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Portfolio Optimization with Cvar Budgets

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Portfolio Optimization with Cvar Budgets Portfolio optimization with CVaR budgets ∗ Kris Boudt - Peter Carl - Brian G. Peterson R/Finance 2010 April 16th, 2010 ∗ K.U.Leuven and Lessius 1/42 History Outline Risk budgets: A primer on risk budgets CVaR budgets as objective or constraint in Standard tool to quantify risk allocation; portfolio allocation Dynamic portfolio allocation Previous research: non-normality return Conclusion series, CVaR: PerformanceAnalytics. Appendix Instrument to adjust marginally portfolios; This research: Use of risk budgets as ob- jective and/or constraint in portfolio alloca- tion styles: PortfolioAnalytics; Collaborative: Peter Carl & Brian Peter- son, David Ardia, Christophe Croux. 2/42 Motivation equity/bonds portfolio Outline Bender, Briand, Nielsen, Stefek (JPM, Winter 2010): A primer on risk budgets CVaR budgets as “Traditional approaches to structuring policy portfolios for strategic objective or constraint in portfolio allocation asset allocation have not provided the full potential of diversification. Dynamic portfolio allocation Portfolios based on a 60/40 allocation between equities and bonds Conclusion remain volatile and dominated by equity risk.” Appendix Minimum risk portfolios tend to be dominated by bond risk and have lower expected returns. 3/42 Motivation equity/bonds portfolio Outline Bender, Briand, Nielsen, Stefek (JPM, Winter 2010): A primer on risk budgets CVaR budgets as “Traditional approaches to structuring policy portfolios for strategic objective or constraint in portfolio allocation asset allocation have not provided the full potential of diversification. Dynamic portfolio allocation Portfolios based on a 60/40 allocation between equities and bonds Conclusion remain volatile and dominated by equity risk.” Appendix Minimum risk portfolios tend to be dominated by bond risk and have lower expected returns. Optimize the risk allocation directly in the portfolio strategy. Examples: X A 60/40 risk allocation portfolio or an equal-risk portfolio X The most risk diversified portfolio subject to return/risk targets. 3/42 Outline Outline Risk budgets: A primer on risk budgets CVaR budgets as objective or constraint in Review on risk budgets; portfolio allocation Dynamic portfolio allocation Use of risk budgets as objective and/or Conclusion constraint in portfolio allocation styles. Appendix Illustrations: X Static bond-equity portfolio, R code (DEoptim, see also Guy Yollin’s slides RFinance 2009); X Dynamic 4 assets. 4/42 Outline A primer on risk budgets VaR budget CVaR budget Min CVaR portfolio CVaR budgets as objective or constraint in A primer on risk budgets portfolio allocation Dynamic portfolio allocation Conclusion Appendix 5/42 VaR budget on 60/40 portfolio Outline > library(PortfolioAnalytics) A primer on risk budgets > data(indexes) VaR budget CVaR budget > head(indexes[,1:2]) Min CVaR portfolio US Bonds US Equities CVaR budgets as objective or constraint in 1980-01-31 -0.027189977 0.0610 portfolio allocation Dynamic portfolio 1980-02-29 -0.066876898 0.0031 allocation 1980-03-31 0.005275587 -0.0987 Conclusion 1980-04-30 0.099246261 0.0429 Appendix 1980-05-31 0.000000000 0.0562 1980-06-30 0.060511451 0.0296 6/42 VaR budget on 60/40 portfolio Outline > apply(indexes[,1:2],2,’mean’) A primer on risk budgets US Bonds US Equities VaR budget CVaR budget 0.006916187 0.008238420 Min CVaR portfolio > apply(indexes[,1:2],2,’sd’) CVaR budgets as objective or constraint in US Bonds US Equities portfolio allocation Dynamic portfolio 0.01810161 0.04476569 allocation Conclusion Appendix 7/42 VaR budget on 60/40 portfolio Outline > w6040 <- c(0.4,0.6) A primer on risk budgets > library(PerformanceAnalytics) VaR budget CVaR budget > VaR(R=indexes[,1:2], weights=w6040, Min CVaR portfolio + portfolio_method="component") CVaR budgets as objective or constraint in $MVaR portfolio allocation Dynamic portfolio [1,] 0.04336715 allocation $contribution Conclusion US Bonds US Equities Appendix -0.0002303964 0.0435975440 $pct_contrib_MVaR US Bonds US Equities -0.005312695 1.005312695 8/42 VaR budgets Outline ∂VaR(w) VaR A primer on risk budgets Ci = wi ∂wi VaR budget CVaR budget Gourieroux,´ Laurent and Scaillet (2000): Min CVaR portfolio CVaR budgets as objective or constraint in CiVaR = −E[wiri|rp = −VaR] portfolio allocation Dynamic portfolio allocation Conclusion Estimation: Appendix X Simulation X Explicit formulae Cornish-Fisher estimator (Boudt, Peterson and Croux, 2008; Peterson and Boudt, 2008). 