# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## [PGP-I FM] Stock Valuation: I

Now we come to the first application of the theory we have developed so far – stock valuation.

Why should a rational investor buy a stock?

The intuitive answer seems to be that an investor buys a stock as she expects its value to rise – so that she can get the capital gains out of it. But is the expectation warranted? Think about it. For the value of the stock to rise, just her expectation/her isn’t enough. Others also must hope the same. Only if others believe so that the price would go up. Needless to say there is no guarantee that this will happen. So then, why should a rational investor buy a stock?

The answer is that investor would consider what return does stock provide to him/her, irrespective of whether or not other investors change their mind about it. And what does a stock return? Literally, this is just the cash flows paid by the firm to the shareholders. These cash flows mostly take the form of dividends, but can often be proceeds of takeover or liquidation. In short, a stock is worth only what it gives you!

To quote a saying that John Burr Williams cites in his book:

A cow for her milk,
A hen for eggs,
And a stock, by heck,
for her dividends

Which, translated in our language, means:

$\mbox{PV(Stock)} = P_0 = \mbox{PV(Expected Future Dividends)}$

Or,

\begin{aligned} \displaystyle P_0 &= \frac{D_1}{1 + r} + \frac{D_2}{(1 + r)^2} + \frac{D_3}{(1 + r)^3} + ... \end{aligned}

(where $D_i$ denotes the expected dividend at time $i$)

At this point you may say, that this ok, but really, come on, are you saying that a trader on the street buys the stock for dividends? Isn’t it true that most people out there buy stocks for capital gain? Well, it turns out that there is no discrepancy. Let’s ask the question, what determines next year’s price?

What determines next year’s price?

If our ‘model’ / theory of stock valuation is correct, then it should hold next year too, right? And what would be the price next year be? It will be just the expected future dividends starting year 2 onwards. So we have:

\begin{aligned} \displaystyle P_1 &= \frac{D_2}{1 + r} + \frac{D_3}{(1 + r)^2} + \frac{D_3}{(1 + r)^3} + ... \\& = (1 + r) \big [ \frac{D_2}{(1 + r)(1 + r)} + \frac{D_3}{(1 + r)(1 + r)^2} + \frac{D_3}{(1 + r)(1 + r)^3} + ... \big ] \end{aligned}

(In the second line we’ve just multiplied and divided by the fraction $1/(1 + r)$).

This can be written as:

$\displaystyle \frac{P_1}{1 + r} = \frac{D_2}{(1 + r)^2} + \frac{D_3}{(1 + r)^3} + ...$

We had $P_0$:

\begin{aligned} \displaystyle P_0 &= \frac{D_1}{1 + r} + \underbrace {\frac{D_2}{(1 + r)^2} + \frac{D_3}{(1 + r)^3} + ...}_{\displaystyle = \frac{P_1}{1 + r}} \end{aligned}

$\Rightarrow \boxed{P_0 = \frac{D_1}{1 + r} + \frac{P_1}{1 + r} = \frac{D_1 + P_1}{1 + r_1}}$

To generalize then,

\begin{aligned} \displaystyle P_0 &= \frac{D_1}{1 + r} + \frac{D_2}{(1 + r)^2} + \frac{D_3}{(1 + r)^3} + ... \frac{D_n + P_n}{(1 + r)^n} \\& = \sum_{i=1}^{n} \frac{D_i}{(1 + r)^i} + \frac{P_n}{(1 + r)^n} \end{aligned}

That is, saying that value of a stock is the discounted value of expected future dividends is akin to saying that the value of a stock is the discounted value of expected future dividends and the terminal (final sale) price. This has come to be known as the Dividend Discount Model (DDM).

But to solve the problem in practice, as you can see we need the values of $n + 2$ variables – $n$ dividends ($D_i$), $1$ value for the discount rate ($r$) and the terminal share price $P_n$.

If we have a sense of the expected (future) values of these variables, then of course we are done. But that’s asking for too much. Real world is uncertain, and at best we can have reasonable forecasts for the next couple of years – hardly enough. Presented in this form, the problem clearly is daunting. What do we do?

First thing to do is ask the question – can we simplify it somehow, but still learn something about stock valuation? Say, somehow by reducing the dimensionality/number of variables in the problem?

It turns out it is possible, but we need to make some assumptions.

Cash Flows: Dividends and the Terminal Price

Let’s deal with the terminal price first. We noted earlier that the stock valuation formula with an expected terminal price is equivalent to one with an infinite stream of expected dividends. That is, assuming that in the limit the PV of the expected price at infinity is 0, i.e. $\displaystyle \lim_{n \rightarrow \infty} \frac{P_n }{(1 + r)^n}= 0$, we can write DDM as:

\begin{aligned} \displaystyle P_0 &= \frac{D_1}{1 + r} + \frac{D_2}{(1 + r)^2} + \frac{D_3}{(1 + r)^3} + ... \lim_{n \rightarrow \infty}\frac{D_n + P_n}{(1 + r)^n} \\& = \sum_{i=1}^{\infty} \frac{D_i}{(1 + r)^i} \end{aligned}

What do we about dividends? If we look carefully, our DDM formula for stock valuation doesn’t look too different from an annuity. The difference is that unlike an annuity, the cash flows $D_i$ are not same throughout the year. But if the expected dividend were indeed the same every year we could use the annuity formula. Won’t it be wonderful if we could make this assumption. Could we? Well, as Obama famously popularized it back then, yes, we can!

