Introduction to Diversification and the Efficient Frontier

We are currently living times of high complexity: a global recession is emerging on the horizon, credit availability in the market is low, interest rates are high and there are ripples effect coming from the long tail of the COVID pandemic. Due to this scenario, it is critical to ensure that we are aware of the risks we are running when investing. But how to protect our equity against unacceptably losses resulting from the concentration of risks?

In the financial theory, we consider the hypothesis that the market is efficient most of the time, however, it tends to overreact to news or significant pieces of information about individual firms, generating some inefficient periods. This is emphasized, for example, during speculations that emerge in periods prior to elections, significantly increasing the price volatility of stocks in the Brazilian market.

Nobody that opens an account with a broker and starts investing wants to be an ordinary investor. That’s why so many people fall for the fallacy of charlatans who claim they will teach the secrets of success to raise fortunes like from Charlie Munger, Warren Buffet, Peter Lynch etc. And what will happen in reality is that a few number of people will make it to be above average investors.

This means that if the IBOV index falls, the BTC has a same tendency, however, with a weekly volatility 3 times bigger: the BTC standard deviation was 9.26% against 3.10% of the IBOV index.

How can we protect our equity against unacceptably losses resulting from the concentration of risks?

Considering the modern portfolio theory, we face the “rational investor” concept, which is the one that has the objective to maximize the estimated returns with the minimum risk (standard deviation). This concept of “rational investor” is part of the Efficient Frontier theory to be presented below.

The Efficient Frontier theory was developed in the early 50’s by the economist Harry Markowitz, that awarded him the Economy Nobel Prize in 1990, and it shows that the risk of a portfolio is not related only to the individual risks of the assets in its composition, but to how those individual risks contribute to the overall portfolio risk.

Let’s take a special 2 independent assets case, which the returns and variance of the portfolio are described as:

Considering that the standard deviations are identical, and the assets are non-correlated:

It is clear that if we choose to allocate 100% of our resources to a single asset, the return and risk (standard deviation) of this “portfolio” will be exactly the same as that of the asset itself:

To calculate the amount to be allocated in the asset 1 in order to achieve the minimum standard deviation with the highest possible return, we will use the following equation:

We can find this condition setting the first derivative of the function equal to zero:

By substituting the above result into the equation for the variance of portfolio, we calculate the portfolio risk:

Therefore, a combination of 50/50 of these two independent uncorrelated assets with identical returns and standard deviation can generate the same return “R” with a lower risk (standard deviation), once that:

The above result, although simple, is the mathematical proof of the importance of the Efficient Frontier Theory in reducing investment portfolio risk.

In practical terms, investors use the Efficient Frontier to guide their asset allocation decisions once it is possible to conciliate the expected return with their risk appetite. However, it is important to emphasize that these calculations are based on historical information that may change in future.

To address this issue, we use simulations techniques such as Monte Carlo to forecast variables.

The most important part of any investment is understanding the risks involved and being prepared for them. This is more important than estimating the returns you could achieve on your investment. Unless, of course, you enjoy living with an ulcer while watching your equity melt down.