Deriving Dividend Present Values Under Random Walk Models

Summary

The document asks how rational present-value pricing works when dividends follow a unit-root process or a random walk with drift. It focuses on deriving two expressions: the price premium over dividends divided by the discount rate in the no-drift case, and the price level implied by a constant expected dividend increment.

The question supplies the valuation formula as the discounted expectation of future dividends and gives the assumed dividend dynamics, including the conditional expected value under drift. It does not include an answer, derivation, numerical example, or empirical evidence. As a result, the mathematical steps and assumptions needed to obtain the stated expressions remain open. Readers should treat it as a posed derivation problem rather than a resolved explanation, including checking the discounting convention and the treatment of the first future dividend in the summation.

Key ideas

• The setup prices an asset as the discounted sum of expected future dividends.
• A unit-root dividend process is used to ask about the relation between price and current dividends.
• A random walk with drift implies expected dividends rise with the forecast horizon.
• The document poses algebraic derivation questions but provides no solutions or empirical evidence.
• The timing convention in the discounting formula should be checked when deriving the expressions.

Tags

# Cointegration between prices and dividends. How do I get the following expression?


# Cointegration between prices and dividends. How do I get the following expression?












Actually, I have two questions:

- 1.

Let us assume that expected returns are constant. Then, we have the following expression for how the prices should be determined, provided that the operators are rational:

(1) $$P_t=E_t[\sum_{i=1}^{\infty} (1/1+K)^i D_{t+i}]$$

Campbell, Lo, McKinley in 'The Econometrics of Financial Markets' claim that, if the dividend follows a unit root process, such as:

$$D_t=D_{t-1} + \epsilon_t $$

And $$P_t=D_t/K$$

Then, if we subtract $D_t/K$ from (1) we get: (2)

$$P_t - D_t/K = (1/K)E_t[\sum_{i=1}^{\infty} (1/1+K)^i \Delta D_{t+1+i}]$$

I am not able to get (2) from (1). Can you help me, please?

- 2.

Suppose the Dividend follows a RW with drift process:

$$D_t=u+D_{t-1}+ \epsilon_t$$

Than $E_t[D_{t+i}]= u*i + D_t$ In a book, the author claims that, because of (3), the rational valuation formula becomes:

(4)

$$P_t=u(1+K)/K^2+ D_t/K$$

I am not able to prove the result just claimed, these are the steps I have been able to do so far: $$P_t=E_t[\sum_{i=1}^{\infty} (1/1+K)^i D_{t+i}]=E_t[\sum_{i=1}^{\infty} (1/1+K)^i (u*i+D_t)]=[\sum_{i=1}^{\infty} (1/1+K)^i (u*i)]+E_t \sum_{i=1}^{\infty} (1/1+K)^i (D_t)=[\sum_{i=1}^{\infty} (1/1+K)^i (u*i)]+D_t/K$$

I would really appreciate your help. Thank you.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.