Perpetuities are securities that pay a constant stream of identical cash flows for an infinite amount of time. When it comes to perpetuities, the series of cash flows received by the investor is never ending, it is expected to be received forever. The present value of perpetuity formula and calculation is, therefore, used to gain better insight into how much of a return one can expect from certain investments.

An analyst can use the finite PV of perpetuity to determine the exact value of a company or investment if it continues to perform at the same rate. It is one of the many ways to analyze the time value of money, which many investors take a look into to make investment decisions.

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In real estate, for instance, perpetuity can exist if one invests in a rental property because the investor is expected to have a continuous flow of income for an unknown period in the future.

This continuous flow of income is also applicable in the stock exchange market. Company stocks that pay dividends are examples of perpetuities because they do not have a date on which there is a promised maturity or endpoint, hence, as long as the stock owner does not sell his shares, the dividends owed to him will never cease to be paid.

Another example of perpetuity is the British-issued bonds known as consols, which the Bank of England phased out in 2015. Bondholders that purchased a consol from the British government, were entitled to receive annual interest payments forever.

The time value of money allows certain real estate investments, preferred stocks, and bonds to accrue money endlessly, allowing them to become perpetuities for investors. Since such investments pay investors a recurring fixed annual amount, a present value of perpetuity formula and calculation is used to gauge the current value of the specific amount the investor would receive after every period.

See also: Delayed Perpetuity Examples and Types

Present value of perpetuity formula

Despite perpetuities being an infinite series of cash flows, they can have a definable present value. That is, the total value of a perpetuity may be infinite, but it comes with a limited present value. Due to the time value of money, each payment gotten from perpetuity is only a fraction of the last.

The terminal year of operation assumes that an investment or business will continue to generate cash flows at a constant stable rate forever. That is, the PV of perpetuity formula determines the value of the amount of cash flows in the terminal year of operation.

The basic formula used to calculate the present value of a perpetuity or security with perpetual cash flows is expressed by dividing cash flows by discount rate or interest rate (this is known as Flat Perpetuity or Perpetuity with zero growth).

However, for valuation purposes, an advanced formula that factors in long-term growth is used to calculate the terminal value in a stream of cash flows (known as Growing Perpetuity).

The formula used to calculate the present value of a growing perpetuity is a bit more complicated than the first. It is the estimate of cash flows expected to be received in the next year divided by the difference between the discount rate and the growth rate.

The present value of perpetuity formula (with zero growth)

The flat (zero growth) perpetuity formula is straightforward as it doesn’t include terminal value. An example of a zero-growth perpetuity is an investment that comes with terms stating that a $1,000 payment will be paid out at the end of each year with an indefinite end. This means the annual payout of $1,000 remains the same throughout the life of the investment.

That is, to calculate the present value of perpetuity with zero growth, simply divide the regular cash flow amount by the interest or discount rate. Hence, the present value of a perpetuity formula is expressed as:

PV of perpetuity = CF / r

Where,

CF=The identical cash flows

r =The interest rate or the discounting rate

Present value of perpetuity with growth formula

The present value of growing perpetuity formula factors in a growth rate that increases the cash flows received each period going forward.

As mentioned in the prior example for flat perpetuity, the $1,000 annual payment is kept constant throughout the entire duration of the investment. However, for growing perpetuities, that is not the case as there is a continuous growth rate attached to the series of cash flows. As a result, with equal initial payment amounts, a growing perpetuity will be valued higher than one with zero growth.

For instance, if an investment comes with terms stating that $1,000 would be paid out in the following year but at a 2% growth rate, it means that the annual cash flows would increase by 2% year-over-year (YoY). Since the cash flows increase each year, the growth rate will help to offset the discount rate used to calculate the PV of perpetuity.

That is, if perpetuity grows by a constant growth rate, it would be calculated by dividing the cash flow amount expected to be received in the next year by the discount rate minus the constant growth rate. Hence, the formula for the present value of growing perpetuity is expressed as:

Present value of a growing perpetuity = CF / (r – g)

Where,

CF=The identical cash flows

r=The interest rate or the discounting rate

g= Growth rate

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Present value of a perpetuity calculation

According to the ‘time value of money’ concept, the further away from the date when a cash flow payment is received, the greater the reduction in its current value. That is, the present value of the future cash flows of perpetuity eventually reaches a point in the far future where the cash flow payments have a PV of zero.

Many of the assets that are calculated as perpetuities could have actual ends, but these ends are not known in advance, hence the reason for the present value of perpetuity calculation.

Knowing and calculating the perpetuity present value is important in business valuation because, despite the payment from the investment remaining fixed or equal over the years, the value of perpetuity is likely to change with time due to certain factors like inflation or a change in the interest rate.

