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In the book that I am studying, the tangent portfolio was defined as the regular efficient portfolio in the case with $n$ risky assets and 1 riskfree asset with the extra requirement that the portfolio invests fully in the risky assets. So the tangent portfolio can be derived using the solutions to the mean/variance analysis problem: $$w = \frac{\mu_P}{\mu^T \Sigma^{-1} \mu}\Sigma^{-1} \mu$$$$ \sigma_P^2 = \frac{\mu_P^2}{\mu^T \Sigma^{-1} \mu}$$ where one can apply the restrictions on $w$ to obtain weights, mean excess return, and variance of the portfolio.

Yet I know that in other books, this portfolio is actually defined as the one with the highest sharpe ratio. I don't see the connection. How is this proven, if we used the derivation described above? I can calculate the sharpe ratio (it turns out to be the square root of the denominator in the second equation above), but how do I know it's bigger than the ones corresponding to all other investments in risky assets?

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Your question is very important! In formal way to demonstrate it is very interesting ... but a bit complicated ... and boring for non mathematicians. We may move around this demonstration to explain most of portfolio theory. However, to give the idea, if we have N risky assets we obtain, as efficient frontier, a semi-parabola and the weights of the countless efficient portfolio change point by point. If we have N risky asset + a risk free rate, we obtain, as efficient frontier, a straight line. Now every point/portfolio have only 1 risky component ... a tangent portfolio. Other efficient portfolios are linear combinations between tangent portfolio and risk free asset.

This line, as any other, has a slope ... in this framework the slope is the Sharpe ratio! This line is the CML and it is tangent with previous semi-parabola. For the line, to move above the semi-parable is impossible, but if we move below (possible) we have the (inefficient) CAL ... that has a lower slope ... so the CML has maximum slope/Sharpe Ratio. That's all.

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The tangent line has a couple properties:

  1. it is the slope (rise over run, which is Sharpe ratio) of the tangent portfolio
  2. it dominates the efficient frontier. i.e, for any level of risk, return of a portfolio in the tangent line is no less than (>=) the return of the a portfolio on the efficient frontier. In other word, portfolios on the tangent line have higher Sharpe ratio relative to the portfolios on the efficient frontier.

Tangent portfolio is the one intersect with the tangent line, so is has the highest Sharpe ratio than other portfolios sitting on the efficient frontier.

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