CFA Level I Exam · Discounted Cash Flow (DCF) and Growth Models

Multistage Dividend Discount Models: Two-Stage, H-Model and Three-Stage

Updated 7 October 2026 · Fact-checked

A multistage dividend discount model values a stock when growth is temporarily high and later settles at a stable rate. You forecast dividends for the high-growth years, find a terminal value at the end using the Gordon growth formula, then discount everything at the required return. The H-model approximates a linear growth decline.

Understand Multistage Dividend Discount Models

The Gordon growth model assumes dividends grow at one constant rate forever. Many companies do not behave this way. A young firm may grow dividends at 15% for a few years, then slow to the pace of the wider economy. One growth rate cannot describe that path, so you split the future into stages.

In a two-stage model, dividends grow at a high rate gS for n years, then at a stable rate gL forever. You value the stock as the present value of the dividends in the high-growth years plus the present value of a terminal value at the end of year n. The terminal value is the Gordon growth value at that date: Pn = Dn+1 ÷ (r − gL). It is a price at time n, so you must discount it back n years.

The H-model is a shortcut for growth that falls in a straight line from gS to gL over a transition period. It gives an approximate value without forecasting each dividend. H is half the length of the transition period in years.

A three-stage model has a high-growth stage, a transition stage where growth declines toward the stable rate, and a mature stage. It is more flexible than the two-stage model, but needs more forecasts. The difference between two-stage and three-stage is how growth moves between the high and stable rates: a sudden drop in the two-stage model, a gradual decline in the three-stage model.

Every version needs r > gL in the final stage. If it does not, the Gordon formula gives a meaningless or negative value. The terminal value usually makes up most of the stock's value, so small changes in gL or r move the answer a lot.

Key formulas to remember

Two-stage DDM (finite high-growth period): V0 = Σ [Dt ÷ (1 + r)^t] for t = 1 to n + Pn ÷ (1 + r)^n; Dt is the forecast dividend in year t. Pn is the terminal value at the end of year n.
Terminal value: Pn = Dn+1 ÷ (r − gL) = Dn × (1 + gL) ÷ (r − gL); Valid only when r > gL. Use the first dividend of the stable stage, not Dn.
High-growth dividends: Dt = D0 × (1 + gS)^t; Applies during the high-growth stage only. After year n, dividends grow at gL.
H-model: V0 = [D0 × (1 + gL) ÷ (r − gL)] + [D0 × H × (gS − gL) ÷ (r − gL)]; H = half the length of the transition period = (length of the linear decline in years) ÷ 2. The first term is the Gordon value; the second is the extra value from above-normal growth. The result is an approximation.
Value as a sum of stages: V0 = PV(stage 1 dividends) + PV(stage 2 dividends) + PV(terminal value); The general idea for a three-stage model. Discount each dividend by its own year.

How to solve Multistage Dividend Discount Models questions

Use this method for any multistage DDM question. It works whether the question gives two stages, a transition period or an H-model setup.

1Read the growth pattern and mark the timeline: how many years of high growth, whether growth declines, and when stable growth begins.
2Note whether the given dividend is D0 (already paid) or D1 (next dividend). Many errors start here.
3If the question says growth declines linearly and asks for an approximate value, use the H-model and go to step 7.
4Forecast each dividend during the explicit stages by growing the prior dividend at that stage's rate.
5Compute the first dividend of the stable stage: Dn+1 = Dn × (1 + gL). Then terminal value Pn = Dn+1 ÷ (r − gL).
6Discount each dividend and the terminal value to today at r. Discount Pn by the same n as the last explicit dividend.
7Add the present values. For the H-model, add the Gordon term and the extra-growth term instead.
8Check that r > gL and that the answer is reasonable: the terminal value is often more than half of the total.

Quickest way: Cash flow worksheet shortcut

When to use it: Use it for two-stage or three-stage questions with three to five explicit dividends. It saves time and reduces rounding errors.

Compute the dividends D1 to Dn and the terminal value Pn.
Add Pn to the last dividend Dn so the final cash flow is Dn + Pn.
On the TI BA II Plus press CF, then enter CF0 = 0, then C01, C02 and so on with F = 1 each.
Press NPV, enter I = r (as a percent), press ↓ and CPT to get the value.
On the HP 12C use g CF0 for the initial 0, g CFj for each cash flow, then enter r as i and press f NPV.
For the H-model skip the calculator: two short divisions are faster.

