Modern Portfolio Theory Overview
This article describes the Modern Portfolio Theory, the theory about how to invest a capital with maximal expected return and minimal risk. The investor who behave that way (among many possibilities, he or she chooses portfolio with maximal expected return and minimal risk) is called a rational investor. The procedure of selecting such a portfolio is called portfolio optimization. The method is based on the paper of H. Markowitz from 1952 and a 1959 book.
Instead of investing the whole capital into one financial instrument (those with maximal expected return and minimal risk) the theory suggest capital diversification i.e. investing into a portfolio with maximal return and minimal risk. Expected return is modeled by history average while risk (in economy often called volatility) is modeled by standard deviation.
Although the theory seems intuitively reasonable and it is widespread in practice, nowadays we are aware of disadvantages. Namely, the problem is that it is very imprecise approximation to use standard deviation as a measure of risk. It would holds if the random variable is normally distributed (so cold 'mild' randomness). However, better understanding of market parameters tell us that this is not the case on the market; instead we deal with so cold 'wild' randomness (standard deviation is not constant but changes all the time). Having this in mind, nowadays, 'classical' might will be more suitable term for this theory.
A nice theory, but what about the application
Although there are many phenomena that are successfully modeled by means of simplifying the facts, this is not a case with Modern Portfolio Theory. The fact that standard deviation is not constant within market parameters is neglected in the model. However, there are convincing arguments that this simplification does not lead to the good results. Recently, this obstacle is commented by Andrew G Haldane in his famous speech “The dog and the frisbee”. The talk is given at the Federal Reserve Bank of Kansas City's 36^{th} economic policy symposium, 31 August 2012.
What method you (primarily) use when investing?
Portfolio allocation problem
In today's market one can trade not only traditional financial instruments such as stocks and bonds, and commodities, but also a number of other instruments including derivatives. Having a certain amount of money, an investor face a question how to investt it i.e. which instrumentss to buy. Although investment goals can be very different, we can say that two basic criteria in capital allocation are safety (risk) and expected yield. However, this is general rule and there is still many possible combinations of securities to choose in a portfolio.
Nowadays much is said and written about technical analysis – the discipline dealing with the forecasting of future price movements using mathematical models based on historical data. Although technical analysis and other types of models can achieve limited success, we are still far away of a knowledge how the market behave. Modern portfolio theory, that is based on the work of Harry Markowitz, an American economist was a good approach towards this goal. However, nowadays disadvantages of the modern portfolio theory are obvious: it is suitable for 'mild' randomness, not for 'wild' randomness where market parameters belong to.
The rational investor
It is reasonable to assume that an investor, when forming a portfolio, is manage by the rule 'to achieve maximum yield with minimum risk'. Modern Portfolio Theory calls such an investor the rational investor. But the question is whether this principle should be applied on the level of a single financial instruments or level of the entire portfolio. In the first case, it may be that the investor among all securities available in the market pick just one  those one with the highest expected return and minimal risk. Modern Portfolio Theory says that it is better to form a divesified portfolio as it is formalized in Markowitz's work. Besides suggesting the optimization level of the entire portfolio, this work introduces a quantitative measure of risk or yield.
Furthermore, the investor in selecting the portfolio has a multitude of possibilities for the combination of risk and expected return. For example, an investor who is not prone to risk, may want to invest only in the riskfree instruments. Then it is reasonable, between different instruments (let's say that those are deposits in banks up to the amount under the state guarantee), to choose those that carry the highest yield. Thus, for a given risk a rational investor will choose a portfolio with maximal return.
On the other hand, suppose that the investor wants to realize some return. Then it is reasonable to choose a portfolio with minimal risk out of many possible portfolios. After consideration this two cases, we can precisely define the term 'rational investor'. The rational investor is the investor who invests in a portfolio on the efficient frontier (the term efficient frontier will be explain latter).
Observation
 Company A
 Company A


