Risk–return spectrum

The risk - reward ratio is a term used in portfolio theory and describes the conflict, before a capital market participants is, if he invests capital in a portfolio. The trade-off exists between the risk, which is the capital market participants willing to take with his investment, and the income, the expected capital market participants. With increasing risk thereby increasing the expected return. The highlights of the risk - return relationship is the one RORAC and other RAROC.

• 3.1 Relationship between expected return, relative frequency, risk and standard deviation
• 3.2 Formal derivation of goal conflict
• 3.3 Solution of the target conflict
• 3.4 Effects of different risk settings

Historical classification

In the 50s, Harry Markowitz addressed first in the history of the relationship between risk and return. The result was the Portfolio Selection Model. Based on these findings, developed, among others, William F. Sharpe, Capital Asset Pricing Model. These theories used many researchers, especially Stephen Ross to develop the arbitrage pricing theory. Benefit from the findings of the theories of capital market participants ( particularly asset managers ), by thinking about the trade-off between risk and return and analyze it before making decisions.

Basics

The investment objective

The starting point for considering the risk - reward ratio is that an investor wants to invest its capital in a portfolio of, for example, stocks and / or bonds. The investor is interested in such an investment, because he wants to provide a return that is higher than the inflation rate. The objectives of the investor be to shift the consumption in the future and with the investment income in the future to purchase more goods than at the time of the investment.

Characteristics of a plant

An investment is an investment that promises the investor a cash flow or a power flow. For a power stream is, for example, apartment services to the owner of a house. This output current can be converted by a rental of the house in a cash flow. In a cash flow explicit and implicit payments are possible. Explicit payments are for example dividends. Implicit, however, payments are for example the exchange rate. The latter is capital gains or capital losses that remain not recovered until the sale of the plant, but is only realized with the sale. It is between risky and risk-free investments (also known as risk-free investments ) are distinguished, in which the investor can invest his capital. In a risky investment are reported in money and / or power flow to the investor not sure, but by chance usually and insecure. For example, know the owner of shares not know if there will be a distribution and if so, to what extent this will occur. In a risk-free investment, however, the cash flow and / or performance of power are safe. So the owners of short-term Treasury bills can assume that both the coupons and the amount of money spent will be paid, because it is the issuer, in this case to public bodies and has the German state due to the fiscal monopoly sufficient capital to all to settle debts.

The magic triangle of an investment

Each system is defined by three characteristics: safety, liquidity and profitability. These criteria are not to be considered individually for themselves, as they influence each other. The safety serves as a benchmark for the preservation of invested capital. Risks, such as the creditworthiness of the borrower, but also rate risks and currency risks largely determine whether a system is secure or less secure. The security can be increased by the scattering of capital to be used. This strategy is referred to as risk diversification. The risk can be minimized by a distribution of the assets in various types of securities, as well as by investing in different countries and / or sectors. Liquidity indicates the rapidity with which the capital invested in cash or bank deposits can be converted back. The liquidity of an investment increases with the decrease in the time required for the conversion. The profitability can be derived from the income of an investment. Possible income, dividends, interest, price increases and other distributions. The problem that arises here is that an investor will typically invest in any securities, so that the types of income may vary materially. In order to compare the profitability of different systems that measure return is used. It represents the ratio between the income and the capital employed These three criteria are the magic triangle of an investment. This triangle shows which conflicts can arise in a portfolio with an investment of capital. First, there is a conflict between security and profitability. The more security is desired by an investor, the lower the yield on the investment will be. The higher profitability will be, the more risk must be addressed. Consequently there is a trade-off between risk and return. Secondly, there is a conflict between liquidity and profitability. Plants, which have a higher liquidity, have usually return disadvantages. The ideal case would consist of a high security, high liquidity and high profitability, it is not due to competition between these criteria.

Trade-off between risk and return

Relation between expected return relative frequency, risk and standard deviation

As already mentioned, an investor expects a certain level of income, called a return if he invested his capital in a security. The rate of return expected by the investor shall be, as a rule, the average of all returns from the past and is known as the expected value of the return. The actual yield of the asset reflects the realized income from the investment and is not known at the time of investment. The risk of an investment is that the expected return is not achieved and thus deviations arise from the expected value. Under deviations from the expected value ( upward and downward ) is from the financial perspective, thus, the risk understood. In order for a capital market participants prior to investing know which expected return and risk with which he must expect, the necessary data for these quantities the past are taken. This can be the expected value of return determined by the various forms of returns from the past multiplied by the corresponding relative frequencies and the values ​​obtained are then added. The risk for the deviation from the expected value is expressed by the standard deviation, the deviations of the yield is expected to arise in the future, the deviations correspond that have been implemented in the past, on average. The financial term for Stanadardabweichung is volatility and represents the average deviation from the expected value dar. To determine the standard deviation, the square root of the variance should be considered. With increasing volatility increases the risk that the actual yield is not consistent with the expected yield. If the actual and expected return match, then there is a risk-free investment. To simplify the choice of the investor on a risk-free investment (eg, in short-term Treasury bills ) and a risky investment is limited ( for example, shares) in the following. Thus it is a model in which an investment with a variance and a standard deviation of a different investment with a variance and a standard deviation with a value that is greater than compared. This ensures that the investments are independent. Thematically, you will find yourself within the portfolio theory in an efficient portfolio, which consists of risk-free and risky securities. The relationships can be, however, to many other different systems, for example systems in land, different stocks and corporate bonds transferred.

