# Does variance depend linearly on beta?

Under the CAPM, consider an investment with stochastic cash flows. Does the variance of the return on the investment depend linearly on beta? if not, why so?

• @Dave, the beta-distribution tag is irrelevant here, as none of the variables follow it. Commented Aug 23, 2023 at 11:21

In a simple CAPM model you usually assume that $$r_{i} = \beta_i r_m + \epsilon_{i}$$ with $$\epsilon_{i} \sim N(0, \sigma_i)$$, where $$r_{i}$$ is the return of asset $$i$$, $$r_m$$ is the market return, and $$\sigma_i$$ is the idiosyncratic volatility of asset $$i$$. (Edit: note that the returns used should be net of the risk-free rate as Richard pointed out in the comments).
Since we also assume that idiosyncratic risk is independent of market risk (i.e. $$\text{E}[r_m\epsilon_{i}] = 0$$), we get that $$\text{Var}(r_{i}) = \text{Var}(\beta_ir_m) + \text{Var}(\epsilon_{i}) = \beta_i^2\sigma_m^2 + \sigma_i^2$$, where $$\sigma_m$$ is market volatility. So you get that variance depends linearly on the square of beta. This is because $$\text{Var}(aX) = a^2\text{Var}(x)$$ if $$a$$ is assumed to be known (see https://en.wikipedia.org/wiki/Variance#Propagation)
• You forgot the risk free rate. If you interpret $r_i$ and $r_m$ as returns net of the risk-free rate, then the first equation is fine (though normality of errors is unnecessary). Also, what is $m_t$? Commented Aug 23, 2023 at 11:20