more exposure to market risk, but a very reasonable explanation is that using feature selection results in a higher degree of diversication. In a risk-return plot in gure 7.2 the portfolios are compared either using CVaR or average drawdown as risk measure. The behavior of the ecient frontier known from MPT is somewhat recognized, especially when focusing at the optimization
7.1 Back-testing and performance evaluation 36
portfolios that obviously are optimal, whereas the other portfolios are not (guaranteed to be). Comparing F-STARR-AR with the optimal portfolios, the benet of using an optimization on-top of feature selection is clear. The risk-adjusted returns are higher for the optimal portfolios and F-STARR-AR are below the ecient frontier.
The t-test of zero mean log returns conrms that some of the portfolios produces a signicant positive return.
Ratios are very often used to measure performance and compare portfolios. Sharpe ratio and STARR have been introduced previously in chapter 4, and the Sortino ratio is also asymmetric risk-adjusted return measure that penalizes downside volatility by only considering returns that fall below a target return. Based on Sharpe's ratio BAB-MR performs slightly better than OPT-L followed by OPT-M. Accordingly to the Sortino ratio, the OPT-L portfolio perform better than BAB-MR, with OPT-M and F-beta-AR on the next places. The STARR ratio indicates that BAB-MR and BAB-ACMR perform almost identically with OPT-L a little behind. The STARR ratio punishes in general the optimization portfolios for not earning the expected return (based on the scenarios) quite hard compared to the other ratios, however they are still favored the portfolios with the same risk exposure.
In the next chapter the results found here will included in a broader discussion.
●
●
●
●
●
●
●
●
●
●
0.005 0.010 0.015 0.020 0.025 0.030 0.035
0.020.030.040.050.060.070.08Annual Return
1/N median 1/n
1/zn F−STARR−AR F−beta−AR
OPT−L
OPT−M
OPT−H
BAB−MR
BAB−ACMR
●
●
●
●
●
●
●
●
●
●
1/N median 1/n
1/zn F−STARR−AR
F−beta−AR
OPT−L
OPT−M
OPT−H
BAB−MR
BAB−ACMR
Average drawdown CVaR(5%) Figure 7.2: Scatter plot of risk vs. return.
7.1 Back-testing and performance evaluation 37
Portfolio1 Nmedian1 n1 znF-STARR-ARF-beta-AROPT-LOPT-MOPT-HBAB-MRBAB-ACMR Return Return(ann.)0.0690.0660.0520.0510.0610.0290.0520.0720.0500.034 St.dev(ann.)0.1520.1610.1220.1010.0900.0250.0710.1720.0550.057 Return/St.dev(ann.)0.4530.4100.4300.5010.6781.1620.7250.4200.9160.593 Acc.Return0.8620.8160.6100.5870.7360.3040.6020.9170.5830.360 Alpha(CAPM)5e-053e-0504e-051e-047e-05*1e-042e-041e-04*9e-05 Beta(CAPM)1.1311.20410.7080.4840.0740.2480.6730.1890.100 AverageDrawdown0.0270.0280.0250.0160.0140.0030.0090.0330.0080.011 Maxdrawdown0.4210.4510.4120.2850.2870.0430.1450.2730.0650.066 CVaR0.05-0.023-0.025-0.019-0.016-0.013-0.004-0.012-0.027-0.008-0.008 Log-return t-valueH0:µ=01.3341.2111.2771.4922.008*3.498*2.152*1.2352.730*1.783 Sharpe(ann.)0.3610.3250.3230.3740.5280.6780.5420.3370.6830.378 SortinoRatio(RT=0)0.0380.0340.0360.0420.0590.1040.060.0350.0830.055 STARR0.05-0.009-0.008-0.009-0.010-0.016-0.023-0.010-0.007-0.029-0.028 Table7.1:Summarystatisticsandperformanceratiosoftheportfoliosanalyzed.*indicatesthatthemeanissignicantdierentfromzero,ona95% condencelevel.Onlyindicatedforalphaandthet-test.
CHAPTER 8
Discussion
Previously, dierent portfolios have been studied with the purpose of evaluate a data-driven approach to portfolio management investing in ETFs to see if there is a potential of a simple and cheap way to construct portfolios, that are competitive with both expensive actively managed funds but also the passive managed index funds.
