Portfolio Analytics

1. Introduction: To analyze our portfolio and determine its exposure to risk factors, we assumed that:
Rit+1=a+bi,1F1t+bi,2F2t+…+bi,nFnt+eit+1
First, we calculated z-score for logarithm of b_p and logarithm of cap to determine sensitivities for stock-specific factors. After we got all 13 sensitivities, we did a weighted regression of the 13 sensitivities against the 1000 stock’s forward return in 60 periods, respectively. For each period, we got 13 coefficients, which were the risk premiums for each risk factor. Finally, we regressed the risk premium from our regression against the portfolio return, bench return and active return. Doing so, we get these returns’ sensitivities against risk factors and were able to determine our portfolio’s exposure to various risk factors.

2. Procedures and Analysis
2.1. Working Procedure Nearly all of the calculations in part II were done by VBA Macros. In the Project part2.xlsm, there were 6 worksheets. The Data recorded all the cross-sectional data for 60 periods, having 1000 stocks in each period. StockPriceData recorded the data for stocks in our portfolio. To run the regression, we assumed that for a particular stock, the forward return in a particular period was, rt+1=bisec10λsec10+…+bisec55λsec55+bibetaλbeta+bilnbpλlnbp+bilncpλlncp+e . The equation has no intercept, the risk premium for sector factor has already included the zero beta rate. To prevent stocks with smaller market capitalization from being overweighed by stocks with larger capitalization, we should run a weighted regression. In WeightedFactors, we calculated the z-score for “logarithm of b_p” and “logarithm of cap”. Plus beta and 10 sector factors, we got sensitivities for 13 factors. Then for each period, we weighted each stock’s 13 sensitivities and their forward return by multiplying with the square root of their market capitalization. In LamdaStatistics, we implemented the cross-sectional weighted regression to minimize 11000(wi*ri-wi*bisec10λsec10-wi*bisec15λsec15-…wi*wi*bisec55λsec55-wi*bibetaλbeta-wi*bilnbpλlnbp-wi*bilncpλlncp)2 for 60 months and got 60 groups of risk premium and corresponding statistical details. Each groups contained 13 risk premiums for each risk factor in one month. Then, we counted the number of significant positive, significant negative, and non-significant for risk premium of beta, logarithm of book-price ratio and logarithm of capitalization. In LamdaSummary, we listed risk premiums for the 13 risk factors, our portfolio return, our bench return and portfolio’s active return. Then we drew the chart for various risk premiums in time series. Finally, in Sensitivities, we ran the regression based on data in LamadaSummary to determine the exposure of our portfolio forward return and active return to the 13 risk factors. 3.2. Analysis After cross-sectional regression of the sensitivities against stock forward return, in LamdaStatistics, we got the risk premium for each risk factor and corresponding p-value for 60 months. For beta, “logarithm of b_p” and “logarithm of cap” in each month, if the coefficient was positive and the p-value was less than 0.05, we called it positive significant. If the coefficient was negative and the p-value was less than 0.05, we called it negative significant. All other cases were called non-significant. Then, we calculated the ratio of the number of months in which the risk premium was “positive significant” to “ negative significant”. If the ratio of the factor is larger than 2 or smaller than 0.5, the factor is an alpha factor. Otherwise, it will only be a risk factor. The counting result is as follows:

All the ratios were larger than 0.5 and smaller than 2. So the beta ,bp and mktcap were the only risk factors and could not make alpha factors. We were unable to implement the test for the sector factors. In our model, the premium for sector factors included the zero beta rate, and the beta factor only accounted for the risk premium not covered by sector factors. To identify the trend of each risk premium in market during the consecutive 60 months, we drew a chart of each factor based on time series data.

It seemed that the risk premium for all factors fluctuated randomly. We could not identify a clear trend in the market for any of them. But we found that, for stock-specific factors and beta, the premium just fluctuated above and below zero, while for all sector factors, the premium was slightly greater than zero for most of the period. During the end of 2008 and from September 2011 to present, premium for most factors dropped deeply below zero. This might be caused by financial tsunami in 2008 and the economic recession now. After we got the monthly risk premiums for the 13 factors in 60 months, we could then put our portfolio into the model:
Rt+1=bsec10λsec10,t+bsec15λsec15,t+…+bsec55λsec55,t+bbetaλbeta,t+blnbpλlnbp,t+blncpλlncp,t
where Rt+1 is the forward return of our portfolio. For all 20 stocks of our portfolio were chosen from stocks listed in Russell 1000, we selected R1000 index as our bench to calculate the active return. Based on the risk premiums we found, we implemented a time-series regression to find out the sensitivities of our portfolio return, bench return and active return respectively. The results were summarized in the following table: We found that the model might not fit our portfolio and our bench well, due to the R-square being 0.3871 and 0.2350, respectively. We chose our significance level as 0.05. So, for our portfolio, only the factor of logarithms of cap was significant. The results conformed to our previous time-series chart for factors, the risk premium seemed to fluctuate randomly through time horizon. But for active return, the model became much better. The R-square was 0.58 and most variance of the active return could be explained through our model. The active return was sensitive to logarithm of b_p, logarithm of cap, sector 25, sector 30 and sector 40. The p-value for all beta in three returns was very large and the beta was insignificant, it seemed that the sector membership covered most of the market risk. For active return, it seemed sector 25, sector 25 and sector 40 did a good job to describe our portfolio’s exposure to risk in these memberships. We also found that more than 1400 stocks appeared in the original data sheet and some of them were not the 1000 biggest companies in U.S while all stocks in our portfolio were selected from R1000 list. Then, when we found the risk premium for various factors from the data, we must include some behaviors and features for small-sized companies. Our portfolio sensitivities were regressed out using these premiums that had some distortion. This might be a reason why the model did not fit our portfolio and our bench well. For active return, the portfolio return minus the bench, some features for big sized companies were offset. So the model fit active return much better!

Conclusion In part II of our project, we calculated the risk premium for each risk factor in each time period based on the equation: rt+1,i=bisec10λsec10+bisec15λsec15+…+wiλsec55+bibetaλbeta+bilnbpλlnbp+bilncpλlncp We found none of the beta, log of bp or log of mktcap were alpha factors. The premium for sector factor had included zero beta rate. We were unable to determine whether a sector factor was a risk factor or an alpha factor.
After we found 60 groups of the risk premiums for each month, we ran a regression based on the equation:
Rt+1=bsec10λsec10,t+bsec15λsec15,t+…+bsec55λsec55,t+bbetaλbeta,t+blnbpλlnbp,t+blncpλlncp,t
Our portfolio only had a significant exposure to logarithms of cap while our active return had significant exposures to logarithms of b_p, logarithms of cap, sector 25, sector 30 and sector 40. Based on R-square, our model did not fit our portfolio well. But this might be due to the stocks being used to calculate risk premiums having different features from the stocks in our portfolio. If we use the forward return for stocks listed in Russell 1000 to calculate the risk premium for different factors, our model might work much better!