Risk diversification

In business administration and risk management , risk diversification (also risk diversification ) is when a uniform overall risk is broken down into several individual risks that do not correlate positively with one another and this results in a broad spread of individual risks.

General

The compound risk diversification consists of the words risk and diversification . While risk (in the sense of risk diversification) means the risk of loss that may result from the inadequate predictability ( probability of occurrence ) of future events , the aim of diversification is to expand existing monostructures . The linguistic result of the combination of both parts of the word is that a uniform overall risk is atomized by dividing it into many individual risks.

The strategy of diversification is used in many areas of the economy, for example in sales and procurement . Another example is diversified human capital : instead of founding a sole proprietorship, a company can also consist of several members, so that wrong decisions can be prevented. However, all areas of application are based on a uniform concept, namely the risk is to be reduced through diversification.

history

Naive diversification

The Anglo-Saxon proverb “ Don't put all your eggs in one basket. ", The German proverb " You shouldn't put everything on one card. " And Erasmus' warning in the 14th century " Do not entrust all your goods to a single ship. " Are references to a long-standing knowledge about the possibility of risk reduction through diversification .

Talmudic ⅓ rule

The Babylonian Talmud contains early instructions on how to divide assets into investment forms with different liquidity and different risk. In the German version by Lazarus Goldschmidt it says:

"Furthermore, R. Jiçhaq said : A person always divides his money into three parts: a third in property, a third in goods and a third in his hand."

This instruction , also known as the rule, about one and a half to two millennia old, does not go into the considerations in the background. It is occasionally carried over to the current situation as advice to invest a third in real estate, another third in stocks and a third in liquid or in the form of government bonds . Another transfer is the - rule, the investment without recourse to the characteristics of the model evenly distributed, available asset classes. Experimentally, Benartzi and Thaler found that a substantial proportion of people who invested in Defined Contribution Saving Plans implicitly made decisions according to a rule. ${\ displaystyle \ textstyle {\ frac {1} {3}}}$ ${\ displaystyle \ textstyle {\ frac {1} {N}}}$ ${\ displaystyle N}$ ${\ displaystyle \ textstyle {\ frac {1} {N}}}$

Early modern age

The basic idea of risk diversification can also be found in 1738 with Daniel Bernoulli , who mentions the transport of goods by ship as an example. The importer is aware of the fact that of the 100 ships that sailed between Amsterdam and St. Petersburg , 5 were lost.

In June 1872, the Government & Guaranteed Securities Permanent Trust set a limit of 10% per individual investment for its investments. Francis Galton published the first seminal article on correlations in December 1888, but it was only his nephew Karl Pearson who is considered the father of correlation calculations. The actuary George May of the Prudential Assurance Company presented on 1912 rules as insurance portfolios should be diversified.

Portfolio selection model from Markowitz

The starting point is that an investor can divide his capital into different, risky investments and is always faced with a trade-off between risk and return . The naive diversification does not take into account the expected return and the risk, especially the correlation of risks. In the years 1950–1970 Anglo-Saxon research examined the problem of optimal diversification, from which the classic portfolio theory developed. Harry M. Markowitz published his first ideas on this topic in 1952 in the Journal of Finance . In 1959, in his book Portfolio Selection: Efficient Diversification of Investments, he presented the so-called portfolio selection model, with which efficient portfolios can be derived. AD Roy, William F. Sharpe and James Tobin , among others , then developed the classic portfolio theory from this, which initially solved the problem of correlated risks under a few assumptions.

