§18 Asset Pricing

Stock Prices as Present Values

• Stocks have two key differences with bonds:

  • They pay dividends that vary over time as conditions change (no fixed coupons);
  • They do not have a finite terminal date (no maturity).

• Let’s price a stock by arbitrage, assuming the investor can opt between a 1y bond or buying stocks

  $$1 + i_{1t} + x^{s} = \frac{\$D_{t+1}^{e} + \$Q_{t+1}^{e}}{\$Q_{t}}$$

• Reorganizing we have:

  $$\$Q_{t} = \frac{\$D_{t+1}^{e}}{(1 + i_{1t} + x^{s})} + \frac{\$Q_{t+1}^{e}}{(1 + i_{1t} + x^{s})}$$

  $$\begin{aligned}\$Q_t \ &= \ \frac{\$D_{t+1}^{e}}{(1 + i_{1t} + x^{s})} + \frac{\$D_{t+2}^{e}}{(1 + i_{1t} + x^{s})(1 + i_{1t+1}^{e} + x^{s})} \\&\quad + \frac{\$Q_{t+2}^{e}}{(1 + i_{1t} + x^{s})(1 + i_{1t+1}^{e} + x^{s})} \\&= \ \frac{\$D_{t+1}^{e}}{(1 + i_{1t} + x^{s})} + \cdots \\&\quad + \frac{\$D_{t+n}^{e}}{(1 + i_{1t} + x^{s}) \cdots (1 + i_{1t+n-1}^{e} + x^{s})} + \cdots\end{aligned}$$

• Note also that we can derive this expression in real terms

  $$Q_t = \frac{D_{t+1}^{e}}{(1 + r_{1t} + x^{s})} + \cdots + \frac{D_{t+n}^{e}}{(1 + r_{1t} + x^{s}) \cdots (1 + r_{1t+n-1}^{e} + x^{s})} + \cdots$$

— Apr 26, 2025

§18 Asset Pricing by Lu Meng is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Permissions beyond the scope of this license may be available at About.