GM Mamzato Geometrc Mea Mamzato: Epected, Observed, ad Smulated Performace Rafael De Satago & Javer Estrada IESE Busess School 0/16
GM Mamzato Geometrc Mea Mamzato 1. Itroducto 2. Methodology 3. Evdece 4. Assessmet 1/16
1-Itroducto GM Mamzato Portfolo Approaches Stadard Sharpe rato mamzato (SRM) Mamzato of rsk-adjusted returs (Rsk=SD) May alteratves (HMO, FSO, MSO, ) Geometrc mea mamzato (GMM) Mamze growth of the captal vested or Epected Termal Wealth Has a hstory as log as SRM (Both the 1950s) Key Questo What do vestors really wat to mamze: Rsk-adjusted returs? Growth of the captal vested (termal wealth)? 2/16
1-Itroducto GM Mamzato Whch portfolo, S or G, s more attractve? G S G grows faster ad has a hgher termal wealth G s far more volatle ad has a lower Sharpe rato 3/16
2-Methodology GM Mamzato Methodology: Sharpe rato mamzato (SRM) Ma, 2,..., 1 SR p μ p R σ p f 1 1 j 1 μ R f j σ j subject to 1 1 ad 0 for all 4/16
2-Methodology GM Mamzato Methodology: Geometrc mea mamzato (GMM) Ma 1, 2,..., GM p ep l(1 μ p ) σ 2 p 2(1 μ p ) 2 1 ep l(1 1 μ ) 1 j 1 2(1 1 μ j σ ) 2 j 1 subject to 1 ad 0 for all 1 5/16
2-Methodology GM Mamzato The GM ca be epressed as: Igorg quadratc ad hgher order terms, we get GM(R) 2 σ ep l(1 μ ) 2 2(1 μ ) 1 whch s a good appromato of the GM. Remark: Note the role of volatlty GMM 6/16
3-Emprcal GM Mamzato Evdece Evdece Data 6 asset classes (Through Dec/10) US stocks / EAFE stocks / EM stocks US bods / US real estate / Gold I-sample optmzato Portfolo weghts, dversfcato, characterstcs Costraed GMM (G C ) Out-of-sample performace Rsk ad retur characterstcs Smulatos Rsk ad retur, outperformace, dowsde 7/16
3-Emprcal GM Mamzato Evdece Optmal Portfolos ad Epected Performace Dec/2000 Dec/2005 Dec/2010 S G G C S G G C S G G C Pael A: Weghts (%) US stocks 11.9 0.0 5.0 0.0 0.0 0.0 2.0 0.0 0.0 EAFE stocks 43.4 43.2 47.5 17.1 0.0 27.4 0.0 0.0 26.6 EM stocks 21.5 56.8 47.5 30.7 100.0 47.5 22.3 100.0 47.5 US bods 0.0 0.0 0.0 13.9 0.0 0.0 61.7 0.0 0.0 US real estate 23.2 0.0 0.0 37.5 0.0 25.1 5.9 0.0 25.9 Gold 0.0 0.0 0.0 0.7 0.0 0.0 8.1 0.0 0.0 Pael B: Characterstcs 4 2 3 5 1 3 5 1 3 μ p (%) 1.0 1.2 1.1 1.0 1.3 1.2 0.7 1.4 1.1 GM p (%) 1.0 1.0 1.0 0.9 1.1 1.0 0.6 1.1 1.0 σ p (%) 4.0 5.2 5.0 3.5 6.6 4.5 2.0 7.0 5.1 SR p 0.157 0.142 0.146 0.181 0.149 0.175 0.195 0.154 0.166 Aualzed GM p (%) 12.2 13.0 12.9 11.8 14.4 13.3 8.0 14.2 12.6 Aualzed σ p (%) 13.7 18.2 17.2 12.2 23.0 15.7 6.9 24.1 17.7 TV10 ($) 315 340 338 306 385 350 216 377 328 TV20 ($) 994 1,154 1,142 935 1,485 1,224 468 1,425 1,073 TV30 ($) 3,136 3,921 3,858 2,860 5,724 4,284 1,014 5,378 3,516 8/16
