Kelly and Logarithmic Utility

You talk a lot about utility functions and investor objectives in asset allocation books (see Meucci's Risk and Asset Allocation).  Log utility is defined by the function: U(w)=ln^(-w), where w = wealth.  A good mathematical betting scheme that came to finance as a byproduct of information theory.  John Kelly's formula has been used by Bill Gross at PIMCO and Ed Thorp at Princeton/Newport Partners, to name a few.  In its simplest form: Pct of Capital to Allocate=P - (1-P)/W
    where P = Pr(Win)
              W= E[Win]/E[Loss]

Assumptions: 1) E[Win] and E[Loss] are known (unrealistic)
                     2)Pr(win) is known (also unrealistic)

Still, according to the law of large numbers, if you have lots and lots of trades or investments to sample from, you can use statistics to come up with these values.  Further, if you can enhance anything Bayesian forecasting (or updating), you can further enhance your knowledge for E[Win],E[Loss], and Pr(Win).

Here is a Matlab function with three input parameters to calculate an equity curve:

function [equity] = Kelly(p,PF,initialEquity)
%% Kelly criteria for logartihmic utiliity

optimalF=(p-((1-p)/PF));
equity=zeros(1,1000);
equity(1)=initialEquity;
  
for i=2:1000
    equity(i) = equity(i-1)+ (randn*optimalF);
end

And here is a script that generates 16 equity curves for different E[Win]/E[Loss] ratios and Probabilities for winning.
initialEquity=100000;
p=.35:.10:.65;
PF=.5:.5:2.0;

equityCurves=zeros(length(p)*length(PF),1000);

i=0;
for b=1:4
    for a=1:4
        i=i+1;
        equityCurves(i,:)=Kelly(p(a),PF(b),initialEquity)
        subplot(4,4,i);plot(equityCurves(i,:))
    end
end
   

Lastly, a graph of the plots:

On Process and Implementation:

A friend of mine’s brother co-founded the firm AQR (Applied Quantitative Research) Capital.  This firm is one of the more successful quant funds in the world with $30 Billion AUM.  I recently watched an interview with one of the other co-founders, and most outspoken, Clifford Asness.  Ironically, Dr. Asness’ dissertation advisor was Nobel-Prize winning Eugene Fama, best known for the Efficient Market Hypothesis(EMH).  Asness wrote his dissertation on empirical evidence of out-performance by momentum and value-based strategies in equity markets.  The interview can be seen here:

http://www.youtube.com/watch?v=PcpLuQxidN0

This got me thinking as to what makes a good quant fund, a good hedge fund, a good wealth management firms, and more broadly speaking, what makes a good company.  I have been lucky enough to have several good role models in my life which have imparted some of their wisdom onto me.  Watching the above linked interview with Asness reinforces the ideas of what it takes to run a successful money management firm (or business).  I believe I have identified 6 steps which are the keys to any successful hedge fund, money management firm, or service-oriented business:

1. Reliance on process, and not results. 

Asness talks about how, following the Liquidity Crunch of 2007 (see Khanandi and Lo, 2007), he re-examined his investment processes and models and made an accurate, well-informed, and unemotional decision as to whether those models and processes were still valid.  You can not control how markets behave (in the case of a hedge fund) or how will your customers behave (in the case of a business).  In short, worry about the things you can control and not the things you can’t.

2. Continuous and Never Ending Improvement (CANI)

Edward Deming called this the Continuous Improvement Process or Kaizen. The drive to succeed and bring forth superior risk-adjusted returns to your investors should cause one to continually refine and improve ALL processes.  You should constantly be: focusing on improving your overall investment strategies and asset allocation techniques; reducing commissions via algorithms or leveraging your prime broker relationship; building a strong back-office system;  and maintaining compliance and regulations in order to act in accordance with your investment prospectus. 

3. Communication with your clients and a customer-service orientation

Asness talks about getting on a plane and going to talk to customers after a year of rocky performance.  He explains to customers why, despite some bumps, he believes in his process and why it will continue to work in the future.

One good example of this customer-service orientation is a private wealth management firm by the name of Trust Point in my hometown of LaCrosse, WI with a satellite office in Minneapoolis, MN.  I have two friends that work at this firm and the President told me that their secretary that was memorizing all of their clients by their voices so she could call them by their first name when they phoned.  This is just a minor example of one way in which a once down-trodden investment firm was able to reinvent itself as a process and customer-oriented private wealth powerhouse in the middle of small-town USA. 

