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Sharpe Ratio• The Sharpe ratio is akin to the SINR in communication systems.• It is defined as the ratio of the mean return (excess w.r.t. the returnof the risk-free asset r f ) to the risk measured as the square root ofthe variance (standard deviation):S = wT µ − r f√wT Σw .• The Sharpe ratio can be understood as the expected return per unitof risk.• The portfolio w that maximizes the Sharpe ratio lies on the Paretocurve.Daniel P. Palomar 7
Mean-Variance Optimization (Markowitz, 1956)• There are two obvious formulations for the portfolio optimization.• Maximization of mean return:maximizewsubject tow T µw T Σw ≤ α1 T w = 1.• Minimization of risk:minimizewsubject tow T Σww T µ ≥ β1 T w = 1.Daniel P. Palomar 8
Page 3 and 4: Data Model• Consider M financial
Page 5 and 6: Risk Control• In finance, the mea
Page 7: Daniel P. Palomar 6
Page 11 and 12: Max Sharpe Ratio Portfolio Formulat
Page 13 and 14: √minimize ˜wT Σ ˜ww, ˜w,tsubj
Page 15 and 16: Value-at-Risk (VaR & CVaR) Models
Page 17 and 18: CVaR Portfolio Formulation• Deali
Page 19 and 20: Proof CVaR in Convex Form(a) The mi
Page 21 and 22: • Corollary:CVaR in Convex Form (
Page 23 and 24: • The problem of minimizing the C
Page 25 and 26: • We will consider one particular
Page 27 and 28: • First, consider the worst case
Page 29 and 30: • Finally, the robust portfolio f
Page 31 and 32: Tracking Error• Let c ∈ R M rep
Page 33 and 34: Index Replication with l 1 -Norm Mi
Page 35 and 36: Summary• We have introduced basic

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