This blog is an extension of our previous blog on “Which Cash flow to Discount under DCF Approach

Many of my colleagues have sweated in their shirts while explaining the concept and mathematics behind the Terminal Value (“TV”). Some of them have hard times because they know the formula to calculate but are not able to explain the logic or derivation behind this. While others have sweated because they don’t remember the formula only in the first place and then they start deriving in front of client leading to numerous new queries.

Are you also one of them?

Is the mathematics behind calculation of TV really so difficult that you either can’t derive it, or can derive it but not without introducing new concepts of geometric progression that opens new grounds to client for asking questions?

Is there a simple way out?

1. Any mathematical complexity involved in derivation of TV
2. Any usage of advance concepts of Geometric Progression (“GP”)while deriving the formula
3. Any effort towards memorizing the formula of TV calculation
4. A flexibility to derive it in front of your client effortlessly and simplistically.

What is TV?

1. Define TV = present value of all the cash flows beyond the projection period sitting at the end of projection period. Hence, here in this case, TV will be at the end of year 5 and given by the expression below where r = WACC and g = TGR

2. Now what, have we really simplified the things or complicated it? How should we add it? Let me simplify the things for you, assume

So, the equation changes to

3. Now how should we solve this infinite series problem? Recall how you have solved similar problems in your primary / high school. Do you remember the basic questions from your exams as mentioned below:

a. Convert a recurring decimal into a fraction, say convert 0.111111111………into a fraction

b. Solve:

4. You never required the concept of Advanced Mathematics to solve these things then. Please understand the basic principle – Only an iron can cut another iron. You can get rid of an infinite series only by another infinite series created from the original series itself but in a way such that original series is contained in the new series.

5. Some of you would have already recalled how you used to solve these questions then. I am sure your methods would not be different from what I have illustrated below:

a. Assume, K = 0.11111…. => 10K = 1.1111…. = 1 + 0.11111…. = 1 + K =>10K – K = 9K = 1

K = 1/9

b. For the second one, let’s say the expression equals “X”. So, we get a new equation:

Now square both the sides to obtain a quadratic equation that can be solved to obtain the value of X

6. In both the examples above, we stuck to the basic principle that only way to get rid of an infinite series is to identify an identical series contained inside it. Now think how we can make use of thin our present case. Let’s call this expression as “S”

7. No, substitute the values of A and R in the expression above to get the TV as

Only word of caution here is that R should be < 1 for this mathematics to work. And the answer to this can be found conceptually and mathematically below:

Note that a series of the form we encountered above is called a Geometric Progression (“GP”) where the subsequent term is a fractional or integral multiple of the previous term. A GP is said to be exploding or divergent if the subsequent term is greater than the previous term. Such a series ultimately explodes as number of term “n” tends to infinity. Hence, there is no mathematical formula to add infinite number of terms of a divergent or exploding GP. Only a convergent GP, here the subsequent term is smaller than the previous term (a case here common ratio R < 1) can be added to its infinite terms.

Mathematically, let’s try summing n terms of a GP:

Multiply both sides by r to get the equation:

Subtract the original equation from the resultant equation to obtain:

Now to make this formula work for infinite number of terms, we must take the limiting case when number of terms tends to infinite, hence:

Got a simpler idea for TV?

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Ever wondered why growth rate “g” < discount rate “r”?