How to calculate the present value of a perpetuity

1. Determine the cash flow
2. Divide the cash flow by the interest rate (or interest rate minus growth rate for growing perpetuities)
3. Finalize the valuation of the perpetuity

Determine the cash flow

To calculate the present value of perpetuity, first, determine the cash flow. The stream of cash flows coming in from the specific perpetuity investment has to be determined.

For instance, supposing an investor owns a preferred stock that pays out an annual dividend of $100. This periodic payment of $100 stands in as the cash flow (CF) variable. So when calculating the PV of perpetuity, the formula would read, PV = $100 / r.

Divide the cash flow by the interest rate

For flat perpetuities, divide the cash flow by the interest rate. That is, assuming the interest rate (or discount rate) on the preferred stock that pays out an annual dividend of $100 is 5% a year, the present value of perpetuity will be expressed as, PV = $100 / 0.05.

On the other hand, if the investment is growing perpetuity, the present value will be calculated as $100 / (0.05 – growth rate)

Finalize the valuation of the perpetuity

To get the valuation of the perpetuity, divide the cash flow (CF) by the interest rate (r). For instance, the calculation for the preferred stock investment would be $100 / 0.05 = 2000. This means that the current value of the investment is rated at $2,000.

For further understanding let’s calculate and compare two perpetuities, sharing the following assumptions:

• Cash Flow Amount (Year 0) = $100
• Discount Rate (r) = 10.0%

Let’s assume the $100 annual payment amount from the first perpetuity remains fixed, while the payment from the second perpetuity grows at 2% per year perpetually. This means the first has zero growth rate while the other is a growing perpetuity:

1. Perpetuity with Zero Growth= 0% growth rate
2. Growing Pepertuity= 2% growth rate

For the zero-growth perpetuity, the present value of perpetuity is calculated using the formula:

PV of perpetuity = CF / r

Where,

CF = $100

r =10%= 0.10

PV = $100 / 0.10 = $1,000

The present value of this investment is $1000. This means that if someone comes to you and offers to sell this investment to you, you should only proceed with investing in it if the purchase price is less than or equal to $1,000; if not, the investment would not make much sense economically.

For the growing perpetuity, the first step in calculating the present value is to grow the cash flow amount for Year 0 by 2% once to arrive at the cash flow amount for Year 1. That is:

CF (Year 1) = $100 CF (Year 0) × (1 + 2% Growth Rate)

Year 1 Cash Flow = $102

Using the perpetuity with growth formula;

Present value of a growing perpetuity = CF / (r – g)

Where,

CF=$102

r=10%

g= 2%

Present value of growing perpetuity = $102 / (10% – 2%) = $1,275

From the PV of growing perpetuity calculation, we can see the positive impact that growth has on the value of a perpetuity. The present value of this growing investment is $275 more than the one with a zero-growth rate.

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Uses of the present value of perpetuity formula and calculation

1. Determining the present value of perpetuity is used in real estate valuation.
2. The present value of perpetuity calculation is used in the valuation of businesses.
3. Financial managers also make use of the PV of perpetuity formula to calculate the current values of the dividends for preferred and common stocks.
4. Calculating the PV of perpetuity can be used in retirement planning and endowment schemes.

In valuation, a business or investment is said to be a going concern, meaning that it goes on forever. For this reason, the terminal year of operation is a perpetuity, and analysts use the present value of perpetuity formula to find its value.

We will discuss how the PV of perpetuity formula and calculation is used in the valuation of long-term investments with working examples:

The present value of perpetuity is used for real estate valuation

Through rental income or lease payments, real estate investments can generate a steady cash flow over a long period. Investors can use the present value of a perpetuity formula to calculate the present value of the expected future cash flows from the property and determine the fair value of the property.

How to calculate present value of perpetuity in real estate valuation

Mr Jacob, a businessman, acquires a rental property that will pay him a rental income of $8,000 annually. Given the possible risks, he expects a 16% discount rate of owning the property and a 6% growth rate in the annual payments. Let’s calculate the value of his investment.

Solution

Using the PV of perpetuity formula:

PV = CF / (r – g)

Where,

CF = $8,000

r = 16%

g = 6%

Present value of growing perpetuity = $8,000 / (16% – 6%) = $80,000

The perpetuity value would be $80,000. This means that the current value of the investment is worth $80,000. Hence, with the assumptions of a 6% growth rate and a 16% discount rate of owning the property, selling the rental property for less than $80,000 would not be a good deal.

