9 Derivation of the DDM
9.1 Introduction
The Dividend Discount Model (DDM) is the foundational model of equity valuation: the value of a share equals the present value of all dividends it will ever pay. While this result is often presented as an assumption, it is in fact a derivation — it follows from the definition of investment returns, plus a small number of additional assumptions that deserve careful scrutiny.
This document works through that derivation in full. We begin with a backward-looking identity that is true by definition and requires no assumptions at all. We then identify, one by one, the assumptions needed to convert it into a forward-looking valuation formula.
We use \(P_t\) for the ex-dividend price at time \(t\) (i.e. after the dividend \(d_t\) has been paid), \(d_t\) for the dividend at time \(t\), and \(r_t\) for the realised gross return between \(t-1\) and \(t\).
9.2 The Definition of Returns (No Assumptions)
The one-period gross return on a share between time \(t-1\) and \(t\) is defined as the total payoff received at \(t\) divided by the price paid at \(t-1\):
\[ 1 + r_t \equiv \frac{P_t + d_t}{P_{t-1}} \tag{Def} \]
This is purely definitional — it is true by construction for any asset in any period, regardless of how markets work or what investors believe.
Rearranging for \(P_{t-1}\):
\[ P_{t-1} = \frac{P_t + d_t}{1 + r_t} \tag{1} \]
This says: knowing the future price, the future dividend, and the realised return, I can recover the price today. This is a backward-looking identity — it tells us what past prices must have been, given what we now know happened.
9.3 Iterating Forward
Apply \((1)\) recursively. Start at \(t=0\):
\[ P_0 = \frac{P_1 + d_1}{1 + r_1} \]
Substitute \(P_1 = \dfrac{P_2 + d_2}{1+r_2}\):
\[ P_0 = \frac{d_1}{1+r_1} + \frac{P_2 + d_2}{(1+r_1)(1+r_2)} \]
Continuing for \(T\) periods:
\[ P_0 = \sum_{t=1}^{T} \frac{d_t}{\prod_{s=1}^{t}(1+r_s)} + \underbrace{\frac{P_T}{\prod_{s=1}^{T}(1+r_s)}}_{\text{terminal price term}} \tag{2} \]
This is still a backward-looking identity, valid for any \(T\). Given perfect hindsight of all future dividends, returns, and the terminal price, \((2)\) tells us exactly what \(P_0\) must have been. No assumptions have been made yet.
9.4 From Identity to Valuation Formula
To use \((2)\) as a valuation formula — i.e. to determine what a share should be worth today — we must make two additional assumptions. Consider a one-period investor. The expected holding-period return is defined as:
\[ E(r) = \frac{E(d_1) + E(P_1) - P_0}{P_0} \tag{HPR} \]
This is still just a definition — expected payoff relative to cost. Most notably, the expected return depends mostly on the investor’s expected price in one year. But, so far, we have not said anything about what price the investor will/should expect.
9.4.1 Assumption 1: No-Arbitrage Equilibrium
Equation (HPR) contains both \(E(P_1)\) — which we are trying to form a view on — and \(P_0\) — which is what we are trying to determine. We need a condition that pins down \(E(P_1)\).
This is where no-arbitrage enters. It makes two related claims. First, a risk-adjusted required return \(k\) exists, determined by the asset’s systematic risk (e.g. via the CAPM). Second, and crucially, competition among investors ensures that prices adjust until the expected return equals this required return — no investor can systematically earn more than \(k\) for a given level of risk without that opportunity being traded away.
In equilibrium, the market price adjusts until \(E(r) = k\), where \(k\) is the required rate of return for this level of systematic risk. Equivalently, there are no persistent excess returns available — any mispricing is arbitraged away. This implies a form of rational expectations. If prices are set by optimising agents who can freely trade, then any systematic forecast bias would itself be an arbitrage opportunity: you could trade against the predictable error and earn excess returns.
Rearranging (HPR) with \(E(r) = k\) and solving for \(V_0\) gives the one-period intrinsic value formula:
\[ V_0 = \frac{E(d_1) + E(P_1)}{1 + k} \tag{3} \]
If the market price \(P_0 < V_0\), the stock is underpriced: it offers \(E(r) > k\), attracting buyers who bid the price up. If \(P_0 > V_0\), it is overpriced (see also Bodie et al. (2014)).
Now note that Assumption 2 does more than just the one-period case. Because no-arbitrage must hold at every future date — not just today — the same pricing logic applies recursively. The price \(E(P_1)\) at time 1 must itself satisfy the same equilibrium condition, so:
\[E(P_1) = \frac{E(d_2) + E(P_2)}{1+k}\]
This recursive application of no-arbitrage is what licenses the substitution. Without it, there is no reason why a buyer at \(t=1\) would price the stock using the same \(k\); they might pay something driven purely by sentiment or speculation. Substituting and iterating for \(T\) periods (assuming \(k\) is constant):
\[ V_0 = \sum_{t=1}^{T} \frac{E(d_t)}{(1+k)^t} + \frac{E(P_T)}{(1+k)^T} \tag{4} \]
There is ample evidence that the required return actually varies over time. In that case replace \((1+k)^t\) with \(\prod_{s=1}^{t}(1+k_s)\) where \(k_s\) is the required return for period \(s\). The structure of the derivation is unchanged.
9.4.2 Assumption 2: Transversality
Taking the horizon \(T \to \infty\) in \((4)\):
\[ V_0 = \sum_{t=1}^{\infty} \frac{E(d_t)}{(1+k)^t} + \lim_{T\to\infty}\frac{E(P_T)}{(1+k)^T} \tag{5} \]
The terminal price term persists. Even granting Assumptions 1 and 2, this term need not vanish: if investors expect prices to grow forever at rate \(k\), the discounted terminal price remains positive no matter how far out the horizon is pushed — a self-sustaining rational bubble. Ruling this out requires an additional condition:
Under Assumption 2, the terminal price term in \((5)\) vanishes, yielding the infinite-horizon Dividend Discount Model:
\[ V_0 = \sum_{t=1}^{\infty} \frac{E(d_t)}{(1+k)^t} \tag{DDM} \]
The intrinsic value of a share equals the present value of all expected future dividends, discounted at the required rate of return.
9.5 Remarks on the Economic Meaning of Each Assumption
Assumption 1 (No-arbitrage) is the workhorse of modern asset pricing. It does not require all investors to be rational — only that mispricings are arbitraged away. In frictionless markets with rational investors, it follows from utility maximisation. In markets with frictions, it is an approximation.
Assumption 2 (No bubbles) is the most philosophically contested. Rational bubbles are theoretically permissible in infinite-horizon economies (Tirole (1985)). Empirically, distinguishing a bubble from rapidly growing fundamentals is extremely difficult. The transversality condition is an additional restriction we impose on top of equilibrium pricing — it is not implied by rationality alone.
9.6 Further Reading
The expected holding-period return framing used here closely follows Bodie et al. (2014), which provides an accessible and well-structured treatment of intrinsic value and the DDM. The definition-of-returns approach is standard in asset pricing; a rigorous graduate-level treatment is in Cochrane (2005).