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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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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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    $\begingroup$ 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. $\endgroup$ – SRKX Jun 18 '16 at 13:09
  • $\begingroup$ You are perfectly right. In this case, the efficient frontier is not linear after the tangent portfolio. Nice point! $\endgroup$ – e.mal Jun 19 '16 at 14:11

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