In contrast to the naive diversification, the classic portfolio theory is characterized by an important element: The diversification is examined with the help of the calculation of probability and statistics . The Markowitz portfolio selection model is a stochastic one-period model . The returns of the individual assets in the portfolio are correlated random variables and the risk-averse investor orients himself in the trade-off between expected return and risk when deciding on a portfolio exclusively on the expected value and the standard deviation (also volatility ) of the portfolio return. In essence, it is about the deviation of the random variable (here the return ) from its expected value . The return is calculated as the ratio between the sum of the price development and the dividend and the capital employed. Since the future is uncertain, the expected value and standard deviation are often determined by the empirical expected value and the empirical standard deviation from historical series. The higher the standard deviation, the greater the probability that the expected value will be clearly missed. Alternatively, the expected value and standard deviation can be predicted using expertise (based on research or a financial analysis ). ${\ displaystyle X}$ ${\ displaystyle \ operatorname {E} (X)}$

Markowitz showed in his model that, depending on the investor's risk tolerance, it can be more advantageous to invest in several different investments instead of investing in just one investment with the highest expected return. His theory first examined investment behavior in capital markets , especially in securities such as stocks and bonds . The study distinguished between a systematic and an unsystematic market risk . The systematic risk is that changes in the macroeconomic framework, verified by fundamental data ( e.g. interest rates , unemployment , sales crisis , recession ), can affect the entire securities portfolio. The unsystematic risk has to do with the creditworthiness of the issuer of shares and bonds and affects part of the securities portfolio. Starting from the ideal case of the optimally diversified portfolio, where the unsystematic risk is completely eliminated, an attempt is made to compare the actual portfolios with the ideal portfolio and to adapt it to it.

The model also assumes a perfect capital market . Further premises of the model are:

The market is information efficient . This means that market participants have the same information and process it the same way.
The investor acts rationally and maximizes benefits .
All market participants have market access .
There are no taxes or transaction costs .
All securities can be divided as required. Accordingly, half shares also exist.

The main finding of Markowitz is that the portfolio risk does not correspond to the average risk of its components, but is determined by the correlation coefficients of these components. In summary, it means that the diversification effect shown below is characterized by three parameters: the expected return, the spread and the correlation coefficient.

The beginnings of classic portfolio theory represented a milestone in finance theory, as the calculation of risks and returns enabled recommendations on financial investments to be made. That is why portfolio theory is also known as normative theory. But it also forms the basis for the neoclassical finance theory . Portfolio theory has also become an indispensable part of fund management and insurance.

Possible applications

In portfolio theory, the principle “don't put all your eggs in one basket” applies . Based on securities portfolios, risk diversification is also applied to portfolios such as credit portfolios , which in the best case scenario have a high degree of granularity and low cluster risks . The spread can be based on borrowers , foreign currencies , credit rating classes, sectors , regions and countries (cluster risk) or according to the loan amount (granularity). The limiting regulations for large exposures at credit institutions aim to improve granularity. The exclusion of investments with a high credit risk (speculative creditworthiness, see high-yield bond ) also helps to reduce risk, but does not help to diversify risk.

Within the market risk , risk diversification is particularly relevant for price risks (for stocks , bonds , investment certificates , currencies or types ) and their diversification. Investment companies and capital investment companies are therefore only allowed to invest money according to the principle of risk diversification (e.g. § 110 , § 214 , § 243 KAGB ), which means risk diversification. Hedge funds also proceed in a similar way , whereby risk-limiting investment regulations are also in place for hedge funds of funds according to Section 225 KAGB.

Within a manufacturing company , independent risks can be spread regionally, object-related or person-related through risk diversification:

Regional distribution occurs about by manufacture of the same product in different premises ( parallel production );
object-related diversification takes place, for example, by creating several similar production facilities ( redundancy );
Personal diversification occurs, for example, when several board members travel separately to the same travel destination.

