I am struggling to interpret my mean-variance / efficient frontier / capital market line results. I have no issues calculating the efficient frontier. However, I do increase the risk-free rate from basically zero to higher values. When doing that, the tangential portfolio initially moves further north-east as excpected, but at some point the tangential portfolio drops to the negative part of the efficient frontier.

While I can't get my head around how to interpret this, it might make sense that no tangential point can be found anymore on the positive part of the effcient frontier, hence the max sharpe ratio is the point with "the least negative" Sharpe?

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I optimize to find weights that maximize the sharpe [(mu-rf)/std] under restriction that portfolio weights sum to 100% (short-sales allowed). Maybe I am missig a restriction? IS this solution even feasible?

I would really appreciate help on this.

Enjoy the rest of your weekend. Best regards and stay riskay!


1 Answer 1


Two separate cases were identified by R.C. Merton in 1972:

In the economically more relevant case, where $r_f < b/c$, efficient portfolios are combinations of a long position in [the tangency] portfolio M and lending or borrowing at the risk–free rate.

In the case where $r_f > b/c$, efficient portfolios are generated by short (or zero) positions in the tangency portfolio (which is not efficient) and risk–free lending. The efficient set is above the hyperbola.

The first analysis of these situations appeared in Robert C. Merton (1972): An Analytic Derivation of the Efficient Portfolio Frontier, Journal of Financial and Quantitative Analysis, 7: 1851–1872.

Source: https://www.empiwifo.uni-freiburg.de/lehre-teaching-1/winter-term-10-11/materialien-portfolio-analysis/mvs_riskfree.pdf

As a reminder $b=\mu^T \Sigma^{-1} \mathbf{1}$ and $c=\mathbf{1}^T \Sigma^{-1} \mathbf{1}$

Also the "frontier" of risky assets usually denotes the entire hyperbola but the "efficient frontier" (or the efficient part of the frontier) refers only to the upper portion of the hyperbola. So when the tangency point T occurs on the lower half it is not efficient (and it would be good if the code outputs a message or return code in this case to indicate that this is so).

  • 1
    $\begingroup$ Awesome, thank you very very much for this! $\endgroup$
    – Riskay
    Commented Mar 8, 2020 at 14:17

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