Lesson 1 · Equity AnalysisCFA L1

Foundations of Value and the Dividend Discount Model

All equity valuation rests on one idea: value is the present value of future cash flows. This lesson derives that from no-arbitrage and the stochastic discount factor, builds the dividend discount model and the Gordon growth model, shows how to back out a cost of capital or implied growth rate, examines how to set the discount rate and equity risk premium, and extends to multi-stage and H-models.

On this page

The foundation of value The dividend discount model Gordon growth model Cost of capital & implied returns The discount rate Geometric vs arithmetic Multi-stage & H-model

No-arbitrage — an efficient market self-destructs riskless profit — is equivalent (Cochrane, 2005) to the existence of a stochastic discount factor (SDF) $m$, so that any asset's value is $V_t=E_t[m_{t+1}x_{t+1}]$, where $x$ is the payoff and $m$ reflects investors' beliefs and risk aversion.

Learning outcomes

Explain the no-arbitrage foundation of value and the SDF.
Derive and apply the dividend discount and Gordon growth models.
Back out a cost of capital, implied growth and implied risk premium from price.
Set a discount rate with the CAPM and choose between geometric and arithmetic means.
Value a firm with two-stage and H-models.

The foundation of value

Decompose the SDF pricing equation: $V_t=E_t[m]E_t[x]+\operatorname{cov}_t(m,x)$. With no uncertainty, $V_t=x_{t+1}/R_f$ (riskless cash flows are discounted at the riskfree rate, and $E_t[m]=1/R_f$). With risk, there are two equivalent routes to value:

Two equivalent valuations

$$ V_t=\frac{E_t[x_{t+1}]}{1+r}\quad(\text{risk-adjusted rate}), \qquad V_t=\frac{E_t[x_{t+1}]+R_f\operatorname{cov}_t(m,x)}{1+r_f}\quad(\text{certainty equivalent}) $$

Either discount expected cash flows at a risk-adjusted rate $r=r_f+\text{premium}$, or discount certainty-equivalent cash flows at the riskfree rate. Risk-averse investors price risky assets below equivalent riskfree ones. For equity, the payoff is the next price plus dividend, $x_{t+1}=V_{t+1}+d_{t+1}$, and value = the present value of future cash flows — the intrinsic (fundamental) value, generally different from the market price (which also reflects supply, demand and sentiment).

The dividend discount model

Iterating $V_t=\frac{E_t[V_{t+1}+d_{t+1}]}{1+r}$ forward (a going concern is never liquidated) gives the dividend discount model (DDM):

Dividend discount model

$$ V_0=\sum_{t=1}^{\infty}\frac{E_0[d_t]}{(1+r)^t} $$

Here $d_t$ is net dividends — cash dividends + buybacks − new equity issued. The clean-surplus relation ties this to accounting: net dividends = comprehensive earnings − change in book equity. The DDM is the fundamental pricing model; every other equity model must reconcile to it. Its practical weakness: net dividends are hard to forecast (policy depends on investment opportunities, perceived mis-valuation and capital structure), and the DDM captures wealth distribution, not creation.

The Gordon growth model

Assume dividends grow at a constant $g<r$: $d_{t+1}=(1+g)d_t$. The infinite sum collapses to a growing perpetuity:

Gordon growth model

$$ V_0=\frac{d_0(1+g)}{r-g}=\frac{d_1}{r-g}, \qquad \text{(terminal value) } V_n=\frac{d_{n+1}}{r-g} $$

In practice analysts forecast dividends explicitly for $n$ years and add a terminal (continuing) value $V_n=d_{n+1}/(r-g)$, discounted back. The share is worth the PV of the explicit dividends plus the PV of the terminal price. Because the continuing value is often more than half the total, the valuation is very sensitive to $g$ — a small change moves the answer a lot.

Deriving cost of capital and implied growth

If price is a good proxy for intrinsic value, the Gordon model can be inverted to extract market expectations:

Inverting the Gordon model

$$ r=\frac{d_0(1+g)}{P_0}+g=\text{DY}+g, \qquad g=r-\text{DY} $$

Worked example

Required return on the FTSE 100

2024 dividend yield 3.77%; long-run real GDP and dividend growth 1%; expected inflation 2.5%. Find the required real and nominal return.

