I have just started with Finance and have a physics background. However, I am struggling to understand the formula for the terminal value in a discounted cash flow valuation (DCF valuation) according to the Gordon Growth Method.
To my understanding a firm value ($E_V$) is calculated as:
$$
E_V = \sum\limits_{n=0}^N \frac{C_n}{(1+W)^n} + T\ ,
$$
where $C_n$ are the cash flows at years $n$, where $W$ are the (annual) weighted averaged capital costs, and where $T$ is a terminal value, defined as the sum of all cash flows from the the first year after year $N$ on to infinity. This all makes sense to me, including the terminal value trick, as it is not good to make guesses for cash flows up to infinity. However I am struggling to derive the Terminal Value.
There are a lot of sources (e.g. here) who define the Terminal Value as:
$$
T = \frac{C_N~(1+g)}{W-g}\ ,
$$
where $g$ is the perpetual growth rate, i.e. the interest rate we assume from year $N$ on. Could someone explaine me how that formula is derived? According to my understanding of the Terminal Value I would write it as:
$$
T = \sum\limits_{n=N+1}^\infty\frac{C_N~(1+g)^{n-N}}{(1+W)^n}\ ,
$$
i.e. I would assume the cash flow grow with the perpetual growth rate and I would discount each of these cash flows with the weighted averaged capital costs $W$. By shifting the index like $k\equiv n-(N+1)$ and thus writing the terminal value as a geometrical series I end up with:
$$
T=\frac{1}{(1+W)^N}\cdot\frac{C_N~(1+g)}{W-g}
$$
and thus with a additional factor of $1/(1+W)^N$. Where is my error in reasoning?
Best Answer
The firm value is computed as: $$ E_V = \sum\limits_{n=0}^N \frac{C_n}{(1+W)^n} + \frac{T}{(1+W)^N}\ , $$ instead of $$ E_V = \sum\limits_{n=0}^N \frac{C_n}{(1+W)^n} + T\ , $$ what I assumed when asking this question. I.e. the terminal value is discounted from year $N$ like a regular cash flow of that year. With that in hand the derivation gives the correct result: Only what I have called $T$ in the last two equations of my question should have been $\frac{T}{(1+W)^N}$ instead. I summarise my derivation for those to whom it may concern: $$ E_V = \sum\limits_{n=0}^\infty \frac{C_n}{(1+W)^n} \approx \sum\limits_{n=0}^N \frac{C_n}{(1+W)^n} + \sum\limits_{n=N+1}^\infty\frac{C_N~(1+g)^{n-N}}{(1+W)^n}\ , $$ where in the last step it is assumed that the cash flow of year $N$ grows according to the perpetual growth rate $g$ every year after year $N$. Shifting the index like $k\equiv n-(N+1)$ in the second sum of the above equation yields: $$ E_V \approx \sum\limits_{n=0}^N \frac{C_n}{(1+W)^n} + \sum\limits_{k=0}^\infty\frac{C_N~(1+g)^{k+1}}{(1+W)^{k+(N+1)}}\ . $$ The second sum can be rewritten as follows: $$ \sum\limits_{k=0}^\infty\frac{C_N~(1+g)^{k+1}}{(1+W)^{k+(N+1)}} = \frac{1+g}{(1+W)^{N+1}}~\sum\limits_{k=0}^\infty C_N~\left[\frac{1+g}{1+W}\right]^k = \frac{1+g}{(1+W)^{N+1}}\cdot \frac{C_N}{1-\frac{1+g}{1+W}} =\frac{1+g}{(1+W)^{N}}\cdot \frac{C_N}{W-g}\ , $$ where in the second to last step we made use of the common expression for the geometrical series. Inserting this result into the above approximation for the firm value yields: $$ E_V \approx \sum\limits_{n=0}^N \frac{C_n}{(1+W)^n} + \frac{T}{(1+W)^N}\ , $$ where we have defined: $$ T = \frac{C_N~(1+g)}{W-g}\ . $$ It should be mentioned that this implies that the formula for the terminal value is valid for $(1+g)/(1+W)<1$ and thus for $g<W$. I.e. the cash free cash flows' growth must be smaller than the weighted averaged capital costs – else the formula for the geometrical series must not be applied.