Problem 9 - Advanced Macroeconomics

Question

Consider a stock that pays dividends of \(D_{i}\) in period \(t\) and whose price in period \(t\) is \(P_{r}\). 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 \(P_{t}=E_{t}\left[\left(D_{t+1}+P_{t+1}\right) /(1+r)\right]\) (assume that if the stock is sold, this happens after that period's dividends have been paid). (b) Assume that \(\lim _{s \rightarrow \infty} E_{t}\left[P_{t+s} /(1+r)^{s}\right]=0\) (this is a no-bubbles condition; see the next problem). Iterate the expression in part ( \(a\) ) forward to derive an expression for \(P_{1}\) in terms of expectations of future dividends.

Step-by-Step Solution

Answer

In equilibrium, \(P_1 = E_1 \left[\sum_{s=1}^{\infty} \frac{D_{s+1}}{(1+r)^s}\right]\).

Step 1: Understand the Risk-Neutral Pricing Formula

In this setup, stock prices reflect expected future payoffs discounted at rate \(r\). The equilibrium stock price in period \(t\), \(P_t\), equals the expected present value of next period's price \(P_{t+1}\) and dividends \(D_{t+1}\), discounted by \(1+r\). Thus, \(P_{t} = E_{t}\left[\frac{D_{t+1} + P_{t+1}}{1+r}\right]\).

Step 2: Iterate Price Equation in Part (a)

To derive an expression for \(P_1\) in terms of future dividends, continuously apply the equilibrium condition. Starting with the price equation: \(P_t = E_t\left[\frac{D_{t+1} + P_{t+1}}{1+r}\right]\), replace \(P_{t+1}\) using \(P_{t+1} = E_{t+1}\left[\frac{D_{t+2} + P_{t+2}}{1+r}\right]\). This iteration continues for each future period.

Step 3: Express \(P_1\) via Summed Expected Future Dividends

From step 2 and the assumption \(\lim_{s \to \infty} E_t\left[\frac{P_{t+s}}{(1+r)^s}\right]=0\), all terms involving future stock prices vanish at infinite iteration depth. The remaining terms are solely dividends evaluated each at their discounted future value: \[P_1 = E_1 \left[\sum_{s=1}^{\infty} \frac{D_{s+1}}{(1+r)^s}\right]\].

Key Concepts

Risk-Neutral Pricing

In a risk-neutral environment, consumers do not demand additional returns for bearing risk. Instead, they are content with the expected value of returns, with no risk premium required.
The stock price, therefore, is determined by the expected future payoffs. This includes dividends and the stock's future selling price.
In mathematical terms, the equilibrium price at time \( t \), \( P_t \), is derived from expected future dividends \( D_{t+1} \) and the future stock price \( P_{t+1} \), discounted back to present value using the risk-free rate \( r \) as follows:

• \( P_t = E_t\left[ \frac{D_{t+1} + P_{t+1}}{1+r} \right] \)

This formula reflects the risk-neutral investor's perspective, where the future uncertain payoffs are made comparable to the present through discounting.

No-Bubble Condition

The no-bubble condition is a crucial assumption in finance that ensures a stock's price reflects its fundamental value.
When a stock price is a bubble, it means the price is primarily driven by investors expecting to sell the stock at a higher price, rather than by intrinsic value such as dividends.
To ensure no bubble, it's assumed that the present value of the stock price approaches zero as time approaches infinity:

• \( \lim_{s \to \infty} E_t\left[ \frac{P_{t+s}}{(1+r)^s} \right] = 0 \)

This condition helps focus on fundamental aspects like expected future dividends rather than speculative future price increases.

Equilibrium Stock Price

The equilibrium stock price is the point where the stock's price accurately reflects all known information and expected future returns.
In a risk-neutral world, equilibrium is achieved when the current stock price equals the present value of all expected future cash flows.
This involves iterating the risk-neutral pricing formula forward to eliminate dependence on future stock prices and replace them with expected dividends:

• Start with \( P_t = E_t\left[ \frac{D_{t+1} + P_{t+1}}{1+r} \right] \)
• Substitute future prices continually until dependent only on dividends

By doing so, you derive a present value formula focusing solely on expected dividends.

Discounted Future Dividends

The value of a stock can be conceived as the sum of all its expected future dividends, each discounted back to its present value.
As we iterate the price equation forward under the no-bubble condition, future price terms diminish, leaving dividends as the sole determinants of current price.
Resultantly, for any stock in period 1, its price based solely on expected future dividends is:

• \( P_1 = E_1\left[ \sum_{s=1}^{\infty} \frac{D_{s+1}}{(1+r)^s} \right] \)

This highlights that, in equilibrium, a stock's price today is the present value of its potentially infinite stream of future dividends.