Pricing Markovian Dividend Assets with Ex-Dividend Prices

Summary

This note clarifies how to set up valuation equations for an asset whose dividends follow a Markov process. Its central distinction is between cum-dividend value, which includes a payment made today, and ex-dividend price, which excludes it. The investor’s price equations must place each state’s dividend in the next-period payoff before applying transition probabilities and discounting.

The example uses two states, state-dependent dividends, transition probabilities, and a discount factor. Correcting the placement of the dividend resolves the mismatch with the values reported in the cited paper. The note gives the equations but no independent derivation, broader evidence, or discussion of assumptions such as the discount rate and investor preferences, so it is a focused clarification of price convention rather than a general asset-pricing treatment.

Key ideas

• Ex-dividend prices exclude the dividend paid today.
• Expected continuation value should include dividends associated with the next state.
• Confusing cum-dividend and ex-dividend conventions changes the pricing equations.
• State-contingent prices can be solved as simultaneous equations using transition probabilities and discounting.

Tags

# Pricing a piece of asset whose dividend stream following a Markovian matrix


# Pricing a piece of asset whose dividend stream following a Markovian matrix












I'm trying to calculate the result of an simple example on page 326-327, in Harrison and Kreps(1978). It's pricing a piece of asset whose dividend stream is a simple Markovian process.

Here's my attempt to replicate investor $1$'s evaluation of this asset.

$p^1(0) = 0 + \frac{3}{4} ( \frac{1}{2} p^{1}(0) + \frac{1}{2} p^{1}(1) )$

$p^1(1) = 1+ \frac{3}{4} ( \frac{2}{3} p^{1}(0) + \frac{1}{3} p^{1}(1) )$

But when I substitute the numerical value given in the bottom of second screenshot, they don't match. Do I miss something?

## Answer by pbr142 (score 4, accepted)

https://quant.stackexchange.com/a/10557

Your equations are for cum-dividend prices, i.e. the price plus dividend today. The paper refers to ex-dividend prices. The correct two equations for investor group $a=1$ are \begin{align} p^1(0) =&\ \frac{3}{4} \left(\frac{1}{2}p^1(0) + \frac{1}{2}(1+p^1(1))\right) \\ p^1(1) =&\ \frac{3}{4} \left(\frac{2}{3}p^1(0) + \frac{1}{3}(1+p^1(1))\right) \end{align} When you solve these two equations for $p^1(0)$ and $p^1(1)$ you get the correct results.

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