# Annualized rate of return

## Table of Contents

## Annualized rate of return Or ARR

In the financial world, a return is the profit made on an investment. When an investment is made by an investor, the returns on the investment over time will determine if the investor will continue to expand the business or scale down the business.

If the investor continues to invest when the returns are not in line with his projections or are not proportionate to the level of investment, then the investor risks running the business aground.

The need to ensure any investment is healthy is measured by indicators such as the rate of return. This enables the investor plan for future decisions. Also, projections can be made about the investment over a longer period of time, using data generated within the actual period in review.

That is where an **annualized rate of return** comes into play. It helps the investor make projections based on the present scenario.

An **annualized rate of return** is the rate of return on an investment over time that is less than one year or over one year, but is calculated as if it is for one year.

An **annualized rate of return** is basically an estimate of annual rate of return that has undergone some mathematical extrapolations.

The **annualized rate of return** is helpful in calculating and comparing past performances with the present. It is the equivalent annual return an investor makes on his investment over time. The **annualized rate of return** assists the investor in long haul planning and assessing immediate performances against projected and anticipated performances of the investment or business.

An **annualized rate of return** is a theoretical variable. It does not give you a guarantee that the value obtained in a month will subsist over preceding months to equate with what was predicted. It does not consider the effects of compounding on its value. Rates like inflation and interest rates are generally annualized.

## How to compute an annualized rate of return

The general formulae, which is exponential, considers the effects of compounding for calculating an annualized rate of return. It is presented as:

AP = {(P + G)/P} ^ (1/n)-1

Where P is the principal, G is the losses incurred, n is the number of years, AP is the annual performance rate.

Another formula involves using geometric averages and it is stated as thus:

R_{a }= [(1 + R_{C}/100)^{ P/N}– 1] × 100

Where R_{a }is the **annualized rate of return**, R_{C} is the cumulative or total rate of return, P is the number of time periods in a year or periodicity, N is the number of time periods under review or observation.

In calculating for **annualized rate of return**, care should be taken not to confuse it with annual rate of return. If there is any confusion, then the calculation will become jeopardised and the answer incorrect and inapplicable.

## Annualized Return

An** annualized return** is a periodic return (usually yearly) calculated by extrapolating or projecting returns predicted or measured over periods that may be longer or shorter than a year. It is a backward looking figure that deals with comparing returns from past years with that of the present year.

It is used to monitor and evaluate the performance of an investment, stock, and business over time. It is not a predictive tool, as past returns are not reflected by its values.

It cannot be used to predict or forecast future returns. It is computed as geometric average to depict what an investor would earn over time if the annual return was compounded.

An** annualised return** normally gives a semblance of an investment’s performance or output. One issue with annualised returns is it normally will not help the investor monitor and discern any issues that may negate the stability of the investment.

It is not a good tool to measure the volatility of the investment. **Annualised return** is based on two variables, the returns on the investment over a given period of time, which is depicted by the currency used. Another variable is the period in time the investment was held or effective.

An **annualized return** can be calculated daily. It is not calculated compulsorily on a yearly basis**. Annualized returns** or performance is sometimes confused with annual performance. That should not be the case. While an **annualized return** is the rate of growth of the investment over time to arrive at the final or projected valuation, annual returns is simply the return on the investment in a year.

Calculating for annualized returns is mostly done in days. This is because it will give a more in-depth and clarified picture. The generalized formula for calculating **annualized returns** is stated as thus:

Annualized return = [(1 + r1) × (1 + r2) × (1 + r3) × (1 + r4) ×.....× (1 + r (n))]^{ 1/n}– 1

Where r is the value of the return, and n is the period of time in years.

Assuming an investor gets returns of 2%, 4%, and 6% over 3 years, his annualized rate is

Using the formulae; = [(1 + r1) × (1 + r2) × (1 + r3) × (1 + r4) ×.....× (1 + r (n))]^{ 1/n}– 1

We have; = [(1 + 0.02) × (1 + 0.04) × (1 + 0.06)]^{ 1/3} – 1= [(1.02) ×(1.04)×(1.06)]^{0.33}-1=[1.125]^{0.33 }-1= 1.04 – 1 = 0.04. 0.04 multiplied by 100 will give an annualizing return of 4%.

If the time is in days, then the formula is adjusted to this format:

Annualized return= (1 + cumulative returns)^{ 365/days held }– 1

Using an example of an investor who invested in a mutual fund for 600 days and earned a cumulative return of 25%, the annualized return is;

Formulae for daily calculation is AR== (1 + cumulative returns)^{ 365/days held }– 1

WHERE the cumulative return is 25%, and days held is 600, we now have after converting 25% to its real number value, which is 0.25

(1 + 25)^{365/600}-1 = (1 + 0.25)^{365/600}-1= (1.25)^{0.608}-1=1.145-1 = 0.145

0.145 multiplied by 100 is equal to an annualized rate of 14.5%

**Return formula**

**The return formula** is used to calculate for the return on an investment. The **return formula** is a useful tool in computing the performance of every dollar invested in a stock, business, or strategy. The **return formula** helps to determine the profitability of the investment.

To understand the basis for the **return formula**, some knowledge of the return on investment is needed. The return on investment is a profitability ratio that computes the profit made in an investment or stock or business as a percentage of the original costs accrued in making the investment.

It is used in juxtaposing the performance of contrasting investments irrespective of the size and type of investment.

**Return formula** for return on investment can also be used in measuring the efficiency of an investment. It helps the investor decide whether he should pump more resources into the venture, expand the venture, or scale down operations.

The **return formula** is a simple and basic formula devoid of many mathematical complexities. The return on investment is based on two indices, the profit or returns and the cost of the investment. The return on investment is represented in percentage terms. The return on investment helps us to gauge the performance of an investment.

The return on investment formulae can be adjusted to suit the purpose of the analyst. It is a flexible formula. Its limitation is that there is no generally accepted formula to represent the **return formula**. The formula stated in this work is just for basic understanding.

The return on investment formula has also been modified and expanded to include social impacts of projects. This considers those affected by decisions taken while planning the investment and allocating resources for its execution.

The basic **return formula** is stated as thus:

Return on investment (ROI) = revenue accrued from investment – cost of investment

Cost of investment

For example, John invested in 6000 shares at 2 dollars per share. 3 years later, the value of each share is 4 dollars per share. What is John’s return on investment?

Using the formula = revenue accrued from investment – cost of investment × 100

Cost of investment

We have revenue from share is 4 × 6000 = 24000

Cost of investment (shares) is 2 × 6000 = 12000, we now have

24000 – 12000/12000 = 12000/12000 ×100 = 100% return on investment.

This means John had a 100% return on his investment. This implies his investment is healthy and profitable.

A positive return on investment shows every dollar was efficiently utilized, and the result was profitability. A negative return implies the cost of investment is greater than revenue accrual from the investment. This shows the investment is not efficient and operating at a loss.