# Portfolio theory

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The portfolio theory is a branch of the capital market theory and examines the investment behavior of capital markets (eg. B. stock market ). Modern portfolio theory goes back to a work by the US economist Harry M. Markowitz from 1952. He made certain assumptions about the behavior of investors and thus obtained statements about investment behavior . His work was revolutionary at the time of its publication, and in 1990 he received the Nobel Prize for Economics for it . Later developments such as the single index model , the capital asset pricing model and the arbitrage price theory that is prevalent today are further developments of Markowitz's portfolio selection theory.

The extended question of portfolio selection, while taking account of consumers' decision gives the consumer-investment problem .

## overview

Markowitz was motivated by the following questions:

1. He wanted to scientifically justify and quantify the decision to diversify risk for investors.
2. He wanted to determine which and how many securities should be included in an optimal portfolio .

For the first time, Markowitz provided theoretical evidence of the positive effects of diversification on the risk and return of the overall portfolio. Since the risks of the individual investments are different, they are lower in the entire portfolio (see correlation ).

In order to make the best possible use of returns and risks when selecting investments in a portfolio, Markowitz developed a mathematical method to determine efficient portfolios.

## Goal setting

The aim of portfolio theory is to give instructions for the "best possible" combination of investment alternatives to create an optimal portfolio . In this optimal portfolio, the preferences of the investor with regard to risk and return as well as liquidity are taken into account. This is intended to minimize the risk of a securities portfolio without reducing the expected return. A necessary prerequisite for this is that the securities are not fully correlated .

The portfolio theory is the theoretical framework for the practice of portfolio management .

## Assumptions

Portfolio theory assumes an investor whose behavior is based solely on known financial data (e.g. stock market price , dividend , cash flow ) and who wants to increase his assets. He acts rationally and maximizing benefits ( Homo Oeconomicus ): This means that he informs himself about the realities of the capital market and makes a decision by weighing opportunities and risks against each other. He shies away from risk (one also speaks of risk aversion ). Risk-averse behavior means that a higher risk is only accepted if the expected return increases disproportionately. There has been an intense debate among experts (based on the work of Eugene Fama on information efficiency ) about the question of which information can be obtained from the observable data of the market .

To simplify the analysis, one further assumes that the capital market is perfect .

The core of portfolio theory is the distinction between systematic and unsystematic risk. All securities on the market are subject to systematic risk, so it cannot be diversified and is the risk of the investment itself. The unsystematic risk, on the other hand, is the risk that can be reduced through diversification , i.e. with an increasing number of different securities.

## Efficient portfolios

A portfolio dominates another portfolio if the expected return is greater than or equal to that of the other portfolio and the standard deviation of its value is less than that of the other portfolio, or if the expected return is greater and the standard deviation is the same. It is impossible that a portfolio with the same composition is involved. The standard deviation results from the price fluctuations (spread) and is therefore the measure of the portfolio risk . ${\ displaystyle \ mu}$ ${\ displaystyle \ sigma}$${\ displaystyle \ mu}$${\ displaystyle \ sigma}$

A portfolio is called efficient if it is not dominated by any other portfolio, i. H. if no other portfolio exists which has a lower risk or a higher return with the same expected return.

The efficiency line is the geometric location of all efficient risk-return combinations.

### Efficient portfolios of risk-free and risky securities

An optimal risk-return relationship can be illustrated using two securities. In this situation, the optimal strategy is determined depending on the risk preference of the investor.

