The expected return (or expected gain) on a financial investment is the expected value of its return (of the profit on the investment). It is a measure of the center of the distribution of the random variable that is the return. It is calculated by using the following formula: where is the return in scenario ; is the probability for the return in scenario ; and is the number of scenarios. The expected rate of return 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 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: Discrete scenarios In gambling and probability theory, 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 is thrown and numbers 1 and 2 win $1, but 3-6 lose $0.5, then the expected gain per throw is 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)