Amy started a florist shop. She made an investment of $4,500 on it, including the renovations. How would she analyze the profit she made in 2 years and if the idea of starting this business was relevant enough?

For calculating the net gain or loss an investment brought, we study the rate of return of the complete transaction.

Let us study how it is done.

In this chapter, we will learn about the total rate of return formula, rate of return calculator, internal rate of return formula, and rate of return formula examples in the concept of Rate of Return Formula.

Check out the interactive simulation to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.

**Lesson Plan**

**What Is Meant by the Rate of Return Formula?**

**Definition**

The rate of return formula calculates the total return on an investment over a period of time.

The nature of return can be either profitable or of loss.

If the rate of return formula gives a positive value, that means that there is a gain or profit in the investment. A negative value for the rate of return formula means that a loss has occurred on the invested amount.

It is expressed in the form of a percentage and can be referred to as ROR.

**Formula**

The rate of return on investment can be calculated with the help of the following total rate of return formula:

\( \text {Rate of Return} \ = \)

\(\frac{ \text {Current Value - Original Value } }{ \text {Original Value} } \times 100\)

\( \ \text R \ = \ \frac{\text V_c \ - \ \text V_o}{ \text V_o } \times 100 \) |

where,

\( \begin{align*} \text V_c \ &= \ \text {Current value} \\ \text V_o &= \text {Original value} \end{align*} \)

**How to Use the Rate of Return Formula?**

The rate of return is the net loss or profit on investment when calculated for a specified period of time.

It determines the percentage change that occurred in the investment and cash flows which an investor receives since the beginning of the term.

To compare returns over unequal time periods on an equal basis, we convert each return into a return over a standard period of time. This conversion result is called called the rate of return.

Generally, the period of time is a year, thus the rate of return for this case would be the annualized return and the conversion process is called annualization.

The significance of the rate of return formula can be understood better using the following examples:

**Rate of Return Formula Example**

An investor purchased a share at $10 and he had purchased 500 shares in the year 2017. After one year, he decides to sell them at $15 in the year 2018. Now, he wants to calculate the rate of return on his invested amount of $5,000.

We know,

\( \text {ROR} \ = \ \frac{ \text {Current Value - Original Value } }{ \text {Original Value} } \times 100 \)

Putting the values in the above formula, we get,

\( \text {ROR} \ = \ \frac{15 \times 500 - 10 \times 500}{ 10 \times 500} \times 100 \)

\(\text {ROR} \ = \frac{7500 - 5000}{5000} \times 100\)

\(\text {Rate of Return} = 50 \text{%} \ \)

Rate of return on shares is 50%

Thus, in the above example we saw how the rate of return formula finds application in calculating the net profit obtained on an investment over a period of time.

**Internal Rate of Return**

The internal rate of return (IRR) is that interest rate that makes the Net Present Value zero.

It is a key component of capital budgeting and corporate finance which is used in determining which discount rate will make the initial cost of a capital investment equal to the current value of future cash flows post-tax.

The term internal here signifies the fact that the calculation of rate using this method does not include external factors, such as inflation, risk-free rate, the cost of capital, or any financial risks.

It is also referred to as the discounted cash flow rate of return (DCFROR).

**Internal rate of return formula**

The calculation of IRR includes the trial and error method with the formula of calculation of Net present value(NPV).

Net present value (NPV) determines the total current value of all cash flows generated, including the initial capital investment, by a project.

It finds application in estimating which projects are likely to generate great profits.

The following formula is used to calculate NPV:

\( NPV = \sum_{n = 1}^N \frac{ C_n }{ (1 + r)^n } \)

\( \begin{align*} \text N \ &= \ \text {Total number of time periods} \\ \text n &= \text {time period} \\ \text C_n &= \text {Net cash flow at time period} \\ \text r &= \text {Internal rate of return} \end{align*} \)

That value of \( \text r \) which makes the net present value equal to zero is called internal rate of return.

**Example**

Let us study a real-life example to relate to this:

Anna plans on starting a restaurant. She estimates all the net earnings and investments for 2 years and calculates the net present value.

At 6%, she gets a net present value of $1500. But, the desired net present value should be zero.

So, she keeps calculating for other rates.

At 8%, she gets the net present value at $10, which is close enough to zero.

Thus, the desired internal rate of return is 8%.

**Rate of Return Calculator**

The following rate of calculator can be used to calculate the return obtained on any investment:

- A higher internal rate of return would indicate better profitability of an investment.
- For a net present value (NPV) > 0, profit is expected from the investment.
- For NPV < 0, the investment is expected to suffer loss.
- For NPV = 0, no significant gain or loss is expected for the investment.
- A higher internal rate of return but low NPV would mean substantial growth in returns but it would not add much to the cost bearing entity.
- A lower IRR but high NPV indicates that the return might be slow, but they would add a significant value to the investment.

**Solved Examples**

Example 1 |

Daniel bought a house for $250,000. He plans on selling the house six years later for $335,000, after deducting any realtor's fees and taxes. Calculate the rate of return on the complete transaction.

