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• Henri Caron

# Research Series - Parameters for Modern Portfolio Theory

Updated: May 30, 2018

I have been applying the Modern Portfolio Theory (MPT) on my portfolio for the past years now. In calculating the best risk-optimised portfolio using that theory, one requires to determine how far back to look in historical returns, and how long forward to hold on to that portfolio in order to achieve maximum returns. But how far should one look back, and how long should one hold on to a portfolio before rebalancing?

Markets are in bull conditions for a little over 8 years now, and I wanted to find at which level those time-parameters would have generated me the highest returns. In the model I have developed, I simulate as if I had optimised some portfolios in the past, and see and compare their future results by forwarding that portfolio in time.

Backward optimisation: In the example below, I have determined 1/1/2017 as the date on which I would have optimised the portfolio, but because on 1/1/2017 the markets are closed, a proxy of 30/12/2016 has been chosen. In the case below, I have optimised that portfolio 4 times by using respectively 3, 6, 12 and 24 months of historical returns, and I have been forwarding that portfolio by respectively 3, 6 and 12 months. As such, the portfolio optimised by using 3 months of historical data, would have generated me 67,7% of (annualised) returns after 3 months, 36,8% (annualised) after 6 months and 34,1% after 12 months.

Forward optimisation: Not only did I forward the backward-optimised portfolios, I also looked forward in time and determined what would have been the optimal returns if I were to look from that future point, back in time. Or, as in the example below, if I would have looked respectively 3, 6 and 12 months backwards from 1/4/2017, 1/7/2017 and 1/1/018, I could have constructed a portfolio with 154,6%, 100,7% and 69,4% (annualised) returns on 1/1/2017.

Now, just looking at one portfolio doesn’t generate me enough data to determine which parameters give me the highest potential returns. That is why I have been comparing a set of 15 portfolios between 1/1/210 and 1/1/2017, each at 6 months interval and averaging out the returns.

The results of those averages seem to be straightforward and are displayed below. As it seems, holding on to a portfolio for a shorter amount of time, not only results in higher potential returns. It also is far more accurate method in achieving those ideal, highest potential returns. However, an important point to take into account, is that markets behave positively most of the time, but the rare instances they behave negative, they behave usually far more negative than ever positive. The extremely positive results related to holding on to portfolio’s for shorter periods (i.e. 3 months), can be easily offset by extremely negative results for holding on to portfolio’s for the same shorter periods. It seems from the results, that not only very positive results occur, but also very negative results. And like I mentioned before, 50% profit is far less important than a 50% loss, because with a 50% loss, you lose 50% of your exposure to the market and thus your capacity for rebuilding wealth.

Although those results may seem straightforward and very positive, one has to take into account all the factors in which, and consequences on how this model has been set up and run. First of all, this model has only tested past performances over the past 8 years, in which markets were in an upside trend. As such, these results don’t say anything for instance, about how the model would perform in bear markets. Additionally, this optimisation has been run by maximising mean. Remember from the article, that MPT can be either achieved by maximising returns while maintaining a certain level of volatility (risk), either by minimising volatility (risk) while maintaining a certain level of returns. This time, the model has run on maximising the returns, irrespective of the risk incurred. This leads to the high difference between returns for holding on portfolios for shorter periods.

In any case, those results give me important insights in how to rebalance my portfolio in the future, and more importantly give me a cornerstone to continue my personal research upon. I have built my model in such a way that I can easily remodel it by using different parameters such as downside semi-variance, including sell positions in the rebalancing, minimising the risk instead of maximising the mean, etc.

In any case, there is more to come!

*All returns in this research have been annualised to allow easy comparison