Portfolio theory (portfolio) and financial risk analysis

The investment process consists of two main tasks and they are:

The security and the analysis of the market, by which the risk and the expected benefit of a whole range of investment tools are assessed. The formation of an optimal portfolio of assets; This task involves determining the risk-benefit opportunity that can be found and choosing the best one.

Portfolio theory

Portfolio theory is going to give us a set of rules that prescribe the way in which portfolios with certain characteristics that are considered desirable can be built concretely. For this a method is used; the optimization by means of the Mean Variance (CMV), which indicates the characteristics that those that are efficient should have and advantages of the diversification of investments.

The risk of any proposed investment in an individual asset should not be separated from the existence of other assets. Therefore, new investments must be considered in light of their effects on the risk and return of a portfolio of assets. The financial manager's goal with respect to the company is to either maximize return at a given level of risk, or minimize risk for a given level of return. The statistical concept of correlation underlies the diversification process used to build an efficient portfolio of assets.

From an economic point of view risk aversion, framed within the context of utility theory, is closely identified with the standard deviation. The initial development of the theory of investment portfolios is based on the consideration that the behavior of the investor could be characterized by those types of utility functions for which the standard deviation provided a sufficient measure of risk.

No study of risk in finance is complete without considering risk reduction through diversified holding of investments. Today almost all investments are guaranteed within the context of a set of other investments, that is, a portfolio. If risk can be reduced to any degree by appropriate diversified holdings, then the standard deviation or variability of the returns on a single investment must be an exaggerated assertion of its current risk. Diversification is guaranteed in that the risk expectation of an entire portfolio will be less than the weighted sum of its parts. Portfolio theory shows that the rationale for diversification is correct.

Portfolio Items

The two essential basic elements to make a portfolio choice are:

Characterize the set of packages from which to choose Provide a selection criteria

The first is given by the opportunity region, while the second is provided by the indifference map that defines the utility function. It only remains to unite the elements and see how the selection criteria overlaps the region of opportunities to get to define which package is chosen among all the possible ones.

As has been seen, the portfolio selection problem is a selection problem, it is a choice problem, which handles three different reasons:

The selection criteria that give rise to preference functions are eminently economic and of a rational nature, which allows an explicit mathematical characterization. The investor is limited in his choice by restrictions that can be represented mathematically. The portfolio problem is an optimization problem that can be formulated explicitly, and for which efficient solution techniques exist.

Problems generated by uncertainty

The two facets of uncertainty are:

Subjective appraisals, judgments and valuations that depend on tastes, experiences, style, intuition, etc., but that deep down it is impossible to rationally support with logic in all its aspects.

The problems of scientific, systematic and rigorous treatment posed by subjectivity can be difficult or impossible to solve; However, the choice problem, where the selection criterion is fundamentally economic, subjective assessments can also be translated into economic terms. In general, it is possible to at least measure the cost or benefit of an erroneous or correct assessment.

The second phase of uncertainty comes from the environment or sphere within which the choice must be made, because a large number of forces operate in it outside the control of the subject who must make the choice. The investor is exposed to uncertainty regarding the prices of different assets in the markets, government actions regarding legal and fiscal requirements, regarding their liquidity needs since it is impossible to accurately predict new investment opportunities. profitable than those existing a month before, or simply the occurrence of an unforeseen misfortune that forces you to make an expense.

The portfolio selection problem is not exempt from either of the two sources of uncertainty despite its eminently economic and practical nature. Uncertainty is precisely the factor that makes it conceptually and technically difficult.

If there were no risk or uncertainty, the portfolio problem would be solved; mounting the corresponding mathematical optimization model and solving it mechanically with some numerical method, seeking to allocate more resources to the most profitable instruments, within the imposed restrictions.

The three types of risks in portfolio selection are:

Risk of loss.- not recovering the investment and that there is a decrease or loss of capital.Risk of wasting investment opportunities.- allocating resources to certain assets that are less profitable than others.Risk of liquidity.- committing resources in assets that are difficult to convert into money causing a loss at the time it becomes necessary to make an unexpected payment.

2.1. Risk and return

Concept

Risk

Risk is the probability of losing all or part of what we are investing. Basically, the source of risk is uncertainty, which comes from the fact that you cannot know exactly what will happen in the future. You cannot guess what the price of the dollar or a stock will be. Decisions are made with an expectation of profit, which may or may not be made in the future.

performance

The return is what one hopes to obtain above what is being invested in the market. There is a direct relationship between risk and return, that is, a financial asset that offers greater risk, usually has a greater implicit risk (although it is not perceived). Performance can be seen as the incentive agents must have to overcome natural risk aversion.