9/42 Outline Outline Pearson [2002, p.7]: “Value-at-risk has some well known limitations, A primer on risk budgets and it may be that some other risk measures eventually supplants VaR budget CVaR budget value-at-risk in the risk budgeting process.” Min CVaR portfolio CVaR: 95% Value at Risk and 95% Conditional VaR CVaR budgets as for a Skewed Student t objective or constraint in portfolio allocation Coherent risk mea- 95% VaR Dynamic portfolio allocation sure (most notably: Conclusion subadditive); Appendix Density Less incentive to load 95% CVaR on the tail risk below the VaR used. 0.0 0.1 0.2 0.3 0.4 −10 −5 0 5 10 Portfolio returns 10 / 42 CVaR budgets Outline ∂CVaR(w) CVaR A primer on risk budgets Ci = wi ∂wi VaR budget CVaR budget Scaillet (2002): Min CVaR portfolio CVaR budgets as objective or constraint in CiCVaR = −E[wiri|rp ≤−VaR] portfolio allocation Dynamic portfolio allocation Estimation: Conclusion X Appendix Simulation X Explicit formulae Cornish-Fisher estimator (Boudt, Peterson and Croux, 2008). 11 / 42 CVaR budgets Outline > ES(R=indexes[,1:2], weights=w6040, A primer on risk budgets + portfolio_method="component") VaR budget CVaR budget $MES Min CVaR portfolio [1,] 0.07725177 CVaR budgets as objective or constraint in $contribution portfolio allocation Dynamic portfolio US Bonds US Equities allocation -0.001066194 0.078317964 Conclusion $pct_contrib_MES Appendix US Bonds US Equities -0.01380155 1.01380155 12 / 42 CVaR in portfolio allocation Outline As an objective: Minimum CVaR portfolio (E.g. Rockafellar A primer on risk budgets VaR budget and Uryasev, 2000) CVaR budget Min CVaR portfolio As a constraint: Min CVaR/SD portfolio under CVaR CVaR budgets as objective or constraint in constraints (E.g. Alexander and Baptista, 2004, Krokhmal, portfolio allocation Dynamic portfolio Palmquist and Uryasev, 2002). allocation Conclusion Why? Better risk measure + convex function of portfolio Appendix weights (easier to optimize). 13 / 42 Min CVaR portfolio Outline > library(DEoptim) A primer on risk budgets > obj <- function(w) { VaR budget CVaR budget + if (sum(w) == 0) { w <- w + 1e-2 } Min CVaR portfolio + w <- w / sum(w) CVaR budgets as objective or constraint in + ES(R=indexes[,1:2],weights = w)$MES portfolio allocation Dynamic portfolio + } allocation > out <- DEoptim(fn = obj, lower = rep(0, 2), Conclusion + upper = rep(1, 2), DEoptim.control(itermax=50)) Appendix > wstar <- out$optim$bestmem > wMinCVaR <- wstar / sum(wstar) > print(wMinCVaR) US Bonds US Equities 0.96443348 0.03556652 14 / 42 CVaR budgets Outline > ES(R=indexes[,1:2], weights=wMinCVaR, A primer on risk budgets + portfolio_method="component") VaR budget CVaR budget $MES Min CVaR portfolio [1,] 0.01102894 CVaR budgets as objective or constraint in $contribution portfolio allocation Dynamic portfolio US Bonds US Equities allocation 0.0106366796 0.0003922610 Conclusion $pct_contrib_MES Appendix US Bonds US Equities 0.96443349 0.03556651 15 / 42 CVaR budgets Outline Weight allocation Risk allocation A primer on risk budgets style bond equity bond equity VaR budget CVaR budget 60/40 weight 0.40 0.6 -0.01 1.01 Min CVaR portfolio 60/40 risk alloc 0.84 0.16 0.40 0.60 CVaR budgets as objective or constraint in Min CVaR Conc 0.86 0.14 0.50 0.50 portfolio allocation Min CVaR 0.96 0.04 0.96 0.04 Dynamic portfolio allocation Conclusion Appendix 16 / 42 Outline A primer on risk budgets CVaR budgets as objective or constraint in portfolio allocation CVaR budgets as objective or Risk budget constraints Risk budget objective Efficient Frontier constraint in portfolio Dynamic portfolio allocation allocation Conclusion Appendix 17 / 42 Traditional use Outline ∂CVaR(w) Risk contribution CVaR A primer on risk budgets Ci (w)= wi ∂wi CVaR budgets as objective or constraint in portfolio allocation TM Risk budget constraints Litterman (1999): Hot Spots and hedges. Risk budgets as Risk budget objective ex post instrument to adjust marginally portfolios. Efficient Frontier Dynamic portfolio allocation Keel and Ardia (2009): Conclusion Appendix 1. Only precise for infinitesimal changes, poor approximations for realistic reallocations. 2. Assume changing a single position keeping fixed all other positions >< full investment constraint. 18 / 42 Proposal I: Risk budget constraints Outline Avoid downside risk concentration by: A primer on risk budgets CVaR budgets as CiCVaR objective or constraint in li ≤ %CiCVaR ≡ ≤ ui portfolio allocation CVaR Risk budget constraints Risk budget objective Efficient Frontier Min CVaR portfolio with Dynamic portfolio allocation X 60/40 risk allocation constraint Conclusion X Appendix equal risk allocation constraint. 19 / 42 60/40 risk allocation portfolio Outline > obj <- function(w) { A primer on risk budgets + if (sum(w) == 0) { w <- w + 1e-2 } CVaR budgets as objective or constraint in + w <- w / sum(w) portfolio allocation Risk budget constraints + CVaR <- ES(R=indexes[,1:2],weights = w) Risk budget objective + tmp1 <- CVaR$MES Efficient Frontier Dynamic portfolio + tmp2 <- max(CVaR$pct_contrib_ES allocation + - c(0.405,
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