Portfolio Optimization Theory |
It's very easy to compile a portfolio of the given set of stocks - just buy a certain number of shares in different stocks. If you buy another number of shares in the same stocks, you'll get another portfolio. Apparently, a practically limitless number of portfolios can be compiled of the same stocks. How to select the best of them? The answer to this question is provided by the portfolio optimization theory.
Before describing this theory results, let's consider a question: Why create portfolios at all? The main reason is reduction of risk. It is absolutely clear that probability of 50% price drop in one stock (or even down to 0!) is much higher, than, for example, in five stocks at the same time. Assume that you believe that the stock price will grow 5%. You can invest in this stock and earn 5% if, of course, your forecast turns out true. And what if you were wrong, and the price dropped 10% instead? In that case, you lose 10%. One of the ways to secure yourself from possible mistake is to compile a portfolio of several stocks, so that expected portfolio yield was equal to 5%. Even if you were wrong, the prices of all stocks in your portfolio will not drop 10% at once!
In their portfolio compilation, investors were guided by similar intuitive ideas, until there appeared the portfolio optimization theory. This theory allows making maximally diversified portfolios - such portfolios, whose risk is minimal as compared to all other possible portfolios of the same stocks. As a measure of risk, Markowitz considers a standard deviation describing probability of portfolio yield deviation from expected value. The more is standard deviation, the more risky is portfolio, because probability of deviation from expected portfolio yield is higher. Therefore, any portfolio can be characterized by the two parameters - expected yield and risk. On a plane plotted within yield/risk coordinates, any portfolio could be displayed as a dot on this plane, while all possible portfolios of the given set of stocks form a cloud, as shown in fig. 1.
Figure 1. Effective border and random portfolios.
As figure shows, the multitude of all possible portfolios has a clear-cut border on the left. It means, for example, that at risk level of 6%, the given stocks are impossible to use for making a portfolio with the expected yield less than 5% and more than 9%. From the practical point of view, the upper part of the border, which is referred to as effective border, is interesting.
With risk level predetermined, no portfolio with greater expected yield, than that on effective border, exists. And on the contrary, with expected yield level predetermined, it is impossible to generate a portfolio with lesser risk level.
Besides, there exists a portfolio with minimally possible risk - a so-called minimal portfolio. In fig. 1, it is shown with a red dot. It is impossible to generate any portfolio with risk smaller (at any expected yield), than that of the minimal portfolio.
After the effective border is plotted, all you need to do is select a point reflecting your required yield and acceptable risk preferences. And we'll tell you what portfolio matches it, i.e. how many and what kind of stocks to buy.
A number of useful improvements have appeared since the moment of the classic theory creation. For example, it is possible to form portfolios with short selling possibility in a similar way. The program will determine, how many and what kind of stocks are required to be sold and how to distribute capital among the remaining ones. There are others, more sophisticated, innovations, too. They are associated with method of parameter calculation required for working - expected stock yield and covariance matrix, reflecting their interrelation degree.
We offer a full complex of services in optimal portfolio formation, executed at the highest mathematical level.
List of services within the framework of portfolio optimization theory.
The key to forming a portfolio is an effective border. We could generate an optimal portfolio by an effective border calculated with:
- Classic portfolio parameter evaluation without short selling function;
- Classic portfolio parameter evaluation with short selling function;
- Expected yield forecast;
- Covariance matrix forecast;
- Covariance matrix and expected yield forecast;
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Besides, we can optimize your current portfolio. Since market stationarity is limited, we recommend you to do it regularly.
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