# Stock Valuation

STOCK VALUATION

1. Common stock valuation

A share of common stock is more difficult to value in practice than a bond, for at least three reasons. First, with common stock, not even the promised cash flows are known in a 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. Cash Flows

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 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)

If we were to substitute this expression for P1 into our expression for P0, we would have: |P0= |(D1+P1) |= |D1+(D2+P2)/(1+R) |= |D1 |+ |D2 |+ |P2 | | |(1+R) | |(1+R) | |(1+R)1 | |(1+R)2 | |(1+R)2 |

Now we need to get a price in two periods. We don't know this either, so we can procrastinate again and write: P2 = (D3 + P3)/(1+R)

If we substitute this back in for P2, we have:

|P0= |D1 |+ |D2 |+ |P2 |= |D1 | | |(1+R)1 | |(1+R)2 | |(1+R)3 | |(1+R)2 |

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. 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= |

Some Special Cases

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. Zero Growth The case of zero growth is one we've already seen. A...

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