Prerequisite

Baseline: SPY

Backtest period: 5 Years

Data time frame: Daily

1. Summary

The final sub-portfolio assignment is based on whether the given factors explain the returns or not. In either CAPM or FF5 model, the explained return is represented by the beta coefficients, whereas the unexplained return is represented by the alpha coefficient. Therefore, the alpha and beta resulted from the models only give explanation to the broad market’s overall volatility or risk, rather than finer market risks. It turns out that the alpha probably explain some source of return, which are caused by finer markder risks. Moreover, to obtain a more precise alpha, known as the ‘excess return over the broad market risk’, can be done by adding more systematical risk factors of finer markets, sucha as industrial risk factor, regional risk factor.

In conclusion, I can ensure the accuracy assignment of beta-hedging subportfolio to the cluster with approximately zero aggregated beta, in the broad market’s overall risks, but not sure the precision. To improve the precision, I will need to include more systematical risk factors to remove the extradinary stocks with non-zero-ish beta in the current beta-hedging subportfolio, as well as include stocks with zero-ish beta from the outside, while mantain a approximately zero aggregated beta.

2. Asset Allocation

The percentage of sectors tells the portion of the SP500 for each of sector.

3 Capital Asset Pricing Model: Asset Anslysis

The Asset Anslysis using Capital Asset Pricing Model is regressed by the daily return of each component stocks to the market daily return (baseline: SPY).

With no surprise, the results of CAPM shows The Beta for all SP500 component stock are positive, because the baseline SPY is based on this SP500 index. Besides, more than haft of the stocks has alpha less than zero, where only a few stocks have alpha greater than 0.001 (or 0.1%).

4. Fama French factors

The Asset Anslysis using Five Fama French (FF5) factor s is regressed by the daily return of each component stocks to the Five Fama French factors, including market factor (market risk, MKT), size factor (small minus large, SMB), book-to-market factor (high minus low, HML), profitability factor (RMW, robust minus weak), and investment patterns factor (CMA, conservative minus aggressive). In this model, the Risk Exposure is captured by the Fama French factors, and described using the beta and alpha coefficients.

The risk factor are in unit of presentatge. so, here is an exmample to interpret the coefficient of the risk factor: given the coefficient of beta_x equal to 10, then for 1% increase of risk factor x, the return increase by 1% * 10 =10%.

Besides, mentioned in literature, the FF5 model doesn’t not well explain the return of small size stock with high investment ratio and low profitability.

4.1 Clustering

I first check whether the the SP500 component stocks is clusterable using the hopkins test. As known, a value of the hopkins statistic close to 1 tends to indicate the data is highly clustered, random data will tend to result in values around 0.5, and uniformly distributed data will tend to result in values close to 0. Extracting the six coeficient from the FF5 model of each of stock as the trainning data, the resulting hopkins statistic is 0.0881, which implies there may be more than 1 cluster existed. In the correlation plot of the daily return, we can see there are two obvious clusters. Moreover, there are at less four smaller clusters in the larger cluster of the two obvious clusters. Using hubert index as criterion, cluster number is suggested to be 5.

By putting the 505 stock as data points on the first two principle components, we can see the most of the 5 resulted cluster are barely separated from each other; whereas there are a few clusters have overlap at the center of this two components plot.

## $hopkins_stat
## [1] 0.09704544
## 
## $plot

4.2 FF5 Asset Anslysis

I use a bar chart to represent each the FF5 model coefficient by clusters. The result shows that four coefficients are crucial in clustering such as alpha, Beta_CMA, Beta_HML, Beta_RMV, whereas the Beta_MKT and Beta_SMB vary by magnitude between clusters. Moreover, the next violin plot also represent each the FF5 model coefficient by clusters, but in the sense of distribution.

5. Stock Selection

5.1 Characterizing

The last plot is a scatter plot of executed return by the sum of FF5 coefficients. By masking and unmasking cluster groups, we can see the clusters locate around the origin. Given that the 5 risk factors are weakly correlated, by introducing intrepration of the coeficient of Beta, we make up the sum of Beta coeficient as a aggregated measure of the effect of the 5 risk factor to the return.

As the last step, by considering each cluster as SubPortfolios. I use t-test for the expected_return, Alpha, SumBeta, stand_dev respectively, for each SubPortfolio.

5.2 Beta Hedging SubPortfolio

The Beta-Hedging SubPortfolio has zero sum of beta.

5.3 Value SubPortfolio

The value SubPortfolio has positive expected_return and Alpha, but negative sum of beta, which is potentially undervalued.

5.4 Growth SubPortfolio

The growth SubPortfolio has positive expected_return, Alpha, sum of beta, which is potentially overvalued.

5.5 Underperformance SubPortfolio

The underperformance SubPortfolio has negative expected return.