With insurance , the possibility of risk diversification is that various risks are insured in one insurance company, which are independent of one another. The less the individual probabilities of occurrence of insured risks are positively correlated with one another, the stronger the risk compensation effect in the collective. For example, in health insurance , the illness of Mr. Meier in Wuppertal does not increase the likelihood of an illness of Mrs. Müller in Augsburg. However, things are different with a flu epidemic , which as a systematic risk can also increase the overall risk of health insurance. Spreading the risk across different types of insurance ( household , motor vehicle liability or business interruption insurance ) also reduces the risk of systematic risks and is a suitable measure for risk diversification.

effect

Risk diversification is a strategy in corporate risk management . It serves to limit risks, but does not minimize the likelihood of occurrence of the respective individual risk, but has a reducing effect on the extent of damage . In all cases, a synchronous occurrence of all individual risks is very unlikely due to the risk diversification carried out, because this diversification improves the probability distribution . The systematic risk cannot be eliminated, while the unsystematic risk can be diversified away in the case of negatively correlating individual risks. Risk diversification makes the risk of the overall portfolio smaller than the weighted sum of the individual risks in this portfolio.

Diversification effect

Contrary to the widespread belief that diversification entails foregoing opportunities, portfolio theory illustrates that free diversification is possible. In this context, “free of charge” means that the diversification does not have any negative effects on earnings . The prerequisite is that it is properly diversified. If, on the other hand, an investor takes on risks that he could eliminate through diversification, he will suffer a disadvantage that is associated with an unnecessary loss of benefit.

In the following, the theoretical background of correct diversification is presented and illustrated with the help of an example. Essentially, this involves examining the portfolio risk associated with an investment in two different, risky securities .

Example: For the shares of Dresdner Bank and Volkswagen , the expected value of the return and the standard deviation were estimated. The estimate is based on a sample from April 1978 to March 1998. The returns are normally distributed.

statistical measure	Dresdner Bank	Volkswagen
Expected return (in%)	0.81	1.19
Standard deviation (in percentage points)	7.10	8.41

If a risk- taking investor had to choose between the two investments, he would choose VW shares. A risk-averse investor, on the other hand, would choose Dresdner Bank shares.

However, the question arises as to how an investor would decide if they were to hold both stocks. This is the central question to be pursued below.

In a naively diversified portfolio, the investor would split his capital equally between the two stocks. He would therefore invest 50% of his assets in VW shares and 50% in Dresdner Bank shares.

The portfolio return can be calculated using the following formula: ${\ displaystyle \ mu _ {p}}$

{\ displaystyle \ mu _ {p} = \ sum _ {i = 1} ^ {m} x_ {i} \ mu _ {i}}

With

${\ displaystyle m \,}$ : Number of securities in the portfolio
${\ displaystyle x_ {i} \,}$ : Share of security i in the portfolio
${\ displaystyle \ mu _ {i} \,}$ : expected return on the security i.

The sum of the securities shares in the portfolio must be 1, as a formula:

{\ displaystyle \ sum _ {i = 1} ^ {m} x_ {i} = 1 \,}

.

The expected value of the portfolio return is thus calculated as the weighted sum of the expected values of the individual investments.

For a two-system case, the following applies accordingly:

{\ displaystyle \ mu _ {p} \, = x_ {1} \ mu _ {1} + x_ {2} \ mu _ {2}}

.

The portfolio risk is calculated from the sum of the weighted individual risks. In addition, the stochastic relationship between the returns, the correlation coefficient, must be taken into account. As mentioned above, this was a central finding of portfolio theory.

For the two-plant case, the portfolio risk can be calculated using this formula: ${\ displaystyle \ sigma _ {p} \,}$

{\ displaystyle \ sigma _ {p} = {\ sqrt {{x_ {1}} ^ {2} {\ sigma _ {1}} ^ {2} + {x_ {2}} ^ {2} {\ sigma _ {2}} ^ {2} + 2x_ {1} x_ {2} \ sigma _ {1} \ sigma _ {2} \ rho _ {{1} {2}}}}}

With

${\ displaystyle x_ {1} \,}$ : Share of security 1 in the portfolio
${\ displaystyle x_ {2} \,}$ : Share of security 2 in the portfolio
${\ displaystyle \ sigma _ {1} \,}$ : Standard deviation of the security 1
${\ displaystyle \ sigma _ {2} \,}$ : Standard deviation of the security 2
${\ displaystyle \ rho _ {{1} {2}} \,}$ : Correlation coefficient of securities 1 and 2.