We consider a risk-free (return:) and a risky security (return ). In addition, we want to accept the possibility of short sales (LV). In the cases dealt with, a risky security, which is subject to price and default risk (also: currency risk), is considered. The risk-free investment can be simulated by a government security. The duration must match the planning period. In this way, interest rate and default risks can be excluded for the risk-free instrument. ${\ displaystyle r}$${\ displaystyle \ mu _ {2}}$

Four cases can be distinguished:

#### 1st case: μ 2 > r without short selling

The return on the risky security is greater than the risk-free rate and there are no short sales. The only choice the investor has is the proportion of their funds that they invest in the risky paper. Then the share flows into the risk-free bond. ${\ displaystyle x \ in [0,1]}$${\ displaystyle 1-x}$

The efficient frontier is a straight line from yield-risk combinations, because the total return true ${\ displaystyle \ mu _ {x}}$

${\ displaystyle \ mu _ {x} = (1-x) r + x \ mu _ {2} = r + {\ frac {\ mu _ {2} -r} {\ sigma _ {2}}} \ cdot \ sigma _ {x}}$wherein with .${\ displaystyle \ sigma _ {x} = \ sigma _ {2} \ cdot x}$${\ displaystyle x \ in [0,1]}$

This therefore means:

• The risk-return relationship is linear.
• The factor for the portfolio risk corresponds to a standardized risk premium. This is the excess return on the risky security divided by its risk.

In this case 1 with no short sale, all portfolios are on the route given by the equation; H. all pairs , and efficient. ${\ displaystyle \ mu _ {2}> r ~}$${\ displaystyle (\ mu _ {x}, \ sigma _ {x})}$${\ displaystyle x \ in [0,1]}$

##### Derivation

Searched for is depending on the mixing ratio with . In general ${\ displaystyle (\ mu _ {x}, \ sigma _ {x})}$${\ displaystyle (1-x, x)}$${\ displaystyle 0 \ leq x \ leq 1}$

${\ displaystyle \ mu _ {x} = r (1-x) + \ mu _ {2} x = r + (\ mu _ {2} -r) x}$
${\ displaystyle \ sigma _ {x} = {\ sqrt {(1-x) ^ {2} \ sigma _ {1} ^ {2} + x ^ {2} \ sigma _ {2} ^ {2} + 2 \ sigma _ {1} \ sigma _ {2} (1-x) x \ rho _ {1,2}}}}$, where is the correlation coefficient between the yields of the papers under consideration.${\ displaystyle \ rho _ {12}}$

The special case considered here results from the fact that the first paper, the bond, is risk-free, which is mathematically expressed by, and follows from this . ${\ displaystyle \ sigma _ {1} = 0}$${\ displaystyle \ sigma _ {x} = x \ sigma _ {2}}$

#### 2nd case: μ 2 > r with short sales

The return on the risky security is greater than the risk-free rate and short sales are permitted. Mathematically, the permissibility of short sales means that the proportion of the funds invested in the risky paper is no longer restricted by the interval [0.1]. In terms of the permissibility of short sales, a distinction can be made between the two cases in which the risk-free security or the risky security is sold. ${\ displaystyle x}$

##### Short sale of the risk-free asset

The leverage effect is that when the risk-free instrument is short sold, the expected value of the portfolio increases, but so does the risk in the form of greater diversification. Short selling the risk-free bond means investing more funds in the risky paper. For the total return results with . ${\ displaystyle x> 1}$${\ displaystyle \ mu _ {x}}$${\ displaystyle \ mu _ {x} = r (1-x) + \ mu _ {2} x = \ mu _ {2} + (\ mu _ {2} -r) \ cdot (x-1)}$${\ displaystyle x> 1}$

##### Short sale of the risky asset

The short sale of risky paper means . The total return is therefore less than the required minimum return . ${\ displaystyle x <0}$${\ displaystyle \ mu _ {x} = r + (\ mu _ {2} -r) \ cdot x}$${\ displaystyle r}$

The formal process consists of borrowing a share , then selling it and investing the funds thus obtained in the risk-free paper. Lending means that the party making the stock available for loan will be reimbursed for all payments (dividends) resulting from ownership of the stock, and that the stock will be bought back in the market at the end of the term and returned to that party.

