In Theory

There are two common models of conceptualizing the risk associated with a stock:

1. The Margin of Safety Principal popularized by Benjamin Graham
2. Asset returns volatility

The former can be characterized as a subjective assessment of how much downside risk is associated with a stock. In other words, it’s a consideration of the potential losses if an investor’s thesis is incorrect and/or if the worst case scenario is realized. This can be described with the following probability distribution:

$$ f(x) = \begin{cases} p & \text{if $x = \lambda$} \newline 1 - p & \text{if $x = \alpha$} \end{cases} $$

Where \(p\) is the probability of the worst case being realized, \(\lambda\) is the return in that scenario, and \(\alpha\) is the return if that scenario is not realized. Now,

$$ \operatorname{Var}(X)=\operatorname{E}\left[X^{2}\right]-\operatorname{E}[X]^{2} $$

$$ \operatorname{E}[X]=\sum\limits_{i=1}^{N}p_{i}x_{i} = p\lambda + (1-p)\alpha $$

$$ \operatorname{E}\left[X^{2}\right]=\sum\limits_{i=1}^{N}p_{i}x_{i}^{2} = p\lambda^2 + (1-p)\alpha^2 $$

Therefore, $$ \operatorname{Var}(X)=p\lambda^2 + (1-p)\alpha^2 - \left(p\lambda + (1-p)\alpha\right)^2 = $$ $$ p\lambda^2 + (1-p)\alpha^2 - p^{2}\lambda^{2} - (1-p)^{2}\alpha^{2} - 2p(1-p)\lambda\alpha = $$ $$ p\lambda^2 + \alpha^2-p\alpha^2 - p^{2}\lambda^{2} -\alpha^{2}-\alpha^{2}p^{2}+2p\alpha^{2} -2p\lambda\alpha+2p^{2}\lambda\alpha = $$ $$ p\lambda^2 - p^{2}\lambda^{2} -\alpha^{2}p^{2} + p\alpha^{2} - 2p\lambda\alpha+2p^{2}\lambda\alpha = $$ $$ p\lambda^2 + p\alpha^{2} - 2p\lambda\alpha+2p^{2}\lambda\alpha - p^{2}\lambda^{2} - \alpha^{2}p^{2} = $$ $$ p(\lambda^2 + \alpha^{2} - 2\lambda\alpha) - p^{2}(\lambda^{2} + \alpha^{2} - 2\lambda\alpha) = $$ $$ p(\lambda^2 + \alpha^{2} - 2\lambda\alpha) - p^{2}(\lambda^{2} + \alpha^{2} - 2\lambda\alpha) = $$ $$ p(\alpha - \lambda)^{2} - p^{2}(\alpha - \lambda)^{2} = $$ $$ p(1-p)(\alpha - \lambda)^{2} $$

Moreover, $$ \sigma = (\alpha - \lambda)\sqrt{p(1-p)} < \frac{1}{2}(\alpha - \lambda) $$

Now \(p\) varies primarily based on asset class and is presumably always less than 50%. For example, \(p\) for a T-bill is close to 0%; for a growth stock, it might be 30%. Therefore, we can say that a reasonable proxy for a Sharpe Ratio within a single asset class is: $$ SR \propto \frac{\mu}{\alpha-\lambda} $$

We can also write $$ SR = \frac{p\lambda + (1-p)\alpha}{(\alpha - \lambda)\sqrt{p(1-p)}} = \frac{\alpha-p(\alpha-\lambda)}{(\alpha - \lambda)\sqrt{p(1-p)}} < \frac{\alpha}{(\alpha - \lambda)\sqrt{p(1-p)}} \propto \frac{\alpha}{(\alpha - \lambda)} $$

This means that we can use \(\frac{\alpha}{(\alpha - \lambda)}\) as relative value measure within an asset class (e.g., growth technology stocks), but probably not across asset classes.

In Practice

Within the context of publicly traded stocks, \(p\) is presumably a fairly high value i.e., greater than 20% and less than 50%. This means that \(\sqrt{p(1-p}\) varies between 40% and 50% and the quantity \(\frac{\alpha-\lambda}{2}\) is a reasonable estimate of the long term value of \(\sigma\).

For a buy-and-hold investor, I’d argue that \(\frac{\alpha-\lambda}{2}\) is a better risk metric than \(\sigma\) based on daily price returns.