Stock Valuation

When we developed the formula to price bonds, it was a straight-forward application of the time value of money concepts. The bond produces a series of simple cash flows (fixed interest payments twice per year and a maturity value ($1000) at the end of the bonds fixed life span. However, stocks have no expiration or maturity date, therefore (at least theoretically) the cash flow (dividend) stream extends into infinity. Also, the dividends are MUCH more difficult to project than are the interest payments. Therefore, stocks require a slightly different application of the time value of money concept.

While it is not likely that any person will hold a stock forever (unless that person has discovered the secret of immortality), the valuation process using an infinite cash flow stream remains appropriate. If I buy a stock today based on the present value of the expected cash flows and only plan to hold the stock for three years, why am I concerned with the dividends that will be paid after year three? The answer is pretty simple. If I plan to sell the stock after the three years, I'm going to have to find a buyer. How much will that buyer pay me? According to our framework, the buyer will pay the present value (at the time they buy the stock) of the expected cash flow stream. Therefore, what the buyer will be willing to pay me will depend on the dividends from years 4 and on. Since those later dividends will affect the price at which I can sell the stock, I must factor them into my analysis. By finding the present value of ALL expected cash flows (dividends) that the stock will pay, my holding period becomes irrelevant. Whether I want to hold the stock for one day or twenty years, it is worth the same to me.

Theoretical stock valuation based on the dividend discount model typically take one of three forms depending on what pattern we expect the dividends to follow. These three model variations are (1) The No Growth Case, (2) The Constant Growth Case, and (3) The Non constant (Supernormal) Growth case. There are a couple of other variations, but these three provide a solid foundation. Remember, all three methods do the same thing -- forecast a cash flow stream that will be paid to stockholders (dividends) and then discount that cash flow stream back to the present to see what the stock is worth today.

In all models below, we assume that the current dividend has just been paid (immediately before we buy the stock) and our first dividend received will be one year from today. This makes the application of the time value of money simpler. While it is not realistic, it does not greatly alter the results and therefore is a worthwhile simplification.

NO GROWTH

If we have a stock with no growth in its dividends over time, the infinity issue is solved with a perpetuity. The stockholder will receive the same dividend every year (an annuity) that lasts forever - a perpetuity.

The most common example of a no growth stock is a PREFERRED STOCK. A preferred stock typically pays a fixed dividend (a percentage of its par value), that does not change over time. However, there are some instances where a common stock at least approximates the no growth pattern.

According to the no growth model, to find the value of the stock, we just take the current dividend and divide by the required return (remember, it's just a perpetuity -- an infinite annuity -- since the stock has no maturity date and the dividend is not expected to increase or decrease in value). This is written below

• where P0 represents the current value (price today)
• k represents the required return and
• D1 represents the dividend
• (Note -- while we designate next year's dividend in the formula, this is just to be consistent with the later models. Since there is no growth, all the dividends are the same regardless of which year we are referring to.)

Consider the following example with a preferred stock. Assuming that a preferred stock has a par value of $75, pays a 10% dividend and you have a 8% required return, what is this stock worth to you?

First, find the dividend which is $7.50 ($75*0.10). The next step is to plug it into our model and get the price of $93.75 ($7.50/0.08).

CONSTANT GROWTH

While it is possible for a common stock to have a constant dividend over time, it is not likely. Companies tend to grow and expand, which usually results in dividends growing over time. However, if dividends don't remain constant we can no longer use a perpetuity formula. Also, since the dividend stream doesn't end, we can't use the standard time value of money process. Luckily, as long as the growth rate remains constant over time there is a simple formula we can use to find the present value.

or

• where g is the growth rate in dividends
• P0 represents the current value (price today)
• k represent the required return and
• D0 and D1 represent the dividend paid today (D0) or the forecasted dividend next year (D1) respectively
• (Note -- D0(1 + g) and D1 are the same thing. They both represent the forecasted dividend next year. The only difference is that sometimes you will be given the current dividend and sometime you will be given the forecasted dividend next year. Since the present value formula needs the forecasted dividend next year, D0(1 + g) just gives us that value based on the current dividend and the dividend growth rate.)