The mutual dependency of the returns is measured with the correlation coefficient . Correlation coefficients are always in the interval from −1 to +1. The value +1 means that the returns are completely in the same direction. If, on the other hand, the returns develop in perfectly opposite directions, the correlation coefficient is −1. If the correlation coefficient is 0, the returns are uncorrelated so that there is no systematic relationship between them. If the correlation coefficient assumes a value that is smaller than +1, then the volatility of the portfolio falls below the arithmetic mean of the risks of the portfolio components. The reduction in volatility and thus the minimization of risk is known as the diversification effect. This is different depending on the correlation coefficient. Is diversification carried out in a planned and targeted manner (in the case of asset allocation ), d. H. not naive, the right choice of mixing ratio can almost completely eliminate the risk. This will be the case when the returns move in perfectly opposite directions. ${\ displaystyle \ rho}$

Portfolio possibility curves

The diversification effect can be illustrated with so-called portfolio possibility curves. To do this, portfolios are first entered in the risk-return diagram , which consist of 100% equity. The portfolio risk is plotted on the abscissa axis in the risk-return diagram and the expected portfolio return on the ordinate axis . Point A in the graphic "Effect of the correlation coefficient: portfolio possibility curves " represents the earnings value and the risk of a portfolio that arise when the capital market participant invests his available fixed assets in just one security. The same applies to point B. In the next step, the portfolio return and the portfolio risk are calculated for different mixing ratios and transferred to the risk-return diagram. Depending on which value the correlation coefficient assumes, a portfolio possibility curve of varying strength is created. This is also referred to in the literature as the opportunity line or portfolio line. ${\ displaystyle \ mu _ {A} \,}$ ${\ displaystyle \ sigma _ {A} \,}$ ${\ displaystyle \ rho _ {AB} \,}$

Effect of the correlation coefficient: portfolio possibility curves

In the case of investments whose returns have a correlation coefficient of −1, the portfolio is risk-free if the mix ratio is optimal , as the negative income from one investment is fully offset by the positive income from the other. The return can therefore be regarded as safe. In this case, the portfolio possibility curve is pointed. Point C in the graphic illustrates the resulting maximum diversification effect.
If the returns of the securities are perfectly positively correlated ( = +1), there is no diversification effect, since all possible return-risk combinations of the portfolio lie on the straight line connecting points A and B: If the investor expects a higher return, this increases Risk compulsorily. In this case, the portfolio risk corresponds to the average risk of the investments. The portfolio possibility curves presented represent theoretical special cases. However, the portfolio possibility curve is often hyperbolic . ${\ displaystyle \ rho _ {AB} \,}$

For the portfolio, which consists of VW and Dresdner Bank shares, the history series gives a correlation coefficient of 0.4974. This allows you to calculate different positions in the risk-return diagram by varying the mixing ratio. The following table shows the results of the calculation for some mixing ratios by way of example.

Portfolios of Dresdner Bank	Return ${\ displaystyle \ mu \,}$	risk ${\ displaystyle \ sigma \,}$
0%	1.19	8.41
10%	1.15	7.94
25%	1.10	7.35
50%	1.00	6.72
65%	0.95	6.59
75%	0.91	6.63
90%	0.85	6.85
100%	0.81	7.10

If these values are transferred to a risk-return diagram, the result is a hyperbolic curve that opens to the right and contains all combinations of securities and the associated portfolio risks and the expected portfolio returns.

The hyperbolic shape of the portfolio possibility curve is the rule. Thus, the returns develop basically in the same direction. This is because there are factors that affect all securities. Only the intensity of the influence is different. These factors include, for example, inflation and a severe recession , which has led to a collapse in demand for most companies. This means that even with a diversified portfolio there is always a certain residual risk.
Each hyperbolic portfolio possibility curve has a characteristic point, the vertex. In the graphic "Effect of the correlation coefficient: portfolio possibility curves" this is the point M marked in red.

