Financial Mathematics Text

Thursday, October 24, 2013

Thoughts on the Sharpe Ratio

So this is going to be just a few musings on the Sharpe Ratio. But before that, I want to do a comparison to a technique I utilized because it has similiarities to the Sharpe Ratio. This is also somewhat related to Cullen Roche's question and my response here.

In my blog post, Are Bond Yield Spreads Adequate?, and the subsequent follow-up, Junk Bonds: A Closer Look, I developed a simple model to analyze spreads to see if they were adequate. Today I'll give a more "intuitive" explanation of that model.

The model attempts to estimate the returns on a bond portfolio above treasuries and adjusted for losses for default. So returns will end up being:

$$\text{Returns} = \text{Yield} - \text{Losses due to Default}$$
I then subtracted the yield on treasuries (e.g. the risk-free rate).  This gives an expected value of returns over treasury returns.

But I didn't stop there. I subtracted off one more term: the uncertainty of the model to provide a margin of safety (see my Uncertainty and Margin of Safety for context) . My suggestion was that:
$$E(R_p) - R_{rf} - \sigma_u \ge 0$$
where $E(R_p)$ is the expected returns of the portfolio (yield minus losses due to default), $R_{rf}$ is the risk free rate of return and $\sigma_u$ is the uncertainty estimate.

I can algebraicly manipulate that equation to look like this:

$$\frac{E(R_p) - R_{rf}}{\sigma_u} \ge 1$$
I point this out because it bears a resemblance to the Sharpe Ratio. The main difference is the standard deviation term. In the case of the Sharpe Ratio, it's the volatility of the returns on the portfolio. In my model, it's the uncertainty involved in estimation of the expected returns of the portfolio.

Now, my choice of 1 standard deviation was not necessary; I could have chosen 2 standard deviations or 0.5 standard deviations.

But let's leave that aside for a moment and consider the Sharpe Ratio:
S = \frac{E(R_p) - R_{rf}}{\sigma_p} \tag{Sharpe Ratio}
where $S$ is the Sharpe Ratio and $\sigma_p$ is the volatility of the portfolio returns. I can rearrange this equation as follows:
$$E(R_p) - R_{rf} - \sigma_p S = 0$$
In that sense, you can think of $\sigma_pS$ as the margin of error allowed in determine whether or not the asset will outperform treasuries.

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