That basically means we have two options. Model expected dividends as a:

1. Perpetuity: Assume dividends to be the same forever
2. Growing perpetuity: Assume dividends to grow at a constant rate forever

Dividends as a Perpetuity

This clearly is the simplest case. If we are willing to assume that expected dividends will be the same forever, then we are down to just $2$ ($D_1$ and $r$) variables from our original $n + 2$, and our stock valuation formula simplifies to:

$\displaystyle P_0 = \frac{D_1}{r}$

Dividends as a Growing Perpetuity

The second case of a growing perpetuity is not any more difficult, and in this case we end up with $3$ ($D_1$, $r$ and $g$) variables, and our stock valuation formula is:

$\displaystyle P_0 = \frac{D_1}{r - g}$

where $g$ is the assumed growth rate in dividends. Of course, use of this formula requires that $g < r$.

What if, $g \ge r$? We know that we can’t use the infinite GP formula in this case as the common ratio that is involved in a growing perpetuity $(1 + g)/(1 + r)$ would be greater than $1$. Maths is one thing, but we have to understand what it means.

What we are saying is that the growth rate of expected dividends forever exceeds the opportunity cost of capital – which, loosely speaking, is another way of saying that the company is able to find investment opportunities that return a rate more than the opportunity cost of capital ($NPV > 0$forever. But neither are all companies are like Facebook, nor is everybody like Mark Zuckerberg. (Even Facebook is not growing at a rate that it has for the past 7-8 years, funding from Microsoft and acquisition of WhatsApp notwithstanding.)

So clearly, even economically the case of $g > r$ forever seems unreasonable. But then, are we saying that we cannot consider companies like Facebook in our set-up?

We can consider high growth firms in our set-up as long as we’re willing to allow for multiple growth rates. A firm can enjoy monopoly profits for some time when it has a new product / service that beats everything else, but in a free market eventually the competition catches up, and the margins starts to go down. And it’s reasonable to say, that at some stage when the industry becomes mature (like the passenger car or ‘online messaging’ industry is now), when the challenge is just to sustain one’s existing position in the market (an ‘equilibrium’, when $g < r$). So, yes, if we allow for multiple growth rates in our set-up we can very well think about valuing firms like Facebook.

If you’ve read the Chapter 2 of your book, you may be able to guess where we are heading now. We just consider a growing perpetuity that grows at a high rate $g_1$ for some time, and then at a slower ‘equilibrium’ rate $g_2$ for the remaining. (We can easily include as many stages of growths as we like, as long the last stage is an ‘equilibrium’  stage by which time we say that the competition has caught up.)

Notice that starting with some reasonable first approximations we’ve ended up with a model that has only finite number of variables to estimate. Even if we consider three stages of growth, we still have only $5$ variables to figure out – clearly an improvement over $n + 2$ variables we started out with.

If we consider two stages of growth (a very reasonable assumption in most cases), we just need one extra variable, i.e. $g_2$. That is, we have able been to reduce the complexity of the problem from finding $n + 2$ variables to just $4$. This gives us the following valuation formulas:

• Dividends as Normal Perpetuity:

$\displaystyle \boxed{P_0 = \frac{D_1}{r}}$

• Dividends as Growing Perpetuity:

$\displaystyle \boxed{P_0 = \frac{D_1}{r - g}}$

• Dividends as a Perpetuity with Multiple Growth Rates (working with the in-class example, with $g_H > g_L$):

\begin{aligned} \displaystyle P_0 &= \frac{D_1}{1 + r} + \frac{D_1(1 + g_H)}{(1 + r)^2} + \frac{D_1(1 + g_H)^2}{(1 + r)^3} +\frac{D_1(1 + g_H)^2(1 + g_L)}{(1 + r)^4} + \cdots{} \\& =\frac{D_1}{1 + r} + \frac{D_1(1 + g_H)}{(1 + r)^2} + \frac{D_1(1 + g_H)^2}{(1 + r)^3}\Big[1 + \frac{(1 + g_L)}{(1 + r)} + \cdots{} \Big]\\& =\frac{D_1}{1 + r} + \frac{D_1(1 + g_H)}{(1 + r)^2} + \frac{D_1(1 + g_H)^2}{(1 + r)^3}\Big[\frac{1}{1 - \frac{1 + g_L}{1 + r}}\Big]\\& =\frac{D_1}{1 + r} + \frac{D_1(1 + g_H)}{(1 + r)^2} + \frac{D_1(1 + g_H)^2}{(1 + r)^2}\frac{1}{r - g_L}\end{aligned}

Or, in general, if the growth rate $g_H$ is for $n$ years and $g_L$ thereafter:

$\displaystyle \boxed{P_0 = \sum_{i=1}^{n} \frac{D_1(1 + g_H)^{i - 1}}{(1 + r)^i} + \frac{D_1}{r - g_L}\frac{(1 + g_H) ^{n}}{(1 + r)^n}}$

The last formula above is sometimes also referred to as Multi-stage DDM (just check to make sure that the expression is ok). Of course, the formula becomes complicated as you have multiple growth rates, but the basic PV logic of DDM remains.

