[MUSIC] Ever since the joint stock owners expected to get a portion of earnings from the remarkable Dutch East India Company chartered in 1602, so too have stockholders of all other companies. The universal expectation is to receive dividends, derived from the Latin roots dividendum, meaning thing to be divided. Unlike bond holders who must be paid contractually over a fixed period of time, the firm's board of directors decides on discretionary dividends that stockholders will receive in the future. To find their value today, these must be converted at a rate that compensates stockholders to take on this risk of uncertain stream of income from the future. This in essence is the intuition behind the dividend discount model. It values a stock as the present value of future dividends. Take the example of a firm that expects to earn $3 million annually for its 1 million stocks outstanding. Dividing the two numbers gives us earnings per share, or EPS, which in this case is $3. The dividends per share, or DPS, of the firm would be exactly the same if the firm decided to give all of its earnings back to the stockholders. Many firms do exactly that because they do not have any new value-creating projects to invest in or because they are a subsidiary transferring earnings back to its parent company. Either way, 100% dividend payout ratio, measured by DPS divided by EPS, suggests an absence of any future growth opportunities. Finding the value of a stock in this zero growth case is actually quite simple. From our time value of money discussion we learned that the value of the perpetuity denoted by V is equal to the annuity divided by the discount rate r. Rewriting the formula, the value of a stock is P subscript 0 for the price today, which equals to the dividend in the numerator and again r, which represent the share holders' required rate of return. Let's assume that to be 15% in this example. So what we'll have is the value of a share equal to $20 based on the dividend of $3 divided by the discount rate of 15%. Now, is this a fair price, meaning is it neither over or undervalued? Well, Warren Buffett likely thinks that it is correctly priced assuming the $3 dividend is more or less certain and if the discount rate earns the time value of money. Prices, however, are also influenced by a number of other factors beside a rational model that is based on dividends and discount rates. This is because the stock market is not insulated from changes in other markets including the bond market, credit market, the commodity currency and derivative markets. Nor is it from unanticipated news that will inevitably influence market perceptions about how attractive the stock really is. We explore these factors in detail. But first, let's deal with the engine of growth that markets love to pay for. Firms that are able to sustain growth for long periods of time are clearly more valuable to stockholders as they can potentially benefit from these growth opportunities. Firms that can pay for a growing stream of dividends are known as dividend aristocrats. Their solid records of constant dividend growth make them specially attractive to institutional investors like insurance and pension funds, who are by law bound to invest in these types of companies. If our firm is in an industry that is able to sustain, let's say, a 5% constant growth rate in the future, how does this change its price? Again, it's easy to show that the stock price today will be based on the next period's dividend, which would be divided by the difference between the rate of return and the growth rate. In other words, the price is going to be the next period's dividend, which we're assuming to be $3 and the difference between the discount rate, 15%, and our growth rate of 5%. Remember, it's the next period's dividend that we look at, which we've just assumed to remain at $3. All right, let's pause here for just a moment. First of all, it's worth emphasizing just how valuable growth opportunities are. The 5% growth gives the price a boost from $20 to $30, and that's a 50% increase. Let's also look at the graph. This reveals how price increases dramatically for growth-oriented stocks. And if we add to this, the favorable tax treatment of price increases, which is the treatment of capital gains versus the treatment of dividend income, not to mention the management compensation schemes tied to growth. Well, this might explain the temptation to manipulate financial statements that artificially boost earnings per share and project fictitious growth. The Enron era, marked by deregulation and the financial fraud at the turn of the century, reveals all of the questionable practices across several industries, and a reminder to us to scrutinize this data and work with audited financial statements. By rearranging the constant growth formula, we can also see the two components of return. The first component of return is simply the dividend divided by the price, and this is known as the dividend yield. The second component, which is denoted by g, is the capital gain component, and that reflects the price appreciation. All right, let's get back to our example. So we started off with a pricing model for zero growth. And the price was equal to the dividend divided by the required return minus the growth rate. We can rearrange this formula and see two very distinct components that are in fact very insightful. So the first component is if we isolate r, it is going to be the dividend divided by the price, and that gives us the dividend yield. That's the first part. And then, of course, the second part, which is g, the capital gains portion. If we plug the number into the example that we had, we have a dividend, expected dividend of $3. We have a price today of $30, and we know that our total return, the expected return, was 15%. And since this is 10%, we also know that the capital gains component, or g, is 5%. All right, now let's look at the third case, and in this case, we're going to look at growth rates that vary over time. So we call this the non-constant growth model. Suppose our firm is projecting growth rates of 5% for the next two years that are going to be followed by 2% growth thereafter. How do we price stocks in these cases of variable growth? To find the answer, we'll start with our established method of plotting information on a timeline and then we are going to use three particular steps to get the answer. So what about that timeline? If we draw a timeline, we can see that generally speaking, we have time zero. And then in this case, we're going to have the next period's dividend, the following period's dividend. And until now, we have a growth rate of 5%, which was for the first two periods. Following