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- Explain where the value of a stock is ultimately derived.
- Calculate a stock valuation given a dividend growth rate or a stream of dividends.

The financial value of anything is the present value of all future cash flows. If we knew with certainty what the future dividends of a stock will be, we should be able to determine the value of a share of stock. Hence the dividend discount model (DDM). It is useful for us to consider this method of valuing securities, since, ultimately, this is the driver of value in stock ownership. In practice, however, the uncertainty of future dividend payments, especially with common stock, limits the usefulness of using this method. For preferred stock, where the dividend is fixed when it is paid, this method has a bit more accuracy.

If our dividend stream is constant, we can use the perpetuity formula from chapter 7 to arrive at the financial value:

Figure 10.1 Constant Dividend Timeline

Equation 10.1 Perpetuity Equation

$$PV=\frac{PMT}{r}$$Since the dividend payments are constant, the value of a share of preferred stock should be inversely proportional to our required rate of return.

Equation 10.2 Preferred Stock Price

$${P}_{pref}=\frac{D}{r}$$D and *r* should be in matching time units, so if dividends are quarterly, a quarterly rate of return needs to be used. Note that if the required rate of return doesn’t change, then this implies that the stock price should likewise never change. The corollary to this is: if the dividends are a known constant, then any changes in the stock price must be due to changes in the required rate of return!

Suppose we have a 5% preferred stock and investors require a 6% rate of return. Since par is assumed to be $100, our stock pays $5 in dividends per year. Our expected price would be (.05 × $100) / .06 = $83.33.

If our dividend stream isn’t constant, as is more likely with common stocks, but is growing steadily with a constant growth rate, then we can use another formula from chapter 7:

Figure 10.2 Constant Dividend Growth Timeline

Equation 10.3 Perpetuity with Constant Growth

$$P{V}_{n}=\frac{PM{T}_{n+1}}{(r-g)}=\frac{PM{T}_{n}(1+g)}{(r-g)}\{forgr)$$Equation 10.4 Stock Price with Constant Dividend Growth

$${P}_{n}=\frac{{D}_{n+1}}{(r-g)}=\frac{{D}_{n}(1+g)}{(r-g)}\{forgr)$$Again, D, r, and g should all be in matching time units. Typically we are interested in the price now (that is, at time 0), but this equation could be used to find our expected stock price in a future year by calculating the expected dividend for that year. Also note that *D*_{0} is the dividend that was just paid, and thus is no longer factored into the stock price.

If a company’s most recent dividend (*D*_{0}) was $0.60, dividend growth is expected to be 4% per year, and investors require 10%, we can find the expected current stock price (*P*_{0}). $0.60 × (1 + .04) / (.10 − .04) = $10.40.

If we use the current price (*P*_{0}) and rearrange our equation to solve for returns, we find an interesting result:

Equation 10.5 Components of Stock Returns

$$\begin{array}{l}P0=\frac{{D}_{1}}{(r-g)}\{forgr\}\\ \left(r-g\right)=\frac{{D}_{1}}{{P}_{0}}\\ r=\frac{{D}_{1}}{{P}_{0}}+g\\ r=dividendyield+capitalgainsyield\end{array}$$With this result, we can clearly see the tradeoff between dividends now and growth (which should lead to future dividends). If expected return is steady over time, then a constant capital gains yield (g) implies a constant dividend yield. A constant dividend yield means that the stock price must grow proportionally to the dividends; that is, both should grow by g.

Without constant growth, determining the present value of the stock requires finding the present value of each of the future cash flows. While the most flexible and realistic, this also is the most difficult to execute properly. The best way to think about this method is to imagine holding the stock for a specific number of years, with the intention of selling the stock at the end of the period.

Figure 10.3 Varied Dividends Timeline

If we know the dividends and have an expectation for the future stock’s price, we can discount everything to find the price today. The difficulty, of course, is in getting an accurate expectation for the future stock price. The traditional solution is to assume that, at some point in the future, dividend growth will be steady, and to use the constant dividend growth formula to calculate an expected future price.

Equation 10.6 Stock Price with Constant Dividend Growth

$$\begin{array}{l}P0=PVofFutureDividends+PVofFuturePriceofStock\\ P0=\frac{{D}_{1}}{{(1+r)}^{1}}+\frac{{D}_{2}}{{(1+r)}^{2}}+\dots +\frac{{D}_{n}}{{(1+r)}^{n}}+\frac{{P}_{n}}{{(1+r)}^{n}}\end{array}$$A common mistake is to neglect the discounting on the future price of the stock. Once the cash flows are found, the discounting can also be accomplished using NPV functions on a calculator or spreadsheet, as discussed in chapter 7.

Suppose our stock will pay out $0.50 flat per year for 4 years, and then dividends are expected to grow at 5% afterwards. If investors expect a 9% return, we can find the expected price of the stock:

Figure 10.4 Varied Dividends Example Timeline

Our terminal value (*P*_{4}) should be $0.50 × (1 + .05) / (.09 − .05) = $13.13. Once we add this to the above cash flows and discount appropriately, we arrive at a stock value of $10.92.

We can use this to method value a corporation that is not a going concern (that is, going out of business) or expected to be acquired. In this case, we should use the liquidation value of the shares or the acquisition price as our *P*_{n}.

How do we handle stocks that aren’t currently paying a dividend, like many growth stocks? The assumption is that some point in the future they will need to start paying dividends, so we figure out the price of the stock at that time, and discount it back to today. Note that this is the same as the above equation, using 0 for each dividend until the company begins to pay them.

Because of the extra uncertainty of when to expect a company to begin paying dividends, such companies are typically valued using another approach. Two of the most popular are the market multiples approach and the free cash flow approach, which will be covered in the upcoming sections.

- Every stock must be fundamentally worth the present value of the future cash flows.
- Since many stocks pay in perpetuity, the perpetuity formulas are useful for dividends growing at a constant rate. This formula will only work accurately with a constant rate.
- For stocks with dividends not currently growing at a constant rate, the individual dividends need to be accounted for first, then a future value for the stock can be used for when the dividends are in the constant growth period.

- Investors only require an 8% annual return on a 9% preferred stock. What should the stock’s price be?
- A company’s most recent quarterly dividend was $0.25. This dividend is expected to grow by 3% a year. If investors require an 11% annual return on the stock, what should the stock’s price be?
- A stock is currently not paying dividends. Three years from now, it is expected to start paying a quarterly dividend of $0.10 per share, with growth of 5% per year thereafter. If investors require a 10% annual return, what should the stock’s price be?
- A stock will pay a $1 dividend next year, a $2 dividend the year following, and a $3 dividend in year 3, at which time it will be acquired for $12 per share. If investors require a 12% return, what is the current value of the stock?