**Summary: **Perpetuity is a fundamental concept in finance, particularly in valuing stocks and enterprises. It refers to a constant stream of cash flows expected to continue indefinitely. When valuing a stock with expected constant dividends, the calculation resembles that of a perpetuity. The present value of such dividends is determined by discounting them using the market rate of return, recognizing that future cash flows have a different purchasing power than today’s. For stocks, the discount rate typically reflects the cost of equity, incorporating both the risk-free rate and the additional returns expected for taking on investment risks. The concept extends to growing perpetuities, where cash flows increase at a constant rate, necessitating adjustments in valuation formulas. Delayed perpetuities introduce complexities when cash flows begin after a specified period. In enterprise valuation, the assumption of a going concern is critical; free cash flows are discounted individually until a stable growth rate is achieved, at which point the terminal value—reflecting the growing perpetuity—is calculated. The overall value of the enterprise is the sum of the discounted cash flows and the present value of the terminal value, using the Weighted Average Cost of Capital (WACC) for discounting. Understanding these elements is crucial for accurately assessing the value of investments.

**Valuation **and the role of **–**** Perpetuity**

**Do You Believe **that the perpetuity is the key assumption behind the valuation of a stock or an enterprise? Let me take you on a journey to explore how perpetuity serves as the backbone of valuation.

Valuing a stock that is expected to pay out a constant dividend each year for an indefinite period is just similar to valuating a perpetuity. The word “perpetuity” has Latin origins. It comes from the Latin word “perpetuitas,” which is derived from “perpetuus,” meaning “continuous” or “uninterrupted.” In finance, a perpetuity specifically refers to a hypothetical stream of cash flows that continues forever.

Imagine you have the option to choose an investment that provides a fixed return at the end of each year, forever. Can you guess how to calculate the current value of such an investment, considering it generates returns indefinitely?

Yes, the best suitable way for calculating such value is discounting such returns that are receivable at the end of each year with the current market rate of returns. Because, the purchasing power of a rupee after a year will not be same as that of a rupee today. The very familiar formula below can help us to calculate the present value of the future cash flows.

The formula is ready, now the question comes, if the period at which returns will stop is undefined, how do we discount cash flows for an indefinite number of years? If we observe the current scenario carefully, the return that is expected to receive at the end of each year is constant. So, the above formula can be written as below:

The simplified version of the above formula looks like:

Let’s understand how the above formula works in a practical scenario. Assume the return offered is an amount of Rs. 10,000 at the end of each year and the market rate of return is 10%. To generate the same proceeds by investing at the market rate, we need Rs. 1,00,000 (10,000/10%). So, if we invest Rs. 1,00,000 at the beginning of the year, by the year end it will generate Rs. 10,000. Wealth at the yearend amounts to Rs. 1,10,000. As we will withdraw Rs. 10,000 remaining will be Rs.1,00,000. Same will happen in second year. And the cycle continues for an indefinite period.

Period | 0 | 1 | 2 | 3 | 4 | 5 | n |

Wealth | 1,00,000 | 1,10,000 | 1,10,000 | 1,10,000 | 1,10,000 | 1,10,000 | 1,10,000 |

Drawn | – | 10,000 | 10,000 | 10,000 | 10,000 | 10,000 | 10,000 |

Balance | 1,00,000 | 1,00,000 | 1,00,000 | 1,00,000 | 1,00,000 | 1,00,000 | 1,00,000 |

So, the present value of the perpetuity can be determined like we did in the above scenario.

Now, we will speak about valuating a stock, where it is expected to pay out a dividend for an undefined period. The present value of the future net cash flows (the **future wealth**) will be the intrinsic value of the stock. But, what rate should we use to discount the dividend that is expected to receive?

The appropriate rate would be the Ke (Cost of equity a.k.a. Expected rate of return) which can be determined by using the formula, Rf + (Rm-Rf)*f3.

It can be interpreted as below:

Rf is the Risk-free rate of return that is available in the market.

Rm is the average rate of return that is available in the market.

In addition, to the risk-free return we expect some extra rate of return for the risk involved in the stock. To calculate the same, the excess return that is available in market over the risk-free return (i.e. Rm-Rf) is multiplied with the risk multiplier (i.e. f3) which represents the risk/volatility involved in the stock in relation to the market risk.

The combination of such risk-free return and the rate of return expected to receive for taking the risk results Ke, which is known as the “Expected Rate of Return”.

Finally, the value of the stock can be determined by applying the below formula: D1

The above paragraphs explain you the basic understanding of determining the current value of the stock. However, it all runs upon a very important assumption that it pays out a **constant **amount as dividend at the end of each year.

Let’s take this to the next level. Let’s assume that the company pays a dividend that is expected to grow at a constant rate each year.

Here comes the concept of **“Growing Perpetuity”**

Growing perpetuity is nothing but, there will be a constant growth in the returns that are expected to receive at the end of each year. If we continue with our previous example, at the end of year1 we are expected to receive Rs. 10,000. Now, let the growth be 5%. At the end of year2 the return expected to receive is Rs. 10,500. The value of such investment where there is a growth will be definitely higher than the one without growth in returns. It is very clear that in the current scenario the value of the investment will be higher than Rs. 1,00,000. The future cash flows are not constant now and so the formula used earlier needs a minor modification which adds growth to the future cash flows as shown as below:

The simplified version of the formula looks like:

Let’s understand how the above formula works in a practical scenario. Assume the return offered is an amount of Rs. 10,000 at the end of each year and the market rate of return is 10% and the growth rate is of 5% with the below mathematical representation:

Period | 0 | 1 | 2 | 3 | 4 | 5 |

Wealth | 2,00,000 | 2,20,000 | 2,31,000 | 2,42,550 | 2,54,678 | 2,67,411 |

Drawn | – | 10,000 | 10,500 | 11,025 | 11,576 | 12,155 |

Balance | 2,00,000 | 2,10,000 | 2,20,500 | 2,31,525 | 2,43,101 | 2,55,256 |

The required investment for generating the desired returns with growth is not Rs. 1,00,000, as shown as above it results the value of investment as Rs. 2,00,000 (i.e. 10,000/(10%-5%).

