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November 20th, 2007, 06:37 AM   #11
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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)
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November 20th, 2007, 06:50 AM   #12
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Originally Posted by CRGreathouse

That assumes that return is not correlated with volatility, which seems like a very bad assumption to me.
Not at all. Diversification lowers volatility without effecting expected return - that is the basis of modern finance - just look at the volatility of the average stock in the S&P 500 vs. the index as a whole. There is about a 50% difference, depending on the time period examined.
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November 20th, 2007, 07:53 AM   #13
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Originally Posted by STV
Not at all. Diversification lowers volatility without effecting expected return - that is the basis of modern finance - just look at the volatility of the average stock in the S&P 500 vs. the index as a whole. There is about a 50% difference, depending on the time period examined.
I agree that diversification lowers volatility without reducing return. But this problem does not correlate the two, and they are very related. This is the essential idea of the beta coefficient -- while you can reduce some amount of churn by diversifying, to some extent stocks are correlated with the market as a whole, and that risk cannot be reduced by diversifying into other stocks.* The higher the beta the higher the correlation.

* 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.
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November 20th, 2007, 11:06 AM   #14
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It's log-normal for finite n. The limit of the cdf as n tend s to infinity is uniformly zero.
Yes, what I meant to say is that as the increments n become continuous the process converges to Geometric Brownian Motion



Quote:
I agree that diversification lowers volatility without reducing return. But this problem does not correlate the two, and they are very related. This is the essential idea of the beta coefficient -- while you can reduce some amount of churn by diversifying, to some extent stocks are correlated with the market as a whole, and that risk cannot be reduced by diversifying into other stocks.* The higher the beta the higher the correlation.

* 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.

Yes - MPT differentiates between diversifiable and non-diversifiable risk. Individual company risk is diversifiable and responsible for some portion of an individual stock's volatility. The non-diversifiable 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 non-diversifiable risk is warrants a risk premium.

But given a 2-asset 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%.
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November 20th, 2007, 11:37 AM   #15
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Originally Posted by STV
(however the empirical evidence for ex ante beta being correlated to return is extremely weak)
Really? Looking over the prospectus for my retirement funds last month I saw what I thought was remarkable correlation. I'd like to read up on this, if you have any suggestions. I just read Kocherlakota's 1996 "The Equity Premium: It's Still a Puzzle" which likewise deals with risk vs. return, in this case the equity premium issue.

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Only non-diversifiable risk is warrants a risk premium.
Agreed.

Quote:
Originally Posted by STV
But given a 2-asset 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%.
That sounds reasonable.
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November 20th, 2007, 11:49 AM   #16
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Quote:
Originally Posted by CRGreathouse
Quote:
Originally Posted by STV
(however the empirical evidence for ex ante beta being correlated to return is extremely weak)
Really? Looking over the prospectus for my retirement funds last month I saw what I thought was remarkable correlation. I'd like to read up on this, if you have any suggestions. I just read Kocherlakota's 1996 "The Equity Premium: It's Still a Puzzle" which likewise deals with risk vs. return, in this case the equity premium issue.
Fama & French's famous 1992 paper is discussed below:

The Cross-Section 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 investment-account statements.

The projectile in question was a 39-page 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 month-to-month 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 small-cap stocks, and to value orientation (the latter defined by price-to-book 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 price-to-book 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. all-stock portfolio, the market loading—Sharpe’s famous "beta"—did not explain return. In other words, portfolios of high-beta stocks did not have higher returns than portfolios of low-beta 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 value-weighted 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, beer-swilling, Janus fund-buying 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 efficient-marketeer observation that market-beating strategies have a nasty habit of disappearing the moment they are described.

Academicians raised a more serious objection—Fama and French were guilty of data-mining; 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 value-oriented 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 data-mining 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 tech-led large-growth 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/702/3FM-10.htm
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November 20th, 2007, 11:58 AM   #17
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Very interesting. I'll read the paper and mull it over.
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