What is a Stock Worth? Part 4 – Discounted Cash Flow Models

Originally written as a guest spot on Blueprint for Financial Prosperity, reprinted here as vacation fodder.

In Part 1 we showed how to calculate the present value of a cash flow that is expected to be received in the future: divide it by (1 + r)n. In Part 3 we concluded by saying the value of a stock is the present value of all the dividends the shareholder will receive, plus the present value of whatever it will be worth when the investor decides to sell it. Simple, eh?

Sure. Until you consider that an investor may hold the stock for many years (we used 35 years as an example in Part 2) and both the dividend and the stock price will grow at an uncertain rate in the future. For our investor, that means first forecasting each dividend for the next 35 years, as well as figuring out what the stock price will be in 35 years, then calculating their present values as follows:

D1/(1+r) + D2/(1+r)2 + D3/(1+r)3… + D35/(1+r)35 + (Ending Stock Price)/(1+r)35. At this point, you are probably thinking “never mind. I’ll take my chances with the lower return on bonds.”

Fortunately, there is a formula that simplifies all of these calculations. It is called the discounted cash flow model. First, with a little algebra you can prove that, as long as the dividend stays the same over time the value of a stock today is equal to the current dividend divided by the required return. (V = CF/r) In Part 3 we used the example of Verizon (VZ), which pays a $1.62 dividend. If investors want to earn a 10% return on stocks (the current bond yield plus the historic average risk premium of about 5%) the stock would have to sell at $1.62/0.10 = $16.20 to be fairly priced. $32.24 sounds overvalued.

But didn’t we say that the dividend tends to grow? No problem. As long as we make the assumption that the rate of growth stays constant (or at least averages a certain rate) we can express the formula this way:

V = CF/(r-g) where g is the rate of growth. So if we make the assumption that Verizon’s dividend can grow 3% per year, the stock’s value climbs to $1.62/(0.10-0.03) = $23.14. By some algebraic manipulation, we can also figure out that at the current stock price of $32.24 investors on average expect 5% growth in the dividend.

There are even other variations of the model that allow you to assume different growth rates at different periods, but for most people the single-growth-rate model (also called the Gordon Growth Model) is sufficient.

One final note: We used the abbreviation “CF” in the model rather than “D” because the model works for any definition of cash flow. The most common variants are dividends, earnings, and free cash flow. The advantages and disadvantages of these measures are summarized in the following table:

Comments

[User 5682]

This formula, the Gordon Dividend Discount Model, is the bedrock of theoretical stock analysis. Bear in mind,though, it's unusable in the real world. Assuming we could negotiate the difficulties of estimating r (diffcult, but do-able), estimating g reliably is out of the question. This is not the sort of growth-rate expectation investors are used to seeing. It must be low (to avoid having a negative denominator and, hence, a negative fair value), so the logic is tortured to present g as a "normal" "mature" growth rate a company can be expected to sustain through infinity. Unless you really get serious about ivory-tower stuff like that (as if you could!), the valuations you'll see from this formula are going to be very whacky!

2006 Aug 13 09:46 AM

[William Trent_]

It's true that estimating r and g are nearly impossible. However, reversing the model can tell you exactly what <i>spread</i&... between r and g is implied by the current share price. That spread can be compared to similar companies and the market as a whole to get a sense of the relative riskiness of the stock.

I have also found some of my most successful investments by identifying companies with imbedded expectations that are far out of whack with what I believe is viable - companies either pricing in no growth (CSG Systems at $16) or high growth in businesses I expected to decline (WorldCom). So while it may be of little use in determining the true value of the typical stock, I have found it far from "unusable"

2006 Aug 13 11:03 PM

[NO DooDahs]

Playing with different R and G is the basis of "stupid analyst tricks" like the much-balleyhooed $600 call on GOOG earlier this year. Every investor should know that analysts' price targets usually come from some mast, er, manipulation of this formula.

I treat the DCF on earnings as a sentiment measure, which is how I interpret the phrase "identifying companies with imbedded expectations." I compare the current P/E of the stock to the "assumed growth" that such a P/E would correspond to given my (very conservative) discount rate, and if the two are really far apart, I know that there's potential there (either long or short or both in alternation).

In practice, I do two- or three-stage models with the growth reverting to the S&amp;P 500 historical mean after some length of time. This creates some math, but just about everyone has a spreadsheet program on their computers nowadays ...

2006 Aug 14 12:14 PM

[William Trent_]

I think we're on the same page here.

2006 Aug 14 04:45 PM

[NO DooDahs]

It has occurred to me ... The problem with the question "what is a stock worth?" is that stocks DON'T HAVE a "true value." Value is subjective. There is only the value YOU think it has, and the price that it's trading at - which is somewhere BETWEEN the value that the last seller and the last buyer think it has. That discrepancy is typically viewed as "irrational" by the value investor.

The two opportunities for profit in the irrationality of markets are generally (a) wait for a return to rationality i.e. mean reversion and (b) ride the irrationality i.e. trend-following. Both work if properly executed, but most investors prefer one or the other.

2006 Aug 14 12:22 PM

[William Trent_]

Sure they have a true value. It's just that nobody knows what it is. :)

2006 Aug 14 04:46 PM