# Portfolio selection with no risk-free asset

Can someone explain why some papers on portfolio construction assume that there is no risk-free asset? For example, this paper: Machine Learning and Portfolio Optimization. What could be the reason(s) for this assumption of "no riskless asset"?

There are a few reasons the authors may have only looked at risky assets.

First, they are trying to find a faster way to solve a mean-CVaR optimization through relaxations. Therefore, they probably saw handling the risk (CVaR aka ES) as the most interesting part of the problem. Granted, doing so completely ignores that they should be looking at excess returns, but that is not their concern and would not render their approach incorrect. Essentially, they are not wasting time on getting into the argument about what is the risk-free rate and if it is truly risk-free in light of inflation.

Second: while most portfolio construction assumes there is a risk-free asset, we often build portfolios where we know there is not a risk-free asset.

For example, suppose you were building a portfolio for an equity fund in Argentina (or Russia or many other countries). Those countries have defaulted in the recent past, so assuming their (same-currency) government bonds are risk-free is a poor assumption.

However, Black (1972) says that we are fine so long as we can construct a zero-beta portfolio. If we were to update this for a multi-factor APT world, we need to be able to build a portfolio with all-zero factor betas. If we can do that, then we have a proxy for a risk-free (or as-riskless-as-possible) asset which we can use to compute excess returns.

• Russia defaulted on its local-currency sovereign debt in 1998. That's pretty unusual. EM countries printing lots of local currency to pay their debt (i.e. inflation risk) is much more common. – Dimitri Vulis Sep 25 '20 at 18:41
• You might think so (and I did, at the time) but it's apparently more common than that. I recall seeing a paper on defaults and they had about 10 or more defaults by EM countries in about a century! – kurtosis Sep 25 '20 at 20:37
• Sorry I didn't mean that Russia 1998 was the only example. E.g. Peru (Garcia's first term) defaulted in its local-currency debt about the same time. It certainly happens - just that printing/devaluing the local currency and paying off the nominal debt (which Peru did repeatedly) is more common (and less traumatic). – Dimitri Vulis Sep 25 '20 at 20:51

They just want to apply their technique to risky assets that actually have volatility. The risk-free asset has a volatility of $$0$$ so allocation towards it is treated as an after-thought since it's pretty much in an asset class if its own, whereas the risky assets on the risky side of the portfolio might have to be allocated from various risky asset classes or sectors/regional groupings.

They are not saying the risk-free asset doesn't exist. They just want to address the risky side of the portfolio problem while excluding, or assuming away, the riskless asset for the particular demonstration at hand.

• Thank you for the response! One reason is clear: focus on the risky side of the investment. However, you also mention "...for the particular demonstration at hand". What could be demonstrated (hypothetically)? Why would someone want to focus on the risky side of the investment? To demonstrate/check/verify what? – Qwerty Sep 25 '20 at 10:53
• explained in the answer above – develarist Sep 25 '20 at 14:34
• Still not clear. Why can't you analyze the risky side while also holding a risk-free asset? Having a risk-free asset does not make the entire portfolio riskless. – Qwerty Sep 25 '20 at 14:59
• because many strategies are just programmed not to include the riskless asset. Take for example the mean-variance efficient (frontier) portfolios. None of them include the risk-free rate in their objective functions, only the maximum Sharpe ratio portfolio does. The allocation towards risky assets is naturally an independent problem from the remaining weight given to $r_f$. In fact Markowitz makes this distinction in his 1952 paper since adding in the $r_f$ is just a linear combination with the risky assets. Sharpe ratio was not invented yet until 1960s – develarist Sep 25 '20 at 15:10
• Exactly! But "Black (1972). Capital market equilibrium with restricted borrowing" explicitly admits that this is an unrealistic assumption. Tobin added risk-free asset in 1958 after Markowitz developed his model in 1952. Thus, it seems like there is no benefit from excluding the risk-free asset these days. However, people do it without saying why (the paper that I mentioned in my original question). Thus, I don't see what exactly one can gain from analyzing only risky assets. It seems like people do because "why not". – Qwerty Sep 25 '20 at 15:21