Are Smart Beta indexes valid for hedge fund portfolio allocation?

Are Smart Beta indexes valid for hedge fund portfolio allocation? Asmerilda Hitaj Giovanni Zambruno University of Milano Bicocca Second Young researchers meeting on BSDEs, Numerics and Finance July 2014 A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 1 / 42

Outline 1 Investor Problem according to Markowitz 2 Portfolio allocation beyond MV 3 Smart Beta (Equity Benchmarks) 4 Shrinkage estimator 5 Empirical analysis 6 Conclusions 7 References A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 2 / 42

Investor Problem according to Markowitz Investor Problem and Markowitz model Markowitz (1952) formulated the portfolio problem as a trade-off between mean and variance of portfolios. Maximize µ p for a given σ p maxµ p = N i=1 w i R i s.t. w Σ w = c N i=1 w i = 1. Minimize σ p for a given µ p minσ p = w Σ w s.t. µ p = c N i=1 w i = 1. A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 3 / 42

Investor Problem according to Markowitz Investor problem cont... Markowitz (1959) also proposed the semivariance as a measure of risk. Athyde and Flores (2002) constructed the efficient frontier based on the first four moments of the portfolio return distribution. Jondeau et al. (2007), Martellini and Ziemann (2010), Hitaj et al.(2012) etc, used the EU to introduce higher moments in portfolio allocation. Davies et al. (2009) adopted a multi-objective optimization problem to introduce higher moments in portfolio allocation. Kahneman and Tversky (1979, 1992) developed the Prospect theory which incorporates real human decision patterns and psychology, in explaining how individuals make economic choices. In this work we consider two approaches for introducing higher moments into portfolio allocation. A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 4 / 42

Portfolio allocation beyond MV Expected Utility Theory (EUT) According to EUT the objective of the investor is to maximize his expected utility, under certain constraints: max EU( X) w s.t. N w i = 1 i=1 0 w i 1. In the empirical part we consider negative exponential utility function: U( X) λ X = e A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 5 / 42

Portfolio allocation beyond MV EUT cont.. In order to account for higher moments in portfolio allocation the Taylor expansion of the utility function up to the fourth order is used. The investor problem can be written as: ] max EU( X) = e λ(µw ) [1+ λ2 w 2 wm 2w λ3 6 wm 3(w w ) s.t. N w i = 1 i=1 0 w i 1. A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 6 / 42

Portfolio allocation beyond MV Multi Objective Portfolio Optimization Davies et al. (2009) used the multi objective approach to introduce higher moments to portfolio allocation. Their approach is based on a two step procedure: First step: solves separately each single optimization problem, that is maximize the mean and skewness and minimize the variance: P 1 max w w R = µ P s.t. N i=1 w i = 1 0 w i i. P 2 min w w M 2 w = ( σp 2 ) s.t. N i=1 w i = 1 0 w i i. A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 7 / 42

Portfolio allocation beyond MV Multi objective approach cont... P 3 max w w M 3 (w w) = S P s.t. N i=1 w i = 1 0 w i i. Solving these three problems separately we find the aspiration levels of the investor for the mean, variance and skewness (µ P, ( σ 2 P), S P ) A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 8 / 42

Portfolio allocation beyond MV Multi objective approach cont... Second step: construct a multi objective problem (MO), where the portfolio allocation decision is given by the solution of the MO that minimizes the Minkowski-like distance from the aspiration levels, namely: _ γ min Z = µ P w R 1 w w µ + M 2w (σp) 2 γ 2 i P (σp) 2 + S P w M 3 (w w) S γ3 P s.t. N i=1 w i = 1 0 w i i where γ 1, γ 2 and γ 3 represent the investor s subjective parameters. A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 9 / 42

Smart Beta (Equity Benchmarks) Equally weighted and Global Minimum Variance Equally weighted Equally weighted strategy consists in holding a portfolio with weight 1/N in each component, where N is the number of assets. This strategy does not involve any optimization or estimation procedure. Global Minimum Variance GMV portfolio is the one that has the minimum variance in absolute, without taking into account the expected return of the portfolio. { min w σ2 P = w M 2 w N s.t. i=1 w i = 1, 0 w i i, A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 10 / 42

