# 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?

• Usually it is $r_R-d$ in the exponent where $r_R$ is the required return and d is the dividend. If $r_R$ is chosen according to the CAPM then it is replaced by $R_F+\beta(R_M-R_F)$ which is basically what you have except for notational issues, i.e. $\lambda$ represents $R_M-R_F$ and $R_F$ is written as $r$ – noob2 Nov 28 '16 at 16:31
• @noob2 is there any notes on the web for this which you can point me to? Thanks! – Danny Nov 28 '16 at 19:26
• The way I look at it, the only way to make the exponential term appear is if we assume the required return is continuously compounded? – Danny Nov 29 '16 at 8:21
• What i mean is, we assume that CAPM gives us the required rate of return for the stock as $\mathbb{E}[r_i] = \bar{r}_i = r + \beta\lambda$. Then assuming we use continuously compounded rate of return we get $\mathbb{E}[S_T] = S_t e^{\bar{r}_i (T-t)} = S_t e^{(r+\beta\lambda)(T-t)}$. Does this look correct? – Danny Nov 29 '16 at 12:32
• Yes, we are using continuous time rates of return (or if you prefer we are assuming continuous compounding). Also we are assuming that when the dividend is paid the (ex dividend) stock price drops by $d$ which is, again, a c.t. rate of dividend payment. That is why there is a $-d$ in the equation that forecasts the stock price at maturity. – noob2 Nov 30 '16 at 13:39