Geometric Mean Maximization: Expected, Observed, and Simulated Performance

Transcription

1 GM Mamzato Geometrc Mea Mamzato: Epected, Observed, ad Smulated Performace Rafael De Satago & Javer Estrada IESE Busess School 0/16

2 GM Mamzato Geometrc Mea Mamzato 1. Itroducto 2. Methodology 3. Evdece 4. Assessmet 1/16

3 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

4 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

5 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

6 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

7 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

8 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

9 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 EAFE stocks EM stocks US bods US real estate Gold Pael B: Characterstcs μ p (%) GM p (%) σ p (%) SR p Aualzed GM p (%) Aualzed σ p (%) TV10 ($) TV20 ($) 994 1,154 1, ,485 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

10 3-Emprcal GM Mamzato Evdece Optmal Portfolos ad Observed Performace 9/16

11 3-Emprcal GM Mamzato Evdece Optmal Portfolos ad Observed Performace S G G C World μ p (%) GM p (%) σ p (%) Σ p (%) β p M (%) Ma (%) SR p N p Aualzed GM p (%) Aualzed σ p (%) TV ($) /16

12 3-Emprcal GM Mamzato Evdece Optmal Portfolos ad Smulated Performace G G S S Ja 2011 Dec 2020 G S S G 11/16

13 3-Emprcal GM Mamzato Evdece Optmal Portfolos ad Smulated Performace Pael A S G G C Pael B (%) G G C μ p (%) GM p GM p (%) SR p σ p (%) N p Σ p (%) Pael C ($) S G G C M (%) Avg Ma (%) M SR p Ma 2,255 9,306 5,034 N p Avg Q Aualzed GM p (%) Avg Q , Aualzed σ p (%) Avg D TV ($) Avg D ,569 1,081 G C Pael A (%) S G G C Pael B (%) S G Loss > 0% Loss > 0% Loss > 10% Loss > 10% Loss > 20% Loss > 20% Loss > 30% Loss > 30% /16

14 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

15 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

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