The present value of perpetuity calculation is used for business valuation

Like real estate investments, companies can generate a steady cash flow over a long period, usually through selling goods or services; investors can use a PV of perpetuity formula to calculate the present value of the estimated future cash flows from the business to determine the fair value of the business.

How to calculate the present value of a perpetuity for business valuation

Using a trading business as an example, let’s assume the business intends to receive an income of $120,000 for infinite tenure. The cost of capital for the business is at 13% and the cash flows grow at the proportionate basis of 3%. Here is how the management of the business will determine the present value of the cash flow:

Solution

Using the present value of perpetuity formula;

PV = CF / (r – g)

Where,

CF=$120,000

r=13%

g=3%

Present value of a growing perpetuity = $120,000 / (13% – 3%) = $1,200,000

The PV of perpetuity formula is used to determine the value of dividends from bonds and stocks

Companies that trade in the stock exchange market for both preferred and common stocks tend to use the present value of perpetuity calculation to validate and determine the PV of the company’s cash flow. That is, the PV of perpetuity formula determines the number of cash flows in the terminal year of operation.

Financial managers also make use of the formula when calculating the present values of the dividends for preferred and common stocks. The calculated PV of perpetuity helps to determine the current value of the company if it were to continue performing at the same rate.

Present value of perpetuity example in stocks

Mrs Mary, an individual investor, owns preferred stocks in ABC Company. The company intends to distribute preferred dividends of $20 per share for infinite tenure and she currently holds 200 shares of the company. If the required rate of return for the investor is at 8% and the cash flows grow at the proportionate basis of 2%, let’s look at how Mrs Mary would determine the present value of the perpetuity.

Solution

Computing the total value of dividend income would be:

Preferred dividend per share X number of shares = $20 x 200 = $4,000

i.e Total value of dividends = $4,000

Using the present value of a perpetuity formula:

PV = CF / (r – g)

Where,

CF = $4,000

r = 8%

g = 2%

Present value of perpetuity = $4,000 / (8% – 2%) = $66,666.7

Present value of perpetuity example in bonds

Let’s assume you hold a perpetual bond that generates an annual payment of $500 each year. You believe that the borrower is creditworthy and that an 8% interest rate will be suitable for this bond. Calculate the present value for this perpetuity.

Solution

Using the present value of perpetuity formula:

PV of perpetuity = CF / r

CF = $500

r = 8% = 0.08

PV of perpetuity = $500 / 0.08 = $6,250

This calculated value tells us that someone can pay you $6,250 for your bond and receive an 8% return on their money.

Calculating the present value of a perpetuity is used for retirement planning and endowment schemes

Calculating the present value of a perpetuity is also relevant in retirement planning or an endowment scheme. For instance, an individual who expects to receive a fixed income from a pension plan could use the PV of perpetuity formula to determine the PV of those expected cash flows. This can help the person determine how much he/she needs to save to meet their retirement income needs.

The present value of perpetuity example in an endowment scheme

An endowment scheme intends to provide an income of $320,000 for infinite tenure. If the required rate of return is 10%, help the investor determine its present value.

Solution

Using the present value of perpetuity formula:

PV = CF/ r

Where,

CF = $320,000

r = 10% = 0.10

PV = $320,000 / 0.10

Present value of perpetuity = $3,200,000

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Limitation of the PV of perpetuity formula

Using the PV of perpetuity formula to compute the present value of an investment’s cash flow may seem straightforward, but one needs to understand the underlying concerns and limitations of this metric.

Since perpetuities are subject to market risk, the present value of perpetuity formula has its limitations and concerns because the PV of an investment may fluctuate based on changes in inflation, interest rates, or other market conditions.

For example, low discount rates or high growth rates may result in too high present values. Likewise, high discount rates or low growth rates lead to negative PVs. In some instances, a fixed cash flow may not keep pace with changes in market conditions or inflation, which can crumble the value of the investment over time.

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Conclusion

In conclusion, the present value of a perpetuity is the estimated valuation of the total potential stream of cash flows from an investment as of the current date.

To calculate the PV of perpetuity, simply divide the regular cash flow amount by the interest or discount rate (CF / r). For perpetuity growing at a constant growth rate, subtract the growth rate from the interest rate (r-g) and then divide the cash flow by the figure (CF / r-g). The calculated present value helps investors in investment decisions.

Nonetheless, in as much as the present value of perpetuity formula and calculation can be used in determining the value of long-term investments, it should not be used alone for valuation. This is because the present value of a perpetuity may fluctuate due to changes in inflation, interest rates, or other market conditions.

Other valuation methods, like net present value analysis or discounted cash flow, may be more appropriate to use in certain circumstances. Hence, investors should carefully consider an investment’s specific characteristics and choose the more suitable valuation method.