In the sample portfolio, this point represents a portfolio that consists of 65% Dresdner Bank shares and 35% VW shares.

With this mix ratio, the portfolio risk, measured as volatility, is lowest. One speaks of the global minimum variance portfolio, or the safety first portfolio. The closer the correlation coefficient approaches −1, the stronger the curvature of the portfolio possibility curve and the stronger the extent of the diversification effect. The more contrary the returns are, the more risk can be eliminated by mixing the investments. The graph shows that with certain mix ratios, the portfolio risk even falls below the lowest risk of the two securities, in this case below the risk of security A.

So far, the diversification effect for the mixture of two investments has been shown. If an investor chooses more than two investments, it can be shown arithmetically that complete diversification can be achieved. However, there always remains some risk that cannot be eliminated. This risk is called systematic risk. The risk that is eliminated by diversification is unsystematic risk. In practice, it is eliminated when the number of securities is around 15.

Efficient portfolios

As noted above, the portfolio possibilities curve for VW and Dresdner Bank shares is typically hyperbolic. Point M represents the portfolio with the lowest risk. However, there are other efficient portfolios. These are on the upper branch of the hyperbolic curve and thus above point M. These portfolios are referred to as minimum variance portfolios. The part of the curve on which these portfolios lie is called the efficiency line. Only if an investor decides on a portfolio that is on this efficiency line will he properly diversify.

The lower hyperload also consists of minimum variance portfolios. However, these portfolios are inefficient compared to the portfolios on the upper hyperload. This becomes clear when, for example, portfolios D and D 'are compared with one another in the graphic "Effect of the correlation coefficient: portfolio possibility curves": Although both portfolios have an identical risk, portfolio D' is characterized by a higher earnings value. Portfolio D 'thus dominates Portfolio D. Efficient portfolios are found with the aid of solution algorithms. This can be done manually using a certain optimization approach, but it is very time-consuming even with a small number of securities.

Optimal portfolios

All efficient portfolios are on the upper hyperload of the portfolio possibility curve. But not all efficient portfolios are also optimal for an investor. The selection of the portfolio depends on the investor's individual risk attitude. The risk attitude is represented with the help of risk benefit functions . These assign a certain utility value to each risk / return combination . With the addition of the risk benefit function, only the portfolio is selected that provides the investor with the maximum individual benefit.

Every investor has their own attitude towards risk, so each risk benefit function looks different. However, all risk utility functions have the following common connection between the benefits , the rate of return and risk : . The selection of an optimal portfolio is to be illustrated using an example. ${\ displaystyle U}$ ${\ displaystyle \ mu}$ ${\ displaystyle \ sigma}$ ${\ displaystyle U = \ operatorname {h} (\ mu {,} \ sigma)}$

Example: There are two portfolios to choose from with the following risk / return combinations:

	Return ${\ displaystyle \ mu \,}$	risk ${\ displaystyle \ sigma \,}$
Portfolio 1	6.0%	4.5
Portfolio 2	7.5%	9

The attitude to risk an investor is presented with these risk utility function . Which portfolio should the investor choose?

{\ displaystyle U = \ mu -0.02 (\ sigma ^ {2} + \ mu ^ {2}) \,}

First, the utility values must be calculated and then compared with one another: The investor should choose portfolio 1, as this gives him the greatest benefit ( ).

{\ displaystyle U_ {1}> U_ {2} \,}

The risk benefit function can be represented in a simplified two-dimensional manner. It is then referred to as the indifference line , ison utility curve or also as utility indifference curve . All risk-return combinations that lie on an indifference curve create the same benefit, hence the name.