Smart Beta (Equity Benchmarks) Equal Risk Contribution (ERC) Qian (2006) proposed the ERC, where weights are such that each asset has the same contribution to portfolio risk. Maillard et al. (2010) analyzed the properties of an unconstrained analytic solution of the ERC. The marginal risk contribution of asset i is defined as: wi σ P = σ P = w iσi 2 + i j w jσ ij. w i σ P The total risk contribution of the i th asset is: σ i (w) = w i wi σ P. The portfolio risk can be seen as the sum of total risk contributions: σ P = N σ i (w). i A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 11 / 42

ERC cont... Smart Beta (Equity Benchmarks) A characteristic of this strategy is that: w i wi σ P = w j wj σ P i, j The investor problem in this case is: N N min w i=1 j=1 (w i(m 2 w) i w j (M 2 w) j ) 2 s.t. N i=1 w i = 1 0 w i i, A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 12 / 42

Smart Beta (Equity Benchmarks) Maximum Diversified Portfolio The objective of this strategy is to construct a portfolio that maximizes the benefits from diversification (see Choueifaty and Coignard (2008)). Where the diversification ratio is defined as: DR = N i=1 w iσ i w M 2 w The investor problem in the MDP context is used is: N max DR = i=1 w iσ i w i w M 2 w s.t. N i=1 w i = 1 0 w i i. De Miguel et. al (2009), Optimal Versus Naive Diversification: How inefficient is the 1 N Portfolio Strategy? A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 13 / 42

Shrinkage estimator Shrinkage estimator Estimation of moments and comoments is needed for portfolio allocation. In the empirical part we use the shrinkage estimators (see for e.g. Ledoit and Wolf (2003) and Martellini & Ziemann(2010): For the mean we use the shrinkage toward the Grand Mean. µ shrink = (1 φ) µ +φ µ target (0 φ 1) where (Jorion 86): ( ( )) φ = min 1,max 0, 1 (N 2) T ( µ µ target ) Σ 1 ( µ µ target ), and: µ target = 1 Σ 1 µ 1 Σ 1 1 A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 14 / 42

Shrinkage estimator Shrinkage estimator cont... For mean, variance and skewness we use the shrinkage towards the constant correlation approach, Elton and Gruber (1973). M shrink = (1 φ) M +φ M CC A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 15 / 42

Empirical analysis Empirical analysis and Statistics We consider 4 hedge funds portfolio and 1 equity portfolio. The statistics of each index in portfolio are: Dow Jones Credit Suisse Hedge Funds indexes Period under consideration Jan/1994 to Dec/2011 General statistics for each component in HFP 1 portfolio Annual Mean Annual STD Skewness Kurtosis JB-test Hedge Fund Index 0.084 0.076-0.327 5.437 57.295 Convertible Arbitrage H. F. 0.072 0.072-2.997 21.048 3254.979 Dedicated Short Bias H. F. -0.035 0.168 0.449 3.745 12.252 Emerging Markets H. F. 0.071 0.152-1.201 9.714 457.581 Event Driven H. F. 0.088 0.065-2.451 15.221 1560.520 Event Driven Distressed H. F. 0.097 0.068-2.388 15.579 1629.257 Event Driven Multi-Strategy H. F. 0.083 0.070-1.944 11.603 802.213 Event Driven Risk Arbitrage H. F. 0.065 0.042-1.097 8.028 270.850 Fixed Income Arbitrage H. F. 0.051 0.060-4.700 36.613 10963.463 Global Macro H. F. 0.114 0.097-0.246 6.888 138.245 Long/Short Equity H. F. 0.088 0.099-0.218 6.205 94.174 Managed Futures H. F. 0.058 0.117-0.079 2.979 0.227 Table: General statistics for HFP 1 A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 16 / 42

