Finance

# Rates of Return: A Core Concept in Finance

Rates of return are among the central financial ratios, as all other market prices of financial assets are priced through them. It is, therefore, a paramount issue for every investor to learn about and gain the ability to compute. Returns help investors compare one asset to another, measure the performance of their portfolio, and measure a trade-off between risk and reward. In the paper below, we will discuss different types of rate of returns, their importance, and how they are calculated with examples.

**Defining Rates of Return**

A rate of return is the gain or loss on an investment relative to the original amount of an investment, expressed as a percentage. The basic tool used in measuring the performance of an investment over time is known as the rate of return. There are two primary sources of returns:

- Periodic Income: Cash dividends or interest payments.
- Capital Gains or Losses: Changes in the price of the asset.

Some assets produce returns in one of these ways, whereas others do so in a combined fashion. For example, non-dividend-paying stocks return only through price appreciation, and bonds typically return through interest.

**Types of Rates of Return**

## 1. Holding Period Return (HPR)

Holding Period Return is the return earned from holding an asset over a specified period, which could be days, weeks, months, or years. It is calculated as:

**Example** **1**: Suppose you buy a stock for ₹1,000, and after one year, the stock price rises to ₹1,200. During the year, you receive ₹50 as a dividend. The HPR is calculated as follows:

This calculation shows a 25% return over the holding period, accounting for both capital gains and income.

A holding period return can be computed for a period longer

than one year. For example, an analyst may need to compute a one-year holding

period return from three annual returns. In that case, the one-year holding

period return is computed by compounding the three annual returns:

**Example 2:** Consider an investment that yields 8% in the first year, 5% in the second year, and 10% in the third year. The HPR over the three years would be:

This result shows a 24.74% return over the three-year period.

### Application

HPR is often used by investors to evaluate the return of a single investment over a specific period, making it ideal for comparing the performance of different investments. It provides a straightforward way to understand how much an investment has grown (or shrunk) over a particular time.

## 2.Arithmetic Mean Return(AMR)

Investment returns usually get reported on a daily, monthly, or annual basis. In order to make comparisons between performances over time, the return measures have to be standardized so that it would be easier to compare and appreciate. A simple approach involves the calculation of arithmetic mean, which is the average of all returns.

Generally, if you want to calculate the arithmetic mean return for an investment, you’d use the following formula:

**Examples of Arithmetic Mean Return**

**Example 1: Simple Average Return Calculation**

**Scenario**: You have invested in a stock that provided the following annual returns over three years:

- Year 1: 10%
- Year 2: 12%
- Year 3: 8%

**Calculation**:

**Interpretation**: The arithmetic mean return for your investment over these three years is 10%. This means that, on average, the investment returned 10% per year over this period.

**Example 2: Returns with Negative Values**

**Scenario**: An investor has the following annual returns for a particular investment:

- Year 1: 15%
- Year 2: -5%
- Year 3: 10%

**Calculation**:

**Interpretation**: Despite the negative return in Year 2, the average return over the three years is 6.67%. This arithmetic mean gives a simple view of the overall performance, but it does not consider the effects of compounding or the volatility of returns.

**Example 3: Comparing two Investments**

**Scenario**: You are comparing two investments:

**Investment A**has returns: 8%, 12%, 14% over three years.**Investment B**has returns: 10%, 10%, 10% over three years.

**Calculation**:

For **Investment A**:

For **Investment B:**

**Interpretation**: Investment A has a higher arithmetic mean return of 11.33% compared to Investment B’s 10%. However, the returns of Investment A are more volatile, while Investment B has consistent returns.

## 3.Geometric Mean Return(GMR)

It is a measure used in finance to calculate the average rate of return of an investment over multiple periods. It differs from the arithmetic mean, which only provides the average return, as it considers the compounding effect of returns on returns over periods. This makes the geometric mean return an accurate reflection of actual performance by an investment when returns fluctuate significantly.

The geometric mean return is especially useful when you’re dealing with returns that vary over time. It provides a better indication of what was happening with the investment’s performance since it takes into consideration the compounding effect of earning a return on a return.

This would be the case, for example, when an investment lost 50% in one year and then gained 100% in the next; the arithmetic mean return would reflect an average gain, while, in fact, the actual performance, accounting for compounding, would correctly result in no gain or loss over two years, which the geometric mean return would show.

**Formula**

The formula for calculating the geometric mean return is:

E**xamples of Geometric Mean Return**

**Example 1: Simple Calculation**

**Scenario**: Suppose you invest in a stock with the following annual returns over three years:

- Year 1: 10%
- Year 2: 15%
- Year 3: 5%

**Calculation**:

**Interpretation**: The geometric mean return over these three years is 9.76%. This means that, on average, the investment grew by 9.76% per year, taking into account the compounding effect.

**Example 2: Handling Negative Returns**

**Scenario**: Consider an investment with the following returns:

- Year 1: 20%
- Year 2: -10%
- Year 3: 15%

**Calculation**:

**Interpretation**: The geometric mean return for this investment is 7.49% per year. Despite having a negative return in the second year, the average annual return, accounting for compounding, is 7.49%.

## 4.Harmonic Mean

The Harmonic Mean is a statistical concept that enables to calculate the average value of a sequence of numbers, especially if these are rates or proportions. It is extremely useful when high discrepancies exist between values since it gives more weight to lower figures. The approach can also help you control for large outliers.

Now, the Harmonic Mean is a very special kind of average which becomes useful in certain situations, especially when one is concerned with rates or ratios. Unlike the arithmetic mean, wherein one simply adds up all of the numbers and then divides by the count, the Harmonic Mean does things a little differently. It gives greater weight to smaller values, and that turns out to be very useful when the data involve extreme outliers which can skew the average.

**Formula**

where: n is the total number of observations (how many numbers you have).

Xi represents each individual number in your dataset.

**Example:**

Let’s consider the set of numbers 2, 4, 6, 8, 10, 12, and 200. Their arithmetic mean would be about 34.57 and be too strongly influenced by that large number of 200. Using the Harmonic Mean, the average falls significantly to about 5.97. The reason this happens is because the Harmonic Mean places more weight on the smaller values, and does not let the large outlier of 200 disproportionately affect the result.

This is very useful in finance, especially when dealing with ratios such as the price-to-earnings ratio. For instance, take three companies with P/E ratios of 60, 20, and 20, respectively. The arithmetic mean of the three will be 33.3, while the Harmonic Mean-the calculation of which does decrease the influence of the higher P/E of 60-figures to approximately 28.8. This value, being lower, gives a better balance and makes comparison among companies much easier.

### Application

Another easily workable range of applications for the Harmonic Mean is in cost averaging, a strategy often followed by investors. If you regularly invest a fixed amount of money, you may purchase shares at different prices over time. The Harmonic Mean will allow you to find what average price per share you have paid and give a more appropriate view of your investment’s performance as a whole.

## Conclusion

In conclusion, knowing how to calculate returns in different ways is crucial for making smart investment choices. Whether you’re using **Holding Period Return (HPR)** to track an investment’s performance over time or opting for the **Arithmetic Mean Return (AMR)** for a straightforward average comparison, these methods offer valuable insights. When you need to consider compounding, the **Geometric Mean Return (GMR)** comes in handy, while the **Harmonic Mean** is useful for tackling outliers. Each approach serves a specific purpose, helping you evaluate risks, compare various assets, and gain a clearer understanding of your portfolio’s performance. By mastering these techniques, you’ll be able to manage your investments more confidently and strive for improved returns.