From CAPM Expected Returns to Expected Future Spot Prices

Summary

The document asks how the CAPM relates an asset’s beta and the equity risk premium to its expected future spot price, including continuous dividend yield. It contrasts a simple one-period return calculation, which gives a linear expression, with an exponential expression over a longer interval. The question proposes continuous compounding as the reason for the exponential form.

The setup highlights a distinction between a one-period expected return and a continuously compounded expected price path. Translating the CAPM expected return into an expected future spot price requires care about the return convention, time horizon, dividend treatment, and assumptions used to extend the relationship through time. In general, an expected return relationship alone does not necessarily determine the expected future price through exponentiation without additional modeling assumptions. The document contains the question and its algebra, but no answer or supporting evidence resolving those assumptions.

Key ideas

• The CAPM links an asset’s expected return to the risk-free rate, beta, and equity risk premium.
• The document distinguishes a linear one-period return expression from an exponential multi-period price expression.
• Continuous dividend yield affects the relation between expected price growth and total return.
• Exponentiating a return requires assumptions about compounding and how expected returns evolve over the horizon.
• The document poses these issues without providing a resolution.

Tags

# CAPM and expected future spot price


# CAPM and expected future spot price












Let $S_t$ be the current spot price at time $t$ and $S_T$ be the spot price in the future (at time $T$). Assume the stock pays continuous dividends $d$.

How does the CAPM imply that the expected spot price is given by,

$$ \mathbb{E}[S_T] = S_te^{(r + \beta\lambda - d)(T-t)} $$

where $\beta$ is the asset beta and $\lambda$ is the equity risk premium?

Usually from the definition of asset returns we have,

$$ \mathbb{E}[R_i] = r + \beta \mathbb{E}[R_m - r]\\ \Rightarrow \mathbb{E}\left[\frac{S_T - S_t}{S_t}\right] = r+ \beta\lambda\\ \Rightarrow \mathbb{E}\left[\frac{S_T}{S_t}\right] = 1 +r+ \beta\lambda\\ \Rightarrow \mathbb{E}[S_T] = S_t(1 +r+ \beta\lambda) $$

So am I right to say that to get the exponential we have to assume continuous compounding?

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.