Risk-reward ratio

The risk-reward ratio is in the portfolio theory of conflict , before a capital market participants , if it is capital as investors in a portfolio invested.

General

For the investor, there is a trade-off between the risk that he is willing to take with his investment and the return that he expects. In general, an increasing expected return is associated with an increasing risk; both economic variables are positively correlated with one another .

The key performance indicators of the risk-return ratio are RORAC on the one hand and RAROC on the other .

Historical classification

In the 1950s, Harry M. Markowitz was the first in history to deal with the relationship between risk and return. The portfolio selection model was born .

Building on this knowledge, William F. Sharpe , among others, developed the Capital Asset Pricing Model . These theories were used by many researchers, notably Stephen Ross, to develop the arbitrage pricing theory.

Capital market participants (especially asset managers ) benefit from the findings from the theories by reflecting on the trade-off between risk and return and analyzing it before making decisions.

Basics

Goal of an investment

The starting point for the considerations on the risk-return ratio is that an investor wants to invest his capital in a portfolio of, for example, stocks and / or bonds . Investors are interested in such an investment because they want to generate an investment return that is above the inflation rate. The investor's goal is postulated to shift consumption into the future and to use the investment income to acquire more goods in the future than at the time of the investment.

System characteristics

A plant is an investment that promises the investor a flow of money or a power flow . A power flow is, for example, housing services to the owner of a house. This power flow can be converted into a cash flow by renting out the house.

With a cash flow, explicit and implicit payments are possible. Explicit payments are dividends , for example . Implicit payments, on the other hand, are exchange rate changes, for example. The latter are capital gains or capital losses that are not realized until the system is sold, but are only realized upon sale.

A distinction is made between risky and risk-free investments ( also known as risk-free investments ) in which the investor can invest his capital. In the case of a risky investment, the flow of money and / or services to the investor is not certain, but is usually random and uncertain. For example, owners of shares do not know whether there will be a profit distribution and, if so, how much it will be. In the case of a risk-free investment, on the other hand, the flow of money and / or the power flow are secure. The owners of short-term treasury bills can therefore assume that both the coupons and the amount invested will be paid out, since the issuers in this case are public bodies and the German state has sufficient capital to cover all of them due to the tax monopoly debt to pay.

The magic triangle of an investment

Every investment is determined by three characteristics: security , liquidity and profitability . These criteria are not to be considered individually as they influence each other.

The security serves as a measure for the preservation of the invested capital. Risks, such as the creditworthiness of the borrower , but also price risks and currency risks , largely determine whether an investment is safe or less safe. Security can be increased by spreading the capital to be used. This strategy is known as risk diversification . The risk can be minimized by distributing the assets across different types of securities and investing in different countries and / or industries.

Liquidity indicates the speed with which the capital employed can be converted back into cash or bank balances . The liquidity of an asset increases as the time it takes to convert decreases.

The profitability can be derived from the yield of a plant. Possible income is dividends, interest , price increases and other distributions. The problem that arises here is that an investor will typically invest in any security, so the types of income can vary significantly. In order to be able to compare the profitability of the different systems, the key figure return is used. It represents the relationship between the return and the capital employed.

These three criteria form the magic triangle of an investment . This triangle illustrates the conflicts that can arise when investing capital in a portfolio:

First, there is a conflict between safety and profitability. The more security an investor wants, the lower the return on the investment will be. The higher the profitability, the more risk must be taken. There is consequently a trade-off between risk and return.

Second, there is a conflict between liquidity and profitability. Investments that have a higher level of liquidity usually have disadvantageous returns.

The ideal case, which would consist of high security, high liquidity and high profitability, does not exist because of the competition between these criteria.

Conflicting goals between risk and return

Relationship between expected return, relative frequency, risk and standard deviation

An investor expects a certain return, a so-called return, when he invests his capital in a security . In a simplified model assumption, this expected return is often equal to the average of all returns from the past, i.e. the empirical expected value .