Empirical analysis Statistics for portfolio HFP 2 Period under consideration Jan/1997 to Jan/2011 General statistics for each component in HFP 2 portfolio Annual Mean Annual STD Skewness Kurtosis JB-test Convertible Arbitrage 0.079 0.065-2.626 20.019 2550.845 CTA Global 0.066 0.085 0.151 2.795 1.071 Distressed Securities 0.098 0.063-1.504 8.296 298.244 Emerging Markets 0.092 0.126-1.218 8.202 265.339 Equity Market Neutral 0.064 0.030-2.396 18.066 2010.055 Event Driven 0.089 0.062-1.561 8.228 298.111 Fixed Income Arbitrage 0.059 0.045-3.770 25.089 4380.870 Global Macro 0.082 0.056 0.851 4.984 54.942 Long/Short Equity 0.085 0.076-0.408 4.042 14.080 Merger Arbitrage 0.074 0.036-1.493 8.669 330.163 Relative Value 0.079 0.044-1.955 11.523 707.179 Short Selling 0.014 0.181 0.644 5.384 59.057 Funds Of Funds 0.061 0.059-0.410 6.418 99.369 Table: General statistics for HFP 2 A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 17 / 42

Empirical analysis Statistics for portfolio HFP 3 Period under consideration Jan/1990 to Sep/2013 General statistics for each component in HFP3 portfolio Annual Mean Annual STD Skewness Kurtosis JB-test HFRI ED: Distressed/Restructuring 0.116 0.065-1.022 7.795 322.669 HFRI ED: Merger Arbitrage 0.082 0.040-2.065 11.777 1117.400 HFRI EH: Equity Market Neutral 0.066 0.032-0.251 4.570 32.254 HFRI EH: Quantitative Directional 0.121 0.128-0.432 3.818 16.805 HFRI EH: Short Bias 0.005 0.185 0.248 5.261 63.629 HFRI Emerging Markets (Total) 0.125 0.141-0.832 6.604 187.096 HFRI Emerging Markets: Asia ex-japan 0.098 0.135-0.092 3.813 8.240 HFRI Equity Hedge (Total) 0.124 0.091-0.253 4.789 41.021 HFRI Event-Driven (Total) 0.112 0.068-1.294 6.973 266.972 HFRI FOF: Conservative 0.062 0.039-1.691 10.497 803.204 HFRI FOF: Diversified 0.068 0.059-0.446 7.020 201.339 HFRI FOF: Market Defensive 0.073 0.058 0.235 3.891 12.042 Table: General statistics for HFP 3 A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 18 / 42

Empirical analysis Statistics for portfolio HFP 4 Period under consideration Jan/1990 to Sep/2013 General statistics for each component in HFP4 portfolio Annual Mean Annual STD Skewness Kurtosis JB-test HFRI FOF: Strategic Index 0.095 0.086-0.463 6.393 146.916 HFRI FOF Composite Index 0.072 0.058-0.666 6.869 198.828 HFRI Fund Weighted Composite Index 0.106 0.069-0.681 5.425 91.828 HFRI Fund Weighted Composite Index CHF 0.093 0.070-0.711 5.342 89.181 HFRI Fund Weighted Composite Index GBP 0.122 0.070-0.698 5.364 89.498 HFRI Fund Weighted Composite Index JPY 0.082 0.069-0.671 5.108 74.185 HFRI Macro (Total) Index 0.113 0.075 0.559 3.972 26.069 HFRI Macro: Systematic Diversified Index 0.102 0.075 0.150 2.659 2.460 HFRI Relative Value (Total) Index 0.098 0.044-2.112 16.381 2338.090 HFRI RV: Fixed Income-Convertible Arbitrage Index 0.085 0.066-3.049 31.509 10093.109 HFRI RV: Fixed Income-Corporate Index 0.078 0.065-1.336 10.937 832.768 HFRI RV: Multi-Strategy Index 0.082 0.044-2.063 16.151 2255.866 Table: General statistics for HFP 4 A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 19 / 42

Empirical analysis Statistics for Equity portfolio taken from S&P 500 Period under consideration F eb/1985 to May/2013 General statistics for each component in Equity portfolio Annual Mean Annual STD Skewness Kurtosis JB-test AMD UN Equity -0.052 0.645-0.200 3.566 6.820 APA UN Equity 0.102 0.347-0.240 3.658 9.383 CMCSA UW Equity 0.129 0.310-0.186 3.383 4.037 ED UN Equity 0.048 0.174-0.244 3.703 10.387 FDX UN Equity 0.086 0.298 0.077 3.357 2.141 GIS UN Equity 0.099 0.189-0.072 3.304 1.605 JNJ UN Equity 0.125 0.203-0.181 3.598 6.911 L UN Equity 0.091 0.255-0.270 3.600 9.240 NUE UN Equity 0.119 0.340-0.225 3.608 8.098 PAYX UW Equity 0.183 0.299-0.056 3.357 1.981 PFE UN Equity 0.099 0.244-0.259 3.656 9.886 SO UN Equity 0.073 0.171-0.076 3.794 9.256 LUV UN Equity 0.090 0.331-0.109 3.140 0.952 WAG UN Equity 0.121 0.266-0.059 3.611 5.487 Table: General statistics for Equity A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 20 / 42

