Preferred stock usually represents a **perpetuity** or, in other words, has no maturity date. It is valued in the market without any principal payment since it has no ending life. If preferred stock had a maturity date, the analysis would be similar to that of the preceding bond example. Preferred stock has a fixed dividend payment carrying a higher order of precedence than common stock dividends, but not the binding contractual obligation of interest on debt. Preferred stock, being a hybrid security, has neither the ownership privilege of common stock nor the legally enforceable provisions of debt. To value a perpetuity such as preferred stock, we first consider this formula:

where

Pp = |
the price of preferred stock |

Dp = |
the annual dividend for preferred stock (a constant value) |

Kp = |
the required rate of return, or discount rate, applied to preferred stock dividends |

Notice that, unlike a bond, the preferred stock never matures. Because the dividend payments are promised to continue forever, a preferred stock is valued as a perpetuity. A perpetuity is described by a timeline that stretches to infinity as shown here:

The preferred stock is easily valued as

Actually, Formula 10-3 can be used to value any perpetuity, as long as the first payment occurs one year from the valuation date. All we have to do to find the price of preferred stock (*Pp*) is to divide the constant annual dividend payment (*Dp*) by the required rate of return that preferred stockholders are demanding (*Kp*). For example, if the annual dividend were $10 and the stockholder required a 10 percent rate of return, the price of preferred stock would be $100.

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As was true in our bond valuation analysis, if the rate of return required by security holders changes, the value of the financial asset (in this case, preferred stock) will change. You may also recall that the longer the life of an investment, the greater the impact of a change in required rate of return. It is one thing to be locked into a low-paying security for one year when the rate goes up; it is quite another to be locked in for 10 or 20 years. With preferred stock, you have a *perpetual* security, so the impact is at a maximum. Assume in the prior example that because of higher inflation or increased business risk, *Kp* (the required rate of return) increases to 12 percent. The new value for the preferred stock shares is:

If the required rate of return were reduced to 8 percent, the opposite effect would occur. The preferred stock price would be computed as:

It is not surprising that the preferred stock is now trading well above its original price of $100. It is still offering a $10 dividend (10 percent of the original offering price of $100), and the market is demanding only an 8 percent yield. To match the $10 dividend with the 8 percent rate of return, the market price will advance to $125.

**Determining the Required Rate of Return (Yield) from the Market Price**

In our analysis of preferred stock, we have used the value of the annual dividend (*Dp*) and the required rate of return (*Kp*) to solve for the price of preferred stock (*Pp*). We could change our analysis to solve for the required rate of return (*Kp*) as the unknown, given that we know the annual dividend (*Dp*) and the preferred stock price (*Pp*). We take Formula 10-3 and rewrite it as Formula 10-4, where the unknown is the required rate of return (*Kp*).

Using Formula 10-4, if the annual preferred dividend (*Dp*) is $10 and the price of preferred stock (*Pp*) is $100, the required rate of return (yield) would be 10 percent as follows:

If the price goes up to $130, the yield will be only 7.69 percent:

We see the higher market price provides quite a decline in the yield.

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Valuation of Common Stock

The value of a share of common stock may be interpreted by the shareholder as the *present value* of an expected stream of *future dividends.* Although in the short run stockholders may be influenced by a change in earnings or other variables, the ultimate value of any holding rests with the distribution of earnings in the form of dividend payments. Though the stockholder may benefit from the retention and reinvestment of earnings by the corporation, at some point the earnings must be translated into cash flow for the stockholder. A stock valuation model based on future expected dividends, which is termed a **dividend valuation model**, can be stated as:

where

P0 = |
Price of stock today |

D = |
Dividend for each year |

Ke = |
the required rate of return for common stock (discount rate) |

This formula, with modification, is generally applied to three different circumstances:

1. No growth in dividends.

2. Constant growth in dividends.

3. Variable growth in dividends.

**No Growth in Dividends**

Under the no-growth circumstance, common stock is very similar to preferred stock. The common stock pays a constant dividend each year. For that reason, we merely translate the terms in Formula 10-3, which applies to preferred stock, to apply to common stock. This is shown as new Formula 10-6:

P0 = |
Price of common stock today |

D1 = |
Current annual common stock dividend (a constant value) |

Ke = |
Required rate of return for common stock |

Assume *D*1 = $1.87 and *Ke* = 12 percent; the price of the stock would be $15.58:

A no-growth policy for common stock dividends does not hold much appeal for investors and so is seen infrequently in the real world.6

**Constant Growth in Dividends**

A firm that increases dividends at a constant rate is a more likely circumstance. Perhaps a firm decides to increase its dividends by 7 percent per year. The general valuation approach is shown in Formula 10-7:

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where

P0 = |
Price of common stock today |

D0(1 + g)1 = |
Dividend in year 1, D1 |

D0(1 + g)2 = |
Dividend in year 2, D2, and so on |

g = |
Constant growth rate in dividends |

Ke = |
Required rate of return for common stock (discount rate) |

As shown in Formula 10-7, the current price of the stock is the present value of the future stream of dividends growing at a constant rate. If we can anticipate the growth pattern of future dividends and determine the discount rate, we can ascertain the price of the stock.