The rst step was a due diligence analysis of raw dataset consisting a lot of junk, e.g. the dataset included more than 9000 tickers claiming that they all were ETFs, but there exist under 2000 active ETF today so clearly the data set must be cleaned befor use. The need for a systematic and automatic approach to screen data is obvious. The screening let 559 ETFs pass for the selection step, all with acceptable data quality and the desired properties. However, the requirement of sucient historical data for back testing and the fact that the ETF universe is rather new, resulted in equity ETFs being heavily overweighed. If more type of instruments were considered, this overweighting could have been avoided. The hierarchical asset ltering successfully provides an easy and generic way to screen instruments to identify a subset of potential candidates with properties specied by user.
The feature selection algorithm presented in this study consisted of two steps. Clustering provided diversication and a selection criterion was used for the asset picking. The result of the steps, entirely relies on the behaviour in historical data and the strategy chosen. The rationale behind using this aproach, is that the properties observed in recent data will to some extent repeat itself in the future. The clustering provided a lower exposure toward equity ETF's and diversication. The Sharpe ratio and STARR criteria ensured strategies seeking high risk-adjusted returns and the low-beta criterion ensured strategies that reduces the exposure toward market risk.
The STARR and low-beta criteria provides the best results and performs better on a risk-adjusted basis than simple passive market and 1/N strategies after costs. The more actively managed feature selection portfolios did not perform very well for two reasons. They did not benet from the long-term trend and were punished on a high turnover.
The assets in the cluster portfolios are weighted equally without any further considerations, but the idea of the third step in the framework is to add a decision model that handles allocation problem and thereby improve the performance of the portfolio. CVaR optimization is used in this study, but other models exist. Moment matching provided reliable scenarios that can be used to reect the uncertainty in the CVaR model. Using this approach only captures the rst four statistical moments and the correlation structure, but not autocorrelated properties. The performances of tree portfolios reprecenting dierent investors, the risk-averse, risk-neutral and
39
risk-seeking investor, is analysed by back-testing CVaR optimal portfolios. The risk-averse and the risk-neutral portfolios perform well on a risk-adjusted basis due to low risk. They handle the down-side volatility very well and experience almost no losses during the nancial crisis like the equally weighted portfolios. The risk-loving portfolio performs at the same level as its benchmark. This could probably be improved if the scenarios were reecting the autocorrelation structure, especially of the volatility, known to exists. When being highly exposed to the market uctuations, timing is a very important factor. Scenarios based on moment matching captures the general trend in the recent period, hence the algorithm cannot adapt immediately to changes in the marked and is therefore not good at timing. The use of a decision model on top of feature selection seems to provide value.
The CVaR optimization model with moment matching is benecial for medium to low-risk investors, providing positive signicant returns. These portfolios increase the risk-adjusted returns signicantly when compared to the benchmark with equally weighted assets after feature selection and the 1/N type-of strategies. CVaR seems to be a good risk-measure, allowing the portfolios to take on some risk, but avoiding the extreme movements.
Using CVaR or other asymmetric volatility measure rather than a regular VaR is important, because the return distributions are not symmetrical, and extreme event occur more often in the left (loss) tall, hence up- and down side volatility should not be handled equally as they are not guaranteed to be similar. Using a risk measure like maximum drawdown would probably also produce some interesting results, and has the advantage of not rely on any assumptions of the underlying distributions. CVaR can only be mapped directly to happiness of an investor if he or she is a cold-blooded machine without humanity. Utility function are often used to represent the happiness of an investor as a function of risk and reward. Optimizing the utility function would be relevant, if the results were targeting a more specic group of investors with known utility function.
CVaR is beeing criticized for being a probabilistic measure, only guaranteeing that the extreme loss would not appear with a given certainty. Further it has been criticised for hard to back-test (not elicitable), however numerous solutions (like simulation) has been suggested to handles this. The regulatory institution (Basel Committee on Banking Supervision) recommends the use of CVaR under the Basel III (current regulations), which is a big approval and acceptance of CVaR as a risk measure.
The benets of having a data-driven approach are that a large number of instruments can be considered easily, the selection is not aected by behaviours biases and it does not require the manager to be totally updated on what happens in the political and nancial world at any time. On the other side, if the futures behaviour of an instrument is not represented in historical data, then the approach is useless. The portfolio manager might add his or her personal point of view e.g. eects of macro-economic assumptions to be reected in the framework, but it is this type of active decitions that often are assumed to be pure guessing and on average not benecial for the investment.
