Chapter 4.17® - Accounting for Perpetuities & Using the Annuity Present Value Factor Formula

• Part 4.1 - Time Value of Money, Future Values of Compounding Interest, Investing for more than 1 Period & Examination of Original Investment & Growth of Investment
• Part 4.2 - Compounding Interest Homework Problem & Time Value of Money Continued - Future Value Formula, Growth of $100 & Future Value Comparisons
• Part 4.3 - How to Use a Financial Calculator BAII Plus to Perform Time Value of Money & Present / Future Value Calculations
• Part 4.4 - Changing Advanced Function Keys (BGN, C/Y, P/Y), Converting from Nominal Interest to Effective Interest Rates using BAII Financial Calculator
• Part 4.5 - Examples of Interest Rate Calculations & Practice Questions #1 - #7
• Part 4.6 - Nominal to Effective Interest Rate Calculations & Practice Questions #8 - #16
• Part 4.7 - Present Value & Discounting – We Know the Future Value, how do we find the Present Value? Single Period Case & PV Discounting for Multiple Periods
• Part 4.8 - An Example of Present Value Discounting & Saving Up Money & Deriving the Basic Present Value Equation
• Part 4.9 - Determining the Discount Rate using Basic Present Value equation & Finding the Number of Accounting Periods
• Part 4.10 - Summary of Time Value Formulas & Calculations
• Part 4.11 - Discounted Cash Flow Valuations - Future Value of Multiple Cash Flows & Designing the Cash Flows Timeline
• Part 4.12 - Compound the Accumulated Balance Forward One Year at a Time - Discounted Cash Flow Valuation - Determining Present Value of Multiple Future Cash Flows & Designing a Financial Timeline
• Part 4.13 - Determining Present Value of Multiple Future Cash Flows – Homework Example
• Part 4.14 - Calculating Present Value with Multiple Future Cash Flows – Example #2
• Part 4.15 - Valuing Annuity Level Cash Flows – Present & Future Value of Annuities
• Part 4.16 - Calculating Annuity Payments using Annuity Present Value Factor – Example
• Part 4.17 - Accounting for Perpetuities & Using the Annuity Present Value Factor Formula
• Part 4.18 - Accounting for Growing Perpetuities & Simplified Formula for Present Value of Growing Perpetuity - Example of Growing Perpetuity on Rental Cash Flows

From the chapter on present value of steady annuity cash flows, we know that we can find the present value of future steady annuity payments. However, what happens when the level of future cash flows is not for a definite period of time, but forever? Such an asset is known as perpetuity. Thus simply defined, perpetuity is an annuity in which the cash flows continue forever. Some perpetuity is also known as consols.

Since perpetuity has infinite number of cash flows, we can’t find its value by discounting each flow to its present value state. However, valuing perpetuity is very easy; consider this example. Say perpetuity costs $2000 and offers a 15% rate of return with payments at the end of the year. The cash flow each year must be:

To define this in terms of an equation, the present value of perpetuity (PV = $2,000) multiplied by the rate (r = 15%) = $300. The formula then is:

Therefore, given a cash flow and a rate of return, we can easily calculate the present value for the perpetuity:

For instance, an investment offers a perpetual cash flow of $1000 every year. The return you would expect from such an investment is 10% a year. What is the value of this investment? The value of this perpetuity will be:

Consider another example where an investment offers a perpetual cash flow of $4,000 per year and the return rate you should expect is 18% a year. What is the present value of this perpetuity?

Another way of looking at perpetuity’s value and why it is easy to determine is by taking into account the expression for an annuity present value factor:

Let’s revisit our previous example; an investment offers a perpetual cash flow of $1000 every year for 100 years. The return you would expect from such an investment is 10% a year. What is the value of this investment? The value of this perpetuity will be:

Suppose we did this calculation using our BAII Plus financial calculator. The table would look like:

This amount of $9,999.27 is pretty close to the $10,000 we derived using our annuity present value factor formula.