# Rate of Return

It is the percentage change in the value of an investment.

Return in finance means the profit or money made by an investment over a given period. It can also be said it's the profit made by investment.

It includes any change in the investment's value and any cash flows (or securities or other investments) the investor receives due to the investment.

Examples include interest payments, coupons, cash dividends, stock dividends, dividends on cash investments, and the payout from derivatives or structured products.

It can be calculated as a percentage of the total amount invested or in absolute terms.

The holding period return is another term for it.

**Formula**

*Rate of Return = ((Current Value – Initial Value) / Initial Value) x 100*

Many a time, Net Asset Value is used for calculation purposes.

How does this return calculation work? First, it tries to give the returns a standardized mathematical form that enables the comparison of various investment strategies and financial instruments. This arithmetic form gives investors an idea of where their investments are headed.

What is Annualized Rate of Return?

Investors want to manage their finances wisely; to do that, they need a standardized return rate that excludes time factors and gives a return in a specified period. Calculating an investment's annualized return rate is one method for doing that.

You can find the average return rate (or loss) on investment over 12 months by looking at the annualized total return.

It's the process of determining the investment's returns every year. The annualized perspective on the return rate helps the management view gains and losses annually.

The rate shows the profits or losses the investments have procured.

It is frequently stated as a percentage(can also be mentioned in ratio form). It should be noted that the average mentioned in the definition doesn't necessarily indicate arithmetic mean calculation. Here's the formula to evaluate it:

*Annualized Rate of Return =* *(Current Value / Initial Value)^(1/N)-1*

*Where N is the period*

## Common Ways of calculating returns

What is Compound Annual Growth Rate?

The annual increase of your investments over a given period is known as the compound annual growth rate or CAGR. In other words, it is a measurement of the annual return on your investments over a specific time with the effect of compounding taken into consideration.

This is one of the most accurate ways to determine how your investment returns will change over time. As you would have understood, this is also called an Annualized return rate, as it considers the compounding effect for average return calculation.

It is to be noted that Compound Annual Growth Rate works well for lump-sum investments. Systematic Investment Plans (SIPs) do not account for recurring investments; instead, the calculation simply considers the initial and end values.

Let us have a look at how Compound Annual Growth Rate is calculated:

*CAGR = (Current Value/Initial Value)^(1/N) - 1*

What is the Nominal Rate of Return?

The nominal return rate is another term used for a simple return rate. It doesn't take into consideration inflation for return calculations. For example, there is a possibility that someone might earn about a 5% return. However, the inflation in that country might be greater than 5%.

This rate of return doesn't incorporate taxes and investment fees.

*Nominal ROR = current market value - the original value of the investment/ original value of the investment*

Hence, it is important to consider inflation for the investment strategy selection. Therefore, we need a modification to this term. Coming on to the real rate of return.

What is the Real Rate of Return?

The annual return rate taken into account after taxes and inflation is the real rate of return. A nominal return rate, on the other hand, excludes taxes and inflation.

Similarly, the real return rate accounts for taxes or inflation in its computation. After making any investment, the investor will receive the true return rate.

You might receive a higher, more desirable return rate on your investment from other computations. But because they rarely consider what you'll receive after taxes, those do not adjust for inflation.

Formula:

*The real rate of return = Nominal return (%) - Inflation rate (%)*

There are some of the applications of return in Discounted Cash Flow Modeling. Let us dive into this, to understand better.

Discounted cash flow analysis (DCF) is a technique used in finance to value security, project, business, or asset utilizing the ideas of the time value of money.

Discounted cash flow analysis is frequently used in corporate financial management, real estate development, and investment financing.

To use this approach, all future cash flows are calculated and discounted using the cost of capital and risk-free return, and their present values are then determined (PVs). The net present value is the total of all expected future cash flows, both arriving and exiting (NPV).

Formula:

*DCF = CF1/(1+return) + CF2/(1+return)^2 + …..*

## Internal Rate of Return

The internal rate of return (IRR) is a formula for estimating the return rate on investment. The computation does not consider external variables like the risk-free rate, inflation, the cost of capital, or financial risk, hence the name "internal."

Ex-post or ex-ante applications of the approach are both possible. The IRR is a projection of a potential future annual rate of return when used ex-ante. It evaluates the real investment return of a past investment when applied ex-post.

What is the internal rate of return rule?

This rule states that if the IRR exceeds the required return, it is safe to proceed with the investment. This required rate is also known as the hurdle rate. The IRR doesn't represent the dollar amount; rather, it's a percentage.

It's quite different than NPV. It's the rate at which the NPV of all cash flows will equal zero in a DCF Analysis.

However, Investors using financial statements frequently utilize the accounting rate of return (ARR) rather than the economic rate of return (IRR).

This is to evaluate capital investment plans, assess the performance of corporations and public-sector enterprises, and price financial claims such as shares.

Whether ARR indicators are economically significant has been hotly debated for years because they are based on reported accounting statements.