While we’ve managed to reduce the dimensionality of our problem, we still haven’t quite solved the problem yet. We still need to figure how to estimate the variables of interest – $D_1$, $r$ and $g$.

As it happens dividends of firms typically don’t change much year-on-year, and for $D_1$, last year’s dividends is not a bad first estimate.

Where does $g$ come from?

More often than not, in practice, the best way to get a sense of $g$ is to just ‘ask’ the equity research experts in your bank, or if you are not working in one, perhaps just buy a subscription to the sector/firm equity research reports from one, or from those boutique equity research ‘shops’.

But what if you are that equity research professional, or what if the investment-banking desk/consulting firm you work for want you to provide that estimate? (Perhaps because they don’t rely on other’s opinions, or just feel uncomfortable that all assumptions in outside equity research reports are not properly spelled out. This is not just a hypothetical situation, these concerns are sometimes very relevant and important). In this post we try to learn the textbook way of how to think about the growth rate $g$.

So how do we think about finding $g$? First of all, think what $g$ really means? Loosely speaking, when we say that the firm is able to ‘grow’ its dividends at a rate $g$ year-on-year, it’s like saying that the company is able to find just enough $NPV > 0$ investment opportunities to grow at the same rate every year (if even that were not possible, we would have the case of dividends growing as a normal perpetuity – the firm just about surviving to maintain the same dividends every year.)

Consider a firm (selling electric bulbs?) that’s earning a steady-state/equilibrium amount ($E_t = E$ $\forall t$) per year – just about able to recover its depreciation and generating enough revenue to retain its position in the market. Now, say, it finds an $NPV > 0$ investment opportunity (recall our CFL example), which it believes has a positive NPV.

Let’s say that in year $t$ (with earning $E_t = E$), the firm finds this opportunity and retains a portion portion of that, say, amount $I_t < E_t$ to invest, and pays the remaining amount as dividend $D_t = E_t - I_t$. Now since the firm’s steady state operations (electric bulb business) are not affected, it’ll earn the same amount from that even next year. If the investment $I_t$ produces a revenue of amount $I_{t + 1}$ in period $(t + 1)$, then the total expected earnings in period $(t + 1)$ would be:

\begin{aligned} \displaystyle E_{t + 1} = {E_t} + I_t \big( \frac{I_{t + 1}}{I_t} - 1 \big) \end{aligned}

Or alternatively,

\begin{aligned} \displaystyle \frac{E_{t + 1}}{E_t} = 1 + \frac{I_t}{E_t} \big( \frac{I_{t + 1}}{I_t} - 1 \big) \end{aligned}

and if $g$ is the growth in earnings, i.e. $1 + g = E_{t + 1}/E_t$, and we call $b_t = I_t/E_t$ as the ratio of retained earnings (or as it sometimes called the ‘plow-back’ ratio), and $R_t = I_{t + 1}/I_t - 1$ as the return on investment (and if you have not raised any debt, return on equity), then we can write the growth rate in earnings as:

$\boxed{g_t = b_t R_t}$

If we assume that the firm is able to find similar investment opportunities in each period ($R_t = R$ $\forall t$), and the plow-back ratio is constant ($b_t = b$ $\forall t$), we end up with our growth rate formula as:

$\boxed{g = b R}$

Note that if the earnings growth rate $g$ is constant because the plow-back ratio $b$ and the return on investment $R$ are constant, the dividend growth rate would be constant ($= g$) too. To see this, write:

$\displaystyle g_D = \frac{D_{t + 1}}{D_t} - 1 = \frac{(1 - b)E_{t + 1}}{(1 - b)E_t} - 1 = \frac{E_{t + 1}}{E_t} - 1 = g$

In the next post we take a closer look at where this growth is coming from and introduce the idea of NPV of Growth Opportunities.

…

Caveats

1. Note that here we are considering $g$ as the growth in earnings because of the positive $NPV$ opportunities. If the firm is giving extra dividends by investing in opportunities that are growing at a rate lower than the opportunity cost of capital, then the firm is reducing value for the shareholders despite the growth rate in dividends. We take a look at it in a stylized setting in the next post.
2. This should be probably obvious, but worth mentioning nonetheless. When you rely on equity research reports to get a sense of $g$, beware that security analysts are subject to behavioural biases and their forecasts tend to be over-optimistic.
3. Our results above are for a single/constant growth rate (dividends as a constant growth perpetuity) case,  so, again, remember that the analysis will have to modified slightly (though nothing changes much in terms of the logic) if you are considering multiple growth rates.