By Yuriy Smirnov Ph.D.

The expected rate of return is a percentage return expected to be earned by an investor during a set period of time, for example, year, quarter, or month. In other words, it is a percentage by which the value of investments is expected to exceed its initial value after a specific period of time. The expected rate of return can be calculated either as a weighted average of all possible outcomes or using historical data of investment performance.

The rate of return can be calculated as shown below.

Rate of Return = | Amount Received - Amount Invested |

Amount Invested |

The expected rate of return (ERR) can be calculated as a weighted average rate of return of all possible outcomes. In general, the equation looks as follows:

ERR = p_{1}×r_{1} + p_{2}×r_{2} + p_{3}×r_{3} + … + p_{n}×r_{n}

or

ERR = | n | p_{i} × r_{i} |

Σ | ||

i = 1 |

where p_{i} is the probability of *i*th outcome, r_{i} is the rate of return achieved at *i*th outcome, and n is the number of possible outcomes.

Historical data for investment performance can sometimes be used to assess the expected rate of return.

where r_{i} is the actual percentage return of investment achieved in the *i*th period, and n is the number of periods used in a historical data set.

A portfolio is a grouping of several investments, so its expected percentage return is a weighted average of all expected rate of returns of its components according to their proportion.

ERR of Portfolio = | n | w_{i} × ERR_{i} |

Σ | ||

i = 1 |

where w_{i} is a proportion of *i*th investment in a portfolio, ERR is an expected rate of return of *i*th investment, and n is the number of a portfolio’s components.

An investor is considering two securities of equal risk to include one of them in a portfolio. The percentage of return of Security A has five possible outcomes.

The percentage of return of Security B has three possible outcomes.

Let’s put these data in the formula above.

ERR of Security A = 0.10×(-8%) + 0.15×(-2%) + 0.40×4% + 0.30×10% + 0.05×16% = 4.30%

ERR of Security B = 0.25×(-10%) + 0.50×7% + 0.35×25% = 9.75%

Assuming that both securities are equally risky, Security B should be preferred because of a higher expected rate of return.

A financial analyst is considering three securities. Their historical percentage returns during the last 5 years are given in the table below.

ERR of Security A = | 2.3% + 8.0% + (-3.0%) + 5.5% + 4.1% | = 3.4% |

5 |

ERR of Security B = | 6.7% + 2.1% + 4.0% + (-2.0%) + 5.1% | = 3.2% |

5 |

ERR of Security C = | (-3.2%) + 1.5% + 7.5% + (-1.0%) + 2.2% | = 1.4% |

5 |

As we can see, Security A has the highest expected rate of return, and Security C has the lowest.

A portfolio is composed of three securities: Security A, Security B, and Security C. Their weight in a portfolio are 25%, 40%, and 35% respectively. The expected rate of return of Security A is 8.1%, Security B is 4.5%, and Security C is 5.7%.

As was mentioned above, the expected rate of return of a portfolio is the weighted average of the expected percentage return on each security according to their weight.

ERR of Portfolio = 0.25×8.1% + 0.40×4.5% + 0.35×5.7% = 5.82%

The graph below shows the probability distribution of the rate of return of Security A and Security B from Example 1.

The rate of return of both securities is shown in the graph to demonstrate the variability of possible outcomes.

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