The actual return on the investment reflects the realized return on the investment and is not known at the time of the investment.

The risk of an investment is that the expected return is not achieved and thus deviations - up or down - from the expected value arise. The negative deviation represents the risk for the investor. So that the investor knows what expected return and what risk he has to expect before making an investment decision, the necessary data for these values is taken from the past. The expected value of the return can be determined by using the various characteristics of the returns from the past, weighted with their relative frequencies, to determine the expected value and standard deviation . The deviations from the return expected in the future therefore correspond to the deviations observed in the past. The financial term for standard deviation is volatility . With increasing volatility, the risk increases that the actual return will not match the expected return. If an investment has no volatility at all (value zero), it is a risk-free investment.

For the sake of simplicity, the investor's choice is limited to a risk-free investment (for example in short-term treasury bills) and a high-risk investment (for example in stocks). It is therefore a model in which an investment with a standard deviation of zero is compared with another investment with a standard deviation greater than zero. This ensures that the investments are independent. Thematically, one finds oneself within the portfolio theory in an efficient portfolio that consists of risk-free and high-risk securities . The relationships can be transferred accordingly to other forms of investment.

example

An investor wants to invest € 10,000. He has the choice between short-term treasury bills and stocks.

With the risk-free investment in the short-term treasury bills, the investor is promised 4% per year. In this case, the expected return will correspond to the actual return and the capital market participant will receive a secure return of 4% per year. This means that the investor can count on € 400.
The stocks, on the other hand, show fluctuating values for the characteristic “return”. One speaks here of different forms of the characteristic “yield”. These and the associated relative frequencies with which the respective values occurred in the past can be found in the following table:

Expressions of the "yield" feature	40%	20%	15%	−15%	−20%
relative frequency	6%	25%	38%	25%	6%

The expected return on the shares is 8.15%.

In this case, the investor must decide whether he prefers a lower return with greater security or a higher return combined with less security. Investing in the safe short term Treasury bills will bring the investor a safe 4%. The investor who prefers security will always opt for this system. An investor who exhibits this risk attitude is known as a risk averse investor . If the investor is willing to accept a greater risk, then he will invest his capital in stocks and at the same time demand a higher return as compensation. The expected return will increase from 4% to 8.15%. The investor can gain a maximum of 40% from an investment in shares, an amount of € 4,000. However, in the worst case, a 20% loss can be expected. This corresponds to € 2,000. A risk- taking investor will usually opt for this investment .

Every investor has to decide for himself how much of his capital he wants to invest in the respective investment. There are basically three options:

All capital is invested in short-term treasury bills.
All capital is invested in stocks.
The capital is invested in both stocks and short-term Treasury bills.

Formal derivation of the conflict of goals

The following parameters are required for a formal representation of the trade-off between risk and return: The expected return on investment in risk-free securities (here short-term treasury bills) is represented by the variable , the expected return on the high-risk investment in shares by the variable . The actual return on the risky investment is denoted by. The expected return of the high-risk investment is higher than the expected return of the risk-free investment, as a formula: ${\ displaystyle \ operatorname {E} _ {f} \,}$ ${\ displaystyle \ operatorname {E} _ {m} \,}$ ${\ displaystyle r_ {m} \,}$

{\ displaystyle \ operatorname {E} _ {m}> \ operatorname {E} _ {f} \,}

.

If this relationship were not valid, then the risk-averse investors would only buy the risk-free securities, so that the high-risk securities would not be for sale (this is based on the assumption of a perfectly functioning market). The following mathematical relationship applies:

{\ displaystyle \ operatorname {E} _ {p} = b \ cdot \ operatorname {E} _ {m} + (1-b) \ cdot \ operatorname {E} _ {f} \,}

.