Finding an optimal portfolio

The full preference structure of an investor is the family of curves shown indifference curves. The graphic "Finding an optimal portfolio" shows three indifference curves as examples. It always: . This means that the indifference curve that is furthest from the origin has the highest utility. ${\ displaystyle U_ {3}> U_ {2}> U_ {1} \,}$

The portfolio theory assumes a risk-averse investor. With this risk setting, the indifference curves have a convex shape. To find the optimal portfolio, the indifference curve must be determined that is furthest away from the origin and at the same time touches the efficiency line. This procedure is shown in the graphic "Finding an optimal portfolio". The point with the coordinates represents the portfolio that the investor should choose based on its risk benefit function, since this portfolio promises the greatest benefit. ${\ displaystyle (\ sigma _ {opt} | \ mu _ {opt}) \,}$

The investor's risk attitude is determined in the context of investment advice with the help of questionnaires. In these, questions regarding certain hypothetical decision-making situations must be answered, the evaluation of which reflects the individual willingness to take risks.

Industry diversification and international diversification

The appearance and position of the portfolio possibilities curve and thus also the efficiency line depend on the number of risky investments in the portfolio. The greater the number of investments in the portfolio, the further up the portfolio possibility curve is in the risk-return diagram, the more favorable the portfolio is in terms of expected return and risk. In order to realize the risk reduction potential, the portfolio can be enlarged in that the investor not only includes investments from his country but also foreign investments in his portfolio. If the investor's portfolio consists only of domestic investments, he reduces the scope of the diversification effect. This means that by adding foreign investments he would have achieved a more favorable risk-return ratio. Solnik was the first to deal with this so-called international diversification, which is also referred to in the literature as country diversification.

Investments in a country develop homogeneously, as the political framework affects them to a similar extent. These are, for example, monetary , tax and fiscal policy . However, there are differences between countries in this regard, so that plants from different countries have a low correlation coefficient. Thus, with an international diversification, a more pronounced diversification effect is usually achieved.

With international diversification, there are also so-called currency risks in addition to the specific risks from the respective investments . These risks can be through the Secure ( english hedging ), for example by means of currency futures , off. The portfolio can be expanded to include these hedging instruments, so that as a consequence the relationship between risk and return of a portfolio is additionally improved.

However, equity diversification does not necessarily mean that you have to invest in foreign stocks. As long as the price development is heterogeneous, a diversification effect is possible. This can already be the case when investing in domestic stocks from various industries.

Example : If an investor wants to invest in the chemical, biotechnology and pharmaceutical industries and in the process buys shares in Pfizer Inc., Hoechst AG and Novartis AG, the diversification effect is likely to be weak, even though he has chosen investments from different countries. This is because it is only one industry and stocks in one industry face similar risks. The investor would probably have achieved a better diversification effect if he had opted for shares from his country, but shares from different industries, such as BASF , Volkswagen and Siemens shares.

Many empirical studies have investigated the question of whether international diversification or branch diversification is more beneficial. Roll's essay served as a starting point. From this article it emerges that country diversification is in principle always branch diversification, since each country focuses on a specific industrial structure. The correlation between the investments in individual sectors is low, so that a well-developed diversification effect can be achieved.

literature

Peter Albrecht, Raimond Maurer: Investment and Risk Management . Schäffer-Poeschel Verlag, Stuttgart 2008, ISBN 978-3-7910-2827-9 .
Martin Bösch: Finance - Investment, Financing, Financial Markets and Control . Verlag Franz Vahlen, Munich 2009, ISBN 978-3-8006-3634-1 .
Günter Franke, Herbert Hax: Finance of the company and capital market . Springer-Verlag, Berlin 2004, ISBN 3-540-40644-1 .
John C. Hull: Risk Management and Financial Institutions . Prentice Hall, 2007, ISBN 0-13-239790-0 .
Neil Doherty: Integrated Risk Management: Techniques and Strategies for Managing Corporate Risk . McGraw-Hill, New York 2010, ISBN 978-0-13-800617-4 .
Söhnke M. Bartram, Gunter Defey: International Portfolio Investment: Theory, Evidence, and Institutional Framework (= Financial markets, institutions & instruments . Volume 10 , no. 3 ). August 2001, p. 85-155 .
Meir Statman: How Many Stocks Make a Diversified Portfolio? In: The Journal of Financial and Quantitative Analysis. Volume 22, No. 3, 1987, pp. 353-363, doi: 10.2307 / 2330969 .