For a quick example, consider a stock that just paid a dividend (D0) of $5.00 per share with dividends growing at a constant 4% per year. If my required return is 13%, what is the stock worth to me?

P0 = [D0(1 + g)]/(k - g)
P0 = [5.00*(1 + .04)]/(.13 - .04)
P0 = 5.20/(.09)
P0 = $57.78

Three points on this model. First, while it may not look like the present value formulas that we did in Chapter Four, that is all it is. The constant growth model is not magical, its just a special case of present value and could be used to find the present value of any cash flow stream that is growing at a constant rate. Second, growth rates rarely remain constant over time. However if growth rates are relatively stable, this can be a close approximation. Third, this model only works when the required return exceeds the growth rate. This is not usually critical as it is impossible to maintain a growth rate higher than the required return indefinitely.

SUPERNORMAL (NONCONSTANT) GROWTH

This is where things get a little tricky. However, it is the most common situation. The solution is not a simple formula, but instead a three-step process.

The 3-step solution

• Step 1 - Forecast the dividends during the non constant growth period up to the first year at which dividends grow at a constant rate.
• Step 2 - Once a constant growth rate is reached, use the constant growth pricing model to forecast the stock price. This stock price represents the PV of all dividends beyond the non constant growth period.
• Step 3 - Discount the cash flows (dividends found in step one and price found in step two) back to year zero at the appropriate discount rate. This is the current value of the stock.

This is a tricky one, so again, lets do an example. Consider a firm that just paid a dividend of $2.60. They plan to increase dividends by 5% in year one, 10% in year two, 20% in year 3, 20% in year 4, and then 3% per year thereafter. You feel that a 16% required return is appropriate. What is this stock worth to you?

Step One -- Forecast the Dividends
D1 = $2.60*(1.05) = $2.73
D2 = $2.73*(1.10) = $3.00
D3 = $3.00*(1.20) = $3.60
D4 = $3.60*(1.20) = $4.32
D5 = $4.32*(1.03) = $4.45
(Note -- We stop in year 5 because that is the first year of constant growth. There is no need to forecast dividends any further since once they are growing at a constant rate, we can apply the constant growth model discussed above.)

Step Two -- Use the Constant Growth Model to Forecast Price
P4 = $4.45/(.16 - .03) = $34.23
(Note: Be careful here as this is a tough, but critical detail. When we apply the constant growth model we use next year's dividend to get this year's price. Since we are using year five's dividend (the first dividend of the constant growth model), it will tell us the price in year four. This price represents the present value of all dividends paid from year 5 and beyond as of year four.)

Step Three -- Discount Cash Flows Back to Today

Financial Calculator -- HP-10B
1) 2nd Clear All
2) 0 CFj
3) 2.73 CFj
4) 3.00 CFj
5) 3.60 CFj
6) 38.55 CFj
7) 16 I/YR
8) 2nd NPV --> $28.18

Financial Calculator -- TI-BAII+
1) CF 2nd CLR Work
2) 0 ENTER down arrow
3) 2.73 ENTER down arrow down arrow
4) 3.00 ENTER down arrow down arrow
5) 3.60 ENTER down arrow down arrow
6) 38.55 ENTER
7) NPV 16 ENTER down arrow
8) CPT --> $28.18

(Note: A couple of comments here. First, the year 4 cash flow ($38.55) represents both the year 4 dividend and the price in year 4. If you try to enter them separately, the calculator will think the dividend comes in year 4 and the price in year 5, giving you the wrong answer. Second, you may be wondering what happened to the year 5 dividend. The answer is that it is included in the year 4 price. To include it again would be double-counting. Remember what the year 4 price represents -- the present value (as of year 4) of all dividends paid in years 5 and beyond. Third, as with the first two models, this is just another application of time value of money. We forecast the cash flows and then discount them back to today.)