The expected return on all investments corresponds to the weighted average of the two expected returns on the investments, indicating the proportion that is invested in the risky investment. thus indicates what proportion of the total invested capital flows into the risk-free investment. For example, half of the capital could be used to buy stocks and the other half to buy short-term Treasury bills. Assuming the stocks have an expected return of 9% and the short-term Treasury bills have an expected return of 3%, the expected return on the total investment will be 6%. To assess the portfolio risk, the standard deviation of the return is considered, as this is the measure of the risk. The standard deviation of investing in high risk securities is represented by. represents the standard deviation of the entire portfolio. ${\ displaystyle \ operatorname {E} _ {p} \,}$ ${\ displaystyle b \,}$ ${\ displaystyle (1-b) \,}$ ${\ displaystyle \ sigma _ {m} \,}$ ${\ displaystyle \ sigma _ {p} \,}$

The following mathematical relationship applies:

{\ displaystyle \ sigma _ {p} = b \ cdot \ sigma _ {m} \,}

.

The standard deviation of the portfolio corresponds to the product of the proportion that is invested in the high-risk asset and the associated standard deviation. After transforming and inserting the two equations above, the following relationship results:

{\ displaystyle \ operatorname {E} _ {p} = \ operatorname {E} _ {f} + {\ frac {(\ operatorname {E} _ {m} - \ operatorname {E} _ {f})} { \ sigma _ {m}}} \ cdot \ sigma _ {p}}

.

This equation is a budget line because it represents the trade-off between the expected return ( ) and the risk ( ) of a portfolio. The straight line obtained can be drawn in a diagram. ${\ displaystyle \ operatorname {E} _ {p} \,}$ ${\ displaystyle \ sigma _ {p} \,}$

The trade-off between risk and return

${\ displaystyle {\ frac {\ operatorname {E} _ {m} - \ operatorname {E} _ {f}} {\ sigma _ {m}}} \,}$ represents the slope of the function and forms the y-axis intercept. In the graphic "The conflict of goals between risk and return" this line is drawn in black. From the straight line equation it can be deduced that the expected return of the portfolio ( ) increases when the standard deviation of the return ( ) increases. ${\ displaystyle \ operatorname {E} _ {f} \,}$ ${\ displaystyle \ operatorname {E} _ {p} \,}$ ${\ displaystyle \ sigma _ {p} \,}$

${\ displaystyle {\ frac {\ operatorname {E} _ {m} - \ operatorname {E} _ {f}} {\ sigma _ {m}}} \,}$ expresses the price of the risk and describes the additional risk that an investor will accept in order to achieve a greater expected return. If the investor does not want to take any risk with his capital, then he will spend all of his capital on short-term treasury bills. This corresponds to the first possibility mentioned above; here is and only the risk-free return ( ) will be achieved. If the investor wants a greater expected return, then he must expect a greater risk. If he chooses the second option and invests all of his capital in stocks, so is , and the expected return on the portfolio equals the expected return on the investment in stocks ( ). For this, however, the investor has to accept a high risk in the form of the standard deviation . The investor can also choose the third option and invest in stocks as well as in short-term treasury bills. Then the expected return on the portfolio would be between the risk free return ( ) and the high risk return ( ) and the standard deviation between and . ${\ displaystyle b = 0}$ ${\ displaystyle \ operatorname {E} _ {f} \,}$ ${\ displaystyle b = 1}$ ${\ displaystyle \ operatorname {E} _ {m} \,}$ ${\ displaystyle \ sigma _ {m} \,}$ ${\ displaystyle \ operatorname {E} _ {f} \,}$ ${\ displaystyle \ operatorname {E} _ {m} \,}$ ${\ displaystyle 0 \,}$ ${\ displaystyle \ sigma _ {m} \,}$

Solution of the conflict of goals

To resolve the conflict of objectives, it is necessary to determine the optimal relationship between risk and return. This requires determining how much the investor should invest in stocks and how much they should invest in short-term treasury bills in order to get the maximum benefit from the investment. For this purpose, indifference curves are drawn in the diagram. Each indifference curve describes a certain benefit that capital market participants can or will achieve for different combinations of return and risk. Every curve is rising because risk is generally undesirable. The greater the risk, the higher the expected return must be to compensate for the higher risk incurred.

