Present Value of Annuities and Perpetuities

1. Valuing Annuities

An annuity is a sequence of fixed-size cash flows occurring at regular intervals over a specific number of periods. Common examples include mortgage payments, car loan installments, and regular pension income.

While you could find the present value (PV) of an annuity by discounting each cash flow individually and summing them up (as we learned in the last lesson), this would be incredibly tedious for a long series of payments. Fortunately, there's a more elegant and efficient way.

The video below introduces the concept of an annuity and demonstrates how a simple formula, or "annuity factor," can be used to find the present value of the entire stream in one step.

The formula for the present value of an ordinary annuity (where payments are made at the end of each period) is:

Where:

• $PV$ = Present Value
• $C$ = Cash flow per period
• $i$ = Interest rate per period
• $n$ = Number of periods

This formula is derived from the sum of a geometric series, a concept you will be familiar with from your statistics background. We will focus on its practical application.

Example: What is the present value of receiving $1,000 at the end of each year for 5 years, if the discount rate is 6%?

This means you would be indifferent between receiving $4,212.36 today and receiving $1,000 every year for the next five years, given a 6% interest rate.

The following reading provides a more formal presentation of the annuity formula and introduces important terminology. It also includes calculator notes, which are extremely useful for speeding up calculations in the CB1 exam.

Annuity Due

A small but important variation is the annuity due, where payments occur at the beginning of each period (e.g., rent payments). Since each payment is received one period earlier, it has more time to earn interest. The present value of an annuity due is therefore higher than that of an ordinary annuity by a factor of $(1+i)$.

2. Valuing Perpetuities

What if the stream of payments never ends? This is called a perpetuity. While seemingly abstract, this concept is used to value assets with very long-term, stable cash flows, such as certain types of preferred stock or historical government bonds called "consols."

The formula for the present value of a perpetuity is remarkably simple:

Where:

• $C$ = Cash flow per period
• $i$ = Interest rate per period

The intuition behind this formula is straightforward. If you deposit an amount $PV=C/i$ into a bank account that pays interest at a rate of $i$, it will generate interest of $PV\times i=(C/i)\times i=C$ each period. You could withdraw this interest payment $C$ every period forever, without ever touching the original principal.

The next video explains this concept very clearly.

Example: An endowment needs to fund a scholarship of $5,000 every year, forever. If the endowment can earn a reliable 5% return, how much principal is needed?

A one-time donation of $100,000 would be sufficient to fund the scholarship in perpetuity.

Deferred Perpetuities

Sometimes a perpetuity begins at a future date. This is known as a deferred perpetuity.

To find the present value of a deferred perpetuity, we use a two-step process that combines what you've learned:

1. Calculate the present value of the perpetuity as of the start of its payments (e.g., at time $t=n$). This gives you a single lump-sum value at that future date.
2. Discount this single lump sum back to today ($t=0$) using the standard present value formula from our first lesson.

This two-step discounting process is a common technique in financial valuation.

3. Valuing Growing Perpetuities

Our final variation is the growing perpetuity, an infinite stream of cash flows that grows at a constant rate, $g$. This model is famously used in stock valuation as the Gordon Growth Model, where a company's dividends are expected to grow indefinitely.

The formula for the present value of a growing perpetuity is:

Where:

• $C_1$ = Cash flow at the end of the first period
• $i$ = Discount rate
• $g$ = Constant growth rate

A crucial condition for this formula to be valid is that the discount rate must be greater than the growth rate ($i>g$). If the cash flows were to grow faster than the rate at which we discount them, the present value would be infinite.

This final video clip explains the formula and provides a practical stock valuation example.

Example: A company's stock is expected to pay a dividend of $2.00 next year ($C_1$). The dividend is forecast to grow at a steady 4% per year ($g$) forever. If your required rate of return for this stock is 10% ($i$), what is the present value of all future dividends?

According to this model, you would be willing to pay $33.33 for the stock today.

For a deeper dive into the formulas for growing cash flows, including the growing annuity (a finite stream of growing payments), the following resource is excellent.

Conclusion

In this lesson, we have moved from valuing single cash flows to valuing entire streams of them. These tools are indispensable in finance, forming the quantitative backbone for many valuation and investment decisions.

Key Takeaways:

• An annuity is a finite stream of equal payments. Its present value can be found efficiently with the annuity formula.
• A perpetuity is an infinite stream of equal payments. Its present value is calculated with the simple formula $PV=C/i$.
• A growing perpetuity is an infinite stream of payments growing at a constant rate $g$. Its present value is given by $PV=C_1/(i-g)$, but only if $i>g$.
• The timing of cash flows is critical. Distinguishing between an ordinary annuity (end-of-period payments) and an annuity due (beginning-of-period payments) is essential for accurate valuation.

The skills you've developed today are direct inputs for our next topic. In the following lesson, we will learn how to apply these present value calculations to make investment decisions by using the Net Present Value (NPV) rule to compare projects and choose the one that creates the most value.