The short seller bears the same risk as a share holder and, in the present case, generates a lower return than one could get without risk. Therefore, portfolios created by short selling the risky asset are not efficient in this case. ${\ displaystyle \ mu _ {2}> r}$

#### 3rd case μ 2 < r without short selling

The return on the risky security is less than the risk-free rate and there are no short sales. For the total return applies . ${\ displaystyle \ mu _ {x} = r- (r- \ mu _ {2}) \ cdot x, \, x \ in [0,1]}$

In this case, a portfolio that only invests in the risk-free instrument is efficient because by taking an increased risk, that is, by making a choice , the return is reduced. ${\ displaystyle x> 0}$

#### 4th case μ 2 < r with short sales

Short sales are permitted: By short selling the risky instrument, i.e. by making a choice , the portfolio return can be increased at will, of course only with a simultaneous increase in the overall risk due to the short sale. ${\ displaystyle x <0}$${\ displaystyle \ mu _ {x} = r + (r- \ mu _ {2}) \ cdot (-x)}$

### Efficient portfolios of two risky stocks

The following cases can be distinguished:

• The return on the second security is greater than that of the first and the variance of the second security is greater than that of the first.

The lifting of the short selling restriction does not lead to changes in the minimum variance portfolio if the correlation assumes certain values, which result from the ratio of the standard deviations of the two stocks. This means that both securities are represented in the initial portfolio with positive proportions. ${\ displaystyle \ rho}$

#### Iso yield lines

An optimal portfolio based on this criterion is included ${\ displaystyle (x_ {1}, x_ {2}) = (0,1)}$

Budget straight

Iso yield line ${\ displaystyle \ mu _ {1}}$

Iso yield line ${\ displaystyle \ mu _ {2}}$

#### Iso-risk lines

An optimal portfolio according to this criterion is not due to the extreme points.

Budget straight without the possibility of short selling

Iso yield line ${\ displaystyle \ mu _ {1}}$

Iso yield line ${\ displaystyle \ mu _ {2}}$

• The return of the second security is greater than that of the first and the variance of the second security is less than or equal to the first.

### Analytical determination of the global portfolio with minimal variance

#### Uncorrelated Securities

With uncorrelated securities , there is always a diversification effect. The optimal mixing ratio is: ${\ displaystyle \ rho = 0}$${\ displaystyle (x_ {1} ^ {*}, x_ {2} ^ {*} = 1-x_ {1} ^ {*})}$

${\ displaystyle x_ {1} ^ {*} = {\ frac {\ sigma _ {2} ^ {2}} {\ sigma _ {1} ^ {2} + \ sigma _ {2} ^ {2}} }}$
${\ displaystyle x_ {2} ^ {*} = {\ frac {\ sigma _ {1} ^ {2}} {\ sigma _ {1} ^ {2} + \ sigma _ {2} ^ {2}} }}$.

#### Correlated Securities

Two risky securities in the form of stocks (without LV)

Risk diversification depending on the correlation coefficient : ${\ displaystyle \ rho}$

The choice of the portfolio results in the minimum variance portfolio (short: MVP ):

${\ displaystyle x_ {1} ^ {*} = {\ frac {\ sigma _ {2} ^ {2} - \ sigma _ {1} \ cdot \ sigma _ {2} \ cdot \ rho} {\ sigma _ {1} ^ {2} + \ sigma _ {2} ^ {2} -2 \ cdot \ sigma _ {1} \ cdot \ sigma _ {2} \ cdot \ rho}}}$
${\ displaystyle x_ {2} ^ {*} = {\ frac {\ sigma _ {1} ^ {2} - \ sigma _ {1} \ cdot \ sigma _ {2} \ cdot \ rho} {\ sigma _ {1} ^ {2} + \ sigma _ {2} ^ {2} -2 \ cdot \ sigma _ {1} \ cdot \ sigma _ {2} \ cdot \ rho}}}$

If the covariance is known, the formula in the first case looks like this:

${\ displaystyle x_ {1} ^ {*} = {\ frac {\ sigma _ {2} ^ {2} - \ sigma _ {12}} {\ sigma _ {1} ^ {2} + \ sigma _ { 2} ^ {2} -2 \ sigma _ {12}}}}$

