# Return

The return (borrowed from Italian rendita , "income") in finance is the effective interest expressed as a percentage of a reference value that an investor earns for financial products or an investor for investments within a year. Since the return mostly relates to an annual return on capital , it can not be equated with the profitability indicator , which relates to a company's success.

## General

The subject of the return calculation are return objects such as capital investments , investments in tangible and financial assets ( return on capital ), companies or real estate ( rental return ). The different reference values ​​of these yield objects such as interest rate , nominal interest rate , dividends , profit , purchase price , investment costs , market value , land value or income value are not suitable alone as comparison values if different yield objects of the same type are to be compared with one another with regard to their earnings situation . In addition, the risk to which an investment is exposed cannot be derived from these reference values . In this context, risk is understood to mean the risk of partial or total loss of capital for the investor. Both of these tasks are fulfilled by the return because it makes the profitability of an investment form comparable and is a suitable risk measure for determining the investment risk. Investors, regardless of whether they are private investors or institutional investors , generally choose the risk-adjusted (risk-adjusted) return as an essential investment criterion for their investment decision, because investment decisions are considered to maximize risk-adjusted returns.

As a price, the rate of return exercises the important signaling function for capital providers, among other things, for directing capital towards the most advantageous risk / return combinations. However, while high prices signal a shortage of goods and services , returns have a reciprocal signaling function, because high returns signal low scarcity and high risk and vice versa.

## Basic formulas

In general, the return is the difference between an income and an expense in relation to this expense:

${\ displaystyle {\ text {Yield}} = {\ frac {{\ text {Yield}} - {\ text {Expenditure}}} {\ text {Expenditure}}} = {\ frac {\ text {Yield}} {\ text {effort}}} - 1}$

With the return, the overall success of an investment is usually measured as the actual return on the capital employed.

In the basic formula of the return, the profit is set in relation to the capital employed:

${\ displaystyle {\ text {Yield}} = {\ frac {\ text {Profit}} {\ text {Capital employed}}}}$ (Return as a numerical value)

For example, if you put € 50 as an investor and receive € 70 back after the investment has expired, the return is 40%:

${\ displaystyle {\ text {Yield}} = {\ frac {70-50} {50}} = {\ frac {20} {50}} = 40 \, \%}$

The return is given either as a percentage (here 40%) or as a numerical value ( decimal fraction , here 0.4).

## species

With regard to the consideration of costs, a distinction is made between gross and net return . The latter takes into account the transaction costs of an investment ( bank fees for securities orders such as brokerage fees or custody fees ), ancillary costs when buying real estate (such as notary fees and land registry fees ) and taxes . For the investor, the net returns are more meaningful because they reflect the return actually remaining with him .

With regard to the type of investment, one differentiates:

The best-known rate of return is the interest rate . However, the term is not sharply defined, which means that it is hardly possible to classify it in a specific market.

### Return on an investment

The return is used to compare different investments. The background to this is that different types of investment often contain different income and cost components. The yield here provides the answer to the question of what interest rate would be required per year to get the same investment result.

The mean of comparison could be the current yield (it indicates the average yield of a selection of fixed income securities that are in circulation) or a reference interest rate .

In the case of investments with an agreed maturity date (especially in the case of bonds ), the term yield to maturity is used. The prerequisite for their calculation is the assumption that the security will be held until maturity and that it has no option rights.

Often one speaks of return after (income) tax in order to compare investments with different tax treatment with one another.

### Returns on Securities

In the securities market, an interest rate is not determined, but a price is fixed for a security. This price is the price of a security. The return (effective interest rate) can be derived from this price. The return on a security is thus what you get on the security in one year, minus the price paid today, divided by today's price.

${\ displaystyle i_ {B} = {\ frac {{\ text {face value of the security}} \ cdot \ left (1 + i_ {0} \ right) -P_ {B}} {P_ {B}}}}$

With

${\ displaystyle i_ {B}}$: Return
${\ displaystyle i_ {0}}$: Nominal interest
${\ displaystyle P_ {B}}$: today's price of the security
${\ displaystyle i}$: current interest rate

### Return on bonds

The return on a bond is not identical to its nominal interest rate, but also depends on the current price and (remaining) term.

The yield structure curve (also called the yield curve for simplicity) can be used to illustrate the relationship between yield and maturity. It reflects the temporal structure of bond yields; That is, you can tell the difference between short-term and long-term bonds. Normally, a yield curve is rising, so that the yield on a longer-term bond is always higher than the yield on short-term bonds. If the curve falls, the short-term bond yields are higher than the long-term bond yields.

Yield to maturity of an n-year bond is defined as the constant annual interest rate that makes the bond price equal today to the present value of the future bond payment.

Suppose a bond is held for two years. This bond should result in a payment of € 100 at the end of these two years. What the investor is interested in is what percentage of the bond will pay off after the two years have elapsed.