3-Emprcal GM Mamzato Evdece Optmal Portfolos ad Observed Performace 9/16
3-Emprcal GM Mamzato Evdece Optmal Portfolos ad Observed Performace S G G C World μ p (%) 0.9 1.2 1.1 0.4 GM p (%) 0.7 1.0 0.9 0.3 σ p (%) 5.6 6.5 6.3 5.1 Σ p (%) 4.0 4.4 4.3 3.7 β p 1.1 1.2 1.2 1.0 M (%) 25.5 25.6 25.0 19.8 Ma (%) 17.5 15.9 15.4 11.9 SR p 0.097 0.131 0.120 0.019 N p 0.221 0.269 0.253 0.118 Aualzed GM p (%) 9.0 12.3 11.2 3.7 Aualzed σ p (%) 19.5 22.4 21.8 17.5 TV ($) 236 319 289 144 10/16
3-Emprcal GM Mamzato Evdece Optmal Portfolos ad Smulated Performace G G S S Ja 2011 Dec 2020 G S S G 11/16
3-Emprcal GM Mamzato Evdece Optmal Portfolos ad Smulated Performace Pael A S G G C Pael B (%) G G C μ p (%) 0.7 1.4 1.1 GM p 82.9 81.1 GM p (%) 0.7 1.1 1.0 SR p 15.8 16.6 σ p (%) 2.9 7.3 6.0 N p 16.8 17.3 Σ p (%) 1.7 4.4 3.6 Pael C ($) S G G C M (%) 7.2 17.5 14.5 Avg 250 501 395 Ma (%) 8.5 20.2 16.7 M 81 19 26 SR p 0.254 0.186 0.190 Ma 2,255 9,306 5,034 N p 0.457 0.322 0.329 Avg Q1 152 141 142 Aualzed GM p (%) 8.8 14.2 12.4 Avg Q10 399 1,099 791 Aualzed σ p (%) 10.0 25.2 20.8 Avg D1 133 94 102 TV ($) 250 501 395 Avg D10 510 1,569 1,081 G C Pael A (%) S G G C Pael B (%) S G Loss > 0% 0.2 5.2 4.2 Loss > 0% 5.8 15.4 13. Loss > 10% 0.1 4.0 2.9 Loss > 10% 0.5 9.9 7. Loss > 20% 0.0 2.8 1.9 Loss > 20% 0.0 5.9 3. Loss > 30% 0.0 1.9 1.1 Loss > 30% 0.0 3.3 1. 12/16
GM 4-Assessmet Mamzato Our Evdece shows that: G ad S are very dfferet portfolos Relatve to S, portfolo G s much more udversfed, volatle, ad aggressve grows much faster ad provdes much hgher W T does ot ecessarly uderperform terms of RAR G s dowsde potetal s rather lmted: Very ulkely to yeld large termal losses over 10 years Not very lkely to yeld large losses aytme over 10 years G C may be a attractve portfolo 13/16
GM 4-Assessmet Mamzato Besdes low rsk-averso prefereces, what codtos make G the more attractve choce? G becomes more attractve the loger the holdg perod the more certa the holdg perod the less relevat short-term fluctuatos are Short-term results are domated by luck. The loger the holdg perod, the lower the mpact of luck, ad the more lkely G s to outperform Ucerta holdg perods magfy the mpact of rsk the more certa the (log) holdg perod, the less lkely t s for the vestor to bear the cost of short-term rsk 14/16
GM 4-Assessmet Mamzato GMM s a attractve alteratve to SRM Plausble uderlyg dea. Smple to mplemet. o Requres the same formato/algorthm as SRM. Performs well emprcally. o Captal grows fast, dowsde potetal s lmted. Partcularly plausble for log-term vestors. 15/16