## What is the Present Value of Perpetuity?

Perpetuity can be defined as the income stream that the individual gets for infinite time period and its present value is arrived 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.

### Explanation

Perpetuity is a series of cash flows that have an infinite life and such income stream grows with a proportionate rate. The cash flows should be identical.

The formula is basically derived from the dividend growth model. The formula attempts to determine the terminal value 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: –

**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 having discount rate only, 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 assets 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.

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**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 assets 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 dividends 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 dividends= 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 income 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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