${\ displaystyle i_ {2, t} = {\ sqrt {\ frac {100} {P_ {2, t}}}} - 1}$
${\ displaystyle P_ {2, t}}$ : today's bond price for two-year bond
${\ displaystyle i_ {2, t}}$ : expected return on two-year bond

#### Period yield on a bond

The period return on a bond is referred to as return. The return refers to a period of length T with a coupon date in between.

${\ displaystyle r: = {\ frac {K_ {t} ^ {~} + c_ {t} + c-K_ {0} -c_ {0}} {K_ {0} + c_ {0}}} \ cdot {\ frac {1} {T}}}$
${\ displaystyle K_ {t} + c_ {t}}$: dirty price tomorrow
${\ displaystyle K_ {0} + c_ {0}}$: dirty price today
${\ displaystyle c}$: Coupon within the period

Transaction costs and taxes are abstracted here.

The period return is used. a. in the calculation of critical values ​​and scenario analyzes .

### Dividend yield

The dividend yield (dividend-price ratio) shows the relationship between the dividend paid and the share price. This enables a comparison of different forms of investment and the investor can deduce how high the return on his shares is.

• Discrete rate of return (simple rate of return): percentage increase from one point in time to another
${\ displaystyle R_ {s, t} = {\ frac {S_ {t} -S_ {s}} {S_ {s}}} = {\ frac {S_ {t}} {S_ {s}}} - 1 }$
${\ displaystyle R_ {s, t}}$: discrete return
${\ displaystyle S}$: Course
${\ displaystyle s}$: Start time
${\ displaystyle t}$: End time
• Continuous return (difference between logarithmized prices, logarithm return): natural logarithm of the growth ratio
${\ displaystyle r_ {s, t} = \ ln \ left ({\ frac {S_ {t}} {S_ {s}}} \ right) = \ ln S_ {t} - \ ln S_ {s}}$
${\ displaystyle r_ {s, t}}$: steady return

### Time-weighted and capital-weighted returns

• The time-weighted return (geometric average return) shows how an earlier invested amount of money is transformed into a later investment result, assuming that no payments or withdrawals are made during the observation horizon, or, if there are, the return is adjusted for the payments.
• The capital-weighted return ( internal rate of return (IRR)) also shows how an amount of money invested earlier is transformed into a later investment result, but here it is assumed that there are deposits and withdrawals, i.e. This means that the return generated is weighted with the assets invested in each case. It depends on the time of deposits and withdrawals.

Both types are mostly reported as average return (i.e. annualized) rather than total return.

### Promised, expected and achieved return

The promised return is calculated in advance according to certain conventions (ex ante). The actual return, on the other hand, is a retrospective concept (ex-post) that takes into account the actual reinvestment opportunities. Since promised returns often cannot be achieved, a distinction must be made ex ante between promised ("planned") and expected returns (i.e. the expected value of the return). For example, the contractually promised interest rate on borrowed capital differs from the borrowing costs, i.e. the expected return of the lenders, because bankruptcy can occur (see insolvency risk and cost of capital ). Business decisions, e.g. B. with regard to an investment, are based on expected returns, the calculation of which requires forecast values ​​that are true to expectations (i.e. expected values, e.g. of future sales and costs, i.e. values ​​that can be realized "on the average" of possible future scenarios). The calculation of expected values, in turn, requires the analysis of opportunities and dangers that can trigger deviations from the plan, i.e. a risk analysis .

### Term of return for investments

The return on an investment is the ratio of the return on the investment to the amount originally invested. An example of the application of the rate of return is the return on a company's investment in new production facilities.

### Return on a portfolio

The interest formula from Hardy is used to approximately calculate the return on a portfolio achieved in one year , with deposits and withdrawals made in the course of the year .

## Annualization

In order to make the returns of different types of investment with investment periods of different length (e.g. per quarter) comparable, they are usually annualized, i.e. H. based on the period of one year.

In the case of discrete returns, the annualization is carried out using the geometric mean and not the arithmetic mean. That is, the annual return results as: ${\ displaystyle \ mathrm {JR}}$

${\ displaystyle \ mathrm {JR} = {\ sqrt [{n}] {1 + {\ text {Total return}}}} - 1}$, with number of years.${\ displaystyle n}$

To calculate the total return from the individual annual returns, use:

${\ displaystyle {\ text {Total return}} = \ prod _ {t = 1} ^ {T} (1 + R_ {t}) - 1}$, with the respective annual return in year t${\ displaystyle R_ {t}}$

## Return and risk

In the case of risk aversion, the higher the risk, the higher the expected return. The risk-based requirement for a return is called the cost of capital .

