Handling a Risk-Free Asset in Portfolio Covariance Optimization
Summary
The document addresses a singular excess-return covariance matrix that arises when cash earns exactly the risk-free rate. Because that return is constant relative to the chosen benchmark, cash has zero variance and zero covariance with the other assets, which can make covariance-based portfolio calculations fail.
It suggests treating asset allocation and cash allocation as separate decisions, or using a longer return horizon where changing risk-free rates give the cash proxy variation over time. The first approach preserves cash as riskless; the second changes the modeling assumption by treating the asset as exposed to rate changes. The discussion offers conceptual workarounds rather than a detailed optimization procedure, and it does not establish a universally appropriate horizon or quantify the effects on portfolio weights.
Key ideas
- Cash returns equal to the risk-free benchmark produce a zero row and column in an excess-return covariance matrix.
- That zero variance makes the covariance matrix singular and can disrupt optimization calculations.
- One workaround is to optimize risky-asset allocations separately from the cash allocation.
- A longer measurement horizon can introduce cash-return variation when risk-free rates change over time.
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# Excess Return Covariance Matrix is Singular - Cash return and risk free rate are the same # Excess Return Covariance Matrix is Singular - Cash return and risk free rate are the same I've created a three asset excess return covariance matrix. The assets are; equity, bonds, and cash. However, my cash return is the same as my risk free rate ( i.e. 3 month Euribor). This is leaving me with a covariance matrix where the cash asset has 0 covariance with itself, equity and bonds. A singular matrix. There's an example below. This is leading to a lot of complications when I try to use it in my Python code to generate expected returns/weights etc. Unfortunately, I don't seem to have a good alternative to Euribor for historical Euro cash returns (1990 onwards). Is there any way I can adjust how the cash covariance is generated to produce a non zero result without affecting the outcome of my analysis? ## Answer by KaiSqDist (score 2) https://quant.stackexchange.com/a/79111 I believe this answers your question? "Adding" risk-free asset to covariance matrix after the fact but the answer is replacing cash with the riskless asset. The key message is either - - Separate the allocation of assets (equities and fixed-income) from the allocation of cash in the portfolio. or - Consider a longer horizon so you can take the riskless asset as a "risky" asset with changing risk-free rate at different time periods. This generates some variance in its returns, resulting in non-zero entries in your covariance matrix.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.