MARKET EFFICIENCY

Markets are said to be efficient when prices of stocks accurately represent all currently available information. This means that we can not determine which stocks are "good" and which are "bad". All stocks are properly valued given what is known today. If they turn out to be "good" or "bad" in the future, it is due to information that has yet to be revealed. There are three types of market efficiency that are based on what is considered "current" information.

• Weak Form Efficiency - Markets are efficient based on past price data.
• Semi-Strong Form Efficiency - Markets are efficient based on all publicly available information
• Strong Form Efficiency - Markets are efficient based on all public and private information

For a more thorough analysis of market efficiency, see the handout in class

Par Value vs Book Value vs Market Value

Par Value
The face value of each share of stock stated on the stock's charter. The only time this is a meaningful number is when the stock is initially sold for less than par value (which almost never happens). In this case, shareholders are liable for the difference in the event of bankruptcy. In today's markets, newly issued common stocks often are issued with either no par value or a par value of $0.01. Note that this discussion is focused on common stock. Par value for preferred stock is very different as the dividend is often based on par value for preferred.

Book Value
BV = (BV of Assets - BV of Liabilities)/(# of outstanding shares)
This tells us how much each share is worth on an accounting basis. The book value tends to understate the true value of a stock because the balance sheet focuses on historical value and (in most cases) omits the value of intangible assets (such as brand names, intellectual property, etc.) Also, historical value (purchase price less accumulated depreciation) does not take into account the value of those assets in generating cash flows or the riskiness of the assets.

Market Value
This is the most important measure of share value. It is the price at which you can buy or sell a share of stock. To get the market value of a stock at any time, you can use one of the many free stock quote services found online. One that I use frequently is Yahoo! Finance. When doing so, you will need to know the stock's "ticker symbol" which can be found using the Symbol Lookup link on Yahoo! Finance. This is the value that we as managers are trying to maximize.

Rights and Privileges of Common Stockholders

• Common stockholders have a residual right to the income of the firm. This means that any income generated beyond what is required to pay preferred dividends belongs to the common stockholders. This income may be distributed to common stockholders in the form of dividends or it may be reinvested in the firm.
• Stockholders control the firm through the election of the board of directors, but this control is often limited.
• Stockholders have the right to obtain information from management about the firm's operations.
• Common stockholders can usually lose no more than their initial investment, because they have no liability for the debts incurred by the firm beyond the value of the stock that they own.
• Common stockholder may usually transfer ownership of shares to other investors in the secondary market.

Advantages of Issuing Stock

• Flexibility

  Dividends are much more flexible than interest payments. However, shareholders do not like cut or omitted dividends, so managers must be careful when doing so.

• No maturity date means that there is not a large required cash outflow at a specific point in the future.
• Can help lower risk and, in turn, interest costs.

Disadvantages of Issuing Stock

• Dilutes control as each owner now owns a smaller proportion of the company.
• Negative signaling lowers value of shares already outstanding by 2-3%
• Higher underwriting costs than debt means it is often more expensive to go through the process of raising capital with stock than it is with bonds.
• Is usually more expensive on a cost of capital basis than debt

Stock Valuation from a Non-Textbook Viewpoint

The dividend discount approach has a solid theoretical foundation and provides an essential background into how stock prices are determined. However, there are many issues that make it difficult to apply in practice. Because these issues are (for the most part) beyond the scope of this course and to some extent are more judgmental in nature, they are not a part of your "expected" knowledge for exams. In other words, this section is entirely optional. As such, I have separated it from the rest of Chapter Seven. If you are interested in finding out more about some stock valuation practices and issues beyond the dividend valuation model, go to Chapter Six -- Part Two. This portion is entirely optional. However, if you are interested in investing and the stock market, I think (hope) you will find it worthwhile.