1
 0.12
 0.09

2
 0.11
 0.10

3
 0.12
 0.09

4
 0.10
 0.10

5
 0.10
 0.11

6
 0.09
 0.12

7
 0.08
 0.13

8
 0.09
 0.14

9
 0.07
 0.16

10
 0.07
 0.16

Average
 0.095
 0.12

Stand.dev.
 0.0184
 0.0267

Corr.coef.
 0.9506

Author about his theory
Markowitz's model
Capital that the investor will receive in the future as the yield of securities, in general is not precisely known in advance. Even within financial instruments for which the yield is precisely known in advance (bonds, deposits ...), the future value of this amount is not known  because other market variables, such as inflation rates, interest rates, etc. changes. For example, although the bond coupon (interest rate) is fixed, much more relevant information is yield to maturity, which tell us about the difference between coupon bonds and current interest rates on the market. If market interest rates fall, the bond will receive the value (because each investor willing to pay more for a bond that carries e.g. 10% yield, at a time when new bonds are issued with a yield of only 8%); and vice verse.
Thus, the returns that the investor will realize at the end of a specified period of time, usually is not precisely known, but has the characteristics of randomness. Therefore, we can talk about the expected yield, the parameter this is a random variable in a mathematical sense. Accordingly, the terms of yield and risk can be reduced to statistical characteristics of random variables  the expected value and standard deviation. We can also calculate the correlation coefficient between two random variables. Such assignment of measurable size risk and yield illustrates the following example (Table 1). Note that in this point is the crucial obstacle of the theory. Namely, it is too rough to model expected return on the market with Gaussian random variables.
Table 1 shows the performance of the stock of the two companies, in the recent period. At each observation was recorded percentage of changes instead of explicit current value. So, for example, after a second observation of the share price, the price was 11% higher than that recorded at the beginning of the observation. Data shows that in a given period, A company's shares, in average rose 9.5%, while the B company's shares rose in average of 12%. These averages, according to Markovitz model, representing the expected return over a future period (based on a hypothesis 'future is similar to the past').
Furthermore, it is visible that share prices of the company A are more stable i.e. have lower standard deviation than those of company B. As we said, risk of an individual shock will be measured by the related standard deviation (in economy it is often called volatility). Also, it is evident that the prices are very negatively correlated (simplified explanation may be that the companies belong to the 'opposite' sectors: for example, company A produces ice creams whereas firm B produces umbrellas, and is currently the rainy season).
Definition. Having N securities, let r_{i} be expected return and σ_{i} standard deviation of a particular security, and let μ_{ij} be the correlation coefficient between history prices of securities i and j; i,j=1,2,...,N. We need to find those one combination of assets (in a particular security) x_{1},x_{2},...,x_{N} for which it holds:
Thus, for the expected return P, the assets combination x_{1},x_{2},...,x_{N} is required by means that the total portfolio risk V will be minimal.
Efficient frontier
Thus, according to the Markowitz's model, an optimal portfolio can be defined on two equivalent ways:
 For a given level of volatility, all portfolios with this volatility are considered. Among these portfolios, the optimal portfolio is one with maximal expected return.
 For a given expected return, all related portfolios are considered. Among these portfolios, the optimal portfolio is one with minimal volatility.
If some of these definitions is applied for every possible amount of risk and expected return, we receive a set of optimal portfolios. The first definition gives the optimal portfolios for all possible amount of risk, and the second gives the optimal portfolios for all possible amounts of the expected yield. These two sets of optimal portfolios are equal, since the definitions are equivalent.
The set of all optimal portfolios is called efficient frontier. The figure above shows the possible combinations of expected return and standard deviation of a portfolio consisting of multiple assets. Each point in the coordinate system represents a portfolio. As the image suggests, for the given assets of which the portfolio may consists, there are possible only those combinations of risk and return that are bounded by efficient frontier curve.
Thus, for example, points A and B have the same expected return, where point A has smaller risk. In the same time, point A is the point that has the lowest possible risk with the given expected return  since it is on the efficient frontier. Analogously, the points A and C have the same risk, wherein point A has a higher expected return  and the highest possible for this risk since it lies on the efficient frontier.
Summary of the Modern Portfolio Theory
Two assets portfolio
Portfolio that may be formed from only a single asset we call trivial. Furthermore, the simplest possible case where we can talk about portfolio optimization is a portfolio with two possible assets. We will now analyze the optimization of this portfolio on a concrete example. We will use data that are presented in Table 1. Further data needed for the calculation are r_{1}=0.095, r_{2}=0.12, σ_{1}=0.0184, σ_{2}=0.0267.
In case of 2asset portfolio, an investor has possibility to form a portfolio from two financial instruments, let say that these are shares of company A and company B. Expected return of company A shares is r_{1}=9.5% while for B it is r_{2}=12%. Prices of A shares are more stable than prices of B shares; related standard deviations are 0.0184 and 0.0267, respectively. Let consider now how expected return and standard deviation of the whole portfolio depend on capital share in company A and company B. According to the previous equation, it holds x_{1}+x_{2}=1. Such review will be carried out for three assumed, different correlation coefficients, for the purpose to gain an insight into the solution space. The solution space here is the coordinate system: standard deviation  the expected yield.
In the first case, we assume that prices are very poorly correlated (say we deal with an ice cream company and a furniture factory), μ_{a}=0.1. In the second case, we assume that prices are highly positively correlated (considered companies, for example, are two ice cream companies), μ_{b}=1. Finally, in the third case the correlation is strongly negative (ice cream company and the umbrella factory), μ_{c}=0.55.
x1
 x2
 Stand.dev.
 Exp.return


Coef.corr. 0.1
 
1.0
 0
 0.018
 0.0950

0.9
 0.1
 0.017
 0.0975

0.8
 0.2
 0.016
 0.1000

0.7
 0.3
 0.016
 0.1025

0.6
 0.4
 0.016
 0.1050

0.5
 0.5
 0.017
 0.1075

0.4
 0.6
 0.018
 0.1100

0.3
 0.7
 0.020
 0.1125

0.2
 0.8
 0.022
 0.1150

0.2
 0.9
 0.024
 0.1175

0
 1.0
 0.027
 0.1200

Assuming that correlation coefficient between share prices of companies A and B is 0.1, calculated standard deviation and expected return are presented in the Table 2. Obtained pairs of standard deviation and expected return form the efficient frontier (see figure).
As an illustration of the 'rational' behavior, let notice that two portfolios (90%10%) and (50%,50%) have the same risk (0.017). So, rational investor will choose portfolio (50%,50%) since it have higher expected return (0.1075).
The figure below shows efficient frontier for three chosen correlation coefficients. As one can see on the figure, when the prices are 100% positively correlated (i.e. μ_{b}=1), then the efficient frontier is a line between points (σ_{1},r_{1}) and (σ_{2},r_{2}). The smaller coefficient correlation, the more convex efficient frontier curve.
Now we have to explain how expected return and standard deviation are calculated for chosen values of x_{1} and x_{2}. The procedure will be explain on the case x_{1}=0.9, x_{2}=0.1. According to the relation (1), in matrix notation it holds the next calculation.
Variance of the portfolio is noted by V while σp is standard deviation of the portfolio. Furthermore, according to the relation (2), for the expected return of the whole portfolio R we have R=0.0975.
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