Web links

Individual evidence

↑ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l ^m ⁿ ^o ^p ^q ^r Hendrik Garz, Stefan Günther, Cyrus Moriabadi: Portfolio Management - Theory and Application. 2nd Edition. Bankakademie e. V., Frankfurt am Main 1998, ISBN 3-933165-09-1 , pp. 17 f., 34–41, 47 f., 58, 139–141.

↑ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l ^m ⁿ ^o Klaus Spremann: Portfoliomanagement. 2nd Edition. Oldenbourg Wissenschaftsverlag, Munich 2003, ISBN 3-486-27269-1 , pp. 22-26, 184-186, 191-193, 201-204, 302 f.

↑ ^a ^b Investment bankert-finanz.de, accessed on: November 27, 2010.

↑ The Babylonian Talmud. After the first censorship-free edition, taking into account the more recent editions and the handwritten material, translated into German by Lazarus Goldschmidt, Berlin 1929–1936 , Volume VII, p. 575.

^ Shlomo Benartzi, Richard H. Thaler: Naive Diversification Strategies in Defined Contribution Saving Plans. In: American Economic Review. Los Angeles 2000.

^ Daniel Bernoulli: Exposition of a new Theory on the Measurement of Risk. 1738, p. 30 (in the Latin original: Specimen Theoriae Novae de Mensura Sortis - Commentarii Academiae Scientiarum Imperialis Petropolitanae).

^ Francis Galton: Co-relations and their Measurement. In: The Royal Society. December 20, 1888.

^ Karl Pearson: The Life, Letters and Labors of Francis Galton. Volume III, 1930.

↑ George May: The Investment of Live Insurance Funds. In: The Institute of Actuaries. 1912, pp. 136 and 151 f.

^ Harry M. Markowitz: Portfolio Selection. In: Journal of Finance. 1/1952, pp. 77-91.

^ Martin Bösch: Finanzwirtschaft - Investment, Financing, Financial Markets and Control. Verlag Franz Vahlen, Munich 2009, ISBN 978-3-8006-3634-1 , p. 59.

↑ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l ^m ⁿ ^o ^p ^q Horst Gräfer, Bettina Schiller, Sabrina Rösner: Financing - Fundamentals, institutions, instruments and capital market theory. 6th edition. Erich Schmidt Verlag, Berlin 2008, ISBN 978-3-503-10686-8 , pp. 249-260.

^ Neil A. Doherty, Andreas Richter: Moral Hazard, Basis Risk, and Gap Insurance. In: The Journal of Risk and Insurance. Volume 69, No. 1, 2002, p. 10.

↑ Henner Schierenbeck : Fundamentals of business administration. 2003, p. 448 ff.

↑ Reinhold Hölscher, Ralph Elfgen (ed.): The challenge of risk management. 2002, p. 15.

↑ Peter Bofinger: Fundamentals of Economics. 2011, p. 241 f. ( books.google.de )

^ Rudolf Volkart: Corporate Finance. Fundamentals of finance and investment . 2nd Edition. Versus Verlag, Zurich 2006, ISBN 3-03909-046-1 , p. 1249 .

↑ Joachim Süchting : Financial Management - Theory and Policy of Corporate Finance. 1995, p. 378 ff.

↑ Marco Kern, Iva Kroschel, Arno Peppmeier: Return and risk in the portfolio context . In: Opportunities and Risk Aspects of Real Estate Investment Banking. June 2002, p. 43.

