# Application of Portfolio Decisions Within a Generalized Funding Ratio Framework(中文)

調控資產配置是市場之熱門議題，無論是方法亦或是時間點，皆有無數學者爭相研究。本文著重於方法面，建立在現代投資理論上，嘗試加入投資人追求之勝率及要求報酬等較為直觀之因素，達到勝過原先投資組合之夏普比率。除此以外，發現此透過勝率及要求報酬此二者所建置之模型，若加入現金部位，能夠得到更佳之報酬。

Harmonizing portfolios is always trending. Modern Portfolio Theory (MPT), the 1990 Nobel Memorial Prize in Economic Sciences, was proposed by Markowitz, indicating that portfolios on the efficient frontier are efficient investing decisions. Picking a better portfolio on an efficient frontier has better performance is a problem popularly discussed.

This report examines the application of Portfolio Decisions Within a Generalized Funding Ratio Framework, argued by Martin L. Leibowitz et al. He introduces probability and target return into the MPT. He proposed a formula connecting probability, target return, and the efficient frontier. We can get success frontier and assurance frontier through that. If obtaining the peak of both, we get Success Portfolio and Assurance Portfolio, respectively. In the report, we use the daily price data of nine sectors’ ETFs and rebalance our portfolio monthly. We find that these two innovative portfolios have better performances than the Maximum Sharpe ratio Portfolio, a traditional portfolio that depends on the efficient frontier. Surprisingly, if we take not only equities but also cash position into account, these two portfolios can achieve better performance.

According to all the data before the end of the month, we set a target return of 8% (SPY annual return) and a probability of 0.6, as shown in Figure 1. From this, it can be found whether Success Portfolio (given target return of 8%) or Assurance Portfolio (given probability 0.6) is inferior to the asset allocation portfolio selected by the maximum Sharpe ratio.

Since the portfolios selected above are affected by the data range, rolling days are added to change the entire historical information, and rolling days are set to 3 months. The performance after changing the structure is shown in Figure 2.

The portfolio construction method proposed by Leibowitz is verified with sector ETF, which proves that it is significantly better than the traditional maximum Sharpe ratio portfolio. We observe the influence of various variables on the methods and add cash positions into portfolios to avoid a financial recession. There are two special things: first, as the given probability is lower, the Sharpe ratio brought by it is higher. In this case, it may mean that it is not predictive to evaluate the future portfolio by the return of the past; second, from the above research, it is found that the performance of this method adding cash position is greater than that without adding cash position. Whether the cash position can significantly improve the original method proposed by Leibowitz, is worthy of continuous research.

Contact us atinfo@gammaparadigm.comfor the full report.

Ya-Qi, Lin

Quantitative Research Intern, Quantitative Finance, Gamma Paradigm

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