Risk / return positions of a portfolio of two risky assets

In the financial markets , taking financial risks is generally rewarded with returns. Accordingly, return and risk are symmetrical, so that a high return represents high profitability , but also a high risk and vice versa. In scientific terms, there is a positive correlation between risk and return. This connection was first established by the portfolio theory set up by Harry M. Markowitz in March 1952 . The model is based on the aim to achieve the maximum return with a given risk or to take the lowest possible risk with an expected return. It has replaced the previous one-dimensional view of the return as the sole decision criterion for an investment decision and established the risk-adjusted returns. The risk-adjusted return is calculated from the return and the risk associated with the investment (taking into account cluster risk and granularity in portfolios ):

${\ displaystyle {\ text {Risk-adjusted return}} = {\ frac {\ text {Return}} {\ text {Risk of the plant}}}}$

From this it can be deduced that the risk-adjusted return is lower, the higher the investment-related risk and vice versa. The main investment objective of investors is to achieve a high return, but this is limited by the risk. The attribute “high risk” is mostly based on the yield on government bonds , which - along with the best credit rating of AAA by the major rating agencies - are classified as risk-free ( risk-free interest rate ). All higher returns (“ excess returns ”) therefore mean - assuming that the markets are free of arbitrage - also a higher risk (see credit spread ).

### Fluctuations in return

A risk measure is required to measure the fluctuations in return ; often z. B. the standard deviation . The arithmetic mean and the standard deviation the empirical standard deviation are used as point estimators for the expected value .

• Average return  :${\ displaystyle {\ bar {R}}}$
${\ displaystyle {\ bar {R}} = {\ frac {1} {T}} \ sum _ {t = 1} ^ {T} R_ {t}}$
• Standard Deviation :${\ displaystyle {\ sigma}}$
${\ displaystyle \ sigma = {\ sqrt {{\ frac {1} {T-1}} \ sum _ {t = 1} ^ {T} \ left (R_ {t} - {\ bar {R}} \ right) ^ {2}}}}$
${\ displaystyle R_ {t}}$: Return at the time ${\ displaystyle t}$
${\ displaystyle T}$: Number of periods
• Example:
year Course at the beginning of the year Course at the end of the year Return
2002 65 euros 70 euros 7.7%
2003 70 euros 79 euros 12.9%
2004 79 euros 85 euros 7.6%
2005 85 euros 80 euro −5.9%
${\ displaystyle {\ bar {R}} = {\ frac {1} {4}} \ left (7 {,} 7 \, \% + 12 {,} 9 \, \% + 7 {,} 6 \ , \% - 5 {,} 9 \, \% \ right) = 5 {,} 6 \, \%}$
${\ displaystyle \ sigma = {\ sqrt {{\ frac {1} {3}} \ left (\ left (7 {,} 7 \, \% - 5 {,} 6 \, \% \ right) ^ { 2} + \ left (12 {,} 9 \, \% - 5 {,} 6 \, \% \ right) ^ {2} + \ left (7 {,} 6 \, \% - 5 {,} 6 \, \% \ right) ^ {2} + \ left (-5 {,} 9 \, \% - 5 {,} 6 \, \% \ right) ^ {2} \ right)}} = 8 {,} 04 \, \%}$

When considering the risk, a distinction can be made between systematic and unsystematic risk . This distinction is made using the CAPM model . The systematic risk relates generally to capital investments that are subject to ( economic ) fluctuations in the market ( market risk ). The investment can be perfectly planned and still there is this risk. The unsystematic risk arises differently for each investor, since this risk does not depend on market behavior . Every investor must try to keep the risk as low as possible.

The decisive factor when comparing several investment alternatives is therefore the risk associated with the respective form of investment. In order to make the return of different risk investments comparable with each other, they are risk adjusted (risk adjusted). A well-known, but also controversial measure of risk adjustment in terms of informative value is the Sharpe quotient .

## Individual evidence

1. Marc Engelbrecht, Asset Allocation in Private Banking , 2013, p. 289
2. a b Olivier Blanchard / Gerhard Illing Macroeconomics , 4th edition, 2006
3. Gregory Dorfleitner , Continuous vs. discrete yield: financial mathematical considerations for the correct use of both terms in theory and practice , 2002, pp. 216–241
4. WWZ: Home
5. Markus König, Investor Protection in Investment Law , 1998, p. 34
6. Hanspeter Gondring / Edgar Zoller / Josef Dinauer (eds.), Real Estate Investment Banking , 2003, p. 27
7. Joachim Coche / Olaf Stotz, Asset Allocation: Controlling assets and financial investments professionally , 2002, p. 31
8. ^ Lutz Kruschwitz: Financing and Investment . 2nd Edition. Oldenbourg Wissenschaftsverlag, 1999, ISBN 3-486-24884-7 .

## literature

• Peter Albrecht, Sören Jensen: "Financial mathematics for economists", 2nd edition, 2011
• Oliver Blanchard, Gerhard Illing: Macroeconomics . 4th edition, 2006
• Gregor Dorfleitner: Stetige versus discrete return - considerations on the correct use of both terms in theory and practice , credit and capital, 35th year, Heft 2, 2002, pp. 216–241.
• Thomas Hesse: Periodic corporate success between the realization and anticipation principle . 1996
• Lutz Kruschwitz: finance and investment . 2nd edition, 1999
• Jyrki Veranen, Herbert Hensle: Value orientation and yield . 2000