Common mistakes in Multistage Dividend Discount Models

Forgetting to discount the terminal value
The Gordon formula feels like the final answer, so students stop once they have Pn.
Fix: Pn is a value at time n. Divide it by (1 + r)^n and then add it to the present value of the early dividends.
Using Dn instead of Dn+1 in the terminal value
Students forget that the Gordon model needs the next dividend, not the last one.
Fix: Always grow the last high-growth dividend by (1 + gL) first. Write Dn+1 on your page before dividing.
Discounting the terminal value by the wrong number of years
Students count n+1 because the formula uses Dn+1.
Fix: The Gordon value of dividends starting at n+1 sits at time n. Discount Pn by n years.
Treating the given dividend as D1 when it is D0
Wording such as 'recently paid' or 'current dividend' is skimmed over.
Fix: If the dividend has been paid, it is D0 and must be grown once to get D1. If the question says 'expected next year', it is already D1.
Using the transition length as H in the H-model
Students recall 'H' without remembering it is half the transition length.
Fix: H = transition years ÷ 2. If growth falls from gS to gL over 8 years, H = 4.
Applying the model when r is not greater than gL
Students plug numbers in without checking the inputs.
Fix: Check r > gL before dividing. In real-world valuation, a stable growth rate above the required return signals a bad assumption, not a usable result.

Worked examples

Example 1

A company just paid a dividend of $2.00 per share (D0). Dividends are expected to grow at 15% a year for three years and then at 5% a year indefinitely. The required return is 10%. The value of the share today is closest to: A) $50.12 B) $54.55 C) $58.30

Show the solution

Dividends: D1 = 2.00 × 1.15 = 2.300; D2 = 2.300 × 1.15 = 2.645; D3 = 2.645 × 1.15 = 3.04175.
First stable dividend: D4 = 3.04175 × 1.05 = 3.19384.
Terminal value at year 3: P3 = 3.19384 ÷ (0.10 − 0.05) = 63.8768.
Present values: 2.300 ÷ 1.10 = 2.0909; 2.645 ÷ 1.21 = 2.1860; 3.04175 ÷ 1.331 = 2.2853.
PV of terminal value: 63.8768 ÷ 1.331 = 47.9916.
Sum: 2.0909 + 2.1860 + 2.2853 + 47.9916 = 54.5538.
Calculator check (BA II Plus): CF0 = 0, C01 = 2.30, C02 = 2.645, C03 = 67.9185 (3.04175 + 63.8768), I = 10, CPT NPV = 54.55.

Answer: B) $54.55. Options A and C do not match the correct calculation. Always compute the full value, with D4 in the terminal value and the terminal value discounted three years, and pick the option that matches.

Example 2

A company paid a dividend of €1.50 per share (D0). Dividend growth is currently 12% and is expected to decline linearly over 8 years to a long-term rate of 4%. The required return is 9%. Using the H-model, the value per share is closest to: A) €31.20 B) €40.80 C) €48.00

Show the solution

Find H: the decline lasts 8 years, so H = 8 ÷ 2 = 4.
Gordon term: D0 × (1 + gL) ÷ (r − gL) = 1.50 × 1.04 ÷ 0.05 = 1.56 ÷ 0.05 = 31.20.
Extra-growth term: D0 × H × (gS − gL) ÷ (r − gL) = 1.50 × 4 × 0.08 ÷ 0.05 = 0.48 ÷ 0.05 = 9.60.
Add: 31.20 + 9.60 = 40.80.

Answer: B) €40.80. Option A is only the Gordon term, which ignores the extra value from above-normal growth.

Exam tips

First identify the pattern: high growth in stage one, then stable Gordon growth, or a linear decline in growth. A linear decline in growth signals the H-model.
Check whether the dividend given is D0 or D1 before doing any arithmetic. Many wrong options are built on this error.
Expect the wrong options to include the undiscounted terminal value, the value with Dn instead of Dn+1, and the Gordon-only value in an H-model question. Calculate the correct figure and eliminate these.
Use the 90 seconds per question wisely: a two-stage question with three dividends takes longer, so do the quick checks first and skip to the cash flow worksheet only if needed.
Know the conceptual points: the terminal value is usually the largest part of the value, and a multistage model fits firms whose growth is expected to change.

Practice questions from Discounted Cash Flow (DCF) and Growth Models

Multistage Dividend Discount Models in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Multistage Dividend Discount Models: frequently asked questions

How do you calculate terminal value in a multistage DDM?

Grow the last explicit dividend by the stable growth rate to get the first stable-stage dividend, then divide by (r − gL). This gives the value at the end of the high-growth period. You must discount it back to today by the same number of years as the last explicit dividend.

What is the difference between a two-stage and a three-stage DDM?

A two-stage model has a high-growth period followed by an immediate move to stable growth. A three-stage model adds a transition period in which growth declines gradually toward the stable rate. The three-stage model is more realistic but needs more forecasts.

What does H mean in the H-model?

H is half the length of the transition period, which is the number of years over which growth falls linearly from the high rate to the stable rate. For a 6-year decline, H = 3. The H-model gives an approximate value.

When should I use a multistage DDM instead of the Gordon growth model?

Use a multistage model when growth is expected to change, for example a company with a temporary high-growth phase. The Gordon growth model suits mature companies with stable dividend growth below the required return.