 • Table of Contents • Introduction • 1-Time Value • 2-Risk/Return • 3-Accounting • 4-Securities • 5-Business • 6-Regulatory • Case Studies • Student Papers ## 4.3.2 Basic Stock Valuation

Let's look again at the basic DCF stock valuation formulas --

General DCF formula

The value of shares of common stock, like any other financial instrument, is often understood as the present value of expected future returns. Again we return to the discounted cash flow formula:

 P o = D1/(1+i1) + D2 /(1+i 2 )2 + D3/(1+i3)3 + ... + D inf /(1+i inf)inf Po = Dn / (1+i n)n P o = value of common stock Dt = per-share dividends expected at the end of year t it = required return (discount rate) for each year t inf = infinite time period

This DCF formula leads to two particularized formulas in situations of zero growth and constant growth.

Zero growth

The simplest DCF model assumes constant dividends -- zero growth. In this artificial world (no inflation, no variation, no change) the present value of a constant dividend stream is the present value of a perpetuity:

 P 0 = D / i P 0 = value of common stock D = per-share dividends expected at the end of each year i = required return (discount rate) for each year

Constant growth

In his 1937 Ph.D. thesis, John Burr Williams, a Harvard graduate student, developed what he called the Dividend Discount Model. Williams model called for the investor to forecast the dividends a company would pay and then discount them based on the confidence in that forecast (from Capital Ideas, Peter L. Bernstein). His model has been modified many times. One such modification is found in the Gordon-Shapiro Dividend Discount Model. We often expect that cash flows will grow, with inflation and as the company progresses. Thus, with the assumption that dividends will also grow at a constant rate (g), Gordon and Shapiro produced one of the most often-used formulas in stock valuation, known as the Gordon Shapiro Dividend Discount Model, or Gordon Model for short.

 P o =Do(1+g)1/(1+i)1 +D o (1+g) 2/(1+i)2 + ... + Do(1+g) inf /(1+i) inf Po = D1 / (i - g) -- OR -- P o = Do (1 +g) / (i - g) P o = present value of common stock (with constant growth returns) Do = most recent per-share dividend D1 = per-share dividend after one period of growth [D1 = D0 (1+g)] i = required return (discount rate) for each year t g = rate of growth inf = infinite time period

This model makes heroic assumptions about the flat continuity of growth, that extrapolation from past dividends reflects likely future earnings, and the stock's risk can be reflected in a single discount rate. Such is valuation!

Example

Company is growing. It pays most of its earnings as dividends, but retains some earnings for future growth. The practice has worked well, as its history of growth suggests.

What is the value of Company stock, based
on dividend returns?

 Year Dividend/share 1 1.00 2 1.06 3 1.15 4 1.25 5 1.36 6 1.44 7 1.59

Answer: You can use the Gordon model in three steps --

What is the company's growth rate? This is not obvious and requires a judgment call. Do we predict that growth will continue at the average annual growth rate for the last 6 periods? To compute this we could find the average growth rate. Or do we predict that growth will continue at the overall rate of the last seven years? To compute this we would find the growth rate over the 6 years of growth, in which the dividend went from 1.00 to 1.59 -- what was the annual growth rate that led to a compounded return of 59% over 6 years?

 Year Dividend/share Annual growth Growth rate 1 1.00 2 1.06 6.00% 3 1.15 8.49% 4 1.25 8.70% 5 1.36 8.80% 6 1.44 5.88% 7 1.59 10.42% Average 8.05% 8.15%

Often stock valuators use the growth rate (or other measures such as internal rate of growth), since it reflects the growth rate over a period better than an average. In effect, it gives you an idea of the "growth machine" propelling the business.

Professor Palmiter,

I was looking for some information regarding dcf valuation recently, and found your webpage

I'd like to bring to your attention what might be an error (a small, but confusing one), if I'm correct.

When you're discussing the growth rate, you make a distinction between the average annual growth rate and the growth rate over the period. First, the growth rate over the period has a commonly used name - CAGR, the Compound Annual Growth Rate. Second, your CAGR is calculated using an ending value of \$1.60 per share, instead of the indicated \$1.59. The one penny difference causes a significant difference in the CAGR calculation. At \$1.60, the CAGR = 8.15%, as you note, but at \$1.59, it equals 8.04%.

Perhaps I'm in error, and you're calculating the growth rate in a way unfamiliar to me, but I thought I'd point this out, just in case.

Regards,

J.D. Kern

What is the required return (discount rate) for Company's stock? This is even trickier. One method is to use the CAPM -- which predicts the company's expected return (required return or discount rate) based on the stock's expected volatility and the market's expected return. For now, let's assume the discount rate is 14%.

As we saw before in computing value under different capitalization rate assumptions, the discount rate will be one of the most important (and hotly contested) issues in a valuation.

Apply the Gordon model.

 P0 = D 0 (1 + g)/(i - g) D 0 = Most recent per-share dividend \$ 1.59 i = Required return (discount rate) 14.0% g = Rate of growth 8.15% P0 = Value of one share of common stock \$1.59(1 + .0815) (.14 - .0815) = \$29.39

Notice that the formula requires that you compute the return in the first period of growth [D0(1 + g) = \$ 1.72] and then divide this by the difference of the discount rate and the growth rate [.14 - .0815].

 A Simpler Example Company has preferred stock that promises its holders a fixed annual dividend of \$8. Assuming a required return of 12.8% for the preferred stock based on the risk profile of Company, what is the value of the preferred stock? Answer. Divide the annual dividend by the discount rate (\$ 8 / .128). The result is \$62.50.

 This page was last updated on: November 16, 2004