### Efficient portfolios of three risky stocks

2 cases
• Global minimal variance portfolio with negative proportions:
This can be shown in a - diagram, which shows the breakdown between securities 1 and 2 (and thus implicitly on securities 3) and in a - diagram, which shows the efficiency line.${\ displaystyle x_ {1}}$${\ displaystyle x_ {2}}$${\ displaystyle \ mu}$${\ displaystyle \ sigma}$
${\ displaystyle x_ {1}}$- -Diagram: results from the rest between and . The ordinate is then the location of all mixes of securities 1 and 3 and the abscissa is the mix of securities 2 and 3.${\ displaystyle x_ {2}}$${\ displaystyle x_ {3}}$${\ displaystyle x_ {1}}$${\ displaystyle x_ {2}}$
• Global minimum variance portfolio with positive proportions:

This can be shown in a - diagram, which shows the breakdown between securities 1 and 2 (and thus implicitly on security 3) and in a - diagram, which shows the efficiency line ${\ displaystyle x_ {1}}$${\ displaystyle x_ {2}}$${\ displaystyle \ mu}$${\ displaystyle \ sigma}$

#### Derivation

There are two dependent variables. ${\ displaystyle x_ {1} + x_ {2} + x_ {3} = 1}$

${\ displaystyle \ mu _ {x} = \ mu _ {1} x_ {1} + \ mu _ {2} x_ {2} + (1-x_ {1} -x_ {2}) \ mu _ {3 }}$
${\ displaystyle \ sigma _ {x} ^ {2}}$= ${\ displaystyle x_ {1} ^ {2} \ sigma _ {1} ^ {2} + x_ {2} ^ {2} \ sigma _ {2} ^ {2} + (1-x_ {1} -x_ {2}) ^ {2} \ sigma _ {3} ^ {2}}$${\ displaystyle +2 \ sigma _ {1} \ sigma _ {2} x_ {1} x_ {2} \ rho _ {1,2} ~}$
${\ displaystyle +2 \ sigma _ {1} \ sigma _ {3} x_ {1} x_ {3} \ rho _ {1,3} ~}$ ${\ displaystyle +2 \ sigma _ {2} \ sigma _ {3} x_ {2} x_ {3} \ rho _ {2,3} ~}$

### Efficient portfolios for n securities

This can only be determined mathematically with

${\ displaystyle \ min _ {x_ {1}, \ ldots, x_ {n}} \ Sigma \ left [\ rho _ {ij} \ cdot \ sigma _ {i} \ cdot \ sigma _ {j} x_ {i } x_ {j} \ right]}$

The restrictions must be:

• Minimum return
• Budget condition
• possibly also short selling restrictions are taken into account.

### Mix of efficient portfolios

In the case of funds of funds, for example, the question arises as to whether a mix of efficient portfolios results in an efficient portfolio again. This doesn't have to be true because

• in the event that short sales are not permitted, the efficiency line is broken. If you now create a portfolio from two securities on a different part of the line, this portfolio is no longer on the efficiency line.
• in the event that short sales are permitted, a short sale of an efficient portfolio may create inefficient portfolios.

## Optimal portfolio

You try to find an optimal portfolio. This depends on the investor's risk preference. In the case of optimal portfolios, the slope of the investor's indifference curve is the same as the slope of the efficiency line .

The comparative statics shows that the share of the risky security:

• is always greater than zero
• grows with the excess return
• falls as the risk of the risky security increases
• falls with increasing risk aversion of the investor.

Investors who are guided by the expected return and risk never hold a completely risk-free portfolio. This is because the investors in the - diagram have a horizontal tangent to the indifference curve at the point . ${\ displaystyle \ mu}$${\ displaystyle \ sigma}$${\ displaystyle \ sigma = 0}$

## Portfolio theory results

The most important result of portfolio theory is risk diversification : for every investor there is a so-called optimal portfolio made up of all investment options, which best reflects their risk-opportunity profile. This optimal portfolio does not depend on the investor's original assets or his immediate risk attitude. Rather, only the risk-return combinations of the stocks traded play a role. The proof of the statement goes back to James Tobin , after him this theorem is also called Tobin separation .