this, we had predicted the growth rate of 2% thereafter. So all we need to do now is to make sure that we at least account for the next period's dividend, which is D3. And we also want to calculate the price at this point, which is going to be P2. And remember, always the price refers to the next period's dividend, D3, again divided by r minus g. Right, so then let's apply the information in the problem with the three step procedure. Step one. In step one we forecast the future dividends until they become constant, as in this case, or zero growth. So if we do the forecasting, notice for these three periods, one, two, and three, we have a dividend that is forecasted to be $3 in the first period that grows by 5%. So this is going to be 3 times 1.05, and that gives us $3.15. And then from thereon to D3, which is going to grow at 2%. So we're going to take $3.15, and that will grow at 2%, which gives us a value of $3.21. That's step one. Let's move on to step number two. Once we forecasted the dividends, step number two is to calculate that price. So, we compute the price at the point where the growth rate either becomes constant or zero growth. In this example we just have to plug the numbers in. Step two, we calculate the price, in this case P2, which is going to be 3 over r minus g. And as we can see, the numbers we forecasted, D3 is 3.21. The discount rate we already know is 15%, and the growth rate from this point on is 2%. And if we do this calculation, we get the price of $24.72, and that takes care of step two. Do notice, however, that this price takes into account the next period's dividend and all other dividends in the future. And that takes us to the final step, and that final step we simply present value all of the dividends in step one and the price in step two. So generally speaking, what we're doing now is simply present valuing the dividend and the price. If we apply to a problem here, we have a dividend of $3 that we need to bring back for one year at 15%. And we have the dividend for year two, which is $3.15. So we've taken care of this dividend, we've taken care of the next dividend. Now we need to take care of all the future dividends, which is reflected in this price here. So we're going to add to this the price 24.72, and then simply bring these two amounts back into today's dollars at 15%. And that gives us a present value of $23.67. That should be the price of the stock with variable growth. [COUGH] Now, if you're wondering what do you do in the case when the stocks are not paying any dividends at all. And the answer here is typically to use another approach. This approach is known as the multiple approach. And the most common multiple to use in this approach is the price earnings multiple or the PE multiple. Let's apply this to our example here. So we have in this case multiples, and we mentioned the first approach is the price earnings multiple. And what do we have in this example? The price E is denoting earnings per share. So we know the price is 30, we know the earnings per share is 3, and therefore the multiple is 10 times. So using this $30 price, this suggests that this value is worth ten times its earnings. And the multiple could be benchmarked against, let's say, the firm's historical multiple. Or it could be based on some average or median value of a sample of say comparable competitors' multiples. If we did that and we assume that the competitors have a benchmark multiple of 8 times, then what this suggests is that 8 times our earnings of 3 gives us a value of 24. And the value of 24 is suggesting that the current price, if the current price is going to be $30, which we computed right in the beginning, well then, we can see our shares are actually overvalued. Whereas if we used the calculation earlier of 23, we can see that the shares are undervalued. And accordingly, we would buy or we would sell. So let me summarize. The dividend discount model provides the first step to price a stock. Generally the stock price is equal to the present value of future dividends. We looked at several variations that consider assumptions about future growth, and we came up with a number of formulas. So if I have to summarize these formulas for you with the space that I have left on this lightboard, I will start with the zero growth mode. Zero growth model said the price is simply equal to the dividends divided by the discount rate r. [COUGH] We then move to the next case of constant growth. And in the constant growth case, we said the price is going to be equal to the next period's dividend divided by r minus g. And finally, we look at the third variation of the growth model, which is non-constant. And in this case, we compute the price as simply equal to the present value of those future dividends and the price in the future when dividends become constant or zero growth. We can denote that by forecasting those dividends, D1, D2 right up til Dt. And not to forget that at this point when the growth rate becomes constant or zero, we also calculate the price at that point t. And then we have to bring all of these values back, so we would bring back D1 at the appropriate discount rate for one period, bring back D2, discount rate for two periods right down to the period where the growth rate changes. And then our price, and this price at the point t is going to equal to the next period's dividend, t+1, over r minus g. And this too is going to be brought back at 1 + r raised to the power t, which is what we did right here in the calculation. [COUGH] So, while we know that the stockholders' expected rate of return, the r that we've been using throughout these examples, they include two components, which are the dividend yield and the capital gains component, obtaining that rate of return is easier said than done. This is because the valuation models have to account for risk. And given our formulas, risks can be either imputed in two places. So if we look at the formula where we started for the price, we can have the risk imputed either in the numerator or imputed in the denominator. There's only two places to go. [COUGH] So conventionally what we do is we either adjust the cash flows to take care of risk or we adjust more conventionally in the discount rate, which includes both inflation and risk. So really, when it comes down to it, valuation is more of an art than a science. And as mentioned right at the beginning of the segment, if you are Warren Buffet, you probably want to keep everything simple and you're going to account for risk in the numerator. But as we're going to see in course number three, conventional financial wisdom opts to include risk in the denominator. And that's what we're going to explore as we move further.