The Ideal Portfolio Distribution – The Portfolio Theory

In this post we’re gonna cover an extremely important topic, but discussed very little between retail investors. How to distribute your portfolio the right way – The Portfolio Theory.

In 1952, Harry Markowitz changed the financial world when he published his doctoral thesis on Portfolio Selection: Efficient Diversification of Investment, today known as the Modern Portfolio Theory. This piece of work, like other brilliant pieces by Markowitz, was highly criticized when it came out, because it was about dealing with multiple assets in a statistical manner, in order to find the best portfolio possible.

Due to having a lot of statistics associated with this work, many refused to see the theory beyond the calculations. A theory that was recognized in 1990, forty years after the publishing of the thesis and book with that carried the same name, with the Nobel Prize in Economics granted to Harry Markowitz.

The objective of the Portfolio Theory, is mainly, to choose assets for a portfolio that have a good risk vs return ratio. The theory will do this by calculating always the least possible risk for a portfolio as a whole, considering each asset’s participation, their risk and correlation.

Understanding the Difference Between Risk and Return

Let’s begin by understanding the difference between return and risk.

The return of a Portfolio, the one thing everyone wants to increase, is directly proportional to the assets that compose it. For example, if a portfolio possesses two assets, AssetA and AssetB, the portfolio’s return will be the average return weighted for each asset in its relation to their weight distribution in the portfolio. Look:

AssetA return = 10%

AssetB return = 20%

If both assets have the same weight distribution in the portfolio, meaning, the portfolio is distributed 50% to each asset, the return will be:

Portfolio Return = (0,5 * 10%) + (0,5 * 20%) = 15%

Now, if we change each assets participation, so that AssetA has 30% of the asset allocated to it and AssetB has 70%, we will then have:

Portfolio Return = (0,3 *10%) + (0,7 * 20%) = 17%

We can alter each asset’s participation in a portfolio as much as we’d like, and the return of the given portfolio will always be related to the percentage of each stock’s participation in it.

However, the risk is the uncertainty imputed into the expected returns. The risk can be measured as a standard deviation of the daily percentage changes for each asset during a specific period of time. For example, if the Dow goes up 1% today, drops -2% tomorrow, comes back up 0,4% the next day and so on, we can get this sequence of numbers and calculate its standard deviation, thus having the average daily risk of the Dow.

The great news in the Portfolio Theory is that the overall portfolio risk is less than the weighted average of each asset’s individual risk. This is because the risk of each individual assets do not correlate, meaning, the day that AssetA rises, AssetB can drop, thus making the portfolio’s risk close to zero, a lot less than the risk of each individual asset.

Of course this is only possible if the assets that compose the portfolio oscillate independently. When two assets oscillate together, for example, when one goes up 1% and the other also goes up 1%, we can say that the correlation between them is 1. When two assets oscillate perfectly in opposite ways, for example, one asset goes up 2% and the other drops -2%, we can say that the correlation is -1. And all other possibilities of variation makes so that the correlation between the assets will oscillate between -1 and +1.

The correlation between the assets is found using this formula:

Correlation between A and B = Covariation between A and B / (Standard Deviation A x Standard Deviation B)

The Portfolio Theory calculates the overall portfolio risk based on the probability of different assets, with different risks, to oscillate in an independent fashion, or partially independent, between one another. This creates the effect listed above, when one asset oscillates upwards and the other oscillates downwards, the more assets that are different and independent a portfolio has, the lower the efficient frontier of risk will be.

The calculation for a portfolio risk isn’t simple, the more assets in a portfolio, the bigger the formula that calculates the risk, because it’s necessary to cross all assets between each other, one by one, in order to get to the final result. Underneath is the formula for a portfolio with only two assets: 

Calculating Three or More Assets

The standard deviation represents the risk. The formula above can also be written in the form of a diagram, which will then help us incorporate more assets into it. Here is this diagram:

The formula is nothing more than the square root of the sum of the squares of the diagram. Since there are two duplicate squares, since the relation of A and B are equal to the relation of B with A, then we have a part of the formula that is multiplied by two. In order to increase the number of assets, we just need to create more squares in the diagram, which will facilitate the total understanding of the thought process.

Below is the diagram for three assets:

The best portfolio is always the one with the least risk, meaning, all calculation must be to assist in the decision of which assets will be part of the safest portfolio.

Since risk is a measure made by mathematical iterations, meaning, repetitions of the formula with given variations of W, we will have the following graph after evaluating a combination of a assets:

The blue area is formed by infinite risk values of the portfolio, each dot that forms the blue area is a portfolio with a different composition of the same assets. The elliptical border is called the Efficient Frontier, because on it, are the portfolio with the best risk vs return ratio.

By analyzing just these data points, it’s evident that there is a specific portfolio combination that has less risk than the every other one. The same is marked by the red dot on the graph above and has an intermediate return.

But there is missing information on this graph. In the markets there is always a risk free asset (less than the systemic risk), like government assets. A person, business or bank may purchase national treasury assets, like Treasury Bonds, that do not pose a variable risk. In this case, these types of assets are glued to the Y-axis, of the return, because they do not pose risk.

When a risk-free asset is added into a portfolio, we have the following graph:

Where the portfolio with the least variable risk when including a risk-free asset is defined by the letter M, and our final ideal portfolio will be composed of portfolio M and the risk-free assets. In this case, we evaluate the final return in order to define what’s the weight of portfolio M inside the portfolio as a whole.

When the government pays a high interests for its assets, portfolio M must give even higher returns to compensate the risk of this investment. So in some countries, hedge-funds may choose to distribute less weight into government assets simply because the risk isn’t worth it.

Other countries where the interest of risk free assets is lower, the Stock Market ends up being the best alternative to achieve a higher return, hence why these markets are mode developed when looking at the volume of investors that are involved in it.

We hope to have explained a little bit of how large investors around the world decide on their portfolio allocation. But we just got to the good news:

With TranslateStocks, you can use our Portfolio Tool to automatically calculate the best distribution between the assets in your portfolio, in a matter of seconds. You simply add stocks to the portfolio and let our AI automatically calculate the best weight distribution, for a portfolio with the best risk vs return ratio. So no more guessing!

To wrap it up, we must also calculate the expected return of an asset or portfolio. You can see here how to calculate a portfolio’s return with the CAPM – Capital Asset Pricing Model

 
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