Stock Valuation: Dividend Discount Model for Exam FM

Dividend Discount Model Fundamentals

The value of a stock is the present value of all future dividends: P = sum of D_t / (1+r)^t for t = 1, 2, 3, ..., where r is the required rate of return. This principle follows directly from the time value of money concepts covered throughout Exam FM. Unlike bonds, stocks have no fixed maturity date, so the dividend stream extends indefinitely.

The key insight is that even if an investor plans to sell the stock before collecting all future dividends, the selling price itself reflects the present value of the remaining dividends at that point. Therefore, the current price always equals the present value of the entire future dividend stream, regardless of individual holding periods.

This framework assumes that dividends are the only relevant cash flow to shareholders. Share repurchases, liquidation proceeds, and other distributions are not captured unless they are treated as equivalent dividend payments.

The Gordon Growth Model

If dividends grow at a constant rate g forever, the infinite series simplifies to the Gordon Growth Model: P = D_1 / (r - g), where D_1 is the dividend expected next period and r > g. This formula is a geometric perpetuity. The condition r > g is necessary for the series to converge; if g >= r, the stock would have infinite value, which is not meaningful.

Example: A stock pays a dividend of $2 next year, growing at 3% per year. If the required return is 10%, the stock value is 2 / (0.10 - 0.03) = $28.57. If the growth rate increases to 5%, the value becomes 2 / (0.10 - 0.05) = $40.00. Small changes in the growth rate produce large changes in the valuation, which is an important sensitivity to understand.

The Gordon model can be rearranged to find the implied return: r = D_1/P + g. The first term D_1/P is the dividend yield and g is the capital gains yield. Together they represent the total expected return on the stock. This decomposition is tested on Exam FM.

Multi-Stage Growth Models

When dividend growth is not constant, use a multi-stage model. Typically, dividends grow at a high rate for a finite period, then switch to a sustainable long-term rate. The value equals the present value of the high-growth dividends plus the present value of the terminal value, where the terminal value at the switch point uses the Gordon formula with the long-term growth rate.

Example: D_1 = $3, growing at 15% for 3 years, then 4% forever, with r = 12%. The dividends are D_1 = $3.00, D_2 = $3.45, D_3 = $3.9675. The terminal value at time 3 is D_4/(r - g_long) = 3.9675 * 1.04 / (0.12 - 0.04) = 4.1262 / 0.08 = $51.578. The stock price is P = 3.00/1.12 + 3.45/1.12^2 + 3.9675/1.12^3 + 51.578/1.12^3 = 2.679 + 2.750 + 2.824 + 36.706 = $44.96.

The terminal value typically dominates the total valuation. In this example, the present value of the terminal value ($36.71) accounts for about 82% of the stock price, illustrating why the long-term growth rate assumption is so important.

Estimating the Growth Rate

The sustainable growth rate g can be estimated as g = b * ROE, where b is the retention ratio (also called the plowback ratio) and ROE is the return on equity. The retention ratio equals 1 minus the dividend payout ratio: b = 1 - D/E, where D is dividends per share and E is earnings per share.

Example: A company earns $5 per share and pays $2 in dividends. The retention ratio is b = 1 - 2/5 = 0.60. If ROE = 15%, then g = 0.60 * 0.15 = 9%. This implies the company reinvests 60% of its earnings at a 15% return, growing the dividend base at 9% per year.

This approach assumes the company maintains a constant retention ratio and earns a constant ROE on reinvested earnings. If either assumption changes, the growth rate changes, and a multi-stage model may be more appropriate.

Finding the Required Return or Growth Rate

Exam FM often gives the current stock price and asks you to solve for the implied required return or growth rate. From P = D_1/(r - g), solving for r gives r = D_1/P + g. Solving for g gives g = r - D_1/P.

Example: A stock trades at $50, will pay a $2.50 dividend next year, and has a required return of 11%. The implied growth rate is g = 0.11 - 2.50/50 = 0.11 - 0.05 = 6%. If instead you are given g = 4% and asked for r: r = 2.50/50 + 0.04 = 0.09 = 9%.

These problems are straightforward but require careful attention to timing. Make sure D_1 is the next dividend (one period from now), not the most recent dividend D_0. If the problem gives D_0, compute D_1 = D_0 * (1 + g) before applying the formula.

Comparing Valuations Under Different Assumptions

Exam FM may ask you to compare stock prices under different growth rate or required return assumptions. The Gordon model shows that stock price is increasing in the growth rate and decreasing in the required return. The sensitivity to changes in these inputs depends on how close r and g are: when r - g is small, the price is very sensitive to either parameter.

A related application involves comparing the values of two stocks or determining which has a higher expected return. If two stocks have the same price but different dividend yields, the one with the lower dividend yield must have a higher expected growth rate to compensate.

Study Tips and Common Pitfalls

The most common mistake is using D_0 instead of D_1 in the Gordon formula. If the problem states "the stock just paid a dividend of $2," you must grow it by one period: D_1 = 2 * (1 + g). Using D_0 directly undervalues the stock by a factor of (1 + g).

Another frequent error in multi-stage problems is discounting the terminal value to the wrong time. The Gordon formula P = D/(r - g) gives the value one period before the first dividend in the perpetuity. If the stable growth begins at time 4 with dividend D_4, the terminal value D_4/(r - g) is the value at time 3, not time 4. Discount it back to time 0 using (1 + r)^3.

Practice at least 5 Gordon model problems and 3 multi-stage problems. Pay attention to problems that combine the DDM with other Exam FM concepts like net present value or yield calculations.