↑ Thomas Hartmann-Wendels , Andreas Pfingsten, Martin Weber: Bankbetriebslehre. 1998, p. 339.

↑ Robert S. Pindyck, Daniel L. Rubinfeld: Microeconomics. 6th edition. Pearson Studium, 2005, ISBN 3-8273-7164-3 , p. 231.

↑ ^a ^b ^c Söhnke M. Bartram, Gunter Defey: International Portfolio Investment: Theory, Evidence, and Institutional Framework (= Financial markets, institutions & instruments. Volume 10, No. 3). August 2001, p. 101 f.

[Garz-1] ↑ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l ^m ⁿ ^o ^p ^q ^r Hendrik Garz, Stefan Günther, Cyrus Moriabadi: Portfolio Management - Theory and Application. 2nd Edition. Bankakademie e. V., Frankfurt am Main 1998, ISBN 3-933165-09-1 , pp. 17 f., 34–41, 47 f., 58, 139–141.

[Spremann-2] ↑ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l ^m ⁿ ^o Klaus Spremann: Portfoliomanagement. 2nd Edition. Oldenbourg Wissenschaftsverlag, Munich 2003, ISBN 3-486-27269-1 , pp. 22-26, 184-186, 191-193, 201-204, 302 f.

[benkert-3] Investment bankert-finanz.de, accessed on: November 27, 2010.

[4] The Babylonian Talmud. After the first censorship-free edition, taking into account the more recent editions and the handwritten material, translated into German by Lazarus Goldschmidt, Berlin 1929–1936 , Volume VII, p. 575.

[5] Shlomo Benartzi, Richard H. Thaler: Naive Diversification Strategies in Defined Contribution Saving Plans. In: American Economic Review. Los Angeles 2000.

[6] Daniel Bernoulli: Exposition of a new Theory on the Measurement of Risk. 1738, p. 30 (in the Latin original: Specimen Theoriae Novae de Mensura Sortis - Commentarii Academiae Scientiarum Imperialis Petropolitanae).

[7] Francis Galton: Co-relations and their Measurement. In: The Royal Society. December 20, 1888.

[8] Karl Pearson: The Life, Letters and Labors of Francis Galton. Volume III, 1930.

[9] George May: The Investment of Live Insurance Funds. In: The Institute of Actuaries. 1912, pp. 136 and 151 f.

[10] Harry M. Markowitz: Portfolio Selection. In: Journal of Finance. 1/1952, pp. 77-91.

[Bösch-11] Martin Bösch: Finanzwirtschaft - Investment, Financing, Financial Markets and Control. Verlag Franz Vahlen, Munich 2009, ISBN 978-3-8006-3634-1 , p. 59.

[Gräfer-12] ↑ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l ^m ⁿ ^o ^p ^q Horst Gräfer, Bettina Schiller, Sabrina Rösner: Financing - Fundamentals, institutions, instruments and capital market theory. 6th edition. Erich Schmidt Verlag, Berlin 2008, ISBN 978-3-503-10686-8 , pp. 249-260.

[Doherty-13] Neil A. Doherty, Andreas Richter: Moral Hazard, Basis Risk, and Gap Insurance. In: The Journal of Risk and Insurance. Volume 69, No. 1, 2002, p. 10.

[14] Henner Schierenbeck : Fundamentals of business administration. 2003, p. 448 ff.

[15] Reinhold Hölscher, Ralph Elfgen (ed.): The challenge of risk management. 2002, p. 15.

[16] Peter Bofinger: Fundamentals of Economics. 2011, p. 241 f. ( books.google.de )

[17] Rudolf Volkart: Corporate Finance. Fundamentals of finance and investment . 2nd Edition. Versus Verlag, Zurich 2006, ISBN 3-03909-046-1 , p. 1249 .

[18] Joachim Süchting : Financial Management - Theory and Policy of Corporate Finance. 1995, p. 378 ff.