**Solution**

We know, the formula for rate of return is given by:

\( \ \text R \ = \ \frac{\text V_c \ - \ \text V_o}{ \text V_o } \times 100 \)

Here,

Current value of the house \( V_c\) = $335,000

Original value of the house \( V_o\) = $250,000

Thus, rate of return on complete transaction,

\( \begin{align*} \text R \ &= \ \frac{\text V_c \ - \ \text V_o}{ \text V_o } \times 100 \\ \text R \ &= \ \frac{ ($335,000 - $250,000)}{$250,000} \times 100 \\ &= \ \frac{ ($85,000)}{$250,000} \times 100 \\ &= \ 34 \text % \end{align*} \)

\(\therefore\) Rate of return = 34% |

Example 2 |

The stocks of a Spill Drink Company had -8%, 12%, and 23% rates of return during the last three years respectively. Can you calculate the arithmetic average rate of return for the stock?

**Solution**

We know,

\(\text {Arithmetic avg} = \frac{ \text {Sum of all the individual values}}{ \text {Total number of values}}\)

\(\text {Arithmetic avg of all the rates of return}\)

\( \begin{align*} &= \ \frac{-8 + 12 + 23}{3} \\ &= \ \frac{ 27}{3} \\ &= \ 9 \text {%} \end{align*} \)

\(\therefore\) Arithmetic average rate of return = 9% |

Example 3 |

Sofia invested in real estate. The rate of return for her investment is 9%. If the current value of the real estate is $160,000, find the original value she invested.

**Solution**

We know, the formula for rate of return is given by:

\( \ \text R \ = \ \frac{\text V_c \ - \ \text V_o}{ \text V_o } \times 100 \)

Here,

Rate of return for Sofia's investment\( ( R ) \) = 9%

Current value of the real estate \( (V_c)\) = $160,000

Thus, for original invested on the house \( (V_o)\),

\( \begin{align*} \text R \ &= \ \frac{\text V_c \ - \ \text V_o}{ \text V_o } \times 100 \\ 9 \ &= \ \frac{ ($160,000 - \text V_o )}{ \text V_o } \times 100 \\ 9 \text V_o &= \ ($160,000 \ - \text V_o) \times 100 \\ 0.09 \text V_o &= \ $160,000 - \text V_o \\ 1.09 \text V_c \ &= \ $160,000 \\ \text V_c \ &= \ $84,210.526 \end{align*} \)

\(\therefore\) Original invested on the real estate = $146,788.990 |

Example 4 |

Emily invested $3,500 in a bakery. She got a rate of return of 15% on her complete investment, including the maintenance. Calculate the profit she made from the investment.

**Solution**

We know, the formula for rate of return is given by:

\( \ \text R \ = \ \frac{\text V_c \ - \ \text V_o}{ \text V_o } \times 100 \)

Here,

Rate of return for Sofia's investment\( ( R ) \) = 15%

Original value invested on the house \( (V_o)\) = $3,500

Thus, for current value invested on the house \( (V_c)\),

\( \begin{align*} \text R \ &= \ \frac{ \text V_c \ - \ \text V_o}{ \text V_o } \times 100 \\ 15 \ &= \ \frac{ ( \text V_c - $3,500 )}{ $3,500 } \times 100 \\ 15 &= \ \frac{(\text V_c - $3500)}{35} \\ 15 \times 35 &= \ \text V_c - $3,500 \\ V_c &= \ $4,025 \end{align*} \)

Profit = Current value invested on the house - Original value invested on the house

Hence,

\( \begin{align*} \text {Profit} &= $4,025 - $3,500 \\ &= $525 \end{align*} \)

\(\therefore\) Profit = $525 |

Example 5 |

Miley invested $500 in a company and gets back $570 the next year. If the rate of return is 10%. Calculate the net present value.

**Solution**

The amount invested by Miley = $500

The money received after a year = $570

Rate of return = 10%

We know,

\( \begin{align*} \text {PV} &= \frac { \text {cash value at time period }}{ (1 + \text {rate of return})^{ \text {time period}}} \\ \text{PV} &= \frac { \$ 570}{ (1 + 0.1 )^1} \\ \text{PV} &= \frac{ \$570 }{ 1.1 } \\ \text{PV} &= \$518.18 \end{align*} \)

Net Present Value = $518.18 − $500.00 = $18.18

So,for 10% rate of return, investment has NPV = $18.18

\(\therefore\) Net present value = $18.18 |

- Kevin invested $2000 and received 3 yearly payments of $100 each, plus $2,500 in the final year. Find the internal rate of interest.
- How will you compare the different values of IRR and NPV to determine the scope of an investment?
- Write an expression to calculate the IRR using the NPV formula.

**Interactive Questions**

**Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.**

**Let's Summarize**

The mini-lesson targeted the fascinating concept of the rate of return formula. The math journey around the rate of return formula started with what a student already knew and went on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever.

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**Frequently Asked Questions (FAQs)**

## 1. How do you calculate the rate of return on a product?

The rate of return on a product can be calculated with the help of the following formula:

\( \text {R} \ = \ \frac{ \text {CV(prod) - OV(prod) } }{ \text {OV(prod)} } \times 100 \)

## 2. How do you calculate the rate of return on investment?

The rate of return on investment can be calculated with the help of the following formula:

\( \text {R} \ = \ \frac{ \text {TV(invest) - OV(invest) } }{ \text {OV(invest)} } \times 100 \)

## 3. What is a good rate of return?

A good rate of return is that rate of return that adds a substantial amount of profit to the investment made by an investor.

Generally, 15% is considered a good rate of return annually.