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

Each investor must take for themselves to decide how much he wants to invest his capital in the respective system. There are basically three options:

Formal derivation of goal conflict

For a formal presentation of the conflict between risk and return the following values ​​are required: The expected return on the investment in risk-free securities, such as short-term Treasury bills. This is represented by the variable. Also, the expected return on a risky asset, such as shares, are required. This is represented by the variable. The actual outcome of the risky asset return is denoted by. The expected return of the risky asset is higher than the expected return of the risk-free investment. Consequently, the following applies:

If this relationship would not apply, then the risk-averse investors would only buy the risk-free securities such that no risky securities would be sold. We have the following mathematical relationship:

The expected return on any investments equal to the weighted average of the two expected returns of investments, while the share indicates that is invested in the risky asset. thus indicates what proportion of the total invested capital flows into the risk-free investment. For example, half of the capital to purchase the shares and the other half could be used for the purchase of short-term Treasury bills. Assuming that shares an expected return of 9% and the short-term Treasury bills have an expected return of 3%, the expected return of the total investment will amount to 6%. To assess the portfolio risk, the standard deviation of the return is considered, since this is the measure of the risk. The standard deviation of the investment in risky assets is represented by. represents the standard deviation of the total portfolio. We have the following mathematical relationship:

The standard deviation of the portfolio is the product of the proportion invested in the risky asset, and the associated standard deviation. After forming and inserting the top two equations yields the following relationship:

In this equation, there is a budget constraint, because it represents the trade-off between the expected return ( ) and the risk () of a portfolio. The resulting straight line can be drawn in a diagram.

Represents the slope of the function and is the y-intercept. In the graph, " The trade-off between risk and return " this is drawn straight black. From the line equation can be derived that the expected return of the portfolio () increases as the standard deviation of the return () increases.   expresses the price of risk and describes the additional risk that will take an investor to purchase in order to derive a greater expected return. If the investor wants to take no chances with his capital, he will spend all his capital for short-term Treasury bills. This corresponds to the first option above. The variable will take the value in this case and it is only the risk-free yield () can be achieved. Should the investor a greater expected return, then he must expect a major risk. If he chooses the second option and all his capital is invested in stocks, is worth to accept, so that the expected return of the portfolio is consistent with the expected return from investing in equities (). For this purpose, the investor is a high risk in terms of standard deviation must, however, be accepted. The investor can also choose the third option and invest in both equities as well as in short-term Treasury bills. Then the expected return of the portfolio between the riskless rate () and the risky return ( ) and the standard deviation between and would.

Solution to the conflict of objectives

To solve the conflict of objectives is needed to determine the optimal relationship between risk and return. For this purpose, it must be determined how much the investor in stocks and how much to invest in short-term Treasury bills in order to achieve the maximum benefit from the investment. For this purpose, the indifference curves are drawn in the diagram. Each indifference curve describes a particular benefit that the capital market participants can achieve for different combinations of risk and return, or will. Each curve is rising, because risk is generally not desirable. The greater the risk, the higher must be the expected return to compensate for the higher risk incurred.

The indifference curve in the graph " The trade-off between risk and return " represents the maximum benefit, the indifference curve the least benefit: For a given risk, a greater expected return a larger value is obtained with the indifference curve than with the indifference curve and the indifference curve as with the indifference curve. Basically, an investor would prefer that indifference curve that benefits him the highest benefit. In this case it would be the Indifferenzkurve. However, this indifference curve is not available, because this curve is not tangent to the budget line. The indifference curve is indeed achievable, however, a higher benefit may be achieved. The optimal relationship between risk and return is only achieved with the indifference curve is tangent to the budget line. In the graph, " The trade-off between risk and return " touches the indifference curve the budget line.

Consequently, the investor will distribute its capital on the shares and short-term Treasury bills so that an expected income of is realized at a risk of.

Effects of different risk settings

Due to the different settings of the risk investors the indifference curves will adopt different appearance. The graphic " Different risk settings " clarifies this connection. Investor A is an extremely risk-averse investor. His (green ) indifference curve touches the budget line at a point with a low expected return and low risk. The investor will invest principally its capital in short-term Treasury bills and expect as income, a value slightly larger than the riskless rate (). As investor B is a risk-averse investor, he will invest his capital mainly in shares, thereby received much more risk and for that expect a higher return. Therefore His (red ) indifference curve touches the budget line at a higher expected return, ie.