# Calculate Present Value Of Perpetuity

In a previous post I have described what perpetuity is and how we can derive the formula for the present value of perpetuity. In this article I describe how it can be used in combination with the discounted value of cash flows model.

In a previous post I have described what perpetuity is and how we can derive the following formulas to calculate its present value $$PV$$ at a discount rate $$d$$:

$$PV = \frac{C}{(1 + d)} + \frac{C}{(1 + d)^2} + \frac{C}{(1 + d)^3} + \dots + \frac{C}{(1 + d)^n} = \frac{C}{d}$$

or

$$PV = \frac{C}{(1 + d)} + \frac{C(1 + g)}{(1 + d)^2} + \frac{C(1 + g)^2}{(1 + d)^3} + \frac{C(1 + g)^3}{(1 + d)^4} + \dots = \frac{C}{(d-g)}$$

if we can assume that the free cash flows, $$C$$, will continue growing at an assumed constant rate, $$g$$, in perpetuity.

Let us assume you are able to project the cash flows for the first 10 years. If you invest in a company, you are not just entitled to receive the cash flow in the first 10 years, but you are actually entitled to receive the cash flow for as long as the company exists. That is, in perpetuity.

To figure out the total perpetual value of a company, you also need to look at the cash flows in the years beyond the 10th year. In other words we also need to consider the company’s terminal value. Terminal value is the value of a company’s expected cash flow beyond the explicit forecast horizon. Assuming you have estimated the cash flows for the first 10 years $$C_1, C_2, \dots , C_{10}$$, you can use the final estimated cash flow, $$C_{10}$$, to calculate the value of future cash flows after the 10th year.

Let us also assume that the cash flows, $$C$$, will continue growing at an assumed constant rate, $$g$$, in perpetuity after the 10th year. Our cash flow $$C$$, that we are going to receive at the end of year 11 (remember after our 10 year projection point), will be $$C_{11} = C_{10}(1 + g)$$, where $$C_{10}$$ is the cash flow in the last individual year estimated of our 10 year projection period.

As we proved in this post we know that the present value, $$PV$$, of a security with perpetual cash flows is:

$$PV = \frac{C}{(d-g)}$$

In our case $$C = C_{10}(1 + g)$$ for year 11 which will continue to grow at a rate of $$g$$ in the years 12, 13,…

The present value of perpetuity at the end of year 10 or terminal value is:

$$PV_{10} = \frac{C_{11}}{(1 + d)} +\frac{C_{12}}{(1 + d)^{2}} +\frac{C_{13}}{(1 + d)^{3}} + \dots=\frac{C_{10}(1 + g)}{(1 + d)} +\frac{(C_{10}(1 + g))(1 + g)}{(1 + d)^{2}} +\frac{(C_{10}(1 + g))(1 + g)^2}{(1 + d)^{3}} +\dots=\frac{C_{10}(1 + g)}{(d-g)}$$

or simply

$$PV_{10}= \frac{C_{10}(1 + g)}{(d-g)}$$

or

$$PV_{10}= \frac{C_{1}(1 + g)^{10}}{(d-g)}$$

In the later formula I used $$C_{11} = C_{1}(1 + g)^{10}$$ because I assumed that the growth rate $$g$$ was used to estimate the cash flows in the first 10 years as well (see illustration above). And so $$C_{11}$$ just becomes the compounded value of $$C_1$$ over a period of 10 years at a rate $$g$$. If that assumption is not true then just use $$C_{11} = C_{10}(1 + g)$$.

Important note: we just calculated the present value of perpetuity of the company at the end of year 10. We still need to discount it at a rate $$d$$ for 10 years in order for us to figure out what this value is worth today. So the final formula for present value of perpetuity, DPCF, becomes as follows:

$$DPCF = \frac{PV_{10}}{(1 + d)^{10}} = \frac{\frac{C_{10}(1 + g)}{(d-g)}}{(1 + d)^{10}} = \frac{C_{10}(1 + g)}{(d-g)(1 + d)^{10}}$$

Assuming you have calculated the discounted value of cash flows, DFCF (Discounted Future Cash Flows), for each of the first 10 years, you can use DFCF and the perpetual value of future cash flows after 10 years discounted to today, DPCF, to calculate the intrinsic value of a company:

$$Intrinsic\ Value=DPCF+ \sum\limits_{i=1}^{10}DFCF_i$$

Written on August 15, 2021