The indifference curve in the graph "The trade-off between risk and return" represents the greatest possible benefit, the indifference curve the lowest benefit: For a given risk, the indifference curve achieves a greater expected return than the indifference curve and the indifference curve a value greater than with the indifference curve . In principle, an investor would prefer the indifference curve that gives him the greatest benefit. In this case it would be the indifference curve . However, this indifference curve cannot be achieved because this curve does not affect the budget line. The indifference curve is achievable, but a higher benefit can be achieved. The optimal ratio between risk and return is only achieved with the indifference curve that touches the budget line. In the graphic “The conflict of goals between risk and return” the indifference curve touches the budget line. ${\ displaystyle U_ {3} \,}$ ${\ displaystyle U_ {1} \,}$ ${\ displaystyle U_ {3} \,}$ ${\ displaystyle U_ {2} \,}$ ${\ displaystyle U_ {2} \,}$ ${\ displaystyle U_ {1} \,}$ ${\ displaystyle U_ {3} \,}$ ${\ displaystyle U_ {1} \,}$ ${\ displaystyle U_ {2} \,}$

Different risk attitudes

Consequently, the investor will distribute his capital between the shares and the short-term Treasury bills in such a way that an expected return of at a risk of is realized. ${\ displaystyle E ^ {*} \,}$ ${\ displaystyle \ sigma ^ {*} \,}$

Effects of different risk attitudes

Due to the investors' different risk attitudes, the indifference curves will appear differently. The graphic "Different risk attitudes" illustrates this relationship.

Investor A is a very risk averse investor. Its (green) indifference curve touches the budget line at a point with a low expected return and a low risk. The investor will mainly invest his capital in short-term Treasury bills and expect a slightly greater value than the risk-free return ( ). Since investor B is a risk-taking investor, he will invest most of his capital in stocks, taking much more risk and expecting a higher return. Its (red) indifference curve therefore touches the budget line with a higher expected return, namely . ${\ displaystyle \ operatorname {E} _ {A} \,}$ ${\ displaystyle \ operatorname {E} _ {f} \,}$ ${\ displaystyle \ operatorname {E} _ {B} \,}$

literature

David E. Bell: Risk, Return, and Utility . MANAGEMENT SCIENCE / Vol. 41, No. January 1, 1995.
Richard A. Bettis, Vijay Mahajan: Risk / Return Performance of Diversified Firms . MANAGEMENT SCIENCE / Vol. 31, No. July 7, 1985.
Thomas Cloud: Risk Management . Oldenbourg Wissenschaftsverlag GmbH, Munich 2008, ISBN 978-3-486-58714-2 .
Peter Albrecht, Raimond Maurer: Investment and Risk Management . Schäffer-Poeschel Verlag Stuttgart, Stuttgart 2008, ISBN 978-3-7910-2827-9 .
Neil Doherty: Integrated Risk Management: Techniques and Strategies for Managing Corporate Risk . McGraw-Hill, Inc., New York 2010, ISBN 978-0-13-800617-4 .

