Motivation Portfolio approaches Javier Estrada - - PDF document

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Motivation Portfolio approaches Javier Estrada - - PDF document

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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

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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

Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

Motivation

  • Portfolio approaches
  • Standard/Traditional
  • Sharpe ratio maximization (SRM)

 Maximization of risk‐adjusted returns (Risk = SD)

  • 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)?
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Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

Motivation

  • 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

Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

The Kelly Criterion

  • 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?

$100

200% −100% $300 $0

50% 50%

E(R) = 50%

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Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

The Kelly Criterion

  • Kelly (1956)
  • Considers a gambler …
  • that bets a fixed proportion (F) of his capital
  • over a large number of rounds
  • 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

Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

The Kelly Criterion

  • Kelly fraction
  • K = E/O
  • E = 50%
  • O = $2
  • K = 0.5/2 = 25%

 Betting more than 25% lowers E(WT) and increases risk  Betting less than 25% lowers E(WT) and lowers risk

$100

200% −100% $300 $0

50% 50%

E(R) = 50%

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Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

The Kelly Criterion

  • 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 …
  • W100 is fully determined by F
  • the order of ‘good’ and ‘bad’ returns is irrelevant

$100

200% −100% $300 $0

50% 50%

E(R) = 50%

Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

The Kelly Criterion

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Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

The Kelly Criterion

F = K = 25% → $18,055 F = 15% → $9,629 F = 35% → $5,631

Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

The Kelly Criterion

F = K = 25% → $18,055 F = 15% → $9,629 F = 35% → $5,631

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Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

The Kelly Criterion

F = K = 25% → $18,055 F = 15% → $9,629 F = 35% → $5,631

Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

The Kelly Criterion

F = K = 25% → $18,055 F = 15% → $9,629 F = 35% → $5,631

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Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

The Kelly Criterion

Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

  • Kelly (1956) spanned a vast gambling literature
  • Three interesting results from this literature
  • Terminal wealth is almost certain to be higher than

with any other strategy

  • 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(WT) = Max E(GMp)
  • instead of determining how to split money between

a gamble and cash on hand we determine how to split money across different assets

The Kelly Criterion

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Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

  • Sharpe ratio maximization (SRM)

Estimation

  • Geometric mean maximization (GMM)

Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

GMM and Risk

  • It is essential to note the different role that

volatility plays in SRM and GMM

  • In SRM, volatility is synonymous with risk
  • Higher volatility ⇒ Lower Sharpe ratio
  • 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
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Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

Evidence

  • Data
  • Six asset classes
  • US stocks / EAFE stocks / EM stocks
  • US bonds / US real estate / Gold
  • 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

Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

Expected Performance

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Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

Observed Performance

Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

Observed Performance

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Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

Simulated Performance

Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

Evidence

  • Main takeaways from these results
  • G and S are very different portfolios
  • Relative to the S portfolio, the G portfolio …
  • is much more undiversified, volatile, and aggressive
  • grows much faster and provides a much higher WT
  • 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

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Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

Two Final Thoughts

  • Who should use GMM?
  • GMM becomes more attractive …
  • the higher the ability to tolerate risk

 GMM is clearly not for ‘very risk averse’ investors

  • 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

Javier Estrada IESE Business School Barcelona Spain ADFIN Winter/2014

Two Final Thoughts

“If you’re investing for 40 years in 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.”

Charlie Munger