## Decomposing the Dividend Discount Model

The DDM assumes that the theoretical value of a company's stock (P) can be obtained by summing the present values of all future dividend pay-ments.63 The standard DDM price formula is P = d/(k - g) (see Table 4.11 for symbol definitions).

In the absence of growth (that is, with g equal to zero), the fixed annual earnings are paid out as dividends. Price P is simply the value of a perpetual annuity discounted at a nominal market rate (k). More generally, when g is greater than zero, the investor's return will be derived from a growing stream of dividends (d) and the associated appreciation in share price.64

For example, when k = 12 percent, g = 8 percent, and d = \$8 million, the stock price is \$200 million and the dividend yield is 4 percent. Thus, over a one-year period, the 12 percent return comprises a 4 percent dividend yield and an 8 percent growth rate.

 Variable Name Symbol or Formula Example Value Initial book value B \$100 Return on book equity r 16% Initial earnings E = rB \$ 16 Earnings retention ratio b 0.50 Dividend payout ratio 1 - b 0.50 Initial dividend d = (1 - b)E \$ 8 Dividend growth rate g = rb 8% Nominal discount rate k 12% Stock price P = d/(k - g) \$200

To see the sensitivity of P to rate changes, assume that earnings and dividends do not change. Now, consider the effect of a decline of 1 basis point in the value of k, from 2 percent to 11.99 percent. Then, p _ \$8,000,000 _ 0.1199 - 0.0800 _ \$200,501,253

The \$0.50 million price change represents a 0.25 percent increase to the base price level of \$200 million. This computation shows that the duration of the stock price (Dp, the ratio of the percentage change in price to the change in rates) is 25 (the 0.25 percent increase derived from the 0.01 percent rate move).65 This straightforward computation is the cornerstone for the belief that equity duration is very long, but that belief is not supported by the observed statistical duration of equity, which tends to be between 2 and 6 years.

Figure 4.79 plots the DDM price (left scale) and duration (right scale) for a wide range of nominal rates under the assumptions given in Table 4.11. The sensitivity of P to nominal rate changes is reflected in the steepness of the price curve. This steepness (sensitivity) increases at low rate levels and decreases at higher rate levels. Note that the duration curve follows a path similar to that of the price curve. 