Tensors basics of investing

The dimension of the tensor is called its rank. But this description misses the most important property of a tensor! A tensor is a mathematical entity that lives in a structure and interacts with other mathematical entities. Any rank-2 tensor can be represented as a matrix, but not every matrix is really a rank-2 tensor. In such a task it is critical to distinguish the common risks that have general impact on the returns of most securities from the idiosyncratic risks that influence securities individually.

Empirical results suggest that three latent factors, referred to as level, slope and curvature, are required to almost fully reflect the behaviour of the entire term structure…The degree of robustness in these findings has made PCA a fundamental building block for characterizing single-economy interest rate curves. This leaves such strategies unprotected to cross-country risk arising from global macroeconomic events… For this reason, a parsimonious model to describe the co-variation of interest rates at the relevant maturities and in the relevant countries appears necessary… investors to adequately identify and manage risk.

This motivates the development of multilinear techniques, which have eventually found its place in many real-world applications where tensors naturally reside…Accordingly, we have developed a framework that employs the structure-aware multilinear algebra to rigorously model the risk factors shared by an international universe of fixed income returns.

In this way, the proposed analysis: i achieves a significant reduction in the number of parameters required to fully describe the international investment universe; and ii offers a physically interpretable setting for estimating and identifying global risk factors. The resulting maturity-domain and country-domain factors are shown to provide compact and physically meaningful insight into the global macroeconomic environment. From Fig. This result is given by the fact that, even if in periods of business expansion the two strategies perform equal see periods — or — , the signal-optimized investment plan less suffers the drastic drop of stock prices during crisis phases.

Panel b reports the results for the basket of stocks belonging to the FTSE MIB and panel c presents the cumulative sum of returns for the basket of stocks belonging to the Euronext Paris stock exchange Full size image Similarly Fig. The simulations are obtained using the same parameter values of the previous case. Also for these simulations it is clear that, even if in some periods of business expansion the equally weighted portfolio strategy performs better see periods — , the signal-optimized investment plan less suffers from decreasing phases as around period Moreover after that period the cumulative returns obtained with this strategy are larger than the ones obtained with an equal portfolio investment plan.

Finally Fig. As previously illustrated, also in this case the equally weighted strategy is outdated by the signal-optimized portfolio investment plan. Since the parameter space is huge a sensitivity analysis is needed for investigating the robustness of the method against the change in some parameter values. The aim is to investigate how the results change while parameters vary and to study how these changes influence the final returns.

Figure 4 presents, in each subplot, the sensitivity of the methodology in parameter variations by indicating the cumulative returns obtained at the end of the time sample. In particular, Fig. Panel b shows the end-of-sample cumulative performance obtained by the signal-optimized portfolio strategy in forecasting the dynamic of the 59 stocks of the second dataset FTSE MIB.

Panel c encompasses the cumulative returns obtained by the signal-optimized portfolio strategy in forecasting the dynamic of the stocks of the third dataset Euronext Paris stock exchange. The cumulative returns are positive for all parameter configurations suggesting some robustness of the proposed methodology against different setting of the parameters Full size image Results indicate that the cumulative returns are positive for all parameter configurations, this fact reinforces the idea that the proposed methodology is able to correctly predict most of the movements and that it is robust against different setting of the parameters.

Moreover, for most of the parameter configurations the signal-optimized portfolio strategy outperforms the equally weighted portfolio strategy. Discussion and concluding remarks This work proposed a new dynamical approach to financial systems and stressed the systemic importance of empirical signs that can be used to extend the knowledge of financial markets and complex systems in general.

The results show that the complex approach to financial markets produces investment plans superior to simpler strategies. In this regard, I think that the superior performance of the proposed approach rests on two entangled pillars. The first pillar is the quantification of risk from an interconnectedness perspective that, per se, contains a new value-adding information.

This method avoids the losses of crucial information about the system that can be observed only by holding the original time-varying nature of the records. This approach indeed might reduce estimation error issues in the process of portfolio formation, as one focuses on the evolution of the distances and not on its average value. Furthermore, the findings obtained by the application of this methodology might have important consequences for the understanding of other financial systems like CDS markets and other derivative markets.