Shares vs. Bonds:

**Shares:** Consider that you purchased two shares of ABC Ltd for a total of $100 each. In this case, your initial investment would have been worth $200 (=100*2).

Let us assume that ABC Ltd pays $2 per share in dividends over a year, which will total up to $4(2 per share), and the share price rises to $120 after a year.

This indicates that the total value of your investment would be $244 (the present value of the shares plus dividend payments).

Then you would divide the new value ($244) by $200, having subtracted the investment's initial value of $200. To convert this number to a percentage, multiply it by 100. This results in a 22% yearly return rate.

*Return = ((Present Value + dividends)/(Initial Value) - 1)*100*

*=> ((244/200)-1)*100 = 22% return*

Bonds - In contrast, if you own a $100 bond with a 5% interest rate that matures in four years, you will receive $5 in interest each year (bond value multiplied by interest rate).

If you sell the bond for $120 after a year, the bond's growth, or appreciation, was $20 during that time (subtract the original bond value from the new bond value).

The interest plus appreciation divided by the initial bond price, stated as a percentage, is used to calculate the return rate. Twenty-five percent is the return rate after a year ($5000 plus $20,000 divided by $100,000 multiplied by 100).

## AIR

There are more than 18 fallacies in the IRR approach. To overcome those, a new method has been introduced - The average Internal Rate of Return.

A novel method of financial modeling is AIRR. In particular, it makes it possible to evaluate the generation of shareholder value while also analyzing the economic viability of any project and obtaining a variety of economic data.

The method enhances the conventional NPV analysis while avoiding common IRR errors. It has these characteristics:

- It ensures the existence of the return rate and its uniqueness.
- Evaluates both the project AIRR and Equity AIRR
- It works well with Net Present Value
- Works well for changing interest rates and cost of capital
- Works for both - financial as well as real estate properties

Ways of calculating a Project's return rate

- The ratio of profit to invested capital
- Capital weighted arithmetic mean of the return rates
- grossing up the cost of capital by the investors' relative wealth increase
- calculating the market-based interest per unit of overall capital invested

Here are some of the ways of calculating AIRR:

AIRR 1: Ratio of Total Profit to Total Invested Capital

For one year:

*TI/TC = I1/C0*

*For 2 years: *

*AIRR = TI/TC = (I1+I2/(1+r))/(C0+C1/(1+r))*

*And so on*

AIRR 2: Average Return on Investment

*i1 = I1/C0, i2=I2/C1, …..,.*

*Present Value = PV(Ct) = Ct/(1+r)^t*

*AIRR = C0/PV(C)*i1 + (C1/(1+r))/(PV(C))*i2 + …..*

*The terms before "i" are considered as weights.*

As we can see, the project's ROI and project AIRR are the same terms (likewise, considering the equity capital, one, the equity AIRR, which is an average Return On Equity, ROE, evaluating the shareholder's overall return rate).

AIRR 3: Instantaneous AIRR = Ratio of Present Value of Investment to Present Value of Invested Capital

*PV(Ct) = Ct/(1+r)^t*

*PV(It) = It/(1+r)^t*

*Instantaneous AIRR = PV(I)/PV(C) = (PV(I1) + PV(I2) + PV(I3) + … ) / (PV(C1)+PV(C2)+...)*

*AIRR = PV(I)/PV(C)*(1+r)*

*(1+r) -> Conversion factor*

AIRR 4: Starting from cash flows

Let us define the term F as Income less capital used in that particular period, i.e.,

*F(t) = I(t) - (C(t)-C(t-1))*

*AIRR = r + (F0 + F1/(1+r) + F2/(1+r)^2 …)/(C0 + C1/(1+r) + C2/(1+r)^2)*(1+r)*

## Decisions using AIRR

The Decision criteria for AIRR is that you can proceed with the concerned investment if and only if AIRR>r, where r is a risk-free return.

There are two advantages to this return rate. On the one hand, it is a quick and economical computational shortcut because it can be computed without intermediate capitals.

On the other hand, it is based on data from the capital markets; it is derived from fundamental concepts in corporate finance theory and is predicated on the idea of a healthy market where profitable investment possibilities are quickly arbitrated away.

### Why does the AIRR paradigm work in cases where the IRR strategy does not?

The foundational principle, which is in tautological terms, is that a return rate is the amount of return on an investment per unit of capital and that a project is better characterized as a stream of cash flows, and capital is the only explanation.

Epistemologically, this indicates that no rate of return for a project can be determined without choosing the capital basis, either implicitly or explicitly. The IRR approach enables an automatic implicit procedure to choose fictitious capital that falsely mimics the project's actual capital.

As opposed to this, the AIRR model enables the evaluations to be exogenously chosen according to the appropriate capital base based on the type of projects, the economic data that is accessible, and the piece of information that is necessary.

Even though this is one of the most advanced methods, it still has challenges. The main challenge is selecting one AIRR from an endless number of options.

This is an economic and philosophical problem rather than a mathematical one. Indeed, it relates to the well-known issue in the philosophy of science known as the "problem of underdetermination of theory by data."

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