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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 ...


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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 ...


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Just to add to the previous answer, one example of such asset (returning 'risk-free rate') is a money market (or bank) account, but it is only locally risk-free, with value accruing continuously at the risk-free rate prevailing in the market at every instant. It is risk-free only over a short period of time. In long term it is stochastic too. Its SDE is: $$ ...


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The standard way to think about this is that at time $t$ the riskless asset gives you known return of $r_{f,t}$ over a short time period. However, this rate may itself be time-varying and stochastic so that we don't know its futures values, say $r_{f,t+s}$. E .g. a common assumption is that the rate follows an Ornstein Uhlenbeck process (implying that the ...


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