# Markowitz model

The Markowitz model is a model whose objective is to find the optimal investment portfolio for each investor in terms of profitability and risk. This, making a suitable choice of the assets that make up said portfolio. We can affirm without fear of being wrong that the Markowitz model represented a before and after in the history of investment. Before 1952, all investors based their calculations and strategies on the idea of ​​maximizing the return on their investments. That is, when choosing whether to make an investment or not, they answered the question: Which investment generates the most profitability for me?

Of course, Harry Markowitz, a recent graduate of the University of Chicago and in the process of earning his Ph.D., realized that another question had to be answered. A question without which the first would not make sense. What risk does each investment have? Obviously, no matter how profitable an asset or a group of them can generate, if the probability of losing all our money or a large part of it is high, what sense does it make that the expected profitability is very high?

So in 1952 Markowitz published an article in the Journal of Finance titled Portfolio Selection . In it, he not only exposed the importance of taking into account profitability as well as risk, but also highlighted the reducing effect that diversification had on the latter.

## Portfolio formation theory

The theory of portfolio formation is made up of three stages:

1. Determination of the set of efficient portfolios.
2. Determination of the investor’s attitude towards risk.
3. Determine the optimal portfolio.

And it is also supported by the following starting assumptions:

1. The profitability of a portfolio is given by its mathematical or average expectation.
2. The risk of a portfolio is measured through volatility (according to the variance or standard deviation).
3. The investor always prefers the portfolio with the highest return and lowest risk . See relation profitability, risk and liquidity.

## Determination of the set of efficient portfolios

An efficient portfolio is a portfolio that offers the least risk for an expected return value. Through the following graph we will see it more clearly:

As you can see, at the efficient frontier, each portfolio minimizes risk for a given return. So to increase profitability we must necessarily increase risk.

## How do we find the efficient frontier?

The efficient frontier is found by maximizing the following mathematical problem:

Subject to the following restrictions:

• Parametric constraint

The total sum of the weights of each value in the portfolio multiplied by its covariance must be equal to the Estimated Variance of the portfolio. For each value of V * we will have a different portfolio composition.

• Budget constraint

The total sum of the weights of each value in the portfolio cannot add up to more than 1. That is, if we have 10,000 euros, we can buy at most 10,000 euros in shares, we cannot buy more than 100% of the money we have available. The sum is 1 because instead of in% we will work as much for one.

• Condition of non-negativity

We cannot short sell, so the portfolio weights cannot be negative. They will then be greater than or equal to zero.

## Determination of the investor’s attitude towards risk

The investor’s attitude towards risk will depend on his map of indifference curves. That is, a set of curves that represent the investor’s preferences. Thus, each investor will have a different aversion to risk and for each level of risk that he is willing to take, he will demand a certain return.

The higher the curve is, the more satisfaction it will bring to the investor. For the same level of risk, the upper curve will offer more returns. Likewise, any point on the same curve represents equal satisfaction according to the preferences of an investor.

## Determination of the optimal portfolio

An investor’s optimal portfolio is determined by the tangent point between one of the investor’s indifference curves and the efficient frontier. Curves that are below that point will give less satisfaction and those that are above that point are not feasible.

Being a complex and laborious mathematical problem, we will not discuss the analytical solving method. We will take advantage of technology to, through excel, solve it in a much more intuitive way. Next we will see an example:

Suppose we are hired as investment advisers for a capital management firm. The chief investment officer entrusts us with a client request. The client tells us that he only wants to invest in Repsol and Inditex. He does not want to invest in bonds, or in Telefónica, or in Santander, or in any other asset. Only at Repsol and Inditex. We, as experts in the Markowitz Model, are going to tell you, according to the evolution of these assets, what proportion you should buy of each one.

To do this, we obtain historical information data for both securities. Once this is done, we will perform the necessary calculations to obtain the graph presented above. In it we have the set of investment possibilities. For this we have solved the following table in a very simple way:

The table shows the profitability and risk that the portfolio would have depending on the proportion that we buy of each asset. Efficient portfolios are those with 50% of the weight or more in Repsol. Why? Because if we invest less in Repsol and more in Inditex, we are reducing profitability and increasing risk.

Once this calculation is done, we will go on to study the investor’s preferences. For simplicity, let’s say you’re a very risk-averse person who wants a portfolio that has as little risk as possible. Then, according to these preferences, we will go to the third stage where we will choose the optimal portfolio that will be located in the yellow dot (portfolio of minimum variance).