For example, assume the following information:

D0 = |
Last 12-month’s dividend (assume $1.87) |

D1 = |
First year, $2.00 (growth rate, 7%) |

D2 = |
Second year, $2.14 (growth rate, 7%) |

D3 = |
Third year, $2.29 (growth rate, 7%) etc. |

Ke = |
Required rate of return (discount rate), 12% |

Then

To find the price of the stock, we take the present value of each year’s dividend. This is no small task when the formula calls for us to take the present value of an *infinite* stream of growing dividends. Fortunately, Formula 10-7 can be compressed into a much more usable form if two circumstances are satisfied:

1. The firm must have a constant dividend growth rate (*g*).

2. The discount rate (*Ke*) must be higher than the growth rate (*g*).

For most introductory courses in finance, these assumptions are usually made to reduce the complications in the analytical process. This allows us to reduce or rewrite Formula 10-7 as Formula 10-8. Formula 10-8 is the basic equation for finding the value of common stock and is referred to as the constant growth dividend valuation model:

This is an extremely easy formula to use in which:

P0 = |
Price of the stock today |

D1 = |
Dividend at the end of the first year |

Ke = |
Required rate of return (discount rate) |

g = |
Constant growth rate in dividends |

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In this formula, *P*0 is sometimes referred to as the value of a “growing perpetuity” because it is a perpetuity that grows at a constant rate. In order for Formula 10-8 to work, it is critical that the first dividend come at the end of the first year. Based on the current example:

D1 = |
$2.00 |

Ke = |
0.12 |

g = |
0.07 |

and *P*0 is computed as:

Thus, given that the stock has a $2 dividend at the end of the first year, a discount rate of 12 percent, and a constant growth rate of 7 percent, the current price of the stock is $40.

Let’s take a closer look at Formula 10-8 shown earlier and the factors that influence valuation. For example, what is the anticipated effect on valuation if *Ke* (the required rate of return, or discount rate) increases as a result of inflation or increased risk? Intuitively, we would expect the stock price to decline if investors demand a higher return and the dividend and growth rate remain the same. This is precisely what happens.

If *D*1 remains at $2.00 and the growth rate (*g*) is 7 percent, but *Ke* increases from 12 percent to 14 percent, using Formula 10-8, the price of the common stock will now be $28.57 as shown below. This is considerably lower than its earlier value of $40:

Similarly, if the growth rate (*g*) increases while *D*1 and *Ke* remain constant, the stock price can be expected to increase. Assume *D*1 = $2.00, *Ke* is set at its earlier level of 12 percent, and *g* increases from 7 percent to 9 percent. Using Formula 10-8 once again, the new price of the stock would be $66.67:

We should not be surprised to see that an increasing growth rate has enhanced the value of the stock.

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**Stock Valuation Based on Future Stock Value** The discussion of stock valuation to this point has related to the concept of the present value of future dividends. This is a valid concept, but suppose we wish to approach the issue from a slightly different viewpoint. Assume we are going to buy a stock and hold it for three years and then sell it. We wish to know the present value of our investment. This is somewhat like the bond valuation analysis. We will receive a dividend for three years (*D*1, *D*2, *D*3) and then a price (payment) for the stock at the end of three years (*P*3). What is the present value of the benefits? To solve this, we add the present value of three years of dividends and the present value of the stock price after three years. Assuming a constant growth dividend analysis, the stock price after three years is simply the present value of all future dividends after the third year (from the fourth year on). Thus the current price of the stock in this case is nothing other than the present value of the first three dividends, plus the present value of all future dividends (which is equivalent to the stock price after the third year). Saying the price of the stock is the present value of all future dividends is also the equivalent of saying it is the present value of a dividend stream for a number of years, plus the present value of the price of the stock after that time period. The appropriate formula would be Formula 10-7, where the fourth term would be replaced by *P*3 = *D*4/(*Ke* − *g*).

**Determining the Required Rate of Return from the Market Price**

In our analysis of common stock, we have used the first year’s dividend (*D*1), the required rate of return (*Ke*), and the growth rate (*g*) to solve for the stock price (*P*0) based on Formula 10-8.

We could change the analysis to solve for the required rate of return (*Ke*) as the unknown, given that we know the first year’s dividend (*D*1), the stock price (*P*0), and the growth rate (*g*). We take the preceding formula and algebraically change it to provide Formula 10-9.

Formula 10-9 allows us to compute the required return (*Ke*) for the investment. Returning to the basic data from the common stock example:

Ke = |
Required rate of return (to be solved) |

D1 = |
Dividend at the end of the first year, $2.00 |

P0 = |
Price of the stock today, $40 |

g = |
Constant growth rate 0.07, or 7% |

In this instance, we would say the stockholder demands a 12 percent return on the common stock investment. Of particular interest are the individual parts of the formula for *Ke* that we have been discussing. Let’s write out Formula 10-9 again.

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The first term represents the **dividend yield** the stockholder will receive, and the second term represents the anticipated growth in dividends, earnings, and stock price. While we have been describing the growth rate primarily in terms of dividends, it is assumed the earnings and stock price will also grow at that same rate over the long term if all else holds constant. You should also observe that the preceding formula represents a total-return concept. The stockholder is receiving a current dividend plus anticipated growth in the future. If the dividend yield is low, the growth rate must be high to provide the necessary return. Conversely, if the growth rate is low, a high dividend yield will be expected. The concepts of dividend yield and growth are clearly interrelated.

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