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by: Portfolioand.me

## Summary

There are different approaches to determining the weights for different stocks in an investment portfolio.

We used the Monte Carlo method to figure out the most efficient approach.

Markowitz implied return weights is the preferred approach if the investor can withstand a lot of volatility, while equal weights work better for more conservative investors.

There are different approaches to determining what kind of weight to assign to a particular stock in an investment portfolio.

These approaches are best suited for algorithmic trading, since none of these methods will take all of an individual investor's preferences into account.

Nonetheless, different weighting strategies can serve as starting points the investor can use to build a portfolio in the future.

One of the simplest strategies is to assign equal weights to the stocks in a portfolio. The assumption behind this strategy is that the investor does not prefer any of the stocks in the portfolio over others.

Another simple strategy that is used for the majority of indices is assigning weights proportionally to market capitalization. In this strategy, the investor gives preference to bigger companies.

Another popular strategy is assigning weights in accordance with an approach developed by the Nobel Prize winning economist Harry Markowitz. The core idea of this approach is that if we know expected returns and covariance of the securities than we can compose an optimal portfolio. In real word nobody knows expected returns. So usually historical returns are used as proxies for expected returns. But we can also use implied returns (calculated as Price Target/Price) which we've discussed in one of our previous articles. At first glance this has much more sense, because implied returns reflect expectations of analysts about future returns of a stock whereas historical returns imply the assumption that future performance of a stock would be the same as its past performance.

Let's test these four strategies for assigning weights to stocks in a portfolio:

• Equal weights
• Weights according to market cap
• Markowitz historical return weights
• Markowitz implied return weights.

Let's test the different strategies for assigning weights with the help of the Monte Carlo method.

We will conduct 100,000 tests, in which we will select a random 20 securities for a random date from 01/01/2008 to 02/01/2015.

The conditions for these securities are as follows:

The security has to be traded for at least 1 year prior to the date on which the portfolio is put together. The security must have a Price Target on the date the portfolio is put together.

We are going to use the strategies specified above for each of these portfolios.

Weight limits.

In order to obtain results that aren't heavily dependent on the profitability of one security, we set a 10% limit on the weight of a security. That is, if the market cap strategy assigns a value greater than 10% to a security, the weight will be cut down to 10%. The weights for securities with weights of less than 10% will be proportionally increased.

For both Markowitz weighting weight of one stock is also limited to 10% and short selling is forbidden.

Let's calculate the profitability of a portfolio for 1 year. This way, we will get four annual return values for each of the 100,000 random portfolios - one for each weight strategy.

The table below gives a summary of the results for this test. The table illustrates how many times each strategy yielded a portfolio with the highest returns, second highest returns, third highest returns and lowest returns.

An analysis of the table allows us to make the following conclusions regarding different weight strategies.

The Markowitz historical return weights strategy yields the highest return in approximately 32% of cases, and the lowest return in 30% of cases. The strategy results in second highest returns 18% of the time, and third highest returns 18% of the time. These results show that there is a lot of randomness in this method, since the outcomes are concentrated in the lowest and highest ends of the spectrum. These results give credence to a popular phrase - "past performance does not guarantee future results."

The Markowitz implied return weights strategy is more likely to yield the highest returns (37%). At the same time, the worst result also has a high probability (26%). However, the probability of getting the worst outcome is significantly lower than getting the best outcome and doesn't differ much from the probability of getting the second highest or third highest returns. We are able to achieve this favorable result because implied return is much closer to an expected return, which is necessary for implementing the Markowitz approach.

The equal weights strategy yields outcomes that are concentrated in the middle of the spectrum, in the second and third highest return categories.

The cap-weighted strategy yields the worst results. This strategy rarely yields the highest returns, and frequently results in the lowest returns. This can be partially explained by the fact that using market cap values to assign weights tilts the portfolio in favor of companies that posted stock price increases in the recent past. That is, value opportunities are deliberately avoided.

Thus, the best strategies seem to be the implied return method and the equal weights method. The implied return strategy has a high probability of maximum profits compared to other strategies, but entails more risk. The equal weights strategy is more conservative - it rarely yields the best results, but is also unlikely to yield the worst outcomes.

For clarity, let's look at how these weight strategies work for a portfolio of 20 largest companies from the S&P 500 index from 01/01/2008 to 02/01/2015.

The requirements for securities are the same as in the previous test.

The table shows the composition of the portfolio at every rebalancing date.