Indeed, as pointed out by the recent financial crisis, financial systems are increasingly built on interdependencies and relationships that are difficult to predict and control. This feature is calling for more researches and applications of complex network techniques in economics.

Future research could investigate this conjecture by applying similar methodologies to the banking system so to predict future events of crisis and the cause of the crisis by investigating also the systemic importance of each financial institution within the system. Predicting abrupt market down-turns, as a matter of fact, can facilitate the drafting of policies that can reduce the severity of financial crises, by decreasing the risk of global collapses of financial services by making economic networks more robust.

A measure of the dynamics of stock market crises.

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If you view tensors as containers, a rank 3 tensor is one that packs in an additional layer, much in the same way a matrix packs in an additional layer compared to the vector, and the vector packs in an extra layer compared to a scalar. In geometrical terms, a 3D-Tensor is a cube of numbers: Rank 3 tensors and higher differ from matrices in one very important aspect.

If you transform the entities in the structure, then the tensor obeys the transformation law. What is a Tensor Index? An index plural indices is a way to organize quantities of numbers, equations, functions and similar objects. Indices can be written as a superscript called a raised index or as a subscript called a lowered index. For example: Raised index: qi, Lowered index: qi. Index Placement Index placement—the choice of raised or lowered indices—is usually a matter of convention: Superscript indices: coordinates, vectors.

Subscript indices: covariant vectors. Matrices have two indices, which can be raised or lowered. Tensors have two or more indices, which can also be raised or lowered. For example, the following matrix can be written with index notation as Aij: There are many sub-conventions here, which are extremely important to follow. For example, the Kronecker delta forms the components of a tensor.

But switch the placement around, like and you no longer have a tensor. How Indices Work The index usually represents a series of positive integers. You could list all of the different equations: m1. In some cases, you may even be able to buy a fraction of a share, depending on the investing platform you use. This would be an even smaller portion of the company. Stock prices fluctuate with a company's fortunes and also with the economy at large. These investments can be valued and rated, depending on the underlying company's financial stability.

Some stocks pay a regular return of company profits in the form of dividends , and others do not. Investors can realize capital gains if the shares appreciate in value above what they paid for them. Note If you sell an investment for more than you paid for it, you'll be required to pay capital gains tax on the profits if it is held in a taxable account. Best For: Stocks are a great choice for investors who are aiming for higher returns, have a higher risk tolerance, and have faith in the success of companies.

Purchasing Bonds When you buy a bond, you are lending money to the company or institution that issued it. Bonds are debt securities and can be in the form of Treasuries, municipal bonds, corporate bonds, and other types of debt.

Until they pay you back, the borrower will pay you interest on a regular basis. Bonds have to be held for a period of time before they mature. However, you can resell them on the secondary market through your broker.

Best For: Bonds are best for investors who have a lower tolerance for risk and seek out less volatility in their investments. Bonds also offer consistent payments. Putting Money in Mutual Funds One of the most popular ways to own stocks and bonds is through mutual funds. Mutual funds are pooled money investments that will have a primary focus.

In fact, most people are statistically less likely to own individual investments than they are shares of companies through mutual funds held in their k or Roth IRA. Mutual funds offer many benefits to investors, particularly to beginners who are just mastering investing basics. However, mutual funds also have a few serious drawbacks: They charge fees, which can eat into your profits, and with some funds they may boost your tax bill, even in a year when you don't sell shares.

Note In most cases, there is a broker fee to buy or sell mutual fund holdings. Best For: Mutual funds are a good fit for investors who want a diverse portfolio without the hassle of managing their investments. Investing in Real Estate Yes, you can buy a home for yourself or properties to rent, or you can purchase securities such as a real estate investment trust REIT. REITs have a structure much like a mutual fund, where a professional manager handles the individual assets held within the trust's portfolio.

However, with a REIT, all of the investments are only in real estate. Best For: Real estate is best for those investors who are interested in real assets and have the experience to make the right picks. Investing in real estate without knowledge of the asset, location, and regulations could lead to headaches and a poorly performing asset.