Profitability and risk are the 2 sides of the same coin. Risk is inherent in any investment activity, and at a higher level of expected profitability, it is inevitable to also assume a higher level of risk. There is therefore a direct relationship between profitability and risk.

As can be seen in the graph, equity instruments have a higher expected return, but also carry a higher level of risk than fixed income instruments.

Cost effectiveness

The ability to generate returns is known as profitability.

In an investment, future returns are not certain. They can be large or modest, they may not occur, and it may even mean losing your invested capital. This uncertainty is known as risk.

There is no investment without risk. But some products carry more risk than others.

The only reason to choose a risky investment over a risk-free savings alternative is the possibility of obtaining a higher return from it.

At the same risk conditions, you have to opt for the investment with the highest profitability. At the same profitability conditions, you have to choose the investment with less risk.

The higher the risk of an investment, the higher its potential return will have to be to be attractive to investors. Each investor has to decide the level of risk they are willing to take in search of higher returns. We will see this a little later.

Cost effectiveness

As we can see in this graph, investment strategies are usually classified as “conservative”, “aggressive”, “moderate”, etc., terms that refer to the level of risk assumed, and therefore the potential profitability sought.

The more risk is assumed, the more profitability must be demanded. Likewise, the more profitability you want to obtain, the more risk you have to assume.

Watch out! Risk and return go hand in hand, but accepting higher risk is no guarantee of higher returns.

2.2. The portfolio risk measure

The commonly used measure to assess total risk is the standard deviation or volatility of returns.

What does volatility tell us?

Volatility informs us about the average magnitude of profitability fluctuations around its expected value and, therefore, about the uncertainty that exists as to whether or not said return will be achieved. In other words, volatility measures whether a value rises by 50% in one day, or 10% (and when it falls the same).

Low volatility indicates that yield oscillation is low and the portfolio relatively safe, while high volatility corresponds to higher risk.

The standard deviation provides a comprehensive and intuitive measure of risk and can therefore be used to compare different investments, regardless of their heterogeneity (pears and apples, steel and high-tech).

Diversification: a tool to reduce risk

Intuition already warns us that sharing reduces risk…

We are all aware that concentrating all our funds in a single investment ("putting all your eggs in the same basket") is risky. We will obtain a high return if it evolves in our favor, but it will cause us considerable losses if things do not turn out as we expected. Therefore, we usually distribute our money in various investments. Of course, we hope that all of them will go well, but we are aware that the probability that all of them will go wrong is very low. Spreading is diversifying, and it reduces risk significantly.

… And volatility confirms it.

The standard deviation, as a measure of risk, should reflect this intuitive phenomenon. Indeed, the volatility of investing in assets taken individually is greater than that of investing in all of them together.

The expected return, on the other hand, is not reduced by diversification, as is the case with risk.

The key is in the correlation

What is the explanation for the lower risk we get by diversifying? All assets do not evolve the same: while some are going up, others may be going down or staying stable. Thus, the fluctuations of some can be compensated with those of others, giving a more stable overall result.

Correlation is an indicator of the joint evolution of two variables, whose value ranges between 1 (both evolve in parallel) and -1 (the movements of one of them are opposite to those of the other). The closer to 1 it is, the lower the risk reduction achieved through diversification. The maximum diversification is achieved with a correlation of -1, because the fluctuations of one asset would exactly offset those of the other, but in practice the correlations are not usually even negative. This does not preclude achieving a reasonable degree of risk reduction.

Example

Types of risk of an investment

Diversification, although very useful and certainly recommended to reduce the uncertainty of returns, does not work miracles: there is always a certain risk that we will not be able to get rid of even if we distribute our investment among all the available assets.

This persistence allows the total risk associated with an investment to be broken down into two elements:

unsystematic or idiosyncratic risk: it is caused by factors of each asset and can be partially or totally eliminated through diversification. systematic risk: it is associated with economic factors that affect all assets, and it cannot be diversified. For example, it was seen recently in April with the crisis of technological values. The judges condemn Microsoft, which "hits" the technology companies, which in turn, incidentally "hit" the "blue chips." Everything goes down, because the market has a bearish sentiment, they unload equities from their portfolios and include fixed income.

How are they measured?

In a preliminary way, we can quantify the systematic risk by means of the standard deviation of a suitably diversified portfolio, since adding new assets does not reduce its volatility. Idiosyncratic risk is more difficult to measure and can only be approximated by comparing the volatility of the asset in isolation with that of a diversified portfolio that contains it: the difference between the two will serve as a non-systematic risk indicator.

