The expected return (or expected gain) on a

Using Expected Return to Maximize Growth

{{DEFAULTSORT:Expected Return Theory of probability distributions Financial risk

financial investment
Investment is the dedication of money to purchase of an asset to attain an increase in value over a period of time. Investment requires a sacrifice of some present asset, such as time, money, or effort.
In finance, the purpose of investing i ...

is the expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...

of its return (of the profit on the investment). It is a measure of the center of the distribution of the random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...

that is the return. It is calculated by using the following formula:
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where
:: $R\_$ is the return in scenario $i$;
::$P\_$ is the probability for the return $R\_$ in scenario $i$; and
::$n$ is the number of scenarios.
The expected rate of return
In finance, return is a profit on an investment. It comprises any change in value of the investment, and/or cash flows (or securities, or other investments) which the investor receives from that investment, such as interest payments, coupons, c ...

is the expected return per currency unit (e.g., dollar) invested. It is computed as the expected return divided by the amount invested. The required rate of return
The discounted cash flow (DCF) analysis is a method in finance of valuing a security, project, company, or asset using the concepts of the time value of money.
Discounted cash flow analysis is widely used in investment finance, real estate devel ...

is what an investor would require to be compensated for the risk borne by holding the asset; "expected return" is often used in this sense, as opposed to the more formal, mathematical, sense above.
Application

Although the above represents what one expects the return to be, it only refers to the long-term average. In the short term, any of the various scenarios could occur. For example, if one knew a given investment had a 50% chance of earning a return of $10, a 25% chance of earning $20 and a 25% chance of earning $–10 (losing $10), the expected return would be $7.5: :$E;\; href="/html/ALL/l/.html"\; ;"title="">$Discrete scenarios

Ingambling
Gambling (also known as betting or gaming) is the wagering of something of value ("the stakes") on a random event with the intent of winning something else of value, where instances of strategy are discounted. Gambling thus requires three elem ...

and probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

, there is usually a discrete set of possible outcomes. In this case, expected return is a measure of the relative balance of win or loss weighted by their chances of occurring.
For example, if a fair die
Dice (singular die or dice) are small, throwable objects with marked sides that can rest in multiple positions. They are used for generating random values, commonly as part of tabletop games, including dice games, board games, role-playing g ...

is thrown and numbers 1 and 2 win $1, but 3-6 lose $0.5, then the expected gain per throw is
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When we calculate the expected return of an investment it allows us to compare it with other opportunities. For example, suppose we have the option of choosing between three mutually exclusive investments: One has a 60% chance of success and if it succeeds it will give a 70% ROR (rate of return). The second investment has a 45% chance of success with a 20% ROR. The third opportunity has an 80% chance of success with a 50% ROR. For each investment, if it is not successful the investor will lose his entire initial investment.
* The expected rate of return for the first investment is (.6 * .7) + (.4 * -1) = 2%
* The expected rate of return for the second investment is (.45 * .2) + (.55 * -1) = -46%
* The expected rate of return for the third investment is (.8 * .5) + (.2 * -1) = 20%
These calculations show that in our scenario the third investment is expected to be the most profitable of the three. The second one even has a negative ROR. This means that if that investment was done an infinite number of times one could expect to lose 46% of the money invested on the average occasion. The formula of expected value is very straightforward, but its value depends on the inputs. The more alternative outcome scenarios that could occur, the more terms are in the equation. As Ilmanen stated,
"The foremost need for multi-dimensional thinking is on inputs. When investors make judgments on the various returns on investments, they should
guard against being blinded by past performance and must ensure that they take all or most of the following considerations into account".
* Historical average returns
* Financial and behavioral theories
* Forward looking market indicators such as bond yields; and
* Discretionary views
Continuous scenarios

Ineconomics
Economics () is the social science that studies the production, distribution, and consumption of goods and services.
Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analy ...

and finance
Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of ...

, it is more likely that the set of possible outcomes is continuous (any numerical value between 0 and infinity). In this case, simplifying assumptions are made about the continuous distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...

of possible outcomes.
See also

* Abnormal return *Expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...

*Rate of return
In finance, return is a profit on an investment. It comprises any change in value of the investment, and/or cash flows (or securities, or other investments) which the investor receives from that investment, such as interest payments, coupons, c ...

Notes

External links

Using Expected Return to Maximize Growth

{{DEFAULTSORT:Expected Return Theory of probability distributions Financial risk