If I have a portfolio with a Sharpe ratio lower than the Sharpe ratio of the tangent portfolio, can I conclude something about whether or not it is efficient?
If so, how/why?
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$\begingroup$ You can see also this closely related question quant.stackexchange.com/questions/26034/… $\endgroup$– markowitzCommented May 19, 2016 at 9:06
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Sure you can. Sharpe Ratio is defined as: $$ SR=\frac{E(R)-R_f}{\sqrt{Var(R)}} $$ When you have a risk-free asset, the efficient frontier becomes linear (i.e. the line that passes from the $R_f$ and the tangent portfolio), named Capital Market Line (CML) and $SR$ denotes its slope. So lower $SR$ means that your portfolio does not lie on the efficient frontier and hence it is not efficient.
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2$\begingroup$ Note that this is only true if you are allowed to leverage your portfolio (i.e. borrow at the risk-free rate to invest extra). If you have a long-only portfolio, then portfolios more risky than then tangent portfolio will have lower Sharpe ratios. $\endgroup$– SRKXCommented Jun 18, 2016 at 13:09
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$\begingroup$ You are perfectly right. In this case, the efficient frontier is not linear after the tangent portfolio. Nice point! $\endgroup$– e.malCommented Jun 19, 2016 at 14:11