For example, some non-systematic risk factors of a bank's actions can be a change in the management team, the acquisition of a rival or the publication of unexpected information about a business area; systematic risk factors would include the evolution of interest rates or the exchange rate, the march of GDP or a new government.

Do all investments have the same systematic risk?

No. Although systematic risk affects all assets available to invest in an economy, it will not do so identically in all cases.

Variable Income. For equities, characteristics such as the sector in which the company that issues the securities is located, its financial structure or its international presence will determine the effect that changes in variables such as interest rates will have on the value of its shares.. The same is true for fixed income, although in this case interest rates play a predominant role, due to the very nature of these financial assets. The characteristics of the issuer (sector, credit rating) have effects that can be considered second-order in most cases.

Should idiosyncratic risk be remunerated?

The average return on an investment does not change if we consider it in isolation or within a diversified portfolio. However, we have seen that your risk has been reduced. As investors demand compensation in the form of higher returns on riskier investments, should a higher return be expected from investing in isolated securities than from investment in portfolios?

In general, no…

Since diversification is within the reach of all of us and easily eliminates idiosyncratic risk, it should not be offset by higher profitability, which is otherwise very impractical, as it would force the market to distinguish investors who only have one financial asset of which they have several.

… Unless we cannot diversify.

If there were imperfections in the market or legal restrictions on investment and we could not carry out an efficient diversification, we should obtain an additional return, destined to offset the risk

Dynamic models with finite planning horizon

They are dynamic in the sense that they are not limited to deciding on the best investment in the period considered as present, but also pose relationships for various periods in the future. Past decisions are of no interest since "what's done is done" and only serves as input to the model to plan the initial conditions that restrict portfolio choice, with only the decision regarding the current and future composition being of interest.

The planning horizon is finite, because the number of periods into the future may be relevant to determine the optimal composition of the current portfolio.

There will also be dynamic models that with the assumption of certainty will provide the optimal portfolio in each period considered.

Because the future is uncertain, the only solution of the model that interests and can be useful for making decisions is that of the first period, since it is the only one that requires an immediate decision; This characteristic allows corrective measures to be taken since the model is resolved periodically, adapting the solution to the requirements of the events as they arise.

Main elements of the models

These elements refer to the type of restrictions that operate on them, in addition to the decision criteria that are used. Identifying two types of restrictions that are:

Structural.- imposed by the mechanics of the investment process. For example, the amount of resources available for investment in a period depends on how the resources were invested in previous periods. Environmental.- They are imposed by the environment surrounding the problem. For example, legal, fiscal and institutional policy restrictions.

Types of constraints arising from the dynamic nature of the models

Intraperiod restrictions.- are those that must be respected within each period in which the planning horizon has been divided. Each period has its own set of structural and environmental constraints that must be respected.

Restrictions between periods.- They are generally stated in terms of variables that work within a single period. In addition, it is necessary to chain the variables to reflect the dependencies between one period and another; that is, how the decisions of one period influence the other periods within the planning horizon.

Decision criteria.- deterministic models usually use one of expected returns since any risk criterion involves an explicit growth of uncertainty.

The criteria are:

The expected total return on the portfolio over the planning horizon. The expected return on the portfolio in any specific period. The present value of the total expected return on the portfolio over the planning horizon.

The model is solved using various selection criteria; This has the advantage of providing a broader picture of decision alternatives, compensating somewhat for not explicitly including uncertainty in the model.

2.3. Risk impact of the portfolio of individual securities

Management Styles

Passive management. In a strict sense, passive management consists of determining the composition of a portfolio at the time of constitution, without subsequently making any decision to alter this initial composition. The term is currently used to describe a type of management by which a reference for the portfolio is chosen at the moment of its constitution and its composition is replicated at each moment of time. The benchmark is usually a representative index of a given market, and thus is referred to as "indexed funds or portfolios".

An example of passive management would be the creation of a portfolio referenced to the IBEX 35 or the Standard & Poor's 500. At the time of its constitution, the securities included in the index would be purchased in the proportions they have in it, and changes in this composition they would be exclusively due to those that occurred in the index, without the managers performing any operations in accordance with their expectations. Passive management is also the common practice of buying a series of securities and keeping them in our portfolio permanently, selling them only when our investment horizon has ended.

Active management. Active management consists in adjusting the composition of the portfolio at all times to the manager's expectations regarding the evolution of the market as a whole and of certain values, so that, if these are carried out, positive results are obtained.

This management strategy requires constant monitoring of the markets and the individual securities traded in them, as well as the use of analysis techniques (both technical analysis and fundamental analysis) and, in general, of any tool that supports the selection of securities. concrete at all times.

