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Risk-Free Assets, Benchmarks, and Sharpe Ratio Portfolio Choice

Article Quant Q&A · Author: Nikos
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Summary

The document distinguishes portfolio choice among risky assets from allocating wealth between a risky portfolio and a risk-free asset. It explains that adding a tradable risk-free asset creates a capital allocation line from the risk-free return through the tangent portfolio, while risky-asset optimization concerns the efficient frontier. The answer also gives a mean-variance utility expression in which an investor’s risk aversion affects the preferred point along that line.

It notes that Sharpe ratios can compare efficient risky portfolios without using risk aversion as an input. The question references research that evaluates risky-only portfolios using returns in excess of the risk-free rate, but the responses do not fully resolve how constraints, out-of-sample evaluation, or inability to trade the risk-free asset change the optimization. The discussion is conceptual and brief; it offers no empirical test or detailed derivation, and the answer itself expresses uncertainty about Tobin’s separation.

Key ideas

  • Asset allocation between a risk-free asset and a risky portfolio differs from optimization among risky assets.
  • A tradable risk-free asset allows investors to combine it with a tangent portfolio in different proportions.
  • Mean-variance risk aversion affects an investor’s preferred allocation along the capital allocation line.
  • Sharpe ratios compare excess return per unit of volatility and do not themselves encode an investor’s risk aversion.
  • Constraints and evaluation design may affect whether the classical separation result applies, but the document does not analyze these cases fully.

Tags

Portfolio constructionStatisticsRisk management
Full text
# What is the difference between a riskfree-asset class and a benchmark when computing the Sharpe-Ratio?


# What is the difference between a riskfree-asset class and a benchmark when computing the Sharpe-Ratio?












Question regarding the calculation of the Sharpe Ratio: Is my following understanding correct? Assuming I have the standard quadratic utility function with the risk version parameter Is there a structural difference between using the risk-free asset as a benchmark or as an actual asset class to invest in?

If I use the risk-free asset as an actual asset class, Tobin's separation applies and everyone invests in the same risky asset, but only the amount of wealth invested in the risk-free asset class varies. This gives the maximum Sharpe ratio or tangent portfolio.

I am now interested in whether it is not possible to invest in the risk-free asset class, and I use the risk-free asset class as a benchmark. After portfolio optimisation, I calculate the excess returns by subtracting the risk-free asset from the portfolio return and dividing by the standard deviation of the portfolio. Is the optimal portfolio here dependent on the risk aversion parameter and does here then the Tobin's separation not apply? And I can still use the Sharpe-Ratios for comparing risky-portfolios in relation how high the riskoaversionparamter is?

Thanks in advance! (also any good literature regarding this would be helpful!)

I refer to the Paper of DeMiguel et al. (2009), here only risky assets are used but in the sharpe-ratio computation the risk-free rate is subtracted (see appendix of this paper). So is here the rsik-free rate a benchamrk? And the there is no tangent portfolio. (Or are there other reasons? Such as constraints or decision timing, where the risk aversion matters and the tobins separartion is true, or because the analysis is out-of-sample)

## Answer by KaiSqDist (score 1)

https://quant.stackexchange.com/a/82237

Your question is not phrased clearly and is quite segmented. I will try to answer based on my understanding of what you are asking.

Is there a structural difference between using the risk-free asset as a benchmark or as an actual asset class to invest in?

There is a difference. The risk-free asset is used in capital allocation (see capital allocation line in efficient frontier), which is different from the investment of risky assets that is asset allocation (see Markowitz efficient frontier of risky assets) also referred to as portfolio optimization.

Is the optimal portfolio here dependent on the risk aversion parameter and does here then the Tobin's separation not apply?

I don't know about Tobin's separation, but if each individual $i$ has a unique risk aversion $\lambda_i$ say,

$$Utility_i=w'\hat{\mu}-\lambda_i w'\hat{\Sigma} w$$

then the utility of points/portfolios on the capital allocation line (based on the tangent portfolio) is different for each individual and thus there could only be a single optimal portfolio per individual.

And I can still use the Sharpe-Ratios for comparing risky-portfolios in relation how high the risk aversion parameter is?

You could, but that would only be relevant to portfolios that lie on the efficient frontier (and the risk aversion parameter doesn't have a role here). This question seems to me like a step back from the previous question, I don't quite see the point.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.