## criticism

• Both the assumptions and the statements are assessed critically by economic science, but the portfolio theory is still considered to be secure.
• Most forecasts only work with historical data.
• The investor preferences cannot be clearly operationalized .
• Large amounts of data are processed. For 100 securities, 100 mathematical equations would have to be solved; when looking at daily market prices over a year, around 25,000 data records would have to be taken into account. Such calculations can only be carried out by computer programs in a reasonable amount of time , and the results cannot be easily checked.
• More realistic, dynamic models that take other factors into account are difficult to understand.
• Effects that an investment could have on the course are not taken into account.
• A basic assumption of portfolio theory is that one cannot draw reliable conclusions about the future from the past and generally cannot foresee them. Still, a major factor in portfolio theory is estimates of future returns. Estimation errors when evaluating future returns have an enormous impact on mean-variance optimization and asset allocation .
• The underlying theory of efficient markets considers an idealized financial world. For example, Warren Buffett's investment successes testify to this; In his essay, The Superinvestors of Graham-and-Doddsville, he based his success on undervalued companies that by definition do not exist in an efficient market. According to William F. Sharpe , who calls Buffett's success a 6-sigma event , long-term investment successes like Buffett's are statistically possible without refuting the theory of efficient markets.

## literature

• Edwin J. Elton, Martin J. Gruber, Stephen J. Brown, William N. Goetzman: Modern Portfolio Theory and Investment Analysis. 6th edition. John Wiley & Sons, New York NY 2003, ISBN 0-471-23854-6 .
• Thorsten Hagenloch: Value Based Management and Discounted Cash Flow Approaches. A process and task-oriented introduction. Books on Demand, Norderstedt 2007, ISBN 978-3-8334-8376-9 ( series of publications by the Competence Center for Corporate Development and Consulting (KUBE eV) ).
• Kurt M. Maier: Risk Management in Real Estate and Finance. A guide for theory and practice. 2nd revised and expanded edition. Knapp, Frankfurt am Main 2004, ISBN 3-8314-0756-8 .
• Harry M. Markowitz : Portfolio Selection. In: Journal of Finance. 7, 1952, , pp. 77-91.
• Harry M. Markowitz: Portfolio Selection - The basics of the optimal portfolio selection . FinanzBook Verlag, Munich 2007, ISBN 978-3-89879-118-2 .
• Detlef Mertens: Portfolio optimization according to Markowitz. 2nd Edition. Bankakademie-Verlag, Frankfurt am Main 2006, ISBN 3-937519-09-2 ( Banking & Finance currently 16), ( Also : Vallendar, WHU Hochsch., Diss., 2004).
• Stephen A. Ross, Randolph W. Westerfield, Jeffrey Jaffe: Corporate Finance . 7th edition. McGraw-Hill Irwin, Boston MA 2005, ISBN 0-07-282920-6 ( The McGraw-Hill / Irwin series in finance, insurance, and real estate ).
• Thomas Petermann: Portfolio separation. Separation results of the modern portfolio theory (MPT). Importance and implementation in private banking. Dissertation University of St. Gallen . Difo-Druck OHG, Bamberg 1999.
• Klaus Spremann : Portfolio Management. 3rd revised and expanded edition. Oldenbourg, Munich et al. 2006, ISBN 3-486-57939-8 ( International Management and Finance ).

1. Note: If there are two stocks to choose from, it does not mean that one of them has to be efficient. Counterexample: for 2 securities with and applies .${\ displaystyle \ rho <1}$${\ displaystyle \ mu _ {1} = \ mu _ {2}}$${\ displaystyle \ sigma _ {1} = \ sigma _ {2}}$${\ displaystyle \ sigma _ {MVP} <\ sigma _ {i}}$