Stock Valuation
Zero Growth Stock
The case of zero growth is one we've already seen. A share of common stock in a company with a constant dividend is much like a share of preferred stock. For a zero growth share of common stock, this implies that: D1=D2=D3=D=constant. So, the value of the stock is P0=D/(1+R) + D/(1+R)² +D/(1+R)³ + D/(1+R)⁴ + D/(1+R)⁵ + ... Because the dividend is always the same, the stock can be viewed as an ordinary perpetuity with a cash flow equal to D every period. The formula for the PV of an ordinary perpetuity with cash flow equal to D is P0=D/R. For example, suppose the Paradise Prototyping Company has a policy of paying a $10 per share dividend every year. If this policy is to be continued indefinitely, what is the value of a share of stock if the required return is 20 percent? The stock in this case amounts to an ordinary perpetuity, so the stock is worth $10/.20 = $50 per share.
Stock Cash Flows
A share of common stock is more difficult to value in practice than a bond, for at least three reasons. With bonds, we calculate its value by discounting all of its coupon payments and adding those to the discounted face value. With common stock, not even the promised cash flows are known in advance. Second, the life of the investment is essentially forever, since common stock has no maturity. Third, there is no way to easily observe the rate of return that the market requires. Nonetheless, as we will see, there are cases in which we can come up with the present value of the future cash flows for a share of stock and thus determine its value. Imagine that you are considering buying a share of stock today. You plan to sell the stock in one year. You somehow know that the stock will be worth $70 at that time. You predict that the stock will also pay a $10 per share dividend at the end of the year. If you require a 25 percent return on your investment, what is the most you would pay for the stock? In other words, what is the present value of the $10 dividend along with the $70 ending value at 25 percent? If you buy the stock today and sell it at the end of the year, you will have a total of $80 in cash. At 25 percent: Present value ($10+70)/1.25 = $64. Therefore, $64 is the value you would assign to the stock today. More generally, let P0 be the current price of the stock, and assign P1 to be the price in one period. If D1 is the cash dividend paid at the end of the period, then: P0 = (D1+P1)/(1+R) where R is the required return in the market on this investment. Notice that we really haven't said much so far. If we wanted to determine the value of a share of stock today (P0), we would first have to come up with the value in one year (P1). This is even harder to do, so we've only made the problem more complicated. What is the price in one period, P1? We don't know in general. Instead, suppose we somehow knew the price in two periods, P2. Given a predicted dividend in two periods, D2, the stock price in one period would be: P1 = (D2+P2)/(1+R). All we did is come up with an equation that would give us the value of P1 assuming we have P2, the price of the stock two years from now. If we were to substitute this expression for P1 back into our orignal equation, everything simplifies to D1/(1+R)¹ + D2/(1+R)² + P2/(1+R)². Now we have another problem. What is P2? Well, we can write another equation: P2 = (D3 + P3)/(1+R). Plugging this back into our original equation, everything is simplified to D1/(1+R)¹ + D2/(1+R)² + D3/(1+R)³ + P3/(1+R)³. You should start to notice that we can push the problem of coming up with the stock price off into the future forever. It is important to note that no matter what the stock price is, the present value is essentially zero if we push the sale of the stock far enough away. This is because the present value at the very end would be divded by an infinitely large number, making it essentially zero. What we are eventually left with is the result that the current price of the stock can be written as the present value of the dividends beginning in one period and extending out forever: P0=D1/(1+R)¹ + D2/(1+R)² + D3/(1+R)³ + ... We have illustrated here that the price of the stock today is equal to the present value of all of the future dividends. How many future dividends are there? In principle, there can be an infinite number. This means that we still can't compute a value for the stock because we would have to forecast an infinite number of dividends and then discount them all. In the next section, we consider some special cases in which we can get around this problem. There are a few very useful special circumstances under which we can come up with a value for the stock. What we have to do is make some simplifying assumptions about the pattern of future dividends. The three cases we consider are the following: (1) the dividend has a zero growth rate, (2) the dividend grows at a constant rate, and (3) the dividend grows at a constant rate after some length of time. We consider each of these separately.
Nonconstant Growth
Nonconstant growht is when the dividend grows at a constant rate only after some length of time. For a simple example of nonconstant growth, consider the case of a company that is currently not paying dividends. You predict that, in five years, the company will pay a dividend for the first time. The dividend will be $.50 per share. You expect that this dividend will then grow at a rate of 10 percent per year indefinitely. The required return on companies such as this one is 20 percent. What is the price of the stock today? To see what the stock is worth today, we first find out what it will be worth once dividends are paid. We can then calculate the present value of that future price to get today's price. The first dividend will be paid in five years, and the dividend will grow steadily from then on. Using the dividend growth model, we can say that the price in four years will be: P4 = D₄(1+ g)/(R-g) = D₅/(R-g) = $.50/(.20 - .10) = $5 If the stock will be worth $5 in four years, then we can get the current value by discounting this price back four years at 20 percent: P₀ = $5/1.20⁴ = $5/2.0736 = $2.41. The stock is therefore worth $2.41 today. The problem of nonconstant growth is only slightly more complicated if the dividends are not zero for the first several years. For example, suppose that a company pays $1 the first year, $2 the second, and $2.5 the third. After the third year, the dividend will grow at a constant rate of 5 percent per year. The required return is 10 percent. What is the value of the stock today? To calculate the present value, let's first pretend that the third year when dividends start growing at a constant rate is actually the first. If this is the case, how do we calculate its value? To do so, we simply have to use the dividend growth model: P₀=D₀(1+g)/(R-g)=2.50(1.05)/(.10-.05)=$52.50. This is the price if we assumed that this wasn't a nonconstant stock, but instead a constant growth stock. In our problem, the stock doesn't actually start growing at a constant rate until the third year--we also have dividends in our first, second, and third years. So really, our P₀ that we just calculated is really P₃ in our real example. So to get the present value, we simply have to discount all three dividends from years 1-3 and add that sum to the discounted value of P₃, which we know. So this becomes D₁/(1+R)₁ + D₂/(1+R)² + D₃/(1+R)³ + P₃/(1+R)³ = 1/1.10 + 2/1.10² + 2.50/1.10³ + 52.50/1.10³ = $43.88. This makes sense because P₃ accounts for the constant growth rate in perpetuity starting in the third year (calculated with dividend growth model. Remember to discount this to get PV) while each of the other three dividends in years 1-3 have to be discounted back individually since they weren't constant growth dividends (dividend growth model doesn't apply--we have to discount them the slow way).
