Motivation Portfolio approaches Javier Estrada - PDF document

Portfolio Optimization (II): Geometric Mean Maximization Javier Estrada ADFIN – Winter/2014 1. The GMM Criterion • Motivation • The Kelly criterion • Estimation 2. Evidence • Data • Expected performance • Observed performance • Simulated performance • Two final thoughts Motivation  Portfolio approaches Javier Estrada  Standard/Traditional IESE Business • Sharpe ratio maximization (SRM) School  Maximization of risk‐adjusted returns (Risk = SD) Barcelona Spain  Many alternatives exist nowadays • HMO, FSO, MSO, … • GMM is one of those many alternatives  Maximization of the growth of the capital invested Maximization of expected terminal wealth  Ultimate question today  What do investors ( you ) really want to maximize? • Risk‐adjusted returns? • Growth of the capital invested (terminal wealth)? ADFIN Winter/2014 1

Motivation Javier Estrada IESE Business School Barcelona Spain  Which portfolio, S or G, is more attractive to you ?  G grows faster and has a higher terminal wealth  G is more volatile and has a lower Sharpe ratio ADFIN Winter/2014 The Kelly Criterion Javier Estrada 200% $300 50% IESE Business $100 E(R) = 50% School 50% Barcelona Spain − 100% $0  Assume that …  this gamble is played a large number of rounds  the results are cumulative  Question  What fixed proportion of capital should a gambler bet on each round if the goal is to maximize his terminal capital? • Is it clear why 0% and 100% are not optimal? ADFIN Winter/2014 2

The Kelly Criterion  Kelly (1956) Javier Estrada  Considers a gambler … IESE Business • that bets a fixed proportion ( F ) of his capital School • over a large number of rounds Barcelona Spain • with cumulative results  Asks what should F be if the goal is to maximize the gambler’s expected terminal wealth  Kelly criterion (Kelly fraction) • F * = K = E/O  E (Edge): Expected value of the gamble  O (Odds): Potential payoff per $1 gambled ADFIN Winter/2014 The Kelly Criterion Javier Estrada 200% $300 50% IESE Business $100 E(R) = 50% School 50% Barcelona Spain − 100% $0  Kelly fraction  K = E/O • E = 50% • O = $2 • K = 0.5/2 = 25%  Betting more than 25% lowers E( W T ) and increases risk  Betting less than 25% lowers E( W T ) and lowers risk ADFIN Winter/2014 3

The Kelly Criterion Javier Estrada 200% $300 50% IESE Business $100 E(R) = 50% School 50% Barcelona Spain − 100% $0  Assume that …  we start with $100  we play this gamble 100 times  results are cumulative  the 200% and –100% returns occur 50‐50  Note that in this setting …  W 100 is fully determined by F  the order of ‘good’ and ‘bad’ returns is irrelevant ADFIN Winter/2014 The Kelly Criterion Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014 4

The Kelly Criterion Javier Estrada IESE Business F = K = 25% → $18,055 School Barcelona F = 15% → $9,629 Spain F = 35% → $5,631 ADFIN Winter/2014 The Kelly Criterion Javier Estrada IESE F = K = 25% → $18,055 Business School F = 15% → $9,629 Barcelona Spain F = 35% → $5,631 ADFIN Winter/2014 5

The Kelly Criterion Javier Estrada IESE F = K = 25% → $18,055 Business F = 15% → $9,629 School F = 35% → $5,631 Barcelona Spain ADFIN Winter/2014 The Kelly Criterion Javier Estrada IESE F = K = 25% → $18,055 Business School F = 15% → $9,629 Barcelona Spain F = 35% → $5,631 ADFIN Winter/2014 6

The Kelly Criterion Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014 The Kelly Criterion  Kelly (1956) spanned a vast gambling literature Javier Estrada  Three interesting results from this literature IESE Business • Terminal wealth is almost certain to be higher than School with any other strategy Barcelona Spain • The bets may be very aggressive • The ride may be very bumpy (volatile)  These results hold in investing applications  When the Kelly criterion is applied to investing …  the goal, the multiperiod framework, and the cumulative nature of results remain • Goal: Max E( W T ) = Max E( GM p )  instead of determining how to split money between a gamble and cash on hand we determine how to split money across different assets ADFIN Winter/2014 7

Estimation  Sharpe ratio maximization (SRM) Javier Estrada IESE Business School Barcelona Spain  Geometric mean maximization (GMM) ADFIN Winter/2014 GMM and Risk  It is essential to note the different role that Javier Estrada volatility plays in SRM and GMM IESE Business  In SRM, volatility is synonymous with risk School Barcelona • Higher volatility ⇒ Lower Sharpe ratio Spain  In GMM, volatility slows down the growth of capital • Higher volatility ⇒ Lower geometric mean  This is called the variance drag  Hence, GMM does not ignore risk • It accounts for it in a different way than SRM does ADFIN Winter/2014 8

Evidence  Data Javier Estrada  Six asset classes IESE Business • US stocks / EAFE stocks / EM stocks School • US bonds / US real estate / Gold Barcelona Spain  Expected performance  Portfolios and characteristics  Observed performance  Return, risk, RAR, and terminal capital  Simulated performance  Return, risk, RAR, and terminal capital  Focus on downside potential ADFIN Winter/2014 Expected Performance Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014 9

Observed Performance Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014 Observed Performance Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014 10

Simulated Performance Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014 Evidence  Main takeaways from these results Javier Estrada  G and S are very different portfolios IESE Business  Relative to the S portfolio, the G portfolio … School • is much more undiversified, volatile, and aggressive Barcelona Spain • grows much faster and provides a much higher W T • does not always underperform in terms of RAR  The downside potential of G is rather limited • Very unlikely to yield large losses at the end of 10‐year holding periods • Not very likely to yield loses anytime during 10‐year holding periods ADFIN Winter/2014 11

Two Final Thoughts  Who should use GMM? Javier Estrada  GMM becomes more attractive … IESE Business • the higher the ability to tolerate risk School  GMM is clearly not for ‘very risk averse’ investors Barcelona Spain • the less frequently the portfolio is evaluated  Makes it less likely to observe losses (and react) • the longer the holding period  As is the case with any ‘risky’ strategy • the more certain the holding period  Unexpected liquidation may occur at a ‘bad’ time ADFIN Winter/2014 Two Final Thoughts Javier Estrada Charlie Munger IESE Business School “If you’re investing for 40 years in Barcelona Spain some pension fund, what difference does it make if the path from start to finish is a little more bumpy or a little different than everybody else’s so long as it’s all going to work out well in the end? So what if there’s a little extra volatility.” ADFIN Winter/2014 12