**Part 2 – Lesson 6**

**What is a perpetuity and growing perpetuity?**

A perpetuity is a cash flow that is expected to be received every year forever (hence, “in perpetuity”). A growing perpetuity is a stream of cash flow that is expected to be received every year forever but also grow at the same growth rate forever.

For example, if we expect to receive $100 every year forever, this is considered a perpetuity.

On the other hand, if we expect to receive $100 in year 1, which will grow every year at a rate of 3% in perpetuity, this is considered a growing perpetuity.

Another concept in finance is an annuity. An annuity is the fixed amount of cash flow received for a fixed amount of time. For example, if we received $100 every year for 10 years, this would be considered an annuity. We will not go into the formula for calculating the present value of an annuity since it is rarely used in equity valuations, but the present value of an annuity can be calculated using the NPV formula covered in Part 2, Lesson 5.

**Calculating the present value of a perpetuity**

Present Value of a Perpetuity = Cash Flow / Discount Rate

The formula for calculating the present value of a perpetuity is straight forward. Let’s look at a few quick examples:

- PV of a perpetuity of $100 at 9% discount rate = $100 / .09 = $1,111.11
- PV of a perpetuity of $500 at 10% discount rate = $500 / .1 = $5,000
- PV of a perpetuity of $200 at 12% discount rate = $1,666.67

The formula simply calls for taking the cash flow and dividing it by the rate (expressed as a decimal point).

**Calculating the present value of a growing perpetuity**

**Present Value of a Growing Perpetuity = Year 1 Cash Flow / (Discount Rate – Perpetual Growth Rate)**

With a perpetuity that is expected to grow at a specific rate, the formula calls for the perpetual growth rate to be deducted from the discount rate prior to dividing it into the cash flow. It is important to note that the discount rate must be higher than the perpetual growth rate for the formula to work.

Let’s run through a few examples:

- PV of a perpetuity of $100 growing at 3% and discounted at 9% = $100 / (.09 – .03) = $1,666.67
- PV of a perpetuity of $500 growing at 2% and discounted at 10% = $500 / (.1 – .02) = $6,250
- PV of a perpetuity of $200 growing at 5% and discounted at 12% = $200 / (.12 – .05) = $2,857.14

Assuming the same cash flow and same discount rate, the value of a growing perpetuity is always higher than the value of a perpetuity (with no growth).

**Why are we discussing perpetuities and growing perpetuities?**

Below, I’ve copied the simple DCF model that we created in Part 2, Lesson 5.

If you look toward the end of the model, you’ll see that Company B’s final cash flow is in year 10. However, the value of Company B is not determined by just valuing the first 10 years of its cash flow. Rather, Company B’s present value should be based on __all__ future cash flows of the company. In other words, we need to value the cash flow of Company B into perpetuity.

For example, if we were building a valuation for Google, we would not just value the next 10 years of its operations and ignore all the cash flow that Google will generate from year 11 and beyond. Ignoring all this cash flow would significantly under-state the present value of Google’s cash flows.

**Terminal Value**

To account for all the cash flow that Company B will generate beyond year 10, we should use either a perpetuity or growing perpetuity formula in year 10. To use such a formula, we need to make some simplifying assumptions as to what will occur with regards to cash flow after year 10. If we make the simplifying assumption that Company B will earn $155 in free cash flow per share every year in perpetuity beyond year 10, then, we would value the final $155 cash flow stream using a perpetuity formula.

Instead, if we assume that the $155 of cash flow in year 10 will grow at a rate of 3% into perpetuity, then, we will use a growing perpetuity formula. With regards to discounted cash flow models, it is much more common for practitioners to use a growing perpetuity formula since it is assumed that the cash flow of the company will at least grow at the rate of inflation in the long-term.

**The calculation of “all remaining cash flow in perpetuity” within a discounted cash flow valuation is referred to as the terminal value.**

Investors often reference the phrase, “terminal value,” when discussing the long-term growth our outlook of a business.

Let’s insert a terminal value calculation into our valuation of Company B’s stock price, utilizing the below assumptions:

- Year 10 cash flow of $155 will grow 3% into perpetuity
- We will use a 10% discount rate

Note the following about our terminal value calculation:

- We calculate the present value of $155 cash flow growing at 3% and discounted at 10% using $155 / (.1 – .03) = $2,214
- It is important to pay attention to timing. The first year of cash flow for the growing perpetuity
**is not Year 10**. The first year of cash flow for the growing perpetuity is actually Year 11, which is equal to $155 * 1.03 = $159.65 - However, the present value of the growing perpetuity is assumed to be received at the beginning of Year 11, which is the same as the end of Year 10. Therefore, the Year 10 cash flow and the present value of the growing perpetuity (the terminal value) both appear in Year 10.
- To summarize, this means that we are receiving a theoretical terminal value cash flow of $2,281 at the end of year 10 along with the year 10 cash flow of $155
- Since the terminal value cash flow of $2,281 is calculated at the end of year 10, we still need to discount the terminal value back 10 years to arrive at a present value as of today
- We add the regular yearly cash flows with the terminal value and then discount this entire cash flow stream using our NPV formula in Excel

You may download this Excel file here: Part 2 Excel Download

And there we have it – we’ve now built a very simple, but complete DCF valuation. In subsequent lessons, we will dive deeper into discounted cash flow valuation models.