Consider a stock that pays dividends of Dt in period t and...

Consider a stock that pays dividends of $D_{t}$ in period $t$ and whose price in period $t$ is $P_{t}$. Assume that consumers are risk-neutral and have a discount rate of $r ;$ thus they maximize $E\left[\sum_{t=0}^{\infty} C_{t} /(1+r)^{t}\right]$ (a) Show that equilibrium requires $P_{z}=E_{t}\left[\left(D_{t+1}+P_{t+1}\right) /(1+r)\right]$ (assume that if the stock is sold, this happens after that period's dividends have been paid). (b) Assume that $\lim _{x \rightarrow \infty} E_{t}\left[P_{t+s} /(1+r)^{s}\right]=0$ (this is a no-bubbles condition; see the next problem). Iterate the expression in part (a) forward to derive an ex. pression for $P_{t}$ in terms of expectations of future dividends.

Consider a stock that pays dividends of $D_{t}$ in period $t$ and whose price in period $t$ is $P_{t}$. Assume that consumers are risk-neutral and have a discount rate of $r ;$ thus they maximize $E\left[\sum_{t=0}^{\infty} C_{t} /(1+r)^{t}\right]$
(a) Show that equilibrium requires $P_{z}=E_{t}\left[\left(D_{t+1}+P_{t+1}\right) /(1+r)\right]$ (assume that if the stock is sold, this happens after that period's dividends have been paid).
(b) Assume that $\lim _{x \rightarrow \infty} E_{t}\left[P_{t+s} /(1+r)^{s}\right]=0$ (this is a no-bubbles condition; see the next problem). Iterate the expression in part (a) forward to derive an ex. pression for $P_{t}$ in terms of expectations of future dividends.

Key Concepts

Risk Neutrality

Risk neutrality implies that consumers or investors are indifferent to risk and only care about the expected value of future cash flows. This assumption simplifies asset pricing models by removing the need for risk premiums, making it sufficient to discount expected future dividends and prices at the given rate.

Time Discounting

Time discounting is the concept that future cash flows, such as dividends or sale proceeds, are less valuable today than they are in the future due to the time value of money. A constant discount rate is used to weigh future cash flows, reflecting how current decisions trade off against future rewards.

Present Value Model

The present value model is a core principle in asset pricing that states the current price of an asset equals the sum of its expected future cash flows, each discounted back to present value by the appropriate discount rate. This fundamental idea connects the expected stream of dividends and future sale prices to the current asset price.

Iterated Expectations

Iterated expectations involves repeatedly applying the expectation operator over future periods to describe the evolution of an asset's price. By forward iterating the pricing equation, one can represent the current price as an infinite series of discounted future dividends, linking the short-run pricing formula to a long-run valuation framework.

No-Bubble (Transversality) Condition

The no-bubble or transversality condition ensures that asset prices remain bounded and are driven solely by the fundamental cash flows of the asset, rather than by speculative price increases. This condition rules out solutions where asset prices could diverge from the present value of dividends, which is essential for deriving a meaningful and realistic asset pricing model.

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