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November 20th, 2007, 06:37 AM  #11 
Senior Member Joined: Oct 2007 From: France Posts: 121 Thanks: 1 
U(n)=(X1*X2*...*Xn)^(1/n), where P(Xi=1,1)=P(Xi=0,9)=0,5 ln(U(n))=(1/n)*[ln(X1)+...+ln(Xn)] E(ln(Xi))=0,5*ln(1,1)+0,5*ln(0,9)=0,5*ln(0,99)=ln( sqrt(0,99)) By the law of large numbers, ln(U(n)) converges to ln(sqrt(0,99)) and U(n) converges to sqrt(0,99) 
November 20th, 2007, 06:50 AM  #12  
Member Joined: Aug 2007 Posts: 93 Thanks: 0  Quote:
 
November 20th, 2007, 07:53 AM  #13  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Quote:
* Actually, some stocks do well when the market is in turmoil, perhaps discount chains and the like. But if the market increases in the long run then stocks of negative beta are bad buys, so this is not a particularly successful way to balance one's portfolio alone.  
November 20th, 2007, 11:06 AM  #14  
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Yes  MPT differentiates between diversifiable and nondiversifiable risk. Individual company risk is diversifiable and responsible for some portion of an individual stock's volatility. The nondiversifiable risk as you point out is represented by beta (however the empirical evidence for ex ante beta being correlated to return is extremely weak). Only nondiversifiable risk is warrants a risk premium. But given a 2asset investment universe with a risk free asset with zero volatility and a 5% return and stocks with a 5% risk premium (so a 10% expected return) and a 15% standard deviation, the expected geometric mean return on a 100% portfolio of stocks is 8.88% (10%  half the variance). If one can borrow at the risk free rate and lever the stock portfolio 2X the volatility is double (30%) and the portfolio's expected return is 15%. However the expected geometric mean return is 10.5%.  
November 20th, 2007, 11:37 AM  #15  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Quote:
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November 20th, 2007, 11:49 AM  #16  
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The CrossSection of Expected Stock Returns: A Tenth Anniversary Reflection Ten years ago this month, Eugene Fama and Kenneth French fired the shot heard ’round the world. Its echoes still plainly reverberate today in boardrooms and trading floors. And although most investors are unaware, these effects also appear regularly in their mailboxes under the guise of investmentaccount statements. The projectile in question was a 39page piece bearing the above title, published in the June 1992 edition of Journal of Finance. It was no walk in the park; even among the Journal’s rarefied readership, I doubt many grasped the full meaning without multiple readings and hours of peer discussion. Its import lay on three levels: * The monthtomonth performance of a diversified portfolio of U.S. stocks can be explained by only three factors: the portfolio’s exposure to the market itself, to smallcap stocks, and to value orientation (the latter defined by pricetobook ratio). In plain English, "Show me the returns series of any U.S. diversified portfolio and, in almost every case, I can explain nearly all its performance based on these three factors; the precise securities are irrelevant." And, "Oh, by the way, without knowing exactly what’s in this portfolio, I can tell you the median market cap and pricetobook ratio just by looking at its returns." * The corollary of their work was that once one considered the size and value factor "loadings" of a diversified U.S. allstock portfolio, the market loading—Sharpe’s famous "beta"—did not explain return. In other words, portfolios of highbeta stocks did not have higher returns than portfolios of lowbeta stocks. Beta was dead. * Most importantly, the size and value factors, because they were surrogates for risk, had positive returns. Therefore, value stocks should have higher returns than growth stocks, and small stocks, higher returns than large stocks; small value stocks should have the highest returns of all. The one place where the model "didn’t work" was with small growth stocks, which empirically had much lower returns than expected, having the worst performance of the four "style corners" (large growth, large value, small value, and small growth). Heeding these findings, investors (myself included) began to accumulate small and valueweighted portfolios and promptly had their heads handed to them. Suppose it took several months to read the piece, confer with your savviest colleagues, and assemble a portfolio loaded down with small value stocks on January 1, 1993. Here’s how the "four corners" of the equity world, as defined by Fama and French, would have fared over the next seven years. (But first, let me explain the Fama and French definition of the four corner portfolios. In the simplest case, stocks are split into two halves by size: "large" constitutes size deciles one through five of the CRSP (Center for Research in Security Prices) database, and "small," deciles six through ten. "Value" and "growth" are defined as the bottom and top 30% of stocks sorted by price/book.) Annualized Returns, January 1993 to December 1999 Small Value: 13.90% Small Growth: 16.92% Large Value: 15.72% Large Growth: 21.64% Wilshire 5000: 20.47% Mind you, you’d still have done fine, thank you. But not nearly as well as your uncouth, beerswilling, Janus fundbuying neighbors. (And not as well, for that matter, as the folks at Vanguard, who have never bought into the model and still insist that the optimal core equity holding is their Total Stock Market Fund, which tracks the Wilshire 5000 and is heavily weighted towards large growth stocks.) It didn’t help that the biggest proponent of the model was Dimensional Fund Advisors, an institutional fund company that eschews the mass market and does not go out of its way to cultivate journalists. The latter proceeded to have a field day at the expense of multifactor investing, turning back on its creators the old efficientmarketeer observation that marketbeating strategies have a nasty habit of disappearing the moment they are described. Academicians raised a more serious objection—Fama and French were guilty of datamining; their results were an artifact of the U.S. market during the article’s study period from 1963 to 1990. And finally, practitioners raised the most serious objection of all: small and valueoriented strategies could not be implemented. Yes, in a frictionless world, excess returns could be earned. But in the real world, you’d be eaten alive with commissions and transactional costs. In their quiet way, Fama and French disposed of the datamining j’accuse. They examined stock markets abroad, then those in the U.S. before 1963. Value and size premia were found on every hill and under every rock. Then at last, the markets themselves came to their rescue. In 2000, the techled largegrowth dominance began to violently unwind. By early 2002, all the damage to a multifactor strategy was more than undone. So let’s extend the returns of the above hypothetical investor to April 30, 2002. http://www.efficientfrontier.com/ef/702/3FM10.htm  
November 20th, 2007, 11:58 AM  #17 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
Very interesting. I'll read the paper and mull it over.


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