4. Willingness to fly in the face of conventional wisdom and research all possible ways of providing returns

This hits very close to home with Asness.  His mentor, Dr. Fama, created the Efficient Market Hypothesis.  Although Asness has tremendous respect for Dr. Fama, he still went as far as doing his doctorate dissertation on an idea that directly contradicts efficient markets.

5. Willingness to forego short-term gain in the interest of long-term success and doing what is “right”

Despite slumping performance, Asness sticks to his process and models, believing that despite abandoning them for short-term gains, eventually they will come back into favor.

Another local example I have is the Marc Orgel Investment Group, located in Eau Claire, WI. Marc has grown his AUM to over $3 Billion and ranked top 50 Financial Advisor in the country by Barron’s.  Post ’87 crash, Marc did a regression analysis on just how much alpha the average money manager adds.  His study concluded that 50 basis points (bps) was all the alpha that the average money manager provided, and subsequently has only charged that amount to his clients, an amount substantially lower than most advisors.  Orgel also goes as far as never taking a prospective client’s money after one meeting and frequently telling clients that they need to put more money into their child’s 529 instead of investing with him.

6. An absolute passion and love for what you are doing

Lastly, Asness talks about his love for finance and that running money was really an after-thought to being a finance professor.  He questions the 17 year-old that just wants to manage a hedge fund for the money.  Without a real passion for the industry, no matter what that industry may be, it is unlikely that you will be able to comply with the first 5 steps. 

Hedge Funds substantially outperform market index

Over the past ten years, as of Dec. 31, 2010, the S&P has generated a cumulative return of -3.75%. During this same period, the average hedge fund has generated cumulative returns of +92.35% (*figures provided by Hennessee Group). The reason for this is simple. Due to the compensation structure within hedge funds, some of the brightest minds in the world, and certainly many of the brightest minds within the financial industry have gravitated away from banks, mutual funds, investment banks to either join hedge funds or launch hedge funds. The Volcker Rule is certainly helping to expedite that brain-drain.
Please see article:

http://mail.courtesycorporation.com/exchweb/bin/redir.asp?URL=https://village.albourne.com/user/news.pcg?id=77784%26f=d

It's Pete's world, we just live in it...

Morgan Stanley’s Quant prop trading outfit is titled Process Driven Trading.  PDT (soon to be PDT Advisors) is being spun out of the firm by the end of 2012 due to the Volcker Rule.  Peter Muller is not a really well-known name in finance outside of Wall Street’s quant community.  He graduated from Princeton in Mathematics in 1985, worked at BARRA in Berkeley for 7 years where he developed an alpha model, and has since been at Morgan Stanley’s PDT, which he started.  It is estimated that he and his small group has pulled $4 billion in profits, after a 20% haircut for the group, over the past 20 years.  Pete was also smart enough to figure out his models weren’t working in August 2007 and pulled the plug early to limit losses.  This ability to recognize that a model is just a model, prevents Pete from being the brunt of the old quant joke:  “Well how is your fund doing?” To which the other quant says, “We were just killing it before that 27 standard deviation day.” 

I do no know what his strategies entail, but I can only guess that he uses or has used some factor based models in coupling with an optimization algorithm.  Perhaps some sort of CVaR Portfolio Optimization with constraints in real-time? 

Pete is also an accomplished musician, surfer, poker player, and philanthropist.  See this url: http://www.mathforamerica.org .  Hope to meet him someday and grab a drink, jam, or catch nice overhead, glassy swell off a right point break.  Also curious to know what he uses for alpha-generation, but I doubt I will get that out of him.

Gatheral's Stochastic Volatility Inspired (SVI) approach to model vol skews

Recently, I took a financial modeling class with some fellow students at the University of Minnesota.  We worked on modeling the volatility surface of commodity and SPX options (which I used to trade many moons ago).  Jim Gatheral, in his book The Volatility Surface, uses the following parametric model for the skew:

 Gatheral then uses an objective function minimization process compared to the implied vol given by Black-Scholes:

The resulting skew/surface (3D in time) looks like this:

  This is extremely useful in pricing exotics, variance swaps, and trading illiquid vanilla options.

Many thanks to Chris Prouty, Cargill Risk Mngmt Trader (mentor and friend) and fellow classmates (A. Abraham, F. Yang, F. Wan, S. Chiu, S. Bhimireddy, and Y. Li). 
for more info see:
https://sites.google.com/site/fmmodeling11/?pli=1