Empirical analysis Portfolio allocation procedure A static portfolio allocation procedure is used: Different rolling-window lengths are considered: 24 months in-sample and 3 months out of sample. 24 months in-sample and 6 months out of sample. 48 months in-sample and 3 months out of sample. 48 months in-sample and 6 months out of sample. For each portfolio we have calculated the in-sample optimal weights using the different approaches discussed: Smart Beta strategies: EW, MDP, ERC and GMV. Multi Objective approach using 2 and 3 moments, varying γ i = 1,...,5. Therefore in case of MO approach with two moments we have 25 portfolios and when three moments are considered we have 125 portfolios. EUT with 2 and 3 moments with λ = 1,...,30. A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 21 / 42

Empirical analysis Portfolio allocation procedure (cont...) Once the in-sample weights are calculated we keep them constant for the next out of-sample period and calculate the out-of sample returns obtained with the different strategies. From the out-of-sample returns obtained with each strategy we calculate: Sharpe Ratio, with R f = 2% annual Sh R = R P R f σ P Excess Return on VaR Ratio, with R f = 2% annual and α = 5%. EVaR R = R P R f VaR P (α) Information Ratio, where the Smart Beta obtained indexes are used as benchmark. I R = R P R B σ(r P R B ). A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 22 / 42

Empirical analysis To represent the concentration (lack of diversification) of each portfolio we calculate the modified Herfindahl index. H I = N i=1 w2 i 1 N 1 1 N. A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 23 / 42

Empirical analysis Results for HFP 1, using MO approach (24-3) 0.5 124 Sharpe PGP 24 3 222 245 325 533 Sharpe Ratio 0.45 0.4 0.35 0.3 EW ERC 12 13 14 15 11 22 23 24 25 21 33 32 34 35 31 43 44 45 42 41 53 54 55 52 51 113 115 112 111 114 125 122 121 123 132 131 133 134 135 142 141 143 144 145 154 155 151 152 153 215 214 213 212 211 225 224 223 221 232 233 235 231 234 242 241 243 244 252 251 253 254 255 313 315 311 312 314 322 323 324 321 333 334 331 332 335 344 341 342 343 345 353 351 355 352 354 414 413 415 421 422 425 423 424 411 412 432 433 435 431 434 442 444 441 443 445 452 453 451 454 455 514 515 512 513 511 523 522 521 524 525 532 535 534 531 542 543 545 541 544 552 553 551 555 554 0.25 GMV MDP MV MVS 0.2 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 24 / 42

Empirical analysis MO for HFP 1, EVaR with R f = 2% annual and α = 5% EVaR 0.16 0.15 0.14 0.13 0.12 0.11 0.1 0.09 0.08 0.07 EW ERC GMV MDP 12 13 14 15 11 22 23 24 25 21 33 34 32 35 31 43 44 45 42 41 53 54 55 52 51 113 115 121 112 111 114 124 122 125 123 132 131 133 134 135 142 141 145 143 144 154 151 155 152 153 215 213 214 212 211 222 225 223 224 221 EVaR 24 3 232 233 231 235 234 245 242 241 243 244 0.06 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 252 251 253 254 255 313 315 311 312 314 325 322 324 323 321 333 334 331 335 332 344 341 342 343 345 353 351 355 352 354 414 413 415 421 411 412 425 422 423 424 MV MVS 432 433 431 435 434 442 444 441 443 445 452 453 451 454 455 514 515 512 513 511 523 522 521 524 525 533 532 535 531 534 552 555 542 553 543 545 551 541 544 554 A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 25 / 42

