# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## [PGP-I FM] Stock Valuation: II

In this post we take a closer look at the source of growth rate $g$.

In the previous post, we saw that by retaining a part of earnings ($b$) and investing in available opportunities the firm was able  to generate growth in earnings ($g$) which depended on the rate of return on investment ($R$).

We continue to use the same example, but instead of an opportunity presenting itself at time $\mbox{t}$, we say that the firm has been earning at a steady-state earning of $E_1$ per year ‘in the past’, and has been distributing all of it as dividends, but at time $\mbox{t} = \mbox{1}$  the firm finds a market opportunity which it think it can exploit by investing in some new products.

Let’s assume that the investment of $E_1$ in this opportunity will give the firm an extra earning of $E_1g$ per year in perpetuity. We assume that starting year $2$ it distributes all its earnings as dividends.

Pictorially we have a situation something like this:

(Click on the graph to zoom.)

That is, from a steady state of earning $E_1$ per year, by investing in an opportunity the firm has reached a steady state of earning $E_1 + E_1g$ each year from the investment made at time $1$.

The PV of the firm then is easily calculated as:

\begin{aligned} \displaystyle P_0 &= \frac{0}{1 + r} + \frac{E_1 + E_1g}{(1 + r)^2} + \frac{E_1 + E_1g}{(1 + r)^3} + ... \\& = \frac{E_1(1 + g)}{(1 + r)^2} \Big[ 1 + \frac{1}{1 + r} + \frac{1}{(1 + r)^2} + ... \Big] \\& = \frac{E_1}{r} \frac{1 + g}{1 + r} \\& = \frac{E_1}{r} \frac{1 + R}{1 + r}\end{aligned}

where $R$ is the firm’s return on investment/equity. (Because in our example all the earnings have been plowed-back, i.e. $b = 1$, it implies that $g = bR = R$). That is, the firm value depends on how growth in earning from investment opportunities $R$ relates to the opportunity cost of capital $r$.

This means it doesn’t make sense to invest in all kinds of opportunities, but only in those which have $R$ greater than the opportunity cost of capital, $r$. In fact, only if $R > r$ is the firm value more than what it starts out with. Even if $R = r$, the firm value doesn’t increase but stays the same. That is, as far as the shareholders are concerned, the firm might as well ignore opportunities which have $R \le r$.

And why is that? Because in that case the shareholders would rather receive the dividends (‘get their money back’), and invest in assets in stock market/elsewhere which give return equal to the opportunity cost of capital ($R \ge r$). So, if the only opportunities available to the firm are the ones which have $R < r$, the firm is better off giving the dividends.

So what is happening here? Let’s dig a little deeper still.

Rewrite the above PV as:

\begin{aligned} \displaystyle P_0 &= \frac{0}{1 + r} + \frac{E_1 + E_1g}{(1 + r)^2} + \frac{E_1 + E_1g}{(1 + r)^3} + ... \\& = \frac{E_1 + E_1g}{(1 + r)^2} + \frac{E_1 + E_1g}{(1 + r)^3} + ... \\& = \underbrace{\Big[ \frac{E_1}{1 + r} + \frac{E_1}{(1 + r)^2} + \frac{E_1}{(1 + r)^3} + ... \Big]}_{\displaystyle = \frac{E_1}{r}} + \underbrace{\Big[ -\frac{E_1}{1 + r} + \frac{E_1g}{(1 + r)^2} + \frac{E_1g}{(1 + r)^3} + ... \Big]}_{\displaystyle \mbox{NPV of Cash Flow from Investment}} \end{aligned}

(where, note, we have collected the $E_1$ and $E_1g$ terms together, and added the term $E_1/(1 + r)$ in the first bracket and subtracted in the next.)

Look closely at the term:

$\displaystyle \mbox{NPV of Cash Flows from Investment} = \Big[ -\frac{E_1}{1 + r} + \frac{E_1g}{(1 + r)^2} + \frac{E_1g}{(1 + r)^3} + ... \Big]$

The term $\mbox{NPV of Cash Flows from Investment}$ has two parts: the PV of the investment at time $1$, i.e. $-E_1/(1 + r)$, and the PV of the extra cash flows arising out of the investment, i.e. $E_1g/(1 + r) + E_1g/(1 + r)^2 + ...$. We call the sum, $\mbox{NPV of Cash Flows from Investment}$ as $NPVGO$. When $g = r$, you can (and should) do the algebra and check that $NPVGO = 0$, as it should be.

That is,

$\displaystyle \boxed{P_0 = \frac{E_1}{r} + \mbox{NPVGO}}$

In this simple case, thus, the firm value today can be written as a sum of the value that would have been had the firm not invested in any investment opportunity (the steady state $PV = E_1/r$), and the NPV of Growth Opportunities $\mbox{NPVGO}$.

Growth Opportunities

It turns out this result is more general and holds for investment across time periods in the future. We now ‘prove’ it for the case where the firm retains a (constant) portion ($b$) in each time period (that is similar investment opportunities are taken up every year), and thereby adding to expected cash flows in subsequent periods in perpetuity.

As earlier, pictorially this is represented as:

(Click on the graph to zoom.)