At the end of year1 the total wealth in our hands is Rs. 2,20,000 out of which we withdrew 10,000. The remaining wealth left is 2,10,000 which helps to generate a compounded return with a growth of 5% for next year. And the cycle continues for an indefinite period.

Back to the valuation of the stock. If the above formula used for determining the value of the growing perpetuity, converted to determine the value of the stock in current scenario, looks like below:

Where the D1 is the dividend expected to be received at the end of year1 and the “g” refers to the growth in dividend. Finally, we successfully came to know how to determine the value of the stock when there is a constant growth in the dividend expected to be received at the year end.

It’s time to level up.

**“Delayed Perpetuity”**, does it sound strange?

it’s very simple, when the expected date of receiving the first return is not the end of year1 but at a later point of time, for instance, at the end of year3. How do we calculate the present value of such delayed perpetuity?

Simple, calculate the present value as at the beginning of the year3, using the formulas we

discussed above (based on whether there exists growth factor or not), further discount the determined value from such date to the current date using a normal present value formula. The

result will be the present value of the delayed perpetuity.

Imagine, that there is no growth in the dividend expected for first two years, and then it is expected that a growth in dividends with a constant growth rate. So, the first two year’s

dividends will be discounted normally and then we have to determine the value of such delayed perpetuity with a constant growth rate, as at the end of the year2. Then the same can be discounted to present value. The sum of present value of the dividends of first two years and the present value of the delayed perpetuity results the value of the stock as on the current date.

The above discussion about determining the present value of the delayed perpetuity will make more sense now, as this will be the more practical scenario than earlier. Assume, there is growth in the dividend that is expected to be received at the end of each period. However, the same is not a constant growth rate for few years. It takes some time to such growth rate to get stabilized in the long run.

To use the concept of valuation of a growing perpetuity the growth rate must be stabilized/constant. So, till the growth rate of dividend gets stabilized, valuating the stock

becomes difficult. To achieve the desired result, discount the dividends expected to be received at the end of each year individually. To determine the value of the delayed perpetuity, as at the beginning of the period in which the growth rate is expected to be stabilized. Then discount the determined value of the delayed perpetuity (the **future wealth**) to the current date. The

summation of the present values of the dividends determined individually and the present value of the delayed perpetuity results the current value of the stock.

To understand the above clearly, we will go through the following example:

Let Ke be 20% and D1 = 100,

growth rate for first two years be 5%,

growth rate for next two years be 10%,

growth rate is expected to be stabilized @ 13% later.

Year Dividend Dividend Growth | Present Value | ||

1 | 100.0 | NA | 83.3 |

2 | 105.0 | 5% | 72.9 |

3 | 115.5 | 10% | 66.8 |

4 | 127.1 | 10% | 61.3 |

Total |
284.4 |

The present value of dividends at the end of each year amounts to Rs. 284.4

Year Dividend Dividend Growth Value at beginning of year 5 Present Value

5 | 143.6 | 13% | 2051.0 | 989.1 |

The value of the delayed perpetuity at the end of year4 is Rs. 2051, which turned to Rs. 989.1 when discounted to the beginning of the period. Consequently, the total value of the stock on current date (i.e. at the beginning of the period) is the sum of Rs. 284.4 and Rs. 989.1 which results Rs. 1,273.4

Finally, on we understood how the valuation of stock is similar to valuation of perpetuity. Let us now delve into the “**Valuation of an Enterprise” **using the **DCF **method.

The very basic assumption of evaluating an enterprise is going concern. If no going concern is assumed, then the value of the enterprise will be based on the fair value of the net assets of the enterprise, which we do generally at the time of liquidation of such enterprise.

The growth rate of the enterprise in the long run will not be stable. The business cycle is a matter of fluctuation of the growth and returns. However, when an entity reaches the stage of maturity it is expected to have a stable growth. Till the it reached the phase of maturity it faces several ups and downs in terms of business growth. For instance, the growth of the business in the initial phase will be slightly slower and may also causes losses. However, the growth of the business will be very rapid once it reaches the phase where it has a solid customer base and demand. In the further stage, the demand for the products/services of it will become low. The growth will be of a slow but not zero which we consider to be stabilized.

Therefore, once the business reaches its maturity, the growth rate will be expected to be stable and constant. That means it generates stable returns in the future. With an assumption of going concern we can see the growing perpetuity here. The ultimate value of such returns is nothing but the value of the growing perpetuity which we call as the **“Terminal Value” **of the entity. Thus, the valuation of the enterprise became simple with the concept of “Delayed Perpetuity”. The free cash flows of the initial years where the growth rate is not expected to be stable will be discounted individually and once the growth rate is expected to be stable, the terminal value will be calculated. The summation of the individually discounted cash flows and the current value of the terminal value results the present value of the enterprise.

The rate used to discount the free cashflows will be the “Weighted Average Cost of Capital” a.k.a. WACC, as the capital employed by the enterprise to earn the free cash flows each year might be of different components like equity and debt with different weights.

In closing, I extend my heartfelt gratitude to all readers for their patience and keen attention throughout this exploration of valuation of perpetuity, growing perpetuity and delayed perpetuity, and understanding how the concept of valuating perpetuity is being used in valuation of an enterprise or a stock. Your dedication to mastering these financial concepts is truly commendable!

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