Where does dividend growth come from? Well, dividends come from earnings which in turn comes from the firm's profitability. Earnings in a firm have to equal dividends plus retained earnings by definition. So we are going to use the terminology the plowback ratio to denote the proportion of earnings plowed back into the company. The plowback ratio is going to be denoted b. So b, that's our plowback ratio, is retained earnings divided by total earnings. So if we use the notation E for earnings per share, then our earnings are equal to our dividends plus our retained earnings, or b times E. So D here is the dividend, b times E, this is retained earnings, and rearranging D is one minus b, times E. So those are all accounting identities. Now, we need one more definition and that definition is ROE. It's called return on equity. It is also an accounting concept. It's earnings divided by the book value of equity. What it really is, is it's a measure of the profitability of the firm. So here's my claim which I'm going to prove. The claim is that the growth rate of dividends is equal to my profitability return on equity times my plowback ratio. By the way, this requires certain assumptions. The assumptions are that ROE is constant and that b is constant. So why is this? What connects growth to profitability? I'm going to prove it. So equity at t plus one, is equal to equity, this is book equity, at time t plus retained earnings. Retained earnings keep in mind, are just b times E_t. So let's stick this equity into our factory, meaning let's multiply both sides by ROE. So we're going to take ROE times equity at t plus one, and we're going to take ROE times equity at t, plus retained earnings. By the definition of ROE, what we get on the left-hand side is earnings at t plus one. What we get on the right-hand side is earnings at time t, plus b times ROE, times earnings at time t. Wow. What we've just shown is that earnings at t plus one, is equal to earnings at time t, times one, plus b times ROE. This tells us that the growth rate in earnings is equal to b times ROE. Now wait a second, you may say, very nice. So the growth rate in earnings is b times ROE. But we don't want the growth rate in earnings, we wanted the growth rate in dividends. But it's not only the growth rate in earnings, it's the growth rate in dividends. So E_t plus one, is equal to E_t times one, plus b times ROE. So if we multiply both sides of this equation by one minus b, we get dividends. Dividends at t plus one, equals dividends at t times one, plus b times ROE. You'll remember that g was the growth rate of dividends. So we have just shown that g equals b times ROE. That's the end of our proof. Let's do an example. Let's say, ROE is 15 percent. We invest a $100, that's equity at time one, my plowback is 60.6 percent. So my earnings at time one. Well, I put my $100 into my machine which operates at an ROE of 15 percent. So my earnings by definition at time one are $15. What are my retained earnings? Well, of those $15, I plowback 60 percent, so I retain $9. So therefore, my equity at time two is my original $100 plus the $9, or a $109. I take that $109 and put it back into my factory and get out earnings at time two of $16.35. So you'll notice that the growth rate g in earnings equals 16.35 over 15, minus one, or nine percent. That nine percent is equal to b times ROE, 0.6 times 0.15. So how do we use this formula for growth? Well, let's substitute it back in to our constant growth formula. Remember, the price per share of the stock is the dividend per share next year, divided by r minus g. Substituting in, well, my dividend is the earnings that I didn't plowback. That's sometimes called the payout and what's g? G is ROE times b. So now I have a new formula for the price that's in terms of things that are somewhat more primitive, the earnings, the plowback, and the profitability. We also have an equation for the price earnings ratio which is often used as evaluation metric. So what happens to a company if there's no investment, if nothing is plowed back? So consider the case of b equals 0, no investment. What happens to the formula? So remember, P_0 equals E_1, 1 minus b over r minus b times ROE. B equals 0, we get E_1 over r. This is what's known as a cash cow. The cash cow is milked for E_1 every year, forever. It never grows, it just pays E_1, and so its price is simply the perpetuity with growth equal to zero. If we want some growth, we need some investment b greater than zero. But not all growth is created equal. Let's look back at this formula. So keep in mind that the growth is b over ROE. If you want growth, you need to either increase your profitability or you need to plow back more. But the relation between b and price is complicated. The plow back, which is, remember, reinvestment, is both in the numerator and in the denominator. So you might say, is it always better to grow? Well, there's two different effects. Suppose you decide, "Hey, let's grow the company by increasing b," what could happen? Well, you might increase price. That could happen because g is higher, but you might also decrease the price because D_1 is lower. So there's a trade-off. So which affect wins? What determines whether growth increases the price or not? Well, I hate to say it, but the way we figure this out is by taking the derivative, not just the derivative, but the partial derivative. I'm not going to go through the algebra, you can see the notes. I'll just tell you what it is. The derivative of the price with respect to b is E_1 times ROE minus r over r minus b times ROE squared. Now this denominator here is always positive. So growth increases the price if and only if profitability exceeds the discount rate. So we know when growth is good, it's when profitability exceeds the discount rate. Let's do an example. The discount rate is 12 percent, the return on equity is 10 percent, the plow back is 60 percent, and the earnings per share next year are expected to be $10. P_0 is equal to $10 times 1 minus 0.6, that's my expected dividend next year. In the denominator is r minus g, which is b times ROE, or 0.06. This is four dollars over 0.06 or approximately $66.67 per share. Now this company is what you might call a takeover target. Why? By lowering b, you can raise the stock price. So let's use the same number and set b equal to 0. Then P_0 is just my earnings next year, the shareholders get all of the earnings divided by the discount rate, and that's $83.33. I'm going to write it down. This company destroys value by reinvesting its earnings. By growing, this company actually is destroying value, so we don't want that. So to summarize, if ROE is greater than r, P_0 increases with b, namely, growth is good. If ROE is less than r, P_0 decreases with b. If ROE is equal to r, P_0 is equal to E_1 over r, the cash cow value regardless of b. So when ROE is bigger than r, cash inside the company is more valuable than cash outside. That's why it's better to grow the company. But when ROE is less than r, cash outside the company is more valuable. It's better just to return the cash to shareholders in the form of dividends. When ROE equals r, cash inside the company and outside is the same. So it doesn't matter if cash is reinvested or not, that's surprising. This company is a cash cow. It's worth it's cash cow value, whether or not the cash is reinvested or not. Now we can connect this concept back to that of NPV. It turns out that when ROE is greater than r, this firm has positive NPV opportunities. When ROE is less than r, the firm has negative NPV opportunities. So I encourage you to look at the optional sections in the notes that connect this formula directly to NPV.
