 # Expected Rate of Return By Yuriy Smirnov Ph.D.

## Definition

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.

## Formula

The rate of return can be calculated as shown below.

 Rate of Return = Amount Received - Amount Invested Amount Invested

### Probability Approach

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 = p1×r1 + p2×r2 + p3×r3 + … + pn×rn

or

 ERR = n pi × ri Σ i = 1

where pi is the probability of ith outcome, ri is the rate of return achieved at ith outcome, and n is the number of possible outcomes.

### Historical Return Approach

Historical data for investment performance can sometimes be used to assess the expected rate of return. where ri is the actual percentage return of investment achieved in the ith period, and n is the number of periods used in a historical data set.

### Expected Rate of Return of a Portfolio

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 wi × ERRi Σ i = 1

where wi is a proportion of ith investment in a portfolio, ERR is an expected rate of return of ith investment, and n is the number of a portfolio’s components.

## Calculation Examples

### Example 1

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.

### Example 2

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.

### Example 3

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%

## Graph

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.