Empirical analysis MO for HFP 1, Modified Herfindhal index 0.7 Modified Herfindahl 24 3 Modified Herfindahl 0.6 0.5 0.4 0.3 0.2 0.1 13 14 15 GMV 12 11 MDP 23 24 25 21 22 33 34 35 31 32 41 42 43 44 45 51 52 53 54 55 113 114 112 115 111 123 124 125 122 121 132 133 134 135 131 142 143 144 152 153 154 155 145 141 151 213 214 215 212 211 223 224 225 222 221 232 233 234 235 231 242 243 244 245 241 252 253 254 255 251 313 312 314 315 311 322 323 324 325 321 332 333 334 335 331 342 343 352 353 344 345 354 355 351 341 413 414 415 412 411 MV MVS 422 423 424 432 433 434 435 425 431 421 442 443 444 445 441 451 452 453 454 455 513 514 512 515 511 522 523 524 525 521 532 533 534 535 531 542 543 544 541 545 552 553 554 551 555 ERC 0 EW 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 26 / 42

Empirical analysis MO for HFP 1, IR using as benchmark EW 0.15 Information Ratio with EW as benchmark 24 3 Information Ratio 0.1 0.05 0 0.05 0.1 0.15 15 14 13 12 11 25 24 23 22 21 35 34 33 32 31 45 44 43 42 41 55 54 53 52 51 115 113 114 112 111 124 125 122 123 121 135 132 133 131 134 142 141 143 144 145 154 155 151 152 153 215 214 213 212 211 222 225 224 223 221 235 232 233 231 234 245 242 243 244 241 252 251 253 254 255 313 315 314 312 311 325 323 324 322 321 333 334 332 331 335 344 341 342 343 345 353 351 352 354 355 414 415 413 412 411 422 423 424 421 425 MV MVS 442 444 432 433 435 441 443 445 431 434 452 453 451 454 455 514 515 513 512 511 523 524 522 525 521 533 532 535 534 531 542 543 541 544 545 552 551 553 554 555 0.2 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 27 / 42

Empirical analysis MO for HFP 1, IR using as benchmark ERC 0.25 Information Ratio with ERC as benchmark 24 3 245 Modified Herfindahl 0.2 0.15 0.1 0.05 0 0.05 15 14 13 12 11 25 24 23 22 21 35 34 33 32 31 45 44 43 42 41 55 54 53 52 51 123 113 115 121 114 112 111 124 125 122 132 131 133 135 134 142 141 143 144 145 154 155 151 152 153 215 214 213 212 211 222 225 224 223 221 235 232 233 231 234 242 243 244 241 0.1 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 252 251 253 254 255 313 315 314 312 311 325 323 322 324 321 333 334 331 332 335 341 344 342 343 345 351 353 352 354 355 414 413 415 412 411 422 423 424 421 425 MV MVS 432 433 435 431 434 442 441 444 443 445 452 451 453 454 455 514 515 513 512 511 523 522 524 525 521 533 532 535 534 531 542 543 541 544 545 551 552 553 554 555 A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 28 / 42

Empirical analysis MO for HFP 1, IR using as benchmark MDP 0.35 Information Ratio with MDP as benchmark 24 3 0.3 245 Information Ratio 0.25 0.2 0.15 0.1 0.05 13 14 15 12 11 23 24 25 22 21 34 35 33 32 31 45 44 43 42 41 55 54 53 52 51 113 115 121 123 112 114 111 124 132 125 122 131 133 135 134 142 141 143 144 145 154 155 151 152 153 215 214 213 212 211 222 235 232 225 233 231 224 223 221 234 242 241 243 244 252 251 253 254 255 313 315 321 314 312 311 325 322 323 324 333 334 331 332 335 341 344 342 343 345 351 353 352 355 354 414 413 415 421 422 423 424 412 411 425 MV MVS 432 433 431 435 434 442 441 444 443 445 452 451 453 454 455 514 515 512 513 511 523 533 532 542 543 522 535 541 524 521 525 531 534 545 544 551 552 553 554 555 0 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 29 / 42

