## What is the Present Value of Perpetuity?

Perpetuity can be defined as the income stream that the individual gets for an infinite time period and its present value is arrived at by discounting the identical cash flows with the discounting rate. Here the cash flows are infinite but its present value amounts to a limited value.

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### Explanation

Perpetuity is a series of cash flowsCash FlowsCash Flow is the amount of cash or cash equivalent generated & consumed by a Company over a given period. It proves to be a prerequisite for analyzing the business’s strength, profitability, & scope for betterment. read more that have an infinite life, and such an income stream grows with a proportionate rate. The cash flows should be identical.

The formula is basically derived from the dividend growth modelDividend Growth ModelThe Dividend Discount Model (DDM) is a method of calculating the stock price based on the likely dividends that will be paid and discounting them at the expected yearly rate. In other words, it is used to value stocks based on the future dividends' net present value.read more. The formula attempts to determine the terminal valueTerminal ValueTerminal Value is the value of a project at a stage beyond which it's present value cannot be calculated. This value is the permanent value from there onwards. read more of the identical cash flows. Therefore, the present value of the cash flows at basic expression can be derived as follows: –

Present value = D / (1+r) + D x (1 + g) / (1 + r) ^2 + D / (1+r) + D x (1 + g) ^2 / (1 + r) ^3……….

Present value = D / r

### Present Value of Perpetuity Formula

The formula is expressed as follows: –

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For eg:

Source: Present Value of Perpetuity (wallstreetmojo.com)

**PV of Perpetuity = ICF / r**

Here,

- The identical cash flows are regarded as the CF.
- The interest rate or the discounting rate is expressed as r.

If the perpetuity grows by a constant growth rate, then it would be expressed as described below: –

**PV of Perpetuity = ICF / (r – g)**

Here,

- The identical cash flows are regarded as the CF.
- The interest rate or the discounting rate is expressed as r.
- The growth rate is expressed as g.

### How to Calculate Present Value of Perpetuity?

To calculate it has a discount rate only, the following steps should be performed as displayed below: –

**Step #1 –** Choose the financial instrumentFinancial InstrumentFinancial instruments are certain contracts or documents that act as financial assets such as debentures and bonds, receivables, cash deposits, bank balances, swaps, cap, futures, shares, bills of exchange, forwards, FRA or forward rate agreement, etc. to one organization and as a liability to another organization and are solely taken into use for trading purposes.read more or asset that provides sustainable infinite cash flows for its entire life cycle. Such financial assetsFinancial AssetsFinancial assets are investment assets whose value derives from a contractual claim on what they represent. These are liquid assets because the economic resources or ownership can be converted into a valuable asset such as cash.read more or instruments could be rental residential property, rental commercial property, preferred stocks, and bonds.

**Step #2 –** Next, Determine the identical cash flows or the income stream.

**Step #3 – ** Next, determine the discount rate.

**Step #4 – ** To arrive at the PV of the perpetuity, divide the cash flows with the resulting value determined in step 3.

To calculate the PV of the perpetuity having discount rate and growth rate, the following steps should be performed as displayed below: –

**Step #1** – Choose the financial instrument or asset that provides sustainable infinite cash flows for its entire life cycle. Such financial assetsFinancial AssetsFinancial assets are investment assets whose value derives from a contractual claim on what they represent. These are liquid assets because the economic resources or ownership can be converted into a valuable asset such as cash.read more or instruments could be rental residential property, rental commercial property, preferred stocks, and bonds.

**Step #2 –** Next, Determine the identical cash flows or the income stream.

**Step #3 – ** Next, determine the discount rate.

**Step #4 – **Next, determine the growth rate, if any, corresponding to the infinite cash flows.

**Step #5 – **Next, determine the difference between the discount rate and the growth rate.

**Step #6 –** To arrive at the present value of the perpetuity, divide the cash flows with the resulting value determined in step 5.

### Examples

#### Example #1

Let us then take the example of a trading business. The business intends to receive an income of $120,000 for infinite tenure. The cost of capital for the business is at 13 percent. The cash flows grow at the proportionate basis of 3 percent. Help the management to determine it.

**Solution**

**Calculation of PV of Perpetuity **

- = $120, 000 / (13% – 3%)
**= $1,200,000**

#### Example #2

Let us then take the example of an individual investor who owns preferred stocks in company ABC. The business intends to distribute preferred dividendsPreferred DividendsPreferred dividends refer to the amount of dividends payable on preferred stock from profits earned by the company, and preferred stockholders have priority in receiving such dividends over common stockholders.read more of $20 per share for infinite tenure. The required rate of return for the investor is at 8 percent. The cash flows grow at the proportionate basis of 2 percent. The investor currently holds 200 shares of the company ABC. Help the investor to determine it. Compute the total value of dividend income as displayed below: –

**Solution**

**Total Value of Dividends**

The total value of dividendsDividendsDividends refer to the portion of business earnings paid to the shareholders as gratitude for investing in the company’s equity.read more= Preferred dividend per share x number of shares

- = $20 x 200
**Total Value of Dividends = $4,000**

**Calculation of PV of Perpetuity **

- = $4, 000 / (8% – 2%)
**= $66,666.67**

#### Example #3

Let us then take the example of the endowment scheme. The scheme intends to provide an income of $320,000 for infinite tenure. The required rate of return is 10 percent. Help the investor to determine it?

**Solution**

**Calculation of Present Value of Perpetuity **

- = $320, 000 / 10%
**= $3,200,000**

### Uses

- Perpetuity is normally utilized in preferred stocks.
- The preferred stocks tend to provide fixed dividends throughout the company life cycle.
- Since the perpetuity is an infinite amount, its present value helps in arriving at a value that has a limited amount.
- The perpetuity has its applications in real estate as well.
- If the real estate provides a sustainable income stream, then its present value is derived using the relationship of the present value of a perpetuity.
- Additionally, the PV of the perpetuity forms the basis for several endowment schemes and retirement planning.
- Endowment schemes are financial protection plans that provide financial protections as well as cater to a comprehensive saving plan.

- Such schemes, if planned properly, can deliver a fixed incomeFixed IncomeFixed Income refers to those investments that pay fixed interests and dividends to the investors until maturity. Government and corporate bonds are examples of fixed income investments.read more stream for infinite tenure.

### Conclusion

The perpetuity is identical cash flows that are received for infinite tenure. The PV of such income streams is derived by dividing through a discount rate and is termed as the present value of a perpetuity. The perpetuity determined through the discount rate may vary if the financial analyst modifies the discount rate at periodic levels.

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