Web links

Literature on risk-reward ratio in the catalog of the German National Library
Good presentation of important terms from the areas of finance and the stock market
Learn online

Individual evidence

^ ^A ^b ^c John C. Hull: Risk Management and Financial Institutions. Second Edition, Prentice Hall International, 2009, pp. 1-2, ISBN 978-0138006174 .
↑ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l Robert S. Pindyck, Daniel L. Rubinfeld: Microeconomics. 6th edition, Pearson Studium, 2005, pp. 216-218, pp. 239-246, ISBN 978-3827371645 .
↑ Christian Fähnrich, Denise Manns: Concept development in treasury management for banks - with special consideration of the interest book control. Diplomica Verlag GmbH, 2008, p. 12, ISBN 978-3-8366-5905-5 .
↑ Roland Eller (author), Walter Gruber (author), Markus Reif (editor): Risk management and risk controlling in modern treasury management. Deutscher Sparkassen Verlag, 2002, pp. 51–53, ISBN 978-3093012907 .
↑ ^a ^b ^c ^d ^e Martin Bösch: Finance - Investment, Financing, Financial Markets and Control. Verlag Franz Vahlen Munich, 2009, pp. 59-62, p. 224, ISBN 978-3-8006-3634-1 .
↑ ^a ^b ^c ^d ^e HypoVereinsbank: Basic information about investments in securities - basics, economic relationships, possibilities and risks. Copyright 2008 by Bank-Verlag Medien GmbH, November 2008, pp. 9-10, ISBN 978-3-86556-174-9 .
↑ ^a ^b Angela Steiner: Investment funds or life insurance? - Avoid old-age poverty through the right investment. Diplomica Verlag GmbH, 2010, pp. 32-33, ISBN 978-3-8366-8902-1 .
↑ Heinz Griesel, Helmut Postel, Friedrich Suhr with the participation of Andreas Gundlach (editor): LK - elements of mathematics - advanced course stochastics - with orientation knowledge - linear algebra / analytical geometry. Schroedel Verlag, 2007, p. 109, ISBN 3-507-83938-5 .
^ Thomas Werner: Ecological Investments - Opportunities and Risks of Green Investments. 1st edition, Gabler Verlag, 2009, pp. 224-227, ISBN 978-3-8349-0741-7 .

[Hull-1] A ^b ^c John C. Hull: Risk Management and Financial Institutions. Second Edition, Prentice Hall International, 2009, pp. 1-2, ISBN 978-0138006174 .

[Pindyck-2] ↑ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l Robert S. Pindyck, Daniel L. Rubinfeld: Microeconomics. 6th edition, Pearson Studium, 2005, pp. 216-218, pp. 239-246, ISBN 978-3827371645 .

[Fähnrich-3] Christian Fähnrich, Denise Manns: Concept development in treasury management for banks - with special consideration of the interest book control. Diplomica Verlag GmbH, 2008, p. 12, ISBN 978-3-8366-5905-5 .

[Eller-4] Roland Eller (author), Walter Gruber (author), Markus Reif (editor): Risk management and risk controlling in modern treasury management. Deutscher Sparkassen Verlag, 2002, pp. 51–53, ISBN 978-3093012907 .

[Bösch-5] Martin Bösch: Finance - Investment, Financing, Financial Markets and Control. Verlag Franz Vahlen Munich, 2009, pp. 59-62, p. 224, ISBN 978-3-8006-3634-1 .

[HypoVereinsbank-6] HypoVereinsbank: Basic information about investments in securities - basics, economic relationships, possibilities and risks. Copyright 2008 by Bank-Verlag Medien GmbH, November 2008, pp. 9-10, ISBN 978-3-86556-174-9 .

[Steiner-7] Angela Steiner: Investment funds or life insurance? - Avoid old-age poverty through the right investment. Diplomica Verlag GmbH, 2010, pp. 32-33, ISBN 978-3-8366-8902-1 .

[Stochastik-8] Heinz Griesel, Helmut Postel, Friedrich Suhr with the participation of Andreas Gundlach (editor): LK - elements of mathematics - advanced course stochastics - with orientation knowledge - linear algebra / analytical geometry. Schroedel Verlag, 2007, p. 109, ISBN 3-507-83938-5 .

[Werner-9] Thomas Werner: Ecological Investments - Opportunities and Risks of Green Investments. 1st edition, Gabler Verlag, 2009, pp. 224-227, ISBN 978-3-8349-0741-7 .