When a Capital Market Line’s Tangency Falls on the Lower Frontier
Summary
The document explains why a tangency portfolio in mean-variance analysis can fall on the lower half of the risky-asset frontier when the risk-free rate is sufficiently high. In this case, the tangency portfolio is not on the efficient portion of the frontier, even though an optimizer maximizing the Sharpe ratio under a budget constraint may return it. The result therefore does not necessarily mean the numerical optimization is infeasible or incorrectly implemented.
The discussion cites Merton’s analysis of two cases. When the risk-free rate is below a threshold expressed using quantities derived from expected returns and the covariance matrix, efficient portfolios combine a long position in the tangency portfolio with risk-free lending or borrowing. Above that threshold, the efficient set instead involves short or zero positions in the tangency portfolio and risk-free lending. The answer also clarifies that the full risky frontier is a hyperbola, while the efficient frontier means only its upper portion. It recommends that software flag a lower-half tangency result; it does not provide implementation details or explore additional constraints.
Key ideas
- A sufficiently high risk-free rate can place the Sharpe-ratio tangency point on the lower half of the risky-asset frontier.
- A lower-half tangency portfolio is not itself an efficient risky portfolio.
- Merton’s cases distinguish long tangency positions with lending or borrowing from short or zero positions with lending.
- The full risky frontier and the efficient frontier refer to different portions of the mean-variance hyperbola.
- Optimization software should identify when its tangency result lies outside the efficient portion.
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# Can a capital market line have a negative slope?
# Can a capital market line have a negative slope?
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?
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!
## Answer by nbbo2 (score 7, accepted)
https://quant.stackexchange.com/a/51540
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).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.