Passive management

The choice of this management style is fundamentally based on studies on the efficiency of markets and their consequences on portfolio returns, although there are additional reasons that justify it:

Market efficiency

If the stock markets are efficient, it will not be possible to consistently obtain higher returns than the market as a whole. Periods in which the profitability of the market portfolio is exceeded may be followed by others in which it is not reached. Since the return we should expect from a well-managed portfolio over a relatively long horizon is that of the market, pursuing different strategies will not provide us with significant benefits, but we will run the risk of not achieving even that return. That way, the least risky thing is to replicate the market portfolio from the beginning.

In favor:

In the long term, everything reverts to the average. Statistically, it is shown that much more than 50% of fund managers, in the medium term, do not beat the market.

Against:

A famous Fidelity manager outperformed the S & P500 for 11 years in a row. Minimize brokerage fees

Closely related to the previous argument, any management that involves periodically readjusting the portfolio in the face of changes in our expectations or in the evolution of the market carries expenses as a result of the commissions of financial intermediaries. If, in any case, we are not going to obtain a superior performance by performing these operations, the expense in commissions must be reduced to a minimum.

Facilitate risk taking and coverage

Since the most liquid derivative products are those that are traded on certain market benchmarks, it will be easier to hedge the risks of the portfolio if it is adjusted to the composition of the indices. Likewise, taking positions in the market can be done easily and efficiently using these same derivatives.

Active management

The only valid justification for this management style is the systematic obtaining of higher returns ("beat the market") than those of the most suitable benchmark for comparison. As there are numerous academic studies that affirm the efficiency of the markets in different modalities, there are also documented cases of professionals and investment strategies that obtain systematically higher returns than those of the references.

In addition, it is necessary to bear in mind that active management is more risky, since there is uncertainty about whether the market will be beaten or, on the contrary, its performance will not be achieved. With this perspective, obtaining superior returns only compensates this additional risk.

Why choose one style or another?

The choice between the proposed management styles will fundamentally depend on our investor profile, since it essentially consists of choosing between two alternatives with different levels of risk.

Specifically, passive management must provide us with a return very close to that of the portfolio reference (for example the IBEX), obtained by assuming a similar risk. Instead, active management should provide us with superior profitability. Hopefully the risk will also be higher, although a particularly skilled manager could achieve these returns without taking additional risks.

Can we really choose?

In general, professional managers tend to adopt a mixed management style, setting a benchmark for the portfolio and taking positions based on their expectations. In the event that there is no concrete vision of market behavior, the chosen reference is replicated. This is how an attempt is made to reconcile the positive characteristics of both styles: security and the possibility of beating the market, and the need for a professional manager is justified, which is difficult if he is limited to passively following a given index.

2.4. Relationship between profitability and risk, diversification

Risk and Diversification

What do we understand by risk?

The risk of an asset or a portfolio is the uncertainty about its future value. The greater this ignorance, the more risky the investment will present. Thus, a safe asset will be one whose future evolution is perfectly known.

Are there 100% safe assets?

In practice, no:

Even the purchase of public debt from a State rated AAA (highest rating) by the rating agencies is subject to a certain risk of default (although minimal in this case), if it is not held to maturity, losses could be incurred or obtain benefits as a result of variations in interest rates. For example, I buy a government bond for 100 pesetas. to 5.5% in January. On March 31, interest rates rose one point, to 6.5%. Nobody wants my bonds anymore, to sell them, I will have to do it below 100 pesetas. However, if I wait until the end of the bond, the 10 years, the State will give me 5.5 ptas every year, and at the end of the 10 years, my 100 ptas. Finally, since future inflation is not known, the real return on investment, that is,what goods can be bought with the benefits of it is anyone's guess. For example, if inflation is at 3%, the real return on my bond is 2.5% (5.5% - 3%), since we must discount that life has risen by 3%.

Why do we face risks when investing?

Investors are generally risk-averse, that is, we prefer certainty to uncertainty. Therefore we demand compensation in terms of profitability for the purchase of risky assets. We take risks because we expect to obtain a higher return than investing in safe assets.

How can we measure risk?

The commonly used measure to assess total risk is the standard deviation or volatility of returns.

What does volatility tell us?

Volatility informs us about the average magnitude of profitability fluctuations around its expected value and, therefore, about the uncertainty that exists as to whether or not said return will be achieved. In other words, volatility measures whether a security rises by 50% in one day, or 10% (and when it falls the same).

Low volatility indicates that yield oscillation is low and the portfolio relatively safe, while high volatility corresponds to higher risk.

