# How to choose a tangency portfolio without a risk-free rate

How do you choose an optimal portfolio from the efficient frontier if no risk-free rate is given?

I know that if there exists risk-free asset, then you would combine a portfolio from the efficient frontier and the risk free asset and that would be your optimal portfolio.

But if you do not have risk-free asset, how do you choose one from the efficient frontier? In other words, how do you choose the maximum acceptable portfolio volatility?

• Typically, it's the combination of assets which results in the highest Sharpe ratio. I.e., the greatest slope along the capital allocation line. If you do not have a risk free asset, then it is typical to assume that required rate of return is 0%. – David Addison Feb 10 '18 at 7:26
• @DavidAddison So if I do not have a risk free asset, how do I choose which portfolio? – Jun Jang Feb 10 '18 at 14:34
• I'm assuming you're using conventional MPT, where efficient frontier is the greatest amount of return for a given amount of variation. A risk free rate is then used for a starting point to choose the optimal point on the efficient frontier. If you do not have risk free rate, then any point can be considered optimal. If you assume 0% risk free rate, then its the tangency portfolio with the greatest slope (i.e., Sharpe ratio). – David Addison Feb 10 '18 at 18:35
• @DavidAddison Thank you very much for your reply. I see, so any point is optimal. Would you mind if I ask you more questions? I am working on this portfolio construction project, and would appreciate any help/advice! – Jun Jang Feb 10 '18 at 19:07
• Sure! Feel free to PM me. – David Addison Feb 10 '18 at 19:29

In a standard portfolio optimization setting, an efficient frontier is formed for the mix of asset weights which result in the greatest (expected) portfolio return with least amount of (expected) portfolio volatility.

Technically any point on that frontier can be considered efficient in the absense of a risk-free rate. When a zero variance asset (i.e. risk-free rate of return) is introduced, then the optimal point of the frontier becomes less ambiguous. An optimal portfolio is then formed from the "capital allocation line" drawn between the zero-variance asset to the highest point along the frontier, which is thus called the "tangency portfolio".

1. As you and @AlRacoon point out, one way might be to consider an investor's risk appetite (e.g., via maximum acceptable volatility).

2. Another way, as @AlexC indicated, might be to construct a utility curve that represents an investor's risk preferences. The function $\mathcal{U}\left[\mu,\,\sigma \right]$ is then to be maximized. Typically, such a function is concave, .e.g.: $\mathcal{U}\left[\mu,\,\sigma \right] = \mathbb{E}\left[\mu \right] -\frac{\sigma^2}{2}$.

3. A third (non-mutually exclusive) alternative is to introduce the use of benchmarks into the optimization. Mechanically, this is no different from standard approaches except, in this case, the optimization is between tracking error (i.e., $Abs\left[r_a - r_b \right]$) versus excess returns. In this sense, the benchmark is risk-free with respect to itself, and there will almost surely be some combination of constituent assets which achieves a positive active return versus the benchmark. This approach is distinctly advantanged in that no risk free asset may be needed to identify the tangency portfolio. I.e., the capital allocation line is identifiable by the portfolio with the greatest information ratio (IR) (vice Sharpe ratio). Since IR is typically seen as a proxy for skill, an IR optimal portfolio could be considered to contain the most signal per unit of noise. I have also seen approaches which optimize for IR versus tracking error (i.e., $\frac{ \mathbb{E}\left[r_a-r_b \right]}{\sigma^2_{a-b}}$) with some very interesting results (i.e., the Kelly Capital Growth Criterion of a single asset portfolio is nearly identical!!!). Suitable implementations of the efficient frontier of excess return outlined in the following articles from Mathworld:

Given the details of your assignment (i.e., that you are provided with benchmarks), I would attempt method 3 since there is a possibility that the tangency portfolio will be clearly defined. Moreover, the fewer parameters and/or assumptions an approach requires, the more robust it generally is.

I would assess that the L/S TR index is the most appropriate benchmark provided. The individual long-only benchmarks provided in conjunction with the funds' return are -- in my opinion -- mostly worthless as a comparison to L/S funds' performance. Then again, benchmarking is as much art as science; you will find a diversity of opinion regarding benchmark selection.

In the case where the efficient frontier does not intersect with the vertical axis, the tangency portfolio is clearly defined. In this case, the point with the highest IR is optimal.

It may however be that the efficient frontier intersects the vertical axis (i.e., there is a combination of assets which perfectly replicates the index). This will almost surely be the case when the index is considered to be an investable asset and/or when the asset universe is broadly enough defined. In this instance, the tangency portfolio is not defined unless you go back to defining a maximum acceptable risk tolerance and/or utility function.

There may be another special case where there is no combination of assets which exceeds the benchmark's return. In this case as well, the benchmark itself would be the optimal portfolio.

• Hi David, thank you so much for helping me out and putting everything together. – Jun Jang Feb 13 '18 at 3:05
• @JunJang My pleasure. I am interested in how your project works out. Please let me know which approach you decide to go with and if there are any interesting results! – David Addison Feb 13 '18 at 3:21
• David, I will definitely share my report with you in the near future when I am done! May I ask you one last question please? To calculate active return and tracking error, do I have to subtract the benchmark return from the individual fund's return first and then compute portfolio return, volatility, and etc? Or do I first compute portfolio return, volatility, and etc from the 20 funds first?? – Jun Jang Feb 13 '18 at 18:15
• I think you could do either way. If you use the moments of each strategy’s returns, it should be possible to do a parametric mean-variance optimization with the correlation/covariance matrix. But I would think that the second option is much easier: subtract the benchmark’s periodic return from every funds’ periodic return and then proceed as you would have before. – David Addison Feb 13 '18 at 18:51
• Oh I think your suggestion is the first option. I think my wording was confusing. – Jun Jang Feb 13 '18 at 19:35