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Index Replication• Index tracking or benchmark replication is an strategy investmentaimed at mimicking the risk/return profile of a financial instrument.• For practical reasons, the strategy focuses on a reduced basket ofrepresentative securities.• This problem is also regarded as portfolio compression and it isintimately related to compressed sensing and l 1 -norm minimizationtechniques.• One example is the replication of an index, e.g., Hang Seng index,based on a reduced set of assets.Daniel P. Palomar 29
Tracking Error• Let c ∈ R M represent the actual benchmark weight vector and letw ∈ R M denote the replicating portfolio.• Investment managers seek to minimize the following tracking errorperformance measure:T E 1 (w) =√(c − w) T Σ (c − w).• In practice, however, the benchmark weight vector c is unknown andthe error measure is defined in terms of market observations.Daniel P. Palomar 30
Page 3 and 4: Data Model• Consider M financial
Page 5 and 6: Risk Control• In finance, the mea
Page 7 and 8: Daniel P. Palomar 6
Page 9 and 10: Mean-Variance Optimization (Markowi
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: • Finally, the robust portfolio f
Page 33 and 34: Index Replication with l 1 -Norm Mi
Page 35 and 36: Summary• We have introduced basic

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