Show solution

Real: $r_r=\frac{d_0(1+g)}{P_0}+g=3.77\%\times1.01+1\%=4.81\%$. Nominal: $r=(1+r_r)(1+\text{inf})-1=1.0481\times1.025-1=\textbf{7.43%}$. Inverting the model turns an observed yield and a growth assumption into an implied cost of equity — and, at the index level, an implied equity risk premium (useful in regulation).

Real ≈ 4.81%, nominal ≈ 7.43%

The discount rate and the risk premium

The most common way to set $r$ is the CAPM: $\bar r_i=r_f+\beta_i(\bar r_m-r_f)$ — excess return = beta × market risk premium. Since CAPM is a single-period model, valuation use is always a compromise: a popular approach pairs a duration-matched riskfree rate (from the yield curve) with a constant risk premium drawn from capital-market history (Dimson et al.).

Geometric vs arithmetic mean returns

Which average to use

The geometric mean $\sqrt[n]{\prod(1+r_i)}-1$ is the actual compounded return — use it for expected future wealth (an actuarial estimate). The arithmetic mean $\frac{1}{n}\sum r_i$ treats returns as independent draws — use it for cost of capital / discounting. Example: returns of +10.52% then −9.52% take 100 → 110.52 → 100, so the geometric mean is exactly 0% (the real outcome), but the arithmetic mean is $+0.5\%$. On the risk-premium puzzle: banks use ~4–5%, while Dimson et al. forecast a geometric world premium of 3–3.5% over T-bills (~1.5% more arithmetic).

Multi-stage models and the H-model

Firms rarely grow at one rate forever. A two-stage model applies a short-run $g_s$ for $n$ years, then a sustainable $g_l$:

Two-stage model

$$ V_0=\sum_{t=1}^{n}\frac{d_0(1+g_s)^t}{(1+r)^t}+\frac{1}{(1+r)^n}\cdot\frac{d_0(1+g_s)^n(1+g_l)}{r-g_l} $$

Its flaw: growth "falls off a cliff" at year $n$. The H-model (Fuller & Hsia, 1984) assumes a gradual linear decline from $g_s$ to $g_l$:

H-model

$$ V_0=\frac{d_0(1+g_l)}{r-g_l}+\frac{d_0\,H\,(g_s-g_l)}{r-g_l}, \qquad r=g_l+\frac{d_0\big[(1+g_l)+H(g_s-g_l)\big]}{P_0} $$

Worked example

General Mills — two-stage vs H-model

$d_0=\$0.40$; $g_s=9\%$ for 10 years; $g_l=5\%$; $r_f=2.4\%$, MRP 5.2%, $\beta=0.9$. Value it.

Show solution

CAPM: $r=2.4\%+0.9(5.2\%)=7.08\%$. The full two-stage value is $P_0=4.416+24.119=\$28.54$. The H-model (with $H=5$, half the 10-year transition) gives $\frac{0.4(1.05)}{0.0708-0.05}+\frac{0.4(5)(0.09-0.05)}{0.0708-0.05}=20.19+3.85=\$24.04$ — a "normal" value plus an "abnormal-growth" value. The H-model is an approximation (CFA L2), less accurate when the abnormal period $H$ is long or the growth gap is large; with spreadsheets the exact multi-stage model is preferred, but the H-model's decomposition is instructive.

CAPM r = 7.08%; two-stage $28.54; H-model ≈ $24.04 (20.19 normal + 3.85 abnormal)

Key takeaway

Value is the present value of future cash flows, grounded in no-arbitrage and the SDF — discount expected cash flows at a risk-adjusted rate, or certainty-equivalents at the riskfree rate. The DDM $V_0=\sum E_0[d_t]/(1+r)^t$ is the fundamental model; the Gordon model $V_0=d_1/(r-g)$ operationalises it and, inverted, yields the implied cost of capital ($r=\text{DY}+g$) and risk premium. Set $r$ with the CAPM and the right mean (geometric for wealth, arithmetic for discounting), and handle changing growth with multi-stage and H-models — always mindful that the terminal value, and thus $g$, dominates.

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