 01/02/2008 01/02/2009 01/04/2010 01/03/2011 01/03/2012 01/02/2013 01/02/2014 01/02/2015 Exxon Mobil (NYSE:XOM) Exxon Mobil Exxon Mobil Exxon Mobil Exxon Mobil Apple (NASDAQ:AAPL) Apple Apple General Electric (NYSE:GE) Wal-Mart Stores (NYSE:WMT) Microsoft (NASDAQ:MSFT) Apple Apple Exxon Mobil Exxon Mobil Exxon Mobil Microsoft Procter & Gamble (NYSE:PG) Wal-Mart Stores Microsoft Microsoft Alphabet (NASDAQ:GOOGL) Alphabet Microsoft AT&T (NYSE:T) Microsoft Alphabet Berkshire Hathaway (NYSE:BRK.B) Chevron (NYSE:CVX) Microsoft Microsoft Berkshire Hathaway Procter & Gamble General Electric Apple General Electric IBM (NYSE:IBM) Wal-Mart Stores Berkshire Hathaway Alphabet Alphabet (GOOGL, GOOG) AT&T Procter & Gamble Wal-Mart Stores Alphabet Berkshire Hathaway General Electric Johnson & Johnson (NYSE:JNJ) Chevron Johnson & Johnson Johnson & Johnson Alphabet Wal-Mart Stores General Electric Johnson & Johnson Wells Fargo (NYSE:WFC) Johnson & Johnson Chevron JPMorgan Chase (NYSE:JPM) Chevron General Electric IBM Wal-Mart Stores Wal-Mart Stores Wal-Mart Stores Pfizer (NYSE:PFE) IBM IBM Berkshire Hathaway Chevron Chevron General Electric Bank of America (NYSE:BAC) IBM AT&T Procter & Gamble Procter & Gamble AT&T Wells Fargo Procter & Gamble Apple JPMorgan Chase General Electric AT&T AT&T Johnson & Johnson Procter & Gamble JPMorgan Chase Cisco Systems (NASDAQ:CSCO) Wells Fargo Chevron Johnson & Johnson Johnson & Johnson Pfizer JPMorgan Chase Facebook (NASDAQ:FB) Altria Group (NYSE:MO) Coca-Cola (NYSE:KO) Bank of America JPMorgan Chase Pfizer Procter & Gamble IBM Chevron Pfizer Alphabet Pfizer Wells Fargo Coca-Cola Wells Fargo Pfizer Pfizer Intel (NASDAQ:INTC) Cisco Systems Cisco Systems Oracle (NYSE:ORCL) Wells Fargo JPMorgan Chase AT&T Verizon Communications (NYSE:VZ) IBM Verizon Communications Wells Fargo Coca-Cola Philip Morris International (NYSE:PM) Coca-Cola Amazon.com (NASDAQ:AMZN) Oracle Citigroup (NYSE:C) Oracle Coca-Cola Bank of America JPMorgan Chase Oracle Coca-Cola Bank of America AIG (NYSE:AIG) Intel Oracle Citigroup Oracle Philip Morris International Bank of America Coca-Cola JPMorgan Chase HP Inc. (NYSE:HPQ) HP Inc. Pfizer Intel Bank of America Oracle Intel Coca-Cola PepsiCo (NYSE:PEP) Intel Intel Merck & Co (NYSE:MRK) Verizon Communications Citigroup AT&T
Click to enlarge

The table below contains the results of the test.

Because the overall number of tests is a lot lower compared to the table above, the results are also different. However, there are some similarities.

The Markowitz implied return weights strategy produces the best results more often than other strategies, but also yields the lowest profitability more frequently than other strategies as well (same as the Markowitz historical return weights strategy). This is consistent with the previous findings, which suggest that this strategy produces outcomes that are concentrated on opposite ends of the spectrum, with a higher probability of the best outcome.

The Markowitz historical return weights strategy also yielded results on opposite ends of the spectrum, but the probability of getting the worst possible outcome was higher (this is also in line with the previously obtained results).

The significant difference between this test and the previous one is that the cap-weighted strategy produced more stable results than the equal weights strategy. However, as in the previous test, both of these strategies rarely yield the best outcomes.

Conclusion

As noted above, the standardized approach to selecting weights for instruments in a portfolio is unlikely to be the best solution for an investor, with the exception of algorithmic trading.

However, the standardized selection of weights can be a starting point for determining how much to allocate resources between different instruments.

The test we ran illustrates that out of the following strategies:

• Equal weights
• Weights according to market cap
• Markowitz historical return weights
• Markowitz implied return weights.

The Markowitz implied return weights is optimal for investors who can withstand a lot of volatility, while the equal weights strategy is best for more conservative investors.

Disclosure: I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.

I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.