(This is what it means when we say that the dividends are growing at a constant rate $g$ : that the firm is plowing back a constant ratio every year and investing in investment opportunities that make earning/dividends grow by $g$. )

\begin{aligned} \displaystyle P_0 &= \frac{D_1}{1 + r} + \frac{D_1(1 + g)}{(1 + r)^2} + \frac{D_1(1 + g)^2}{(1 + r)^3} + ... \\& = \frac{E_1(1 - b)}{(1 + r)} + \frac{E_1(1 - b)(1 + g)}{(1 + r)^2} + \frac{E_1(1 - b)(1 + g)^2}{(1 + r)^3} + ... \\& = \frac{E_1( 1 - b)}{1 + r} \underbrace{\Big[ 1 + \frac{1 + g}{1 + r} + \frac{(1 + g)^2}{(1 + r)^2} + \frac{(1 + g)^3}{(1 + r)^3} + ... \Big]}_{= \displaystyle \frac{1 + r}{r - g}} \end{aligned}

And what do we end up with? Of course, just our constant growth perpetuity formula:

$\displaystyle P_0 = \frac{E_1(1 - b)}{r - g} = \frac{D_1}{r - g}$

Investments

Upon closer look we see that growing sequence of dividends is coming from a sequence of investments (second row of the above table) whose PV is:

\begin{aligned} \displaystyle \mbox{PV of Investments} &=\frac{-bE_1}{1 + r} + \frac{-bE_1(1 + g)}{(1 + r)^2} + \frac{-bE_1(1 + g)^2}{(1 + r)^3} + ... \\&=\frac{-bE_1}{r - g}\end{aligned}

Earnings

PV of total earnings are given by the following perpetuity (second row of the above table):

\begin{aligned} \displaystyle \mbox{PV of Earnings} &=\frac{E_1}{1 + r} + \frac{E_1(1 + g)}{(1 + r)^2} + \frac{E_1(1 + g)^2}{(1 + r)^3} + ... \\&=\frac{E_1}{r - g}\end{aligned}

It turns out we can we break this total earnings expression into parts as following:

\begin{aligned} \displaystyle \mbox{PV of Earnings} &=\frac{E_1}{r} + \frac{E_1g}{r(r - g)}\end{aligned}

Of course, as the above table shows the stock price (PV of dividends) is nothing but PV of earnings minus PV of investments, so we must have:

\begin{aligned} \displaystyle P_0 &= \mbox{PV of Earnings} - \mbox{PV of Invesments} \\&=\frac{E_1}{r} + \frac{E_1g}{r(r - g)} +\frac{-bE_1}{r - g} \\&=\frac{E_1}{r} + \Big[\frac{-bE_1}{r - g} +\frac{E_1g}{r(r - g)}\Big] \\&=\frac{E_1}{r} + \mbox{NPVGO}\end{aligned}

where $\displaystyle \frac{E_1g}{r(r - g)}$ represents PV of sequence of extra cash flow from all investments. Showing so economically (and mathematically) and where it is coming is a bit tedious algebraically, but I can tell you the outline of it.

The way it is done by noting that investment at each time period generates a perpetuity. So, investment at time $1$ creates a perpetuity of extra earning $E_1g$ starting at time $2$. The value of this perpetuity time $1$ is $E_1g/r$. (Note that investment in each time period generates a perpetuity of extra income starting from the ‘next’ time period.) So, the perpetuity starting at year $3$ because of retained earnings at year $2$ is worth $E_1g(1 + g)/r$, and so on and so forth. The sum of all these perpetuities is itself a perpetuity which sums as follows:

\begin{aligned} \displaystyle \mbox{PV of Future Cash Flows growing at the rate g} &= \frac{E_1g}{r} + \frac{E_1(1 + g)g}{r} + ... \\& = \frac{\frac{E_1g}{r}}{r - g} \\& = \frac{E_1g}{r(r - g)} \end{aligned}

which is what we set out to show.

We can now put it all together by writing:

\begin{aligned} \displaystyle P_0 &= \frac{E_1}{r} + \underbrace{\Big[ \underbrace{\frac{-bE_1}{r - g}}_{\displaystyle \mbox{PV of Future Investments}} + \underbrace{\frac{E_1g}{r(r - g)}}_{\displaystyle \mbox{PV of Future Cash Flows growing at the rate g}} \Big]}_{\displaystyle \mbox{NPVGO}} \end{aligned}

The bottom line is that under some reasonable assumptions, we may interpret value of any common stock as the sum of PV of earnings in absence of growth and $NPVGO$, i.e:

$\displaystyle \boxed{P_0 = \frac{E_1}{r} + \mbox{NPVGO}}$

To conclude, note that the above implies:

$\displaystyle \frac{P_0}{E_1} = \frac{1}{r} + \frac{\mbox{NPVGO}}{E_1}$

which means for any given level of earnings, market ascribes a high price $\displaystyle \frac{P_0}{E_1}$ ratio (called the price-earnings multiple) to any stock only when it expects $\mbox{NPVGO} > 0$. Of course, future realized value of $GO$ could be anything, and if the firm doesn’t deliver on its promises, its price in the future would fall. So, obviously, the Carlsberg India CEO isn’t very happy that GO in Bihar just got changed!

This completes our discussion of stock valuation.