The result of low risk portfolios earning high risk-adjusted returns is also the idea behind testing the betting-against beta hypothesis on index level. The ndings of a zero-beta ETF portfolio yielding positive signicant risk-adjusted return, after cost of trading and borrowing, conrms the previous results inFrazzini and Pedersen [2014]: constrained investors seeks riskier assets and thereby bid up the prices. But are private investors limited from taking advantage of this? Yes, most private investors are constrained investors and must either accept a low risk investment with low return, or be willing to pay too much to get high market exposure, since they are not able to leverage at a reasonable cost. But, they could follow a zero-beta strategy and short sell high beta assets and then use the money along with savings to buy enough low beta stocks. This strategy is more applicable to large institutional investors or pensions funds, that can borrow at a reasonable rate. But the low-beta selection criterion with feature selection can be an alternative and simple approach - maybe with a decision model on top.
The portfolio provided signicant returns and annual returns equal the returns of the the riskier 1/N type-of strategies. As stated earlier CVaR with moment matching is benetting for consistency and long term trends.
The stable behaviour of the low-beta portfolio indicates, that it would be interesting to investigate if CVaR optimization could make improvements to the portfolio by adjust the allocations.
The term market portfolio is very commonly used in the nancial theories. But what is the true market portfolio and how is it constructed? This question has briey been touched earlier, as the market portfolio often denes the minimum required rate of return or benchmark. Often people tends to use a broad stock index as proxy, but it is clearly missing exposure toward other asset classes, geographical regions, company size etc. And what about assets not traded public, how are they included?. The market portfolio holding all assets available in the global nancial market does simply not exist in the real world and is more an academic term. The market
40
portfolio serves as an anchorage point of alternative investments, and should for practical use be dened as a fair benchmark seeking to as market-neutral as possible. In this study, the 559 ETFs dened the global nancial market with a huge weight towards stocks. But if the results should be comparable outside this sub-universe, using an equally or capitalized weighting of the EFTs to dene the market portfolio would be rather misleading.
An approach of weighting the assets by their behaviour relative to each other, are used, hence the market portfolio becomes an average of possible behaviours and exhibit an average behaviour if one asset from each group are picked. This is a fair neutral benchmark. This approach is also used for identifying high and low beta ETFs, but if an equally weighted market portfolios was used, then the classication would have separated the instruments into asset classes and it was not the purpose. Within asset class, low- and high-beta assets exists.
Often investors tend to specialize within an asset class and therefor seek the riskier investment within that asset class. The approach identifying low- and high-beta assets with an asset class follows the work in Frazzini and Pedersen[2014].
According to the EMH an investor cannot expect to consistently earn an abnormal risk-adjusted return. If costs neglected the investor can on average expect the return of the market portfolio, hence holding the market portfolio passively will on average beat an active strategy when adjusting for cost and risk. But what does it mean to be an passive investor? When you hold an index fund you are somewhat active because the constituents of the index changes and weights changes e.g. due to corporate actions, hence active actions are made. If an investor buys assets of an index representing the market and never touches it, he pr she will end up with a portfolio that is certainly not market neutral, hence the investor cannot expect the return of a market portfolio. Obviously holding an index requires frequently rebalancing. Annual or monthly rebalancing has been considered in the portfolios evaluated here, as it reects the behaviour that can be expected from a private or some institutional investor. In this study, an approach lying somewhere between active and passive management is suggested. Index funds used to get a broad and diversication and active decisions are made upon behaviour in data. Some of the strategies analyzed, provided positive signicant returns after adjusting for costs and risk, and some strategies did not. This suggests that it can be benecial to have an active strategy with passive instruments - but under the right condition. The cost of diversication must be low and if the investor either have access to (cheap) funding or if he or she is willing to accept low to medium risk, then the potential of the risk adjusted return are higher. High fees will vanish the return and must be avoided.
The survivor ship bias is not considered to be inuential on the result in this study as it is assumed that ETFs that are removed from the market is due low interest from investors rather than the all of the constitutes of the index defaulting.
The frame work and the ideas behind are presented along with examples of application. The idea of a data-driven approach should ease the process and limit the risk from behavioural and emotional biases. There is obviously room for improvement of the framework and what methods to be used in the steps. Some of the ideas are presented in chapter 10.
There exist competetive alternatives to both active managed funds and passive managed funds, and active investment styles can to some extent be justied. At the moment, only a few players in the market provides products like the bundled ETF-portfolios suggested here and the investor only has a low inuence on how they are created. I think that this business will evolve in the future, and I see great potential in active-passive investing where investors are combining cheap passive managed index funds with active smart strategies.