Empirical analysis MO for HFP 1, IR using as benchmark GMV Information Ratio 0.4 0.35 0.3 0.25 0.2 0.15 12 13 11 14 15 22 23 21 24 25 32 33 31 34 35 42 43 53 44 54 41 52 55 45 51 Information Ratio with GMV as benchmark 24 3 113 115 111 112 114 122 124 125 121 123 132 131 133 134 135 142 141 143 144 145 154 151 155 152 153 214 213 215 212 211 225 222 223 224 221 232 231 233 235 234 252 242 245 251 241 243 244 253 254 255 313 311 315 312 314 322 323 324 325 321 331 333 332 334 335 341 342 343 344 345 351 353 352 354 355 413 414 415 411 412 421 422 423 424 425 432 431 433 435 434 441 442 444 443 445 451 452 453 454 455 514 512 515 513 511 522 523 521 524 525 532 533 535 531 534 542 543 541 544 545 551 552 553 554 555 0.1 0.05 MV MVS 0 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 30 / 42

Empirical analysis EUT for HFP 1, SR with R f = 2% on annual. 0.38 Sharpe PGP 24 3 Sharpe Ratio 0.36 0.34 0.32 0.3 0.28 EW ERC GMV 9 10 10 11 11 15 14 15 13 14 12 13 12 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 0.26 0.24 1 MDP 2 3 4 5 6 7 8 MV MVS 0.22 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930 A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 31 / 42

Empirical analysis EUT for HFP 1, EVaR with R f = 2% andα = 5% on annual. 0.25 EVaR 24 3 EVaR 0.24 0.23 0.22 0.21 0.2 0.19 EW ERC 1 10 11 14 13 14 12 12 13 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 0.18 0.17 0.16 2 GMV MDP 3 4 5 6 7 8 9 MV MVS 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930 A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 32 / 42

Empirical analysis EUT for HFP 1, Modified Herfindahl. 1 Modified Herfindahl 24 3 1 0.8 2 3 4 5 Modified Herfindahl 0.6 0.4 GMV 6 7 8 9 10 11 12 13 14 15 16 17 17 18 19 18 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 0.2 MDP MV MVS 0 ERC EW 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930 A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 33 / 42

Empirical analysis EUT for HFP 1, IR using as benchmark EW 0.16 Information Ratio with EW as benchmark 24 3 Information Ratio 0.14 0.12 0.1 0.08 0.06 1 2 3 4 5 6 7 8 9 10 10 11 12 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 0.04 0.02 MV MVS 0 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930 A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 34 / 42

Empirical analysis EUT for HFP 1, IR using as benchmark MDP 0.35 Information Ratio with MDP as benchmark 24 3 0.3 Information Ratio 0.25 0.2 0.15 0.1 0.05 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 15 16 17 17 18 18 19 19 27 28 29 30 25 26 24 26 27 28 29 30 23 20 21 22 24 25 23 22 21 20 MV MVS 0 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930 A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 35 / 42

Empirical analysis EUT for HFP 1, IR using as benchmark GMV 0.35 Information Ratio with GMV as benchmark 24 3 Information Ratio 0.3 0.25 0.2 0.15 0.1 0.05 1 2 3 4 5 6 7 8 9 10 15 14 13 12 13 14 15 11 11 12 16 16 17 17 18 18 19 19 20 20 21 21 22 22 MV MVS 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 0 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930 A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 36 / 42

Conclusions Conclusions Multi Objective approach For all the considered portfolios and independently from the rolling window strategy including higher moments decrease the portfolio diversification. Equity portfolio Independently from the rolling window strategy, using the Smart Beta indexes is almost always better than the MV or MVS strategies. HF portfolios There is no clear response to whether higher moments are better than Smart Beta indexes, this is probably due to the algorithm used in the optimization procedure. We are still working on this problem. A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 37 / 42

Conclusions (cont...) Conclusions Expected Utility approach For Equity and Hedge Fund the portfolio is highly concentrated for low levels of risk aversion. For Equity portfolio, Higher moments are better than Smart Beta indexes when the in-sample-period is short. In this case almost always introducing higher moments gives better results for low values of risk averse parameter λ. For Equity portfolio, Smart Beta indexes are better than Higher moments when the in-sample-period is long. In this case the higher the risk averse parameter λ the lower is the information ratio with respect to Smart Beta indexes. For HF portfolio,higher moments are always better than Smart Beta indexes independently from the rolling window strategy, but the information ratio obtained when short in-sample period is used is higher than that obtained in case of long-in-sample period. A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 38 / 42

Conclusions Thank You! A. Hitaj and G. Zambruno (Unimib) Smart Beta and Hedge Funds portfolios 2014 39 / 42

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