Preferred vs Common Stock
Preferred stock differs from common stock because it has preference over common stock in the payment of dividends and in the distribution of corporation assets in the event of liquidation. Preference means only that the holders of the preferred shares must receive a dividend (in the case of an ongoing firm) before holders of common shares are entitled to anything. Preferred stock is a form of equity from a legal and tax standpoint. It is important to note, however, that holders of preferred stock sometimes have no voting privileges. A preferred dividend is not like interest on a bond. The board of directors may decide not to pay the dividends on preferred shares, and their decision may have nothing to do with the current net income of the corporation. Unpaid preferred dividends are not debts of the firm. Directors elected by the common shareholders can defer preferred dividends indefinitely. However, in such cases, common shareholders must also forgo dividends. If Nonetheless, preferred stock dividends are often much higher than dividends on common stock and are fixed at a certain rate, while common dividends can change or even get cut entirely.In addition, holders of preferred shares are often granted voting and other rights if preferred dividends have not been paid for some time, but common shares don't have this benefit. Common stock represents shares of ownership in a corporation and the type of stock in which most people invest. When people talk about stocks they are usually referring to common stock. In fact, the great majority of stock is issued is in this form. Common shares represent a claim on profits (dividends) and confer voting rights. Investors most often get one vote per share-owned to elect board members who oversee the major decisions made by management. Stockholders thus have the ability to exercise control over corporate policy and management issues compared to preferred shareholders. Common stock tends to outperform bonds and preferred shares. It is also the type of stock that provides the biggest potential for long-term gains. If a company does well, the value of a common stock can go up. But keep in mind, if the company does poorly, the stock's value will also go down. The reason why a common stock is more volatile than a preferred stock is because common shares are more directly involved with the company. dividends are withheld for preferred stocks, then these dividends must be paid before any other dividends.
Constant Growth Stock/Dividend Growth Model
Suppose we know that the dividend for some company always grows at a steady rate. Call this growth rate g. If we let D0 be the dividend just paid, then the next dividend, D1, is: D1 = D0(1+g) The divident in two periods is D2= D0(1+g)². We could repeat this process to come up with the dividend at any point in the future. In general, we can determine the dividend value n periods into the future: Dn=D0(1+g)ⁿ. This should make sense because the dividend is growing at a constant rate, so all you're doing is multiplying the rate to some power to calculate the value at that point in time. An asset with cash flows that grow at a constant rate forever is called a growing perpetuity. As we will see momentarily, there is a simple expression for determining the value of such an asset. If we take D0 to be the dividend just paid and g to be the constant growth rate, the value of a share of stock can be written as: P0=D0(1+g)/(1+R) + D0(1+g)²/(1+R)² + D0(1+g)³/(1+R)³ + ... We can simplify this expression by recognizing that this is a geometric series. The first term is D0(1+g)/(1+R) and the factor that's being multiplied each time is (1+g)/(1+R). The geometric series formula is A(1-Rⁿ)/1-R where A is the first term and R is the factor. Plugging both values in, we get D0(1+g)/r-g which is the exact same as D1/r-g since D1=D0(1+g). We call this formula the dividend growht model. To illustrate, suppose D0 is $2.30, R is 13 percent, and g is 5 percent. The price per share in this case is: P0 = $2.30*1.05/(.13-.05) = $2.415/.08 = $30.19. We can actually use the dividend growth model to get the stock price at any point in time, not just today. In general, the price of the stock as of time n is: Pₙ = Dₙ(1+g)/R-g = Dₙ₊₁/R-g In our example, suppose we are interested in the price of the stock in five years, P5. We first need the dividend at Time 5, D5. Because the dividend just paid is $2.30 and the growth rate is 5 percent per year, D5 is: D5 = $2.30(1.05)⁵ = $2.30*1.2763 = $2.935. From the dividend growth model, we get the price of the stock in five years: P5=2935(1.05)/.13-.05 = 30822/.08 = $38.53. You might wonder what would happen with the dividend growth model if the growth rate, g, were greater than the discount rate, R. It looks like we would get a negative stock price because R-g would be less than zero. This is not what would happen. Instead, if the constant growth rate exceeds the discount rate, then the stock price is infinitely large. Why? If the growth rate is bigger than the discount rate, then the present value of the dividends keeps on getting bigger and bigger since we keep multiplying each term by a ratio (1+g/1+R) greater than 1. Essentially, the same is true if the growth rate and the discount rate are equal (meaning the ratio=1) because then it would just be an infinite stream of constant dividends (it is constant since growth rate and discount rate cancel each other out if they are equal).