Diversification: a tool to reduce risk

Intuition already warns us that sharing reduces risk…

… And volatility confirms it.

The expected return, on the other hand, is not reduced by diversification, as is the case with risk.

The key is in the correlation

Types of risk of an investment

This persistence allows the total risk associated with an investment to be broken down into two elements:

unsystematic or idiosyncratic risk: it is caused by factors of each asset and can be partially or totally eliminated through diversification. systematic risk: it is associated with economic factors that affect all assets, and it cannot be diversified. For example, it was seen recently in April with the crisis of technological values. The judges condemn Microsoft, which "hits" the technology companies, which in turn, incidentally "hit" the "blue chips." Everything goes down, because the market has a bearish sentiment, they unload equities from their portfolios and include fixed income.

How are they measured?

Do all investments have the same systematic risk?

No. Although systematic risk affects all assets available to invest in an economy, it will not do so identically in all cases.

Variable Income. For equities, characteristics such as the sector in which the company that issues the securities is located, its financial structure or its international presence will determine the effect that changes in variables such as interest rates will have on the value of its shares.. The same is true for fixed income, although in this case interest rates play a predominant role, due to the very nature of these financial assets. The characteristics of the issuer (sector, credit rating) have effects that can be considered second-order in most cases.

Should idiosyncratic risk be remunerated?

In general, no…

… Unless we cannot diversify.

If there are imperfections in the market or legal restrictions on investment and we are unable to carry out efficient diversification, we should obtain an additional return, destined to compensate for the unsystematic risk we bear. For example, if someone sold me an asset on the condition that I only had that asset.

Systematic (or market) risk and return on investment

Since idiosyncratic risk does not provide any return to the investor, the return on risky assets compensates us exclusively for taking systematic risk. How do we measure the systematic risk of individual assets?

On the other hand, not all portfolios have the same level of this risk. Depending on their composition, they will be more and less volatile. Can it be determined then what profitability corresponds to each one, according to the risk involved? We need a reference that allows us to relate the expected profitability of the investment and its systematic risk. How can we choose it?

This concept only makes sense if we add them to a diversified portfolio, since otherwise we assume their total risk, not just the systematic one.

Adding an asset to a diversified portfolio will change its systematic risk, increasing or reducing it according to its characteristics and its relationship with the other components. In reality, it is only this change in the risk of our portfolio that should matter to us, since we should only demand a return on our investment in accordance with it.

Formula

In this context, a measure of the risk that each individual asset has when added to a diversified portfolio has been developed: beta (b), which is calculated according to the following expression:

Where the subscript R indicates the profitability of a reference portfolio and cov () and var () refer to covariance and variance, respectively.

The beta of the reference portfolio is unity, so an asset with a beta of 1 will have the same systematic risk as that; an asset with a beta greater than 1 will have a higher risk; and another with beta less than 1, lower. Thus, analyzing how the risk of our portfolio will change when adding a new asset is simple: just compare its beta with that of the portfolio we already have.

However, it is necessary to unify the benchmark portfolio, so that we can homogeneously compare the betas. For this, the market portfolio is defined.

The market portfolio

The set of all investments available in an economy, called the market portfolio, can be the reference we need to relate expected return and risk, since:

has reached the maximum possible diversification, so it can be seen as a pure systematic risk indicator; its profitability will be taken as an indicator of the remuneration that risk has in that economy; comparing the level of systematic risk of the remaining investments with that of the market portfolio we will be able to estimate the returns that we should expect from them.

Formula

Although this may seem very simple, its practical application has occupied a large number of academics for more than four decades, without being able to obtain a stable and universally accepted relationship between profitability and systematic risk. Here is an expression commonly used by analysts and companies:

Where the sub-index M indicates the profitability of the market portfolio. Usually, some equity index that is considered representative is used as an approximation to the market portfolio. For example, the Spanish market portfolio could be approximated by the IBEX 35 or the General Index of the Madrid Stock Exchange.

After all this complicated exposure, perhaps the best way to understand a beta is to think about how much more risk a value has individually compared to the market. If the market beta is 1 (if my portfolio has all the IBEX 35 values), and Telepizza's is 1.5, we can say (although it is not totally correct) that Telepizza has 50% more risk than the market. For this reason, Telepizza requires a higher return than the IBEX as a whole.

Bibliography

_____________

To finish, we suggest the following couple of video-lessons, which are part of the master's degree in stock market and capital markets at the ENyd Business and Management School, in which the topics of risk - profitability and diversification are discussed, and the procedure is also taught portfolio management. A good complement to continue learning about portfolio theory